HPSCHD-L discussions

This is a digest of my responses to various questions about tuning, in the online forum HPSCHD-L. In each posting I try to quote back the point to which I am responding directly. These are explanations and defenses of my reasoning, perspective, priorities, and methods of testing the evidence.

Understandably, a reading of this digest may be like listening to one side of a telephone conversation; but the other respondents' comments are available in chronological sequence through the search function at

An especially good posting (IMO) is the March 6th 2006 "What's the squiggle squabble about?", summarizing the main thrust of discussions so far. That same date, 6 Mar 2006, also had my presentation of Neidhardt's "fifth-circle #11" temperament from 1732, the one that features an E-G# wider than Ab-C.

There are other relevant comments before the earliest date, too; I continue to work chronologically backwards through the archive, toward the beginning of 2005 to coincide with the release of the Oxford article.

Bradley Lehman

[Feb 05] [Mar 05] [Apr 05] [May 05] [Jun 05]
[Jul 05] [Aug 05] [Sep 05] [Oct 05] [Nov 05] [Dec 05]
[Jan 06] [Feb 06] [Mar 06] [Apr 06] [May 06] [Jun 06]
[Jul 06] [Aug 06] [Sep 06] [Oct 06] [Nov 06] [Dec 06]

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Date:         Fri, 11 Feb 2005 11:47:08 -0500
From:         Brad Lehman  
Subject:      JS Bach's tuning

The first half of my article "Bach's Extraordinary Temperament: Our Rosetta
Stone" is now available in the February issue of _Early Music_, and at

It describes Bach's specific keyboard tuning, its basic sound, and the
historical background.  The second half of the article (May 2005) gets more
deeply into the musical and mathematical analysis.

I am also releasing today a new web site to accompany that article:

That includes by-ear and electronic instructions to set up this tuning,
along with various other spin-off essays.  For comparison with the Bach
sound I have included by-ear instructions to set up Vallotti, three of the
Neidhardts, and two of the Sorges.

The Oxford web site will soon include downloadable supplementary materials,
comparing Bach's with forty other temperaments.

I know that several HPSCHD-L members have already served as tuners during
the testing of this temperament, setting it for concerts by The English
Concert and Apollo's Fire.  And others here have assisted me more directly
during the writing of the article.  Thank you!


Bradley Lehman


Date:         Sat, 12 Feb 2005 09:09:17 -0500
From:         Brad Lehman  
Subject:      Re: JS Bach's tuning CORRECTED POST

Excellent posting by Daniel Jencka, at

The issue here is starting from C fork, or starting from A fork.  And the
sub-issue is getting the regular 1/6 set of fifths going by listening to
some measured fifth/fourth, vs listening to some measured major third.

Indeed there are several ways to start all this, to get those first five
regular 1/6 fifths F-C-G-D-A-E.

And in practice I do use a C fork just as often as I use an A fork.  In
that case I listen to the beat rate of the major third C-E instead of the
F-A (with a different rate, of course).  That set of rates is stashed into
one of my footnotes.

The geometric exercise of tuning and retuning a couple of notes, to get the
fifths and fourths exactly into 2 to 3 relationship (on my "practical
instructions" web page to do this by ear), isn't due to starting from any
particular fork.  Rather, it's a process of splitting things in half and
then half again to do an exact quarter "fold" of whatever major third is my
measurement, instead of three cumulative fifths/fourths.  All the major
third start does for me is to set the limit exactly for the total of the
four fifths, so I'm not introducing any cumulative errors into those
regular fifths.  That is, it's irrespective of which fork I've started
from.  I set the endpoints first, whether one of those endpoints is a C or
an A.

All those resulting beat rates of every interval are at my "mathematical
analyses" web page there at www.larips.com, for anybody wishing to devise
some other method of getting the regular fifths (or indeed any other method
of doing the whole thing in some different sequence).  For example, C down
to F as a fifth is -1.19 if we're in "A 440", or -1.12 if we're in "A
415".  That's metronome 71 or 67.  Might be even easier to go from middle C
up a fourth to the F, in which case it's 142 (at "A 440" from a C fork of
520whatever), or 134 (at "A 415" from a C fork that's really a B fork from
an old equal-tempered 440 set....).

Whatever.  The broader point is simply to get cleanly regular 1/6 fifths
from F-C-G-D-A-E, somehow, as the ground plan.  Daniel's way to do so works
well too.  These C to F numbers here today supply his "X" value.

Then the whole Bach recipe (for reference), regardless of the baking steps
or sequence, is:

- F-C-G-D-A-E regular 1/6
- E-B-F#-C# pure
- C#-G#-D#-A# regular 1/12
- residual 1/12 wide A#-F

Any reasonable method to deliver that accurate result is fine!  I suspect
that Bach himself just went straight across like that, at least some of the

(As I noted inside the article, the only reason I do the F-Bb-Eb pure step
along my way is to stick the D# exactly where it belongs, early.  That's
just my personal penchant to establish accurate endpoints wherever I can.)

Bradley Lehman


Date:         Sun, 13 Feb 2005 15:11:21 -0500
From:         Brad Lehman  
Subject:      cent charts, and Temperament Units etc

WRT: Ibo's chart of cent values for my proposed Bach temperament, in the

Yes, that is a handy supplement laying out the fifths and the major/minor
thirds with cents.  I deliberately did *not* use cents, for the reasons I
explained both in the article and at my web site www.larips.com (my several
pages about mathematical analysis).  I presented the fifths with the
fractions of Pythagorean comma adjustments: -1/6, -1/12, +1/12, or 0.  And
I presented the major and minor thirds as percentages of the *syntonic*
comma error, since that is the comma that matters in judgment of thirds.

But, it's good to see this complementary method of measurement (cents)
presented as well, for completeness.  Thanks, Ibo!

There are also some 50 pages of additional mathematical analysis by me that
should be available from Oxford's web site, as soon as they get it put up
there.  (I believe it will be in their free area where they normally post
recordings and facsimiles that support other articles of this journal.)

But again, I deliberately did *not* use cents in there either because I
believe they present a skewed way of thinking about the basic issues of the
comma fractions.  The cents give us the inserted modern bias of equal
temperament as a norm, viewing history backwards...through the rose-colored
lens of Ellis, Barbour, et al.  The whole Barbour book presents such a
skewed way of assessing temperaments, measuring their deviation from equal,
as if that's what really matters!  Cents merely steamroll over the comma
relationships and turn the whole thing into a bunch of arbitrary numbers,
based on logarithmic metrics (based on 100) that do not really apply to the
18th century.

So, part of my decision here to disuse cents is obviously a
didactic/pedantic one!....  :)  I myself could not see this proposed Bach
temperament clearly in my mind last spring, until I threw all the cents
stuff out the window first as too distracting, and did the ground-work
analysis with the "Temperament Units" measurement I describe along with the
percentages of commas.  With *those* metrics and that geometric method of
paperwork, the whole thing became perfectly clear to me, and so I pass that
along as the standards I have employed in the paper.  Mr Brombaugh is
pleased, too, that I have been able to bring his TU metrics to a wider
audience through this.  Those show up especially in the Oxford
supplementary files, showing clear differentiation between syntonic comma
temperaments and Pythagorean comma temperaments.

Brad Lehman

Date:         Sun, 13 Feb 2005 16:45:42 -0500
From:         Brad Lehman  
Subject:      the WTC calligraphy

 >When I was thinking about this ornament last year I was wondering, as
 >Bach obviously marked the c, and connected it to the letter C in the
 >word "Clavier", if not the D of the word "Das" could mean the position of
 >the Dis (D-sharp).

Actually, the big C and the little C do not touch each other at all.  They
appear to touch, in modern reproductions after the page has deteriorated
during the 20th century, but they clearly do not in the 1911 photograph for
the old Grove dictionary (see the reproduction from that on my site, in the
FAQ page).  Direct address to that graphic,

The article has some suggestive bits about the way there appears to be a
big "C D E" on the page among the capital letters of the title, and also a
big "C D Es" nearby....both the Ut Re Mi and the Re Mi Fa there.  Note the
way that the "s" of "Es" gets slashed midway, and the equidistant dots
around that line.  (This is all *in* the big putative capital D of
"Das"!)  Sure enough, the temperament has some symmetries with mean
placements of the accidentals...but most of those symmetries happen
specifically around the note C#, i.e. "Cis".

That's all to suggest, the page as a whole probably has a bunch of other
calligraphic stuff in it that I don't know about yet.  I have a hunch that
the bottom flourish might have something to do with the way of listening to
tempered fifths at all, on harpsichord: spiral of noise, followed by the
evening out of a regular series of beats.  But, other interpretations would
also be possible.  And why is that closing flourish so far off center,
unless it's merely balancing out the word funnel that has drifted too far
to the right for the page's gross symmetry?

Also, I have a remark in the article somewhere about copying the spiral as
a right-handed person (re David Pickett's question).

Here at home yesterday (with my wife and our 2-year-old) we were playing a
bit more with a printout of the page, trying things upside down and
backwards with mirrors.  With or without the mirror, the whole page
upside-down looks a little bit like a Christmas tree with an elaborate
crowning ornament on it....  Then there's also the more obvious dismantled
treble clef of the big capital "P", looking at it upside-down and in a
mirror.  My wife was asking yesterday if there's anything special about
JSB's signature down in the shaped portion of the words, with its
odd-looking capitals as well.  I still believe that a good way to find
hidden shapes is to ask a 2-year-old, viewing it at various
distances.  "Pismas pree!"

Brad Lehman

Date:         Mon, 14 Feb 2005 08:59:25 -0500
From:         Brad Lehman  
Subject:      Sorge 1758

Ibo asked:
 >Btw, has someone a description of one of Sorge's temperaments in 1758
 >(the other was equal temp.)

Yes, that Sorge 1758 is the big one featured in my article; it's also in
Lindley's "Stimmung und Temperatur" and his later Michaelstein article, and
in Dominique Devie's book _Le temperament musical_.  Its recipe is on both
my page of practical by-ear instructions and my "comparisons" page, and in
the (forthcoming) Oxford web appendix to all this.

Sorge 1758 is the one that (I believe) Sorge copied off the Leipzig organs
himself, modified to make it a little closer to equal temperament than
Bach's had been (his changes to it are quite logical ones...), and then
published as his own.  Again, that's explained in the first half of my article.

Brad Lehman

Date:         Mon, 14 Feb 2005 17:38:02 -0500
From:         Brad Lehman  
Subject:      Re: the WTC calligraphy plus Alternate Interpretation

At 12:38 PM 2/14/2005 -0800, Daniel Jencka wrote:


But, a few counter-questions so far:

- In your interpretation, what happens to the place where there are
supposed to be three "single" temperings consecutively?  You've ended up
with only two.  If you're starting the line (turned upside down) with C,
shouldn't you follow it all the way through to a tempered E# coming from
A#, at the right side?  And what about the flourish out beyond that?  [Or
perhaps I'm simply not understanding your presentation clearly enough,

- The idea of the notes being the loops rather than the valleys is
attractive; but then, why would Bach have notated the pure F and Bb (in
your interpretation) by drawing them so extraordinarily differently from
the other notes, hanging out the other side and so tiny?  Wouldn't they
just be drawn as big empty loops like the others?  As I suggested inside
the paper, figure 2 on page 7, I've been taking that little cauliflowery
thing with its three humps as a reminder that the F-A major 3rd is 3 beats
per second; it's the *only* constant rate that a tuner by ear really needs
to know as a memorized speed, and therefore part of Bach's diagram.  But I
also like your suggestion a few days ago that maybe that part is the little
temporary step to go F-Bb-Eb to stick Eb exactly where it belongs, ahead of
time...as I described in the practical instructions.  In any case, I have
not ignored that particular doodad.  It means *something*.

- The important principle here, though, is twofold: (1) as you mentioned,
take *all* of the evidence, and (2) think about what Bach's other options
might have been, to draw an expression of whatever interpretation you wish
to test.  That is, starting from your proposed reading today and thinking
backward into it, wouldn't Bach have found a simpler way to draw yours on
the page, representing your pure C-F-Bb fifths and the presence of only two
1/12 fifths?

- Also important here is the transposing situation, Chorton/Cammerton by a
whole step.  See especially my graph at the bottom of p16 in the article,
doubly labeled as the organ would sound to the organist, vs how it would
sound to the singers and players reading their parts a step higher.  In
that case, from the orchestra's perspective, F# major is the scharffest
key, and it ends up being pretty much what we'd expect from our familiarity
with the other circulating temps where the transposition Chorton/Cammerton
is not taken into account.  My related discussion of that is on pp
17-18.  Within that orientation, it becomes less surprising that *in solo
repertoire* the brightest key ends up being E.

- There's much more in part 2 of my article (May '05) supporting my current
interpretation.  That's where we *really* get into the fun
stuff.  Obviously I can't tip off all my points there ahead of time.

- Your proposed layout gets pretty close to the Neidhardts.  I don't have
all of them on my desk at the moment, to see if it coincides exactly with
one of them.  But, the biggest and most consistent practical flaw in the
Neidhardts that I've been noticing (playing through all of Bach's music) is
that the temperaments that start with C-F-Bb pure all end up sounding
rather raunchy in flat-key music.  The accidentals end up too low to serve
really well as flats, in their enharmonic duties that are complex in tonal
music.  This too is discussed in part 2.

- As for E major being an endpoint among normal tonic keys (outside the WTC
itself, I mean), like the top of a mountain or something, beyond which we
normally do not cross in the key signature, check out the Bach repertoire
for hpsi and organ.  Among other things, I'm impressed by the "Big Brother
Chris" capriccio in E.

I'll look into generating one of those Beat Rate Charts for your reading,
when I get the chance.  Or, pick up my old (1997) spreadsheet from
http://www-personal.umich.edu/~bpl/temper.html and plug in your layout
yourself, to get as many of those charts as you want for that and any other
layouts you want to check.

Brad Lehman

Date:         Wed, 16 Feb 2005 10:18:32 -0500
From:         Brad Lehman  
Subject:      Re: More Bach WTC tuning scrip interpretations: 18ths and 24ths

At 09:19 PM 2/15/2005 -0800, Daniel Jencka wrote:
>(...) I am sure that Bradley must have considered this other
>tones-in-the-valleys interpretation, which is simply that the three 5ths
>from C# to G# to D# to Bb would divide the remaining 1/6th comma into
>1/18th each. Just very slightly narrower than perfect 5ths. The last 5th
>from Bb to F would then be perfect, corresponding to that single loop at
>the very end.
>So that makes five 5ths of 1/6th comma each, which is the same as 3/18ths,
>and then three 5ths at 1/18th each. 18/18ths total.
>This interpretation makes the most sense to me because that last Bb to F
>interval in actual practice does not get set as part of the listening and
>tuning process. It is just the result of the tuning process that starts at
>the beginning with F and C.  It wouldn't have to bi given in the script at
>all, but showing it as a smaller single loop completes the scheme

So, why expect it to be pure, then?  Wouldn't that be an astounding piece
of luck to end up with a pure fifth residually after setting up eight
tempered fifths?  And, wouldn't he have drawn the diagram differently if
expecting a pure F-Bb?  (Occam's Razor again.)

In fact, as explained in the paper, if one reads the whole line as a 1/13
PC division (!) since there are 13 little jots in total, it comes out in
practice being a 1/12 SC division off by only a couple of microns, and
there's indeed a residual *pure* Bb-F fifth.  This was my own preferred
reading for a couple of weeks, until I was convinced that it does not suit
either the diagram or the milieu as well as the 1/12 PC reading as
presented.  I have a footnote about that.  My 1/13 PC reading (i.e. 1/12
syntonic comma), for completeness, *is* presented in the supplementary web
files for Oxford's site.

>(...)An interpretation applying a strictly proportional approach would
>necessarily have you divide the comma into 24 parts, because that is how
>many loops there are in total when you add them all up.. One would then
>assign 1/24th of a comma to the three single loop, 2/24ths to each of the
>three double loops, and 3/24ths to each of the triple loops.
>Could that be? How would one actually tune such a temperament? Could this
>be some surprising interpretation? The 2/24ths solution? Charts anyone?

That's getting into the realm of Owen Jorgensen's "Handel" temperament,
theoretically working with 1/25th PC portions and spraying them around
according to a vague set of rules.

>(...)That's it for the moment, though still wondering about how to make a
>beat chart on a Mac.

Good exercise is to develop your own spreadsheet, which lots of people have

Or, did you try picking up the Mac version of OpenOffice.org (free from
Sun) and opening mine that way?  I haven't tried any of this on Mac myself,
but theoretically it should go fine....  I have a link over to
OpenOffice.org on my "system requirements" section at
( or http://how.to/tune )

Brad Lehman

Date:         Wed, 16 Feb 2005 11:40:39 -0500
From:         Brad Lehman  
Subject:      The WTC Scrawl

>From:    Gordon Collins  
>Subject: The WTC Scrawl
>Here's this elegant title page (look at that "P" in "Praeludia"!), neatly
>drawn up, symmetrically arranged, with a fine calligraphic flourish at the
>end - and then this rather careless-looking Scrawl running across the
>top.  To me it looks much like someone trying out a new pen 100 or so
>years later without realizing what that piece of paper is that is sticking
>out from the bottom of a stack of forgotten manuscripts.....  The notion
>that the Scrawl is a deliberate attempt to convey *any* meaning certainly
>requires some justification.  It's clearly an afterthought at best,
>crammed into and overflowing the margin. (...)

The notion that it's anywhere near "crammed into and overflowing the
margin" is from looking at modern facsimiles, after the 20th century
deterioration of the original.  That's documented in the article with
footnote out to info about the source, plus a photographic reproduction
from 1911.  That 1911 photo is also the one I used on the web site: click
on the little one from the first page, and it pops up to a second window
full-size.  (Marginal space is still an open question, of course; but my
point is that the reproductions in the NBA and the _New Bach Reader_ with
crammed/cropped margins don't necessarily tell the story, in themselves.)

If Bach wanted to convey a hidden meaning and at the same time make it
*appear* careless and meaningless, with a sprezzatura-like disorder to it,
isn't this drawing a possible manifestation of it?  The English word for
this is "steganography" - hiding meaning in an otherwise innocuous-looking
carrier.  See especially the papers by Neil F Johnson, on that.

Meanwhile, my article presents both a motive for Bach to have done this,
and lots of other corroboration that he did: especially (in the second
half) in the music itself.

All these are interesting questions/speculations, but isn't it better to
read the article (BOTH halves, plus the 50+ pages of supplemental web
material that I submitted for Oxford's site) first *before* trying to rip
apart what might be in it, by hearsay?  The first printed half of the
article, February, is less than 1/3 of the presented case.  We're in the
unfortunate situation at the moment of having court in recess for another
three months; but at least these three months give everybody the
opportunity to get familiar with the proposed resulting sound, in a
practical way.


Look at it from the opposite angle, the contrapositive.  *If* my proposed
solution is indeed the sound that Bach wanted/expected, how would he convey
it through the means available to him, and why, and does it agree with a
close reading of all available materials in the historical record as to his
preferences and performances?  I believe that, having looked at all that
record (to the best of my ability using the _Bach-Dokumente_ etc) *and* his
music, playing through it tuned this way, he's guilty (as I charge) of
writing down this specific temperament and making full use of it through
his career.  So is CPE, although he was more vague in *his* writing-down of
same.  So is the corroboration especially by Bach's Mizler-buddy Sorge.  So
is the corroboration in the resulting sound of both CPE's and Friedemann's
music.  So is the way it solves all outstanding intonation problems in all
the Leipzig music by JSB, plus the Musical Offering ricercars for king Big
Fred and his court keyboardist.

[Contrapositive construction: "if {A} implies {B}, then {not B} implies
{not A}."
- My {A} is: "My proposed solution is correct; and indeed there's something
there at all to be known/knowable, that Bach had *some* specific
preference/expectation that's merely been lost."
- My {B} is: "All of Bach's music sounds [in]credibly beautiful this way
and all outstanding intonation problems are solved by this; and it fits all
the extant historical record about Bach's abilities and
preferences/expectations, working in situations where he had precise
control over intonation, as in harpsichord tuning done by himself."
- Study all of {B}, and (I believe) you will see/hear that there are no
substantial contradictions.
- This doesn't prove absolutely that {A} is correct, of course, but it
demonstrates that {not A} is very unlikely.]

The broader problem to be solved here is that of plausible *musical*
practice in all of Bach's music: practicality, plus beauty (as far as we
can trust the slippery slope of 18th century aesthetics), plus a plausible
expressive range within the music (i.e. does _Affekt_ do what the 18th
century people said it did?).  The WTC title page drawing is just the major
clue as to what that solution is.  Obviously it's paramount to interpret
the clue correctly.  That interpretation and testing is done by a LOT more
than just staring at the drawing and trying out a boatload of possibilities
at keyboards and in spreadsheets.  That's why, as the article presents, I
believe that the broader solution is in fact this one: and it's
corroborated by the WTC title page drawing and the rest of the historical
record, as to the body of music that Bach actually wrote within his
preferences/expectations of intonation.

That's why this is huge.  It's really a book, slimmed down to a two-part
article and 50+ pages of mathematical analysis, and nearly a year of trying
the music in practice.  It's properly a lifetime of work beyond that,
checking out item {B} noted above to see how solid {A} is.  The clincher,
at least for me, is that *all* the other known stand-ins for {A} over the
past 200 years *do not* fit the evidence {B} as well as my proposed {A}
does.  Not even the Neidhardts and Sorges, and Lindley's otherwise
excellent averagings of them (in his Michaelstein article).  That is: plug
in whatever other {A} you want to, run it all forward, and you'll run into
contradictions in {B}, some pieces by Bach that just don't work very well.


p.s. Gordon, I've taken your off-list notes and tidied up that bit on my
web site.  Thanks!

Brad Lehman

Date:         Wed, 16 Feb 2005 16:59:17 -0500
From:         Brad Lehman  
Subject:      Re: The WTC Scrawl

I believed for about 18 years that Bach probably switched around a bit,
changing his mind variously during his career, as suggested here in Ray's
remarks.  But now I don't believe that anymore.  This was a result I didn't
expect or see coming, that there should be any constant through all or most
of his career, or into the next generation.  (Hence, part of the "Rosetta
Stone" allusion....)

Others' mileage may vary.....

Brad Lehman

At 04:31 PM 2/16/2005 -0500, Ray Lurie wrote:
>Quoting Brad Lehman  :
> > The
> > clincher,
> > at least for me, is that *all* the other known stand-ins for {A} over
> > the
> > past 200 years *do not* fit the evidence {B} as well as my proposed
> > {A}
> > does.
>Brad Lehmann is certainly correct in asking for some forebearance until
>the whole of his article is published, and I'm sure that it must be
>frustrating for him to see questions raised about Pt. 1 that he feels
>he's already answered in Pt. 2 or on the forthcoming web supplement.  I
>don't want to join in the free-for-all, especially since I have only
>glanced at the published portion of his article, but I do want to
>question the logic of the argument he presents in the snippet included
>above.  If I've understood this passage and Brad Lehmann's posts
>correctly, he is arguing that one argument in favor of his thesis is
>that the proof is in the pudding, and that the pudding is not just the
>WTC but the whole of Bach's keyboard output (with the cherry and
>whipped cream being the whole of C.P.E.'s and W.F.'s keyboard output as
>well).  Why, though, should a single temperament fit all of it?
>Wouldn't we be surprised if Bach hadn't experimented and changed his
>decisions about temperament (in much the same way as his contemporary,
>Rameau)?  Whereas Brad Lehman finds a single solution to be
>the "clincher," to me it seems highly suspect.
>Ray Lurie

Date:         Sat, 19 Feb 2005 08:47:03 -0500
From:         Brad Lehman  
Subject:      the big/little C

Ibo Ortgies wrote:

> > http://www-personal.umich.edu/~bpl/larips/wtc-1722-from-1911-grove.gif
>There is no little C
>What has been mistakenly read as little C is a typical ornament which
>was applied frequently to some capital letters especially C, S, E, F -
>sometimes K   you can find easily similar examples in Bach's handwriting.
>For example the C in "Concerto" in the heading of the first Brandenburg
>Concerto (facsimile of this page in Alfred Dürr: Johann Sebastian Bach.
>Seine Handschrift  Abbild seines Schaffens. Wiesbaden: Breitkopf &
>Härtel, 1984. Blatt 16)
>"K" in "Kirchbach" in the title page of BWV 198 ("Laß, Fürstin, laß noch
>einen Strahl") op.cit., Blatt 35
>or "C" in "Concertino"; heading of the "Violino Concertino"-part of the
>a-minor violin concerto (BWV 1041)  op.cit., Blatt 40
>It is therefore an ornament (no little C) of the capital letter C in
>"Clavier", which does hardly touch one of the loops in the calligraphic

Also the title page of the double violin concerto 1043: see facsimile on
p235 of Wolff's _Essays_.  Prominently overlapping the big C of "Concerto".

And (maybe) part of the big F in "Friedemann" on the 1720 title page of his
little book.

And part of the big K at the top of the _Entwurff_.

In all three of those, the little hook-C-ornamenty thing touches its
capital.  On the WTC title page it doesn't touch the capital, but it
touches the spiral drawing *instead*.  (Merely an observation....)  That
bit of the detail was revealed to me when I got the photo reproduction from
Brombaugh and Leedy, scanned from the old 1911 Grove dictionary.

Brad Lehman

Date:         Wed, 23 Feb 2005 09:29:09 -0500
From:         Brad Lehman  
Subject:      BachScholarJokes

 >Ja ja. 'did you ever tune your claviharpsitoot in equal temperament?'
 >Answer: 'I stopped after I read Early Music' - sorry, couldn't resist


So, he stopped doing it after reading Barnes (1979) in EM, and then started
doing it again after Williams (1983) in EM, and now has stopped again in 2005?

Anyway, have a Spiel at the f# toccata 910, the g English suite's sarabande
808, the ouverture (all of it) 831 in both c and b, and both the fraternal
capriccii in E 993 and Bb 992, and the praeludium/toccata 566 in both C and E.

Brad Lehman

Date:         Wed, 23 Feb 2005 12:36:43 -0500
From:         Brad Lehman  
Subject:      paper organists playing pre-composed music

Ibo wrote:

But what about Sorge's report?  I believe that--at least by implication,
tied to your remarks here about "paper organists"--quite a bit of playing
of published music (and manuscript music by masters) did go on, in
practice...by well-respected organists, not only beginners.

Here's the preface of Sorge's 1st book of chorale preludes, 1750, as
explicated in Wolff's _Essays_ p113ff.  (His broader discussion here in
this essay is the various chorale preludes by JSB, Sorge, and others
collected together in the Neumeister MS, having concordances
elsewhere.)  Wolff presents a facsimile of the title page and one of the
compositions in Sorge's separate print, and gives the following translation
and commentary:

WOLFF]]Sorge's preface discusses the function of the three-part manualiter
chorale settings, compares them in general with the difficult and demanding
organ chorales in Johann Sebastian Bach's Clavier-Ubung III (1739),
suggests the possibility of performing them on both organ and clavier
(harpsichord, or another keyboard), and by analogy refers to the overall
purpose of the Neumeister collection:

"Next to the knowledge of figured bass, to which my 'Vorgemach der
musicalischen Compositionen' [Lobenstein, 1745-1747] gives sufficiently
comprehensive and detailed instructions, nothing is more important to the
organist than he be adroit in preluding to the various chorales, according
to their particular content, so that the congregation will be stimulated to
sing the subsequent chiorale with appropriate devotion.  The preludes on
the Catechism Chorales by Herr Capellmeister Bach in Leipzig are examples
of this kind of keyboard piece that deserve the great renown they
enjoy.  But because works such as these are so difficult as to be all but
unusable by young beginners and others who may lack the considerable
proficiency they require, I have prepared, at the suggestion of my good
friends as well as my own pupils, the following eight simple preludes, to
be played only on the manuals, and I herewith publicly present them to
those members of our musical youth who are eager to learn and to all
devotees of this type of playing."[[WOLFF

Now, why would Sorge bother to write any of that if organists really
weren't using pre-composed pieces in church, but *only* used pre-composed
pieces pedagogically to learn how to improvise?  (Ibo, I apologize if I'm
over-simplifying or misunderstanding your thesis on this; I very much would
like to read your diss. to understand better what your position is, in this

And, why couldn't there have been some change in common practice between
Niedt (first decade of 00's) and mid-century, as to the respectability of
playing from scores?


In _Early Keyboard Journal_ 21 (2003 - MHKS and SEHKS joint publication)
there's an excellent review by David Yearsley of three books: Russell
Stinson's books about the _Orgelbuechlein_ and the Great 18, and the
facsimile edition of G F Kauffmann's _Harmonische Seelenlust_.  On the
strength of this review I'm eager to go read Stinson's books directly.

And I like Yearsley's remark on p110 about the Bach/Kauffmann connection:
"Looking west out of the window of the second floor composing room in his
apartments in the St. Thomas School building in Leipzig, Bach could, on
clear days, see the distant spire of Merseburg Cathedral where Kauffmann
was organist and, later, director of church music to the Duke of
Saxe-Merseburg.  Kauffmann is now chiefly remembered, to the extent he is
remembered at all, as one of Bach's competitors for the cantor's position
in Leipzig."  Yearsley's point here is that Stinson probably should have
mentioned GFK as part of his discussion of playing pre-worked chorale
prelude settings.

Brad Lehman


Date:         Wed, 23 Feb 2005 13:11:15 -0500
From:         Brad Lehman  
Subject:      paper organists playing pre-composed music

An addendum to:

And there are several Froberger toccatas that explicitly say they're for a
liturgical function: in communion at the elevation of the host.  Why
shouldn't we take that title at its word, and assume that respected
organists (at least as early as the 17th century) DID really play from
written-out music like this, at least occasionally?

One of those Froberger elevation toccatas even visits harmonies all the way
from  A-flat major to B major, inclusive.  That piece is part of my
argument that Froberger's organ(s) probably had some manner of temperament
that allows such wide-ranging music to sound good.

Similarly, there are a Pachelbel fantasia in E-flat and ricercare in f#
minor that are harmonically adventurous (having lots of Dbs and Abs and a
Cb, and B#s and E#s, respectively); and why wouldn't these be liturgical
music?  [And Pachelbel was JSB's grand-teacher, through the proxy of his

Again, I'm trying to understand Ibo's assertions that such music as this
wasn't as practically liturgical as it claims/appears to be.

Brad Lehman










Date:    Wed, 11 Jan 2006 13:20:09 -0500
From:    Brad Lehman 
Subject: dentists who tune keyboards

> >http://groenewald-berlin.de/
by Dr Grönewald, DDS.

On his page
he gives away a free PDF photocopy of pages 194-5 from Tessmer's 
article, and compares only those four "Bach" proposals from the 
1960s-70s.  Tessmer's article (of which those pages are part of the 
appendix) actually compares 18 temperaments.  Furthermore, it's from 
the "1996" dated volume 25 of _Acta organologica_ (published 1997), 
not "1994" as claimed on that page.

Probably just a pseudo-3D representation of cent deviation from equal 
temperament, nicht wahr?  Everything is boiled down to cent 
measurements, for example in this 1677 Trost temperament he gives:

Measure something with many significant digits of mathematical precision, 
and make diagrams, in lieu of explaining why any of the numbers would 
matter in the actual performance of music.......  Vertical intervals 
are measured with as much theoretical accuracy as anyone could reasonably 
want.  But what about melodic thinking, in the way scales are put 
together with varied sequences of tones and semitones?  (What 
brush-strokes and sweeping lines in the painting make the resulting 
piece look emotionally compelling, or sexy, or even merely *interesting* 
to contemplate?  No, rather, just analyze the atomic structure and 
reflective characteristic in each individual pigment and hope that 
sufficient meaning will emerge from a sufficient mass of measurements...?)

Here's the index of this online book:

The more interesting book by another dentist, that I've seen, is the 
forthcoming one by Thomas Donahue.  He stresses a hands-on approach 
to trying things out at the keyboard (both a musical keyboard and a 
computer keyboard), and learning how to do any procedures *oneself* 
rather than just reading them off somebody else's charts.  His book 
also gives a step-by-step procedure to construct one's own spreadsheets 
from scratch.  Likewise he suggests some procedures to convert any 
on-paper temperament into a set of by-ear tuning instructions, by 
looking for any similar beating coincidences and whatnot.  There's 
some good pedagogical value in that: learn how to build something 
from scratch, and analyze it creatively hands-on, as a way to start 
understanding how it works.


Obligatory non-dentist material:

I have a chart ready for Lindley's 1994/7 Michaelstein proposal, the 
one where he prepared a conference tape of various Bach repertoire 
performed by Peter Sykes.  That's the article where he suggested some 
"ideal" shaping of a Bach organ temperament (according to his own, 
Lindley's, premises!) and then provided some algebraic justification 
of his various "lucubrations" (his $50 word).  But I want to finish 
formulating my commentary first, in response to his musical comments 
within that article.  Then it will be added to my roster at

Bradley Lehman 


Date:    Tue, 17 Jan 2006 18:17:10 -0500
From:    Brad Lehman 
Subject: new organ and harpsichord recordings released

This is to announce two new CD sets released January 4th 2006,
recorded in March 2005.  I am the performer and producer of
both of these sets.  They use the specific keyboard tuning that
I believe was Bach's own, which evidence I have explained at
and in various publications during 2005 (Early Music, The
Diapason, Clavichord International, BBC Radio 3 broadcast,
et al).

Press release about the two recordings:
Ordering information:

The organ set "A Joy Forever: Opus 41 at Goshen College" demonstrates
the new two-manual pipe organ at Goshen College (Indiana), built by
Taylor & Boody Organbuilders (Virginia).  The music is by Bach,
Brahms, Walther, Fischer, Erbach, Zachow, and some others.  To my
knowledge this is the first *complete* recording of Fischer's
_Ariadne musica_ including its five ricercars: a book that inspired
Bach's composition of the Well-Tempered Clavier.  The details of this
set are at
3 CDs, $30.00 USD plus shipping.  Total time slightly over 3 hours.

The single disc "Playing From Bach's Fancy" has nearly an hour of
harpsichord music by JS Bach and WF Bach, and 20 minutes on the
organ.  Preludes, fugues, sinfonias, polonaises, duetti, chorale
preludes, excerpts from the Musical Offering and Art of Fugue, and
several other tidbits.  The harpsichord is a Franco/Flemish style
double by Knight Vernon, and owned by Goshen College.  The album
details are at
1 CD, $15.00 USD plus shipping.  Total time 77 minutes.

The Taylor & Boody organ Opus 41 used in these recordings:


The trumpet + organ album "In Thee is Gladness" from January 2005
(recorded 1997) is also still available:
My colleague on that, Dr Martin Hodel, is a member of the Minnesota
Orchestra, and teaches trumpet and music theory at St Olaf College.
We recorded this album on two equal-tempered organs in northern
Germany.  It includes a variety of compositions by Buxtehude,
Brahms, Bach, Viviani, Baldassare, Pachelbel, Cellier, Bernstein,
Starer, and Lehman.  1 CD, $15.00 USD plus shipping, ordered by
e-mail inquiry to hodel@stolaf.edu .

Enjoy the music,

Bradley Lehman
Dayton VA
17 January 2006 


Date:    Tue, 24 Jan 2006 10:34:45 -0500
From:    Brad Lehman 
Subject: Young #1

> Date:    Mon, 23 Jan 2006 15:35:43 -0500
> From:    David Jensen 
> At least five days a week, I tune at least one harpsichord here at IU I tune
> in W-III, and two in K-III, and a couple in Vallotti, and one in (EGAD!!!)
> Vallotti&Young.

Why not use Young's first one, that he himself claimed to prefer ahead 
of his second one?  It's smooth in result, symmetrical in the same way 
the physical layout of a keyboard is (all intervals reflected exactly 
across D, or across G#), and very easy to prepare.

To set it up, do a complete Vallotti (F-C-G-D-A-E-B in 1/6 comma, and 
all other 5ths pure); then tweak your F so it's averaged out between 
Bb and C (1/12 each), and likewise the B averaged between E and F#.  
That's all there is to it in practice: making those transitional 
half-tempered 5ths at each boundary point with the pure 5ths.

He *described* #1 in the rather obscure way of 3/16 syntonic comma etc; 
but as Barbour pointed out more than 50 years ago, the cent values do 
not differ from using 1/6 PC on the natural 5ths.

And then Young's #2 is just his concession version for people who 
couldn't figure out #1.  That's the one that has the tempered 5ths 
C-G-D-A-E-B-F#, i.e. six entirely different notes on F and all the 
accidentals, as compared with Vallotti.  EGAD!!!  Because F and the 
accidentals are all lower (as compared with Vallotti or with Young's 
preferred #1), this temp doesn't work as well for flat-key music.

N.B. Watch out for references to Vallotti, because there is a bowdlerized 
PC version that supplanted the real one.  Vallotti himself used syntonic 
comma, not Pythagorean comma; and in his the schisma gets absorbed 
imperceptibly among the pure 5ths, giving each pseudo-pure 5th the 
slightest bit of fringing.  The biggest major 3rds then come out to be 
slightly narrower than Pythagorean, instead of being a whole syntonic 
comma sharp.  Reference: Dominique Devie's book.  I like Devie's quip 
that the dumbing-down of syntonic comma temperaments to theoretical PC 
modern temperaments is like getting a bad haircut.  Big brutal chops 
with the shears, instead of refined trimming.

My chart of real Vallotti is at page 15 in the "supplementary data" PDF 
file #2, downloadable through
Bowdlerized Vallotti is at page 18 in that file.

My Young #1 chart is viewable through the page
where I put the distinction between the theoretical version and the 
easier practical version.

Brad Lehman


Date:    Tue, 24 Jan 2006 10:55:47 -0500
From:    Brad Lehman 
Subject: geometric 5ths (any reasonable size) by ear

Paul Poletti wrote:  (about Kirnberger's scheme)

>Surprisingly (or not),
>the irregular division is very easy to tune, which is probably exactly why
>he proposed it in the first place. This makes perfect sense when we remember
>than one of Kirnberger's basic arguments was that a good system must be easy
>to do by ear. In fact, his real alternative to the half comma fifths is much
>easier to do than the ubiquitous modern misinterpretation: 4 fifths all
>tempered by 1/4 Syntonic comma.
>The process is as follows (precise beat rates are for c1 = 249.6 Hz):
>(1) Tune c-e pure.
>(2) Tune the e1 above e.
>(3) Temper c-g so that it beats not quite once a second (0.835 to be exact).
>(4) Temper g-d1 so that it beats 1 1/2 times a second, or three times in 2
>second (the actual beat rate is 1.6/sec).
>(5) Tune d an octave below d1.
>(6) Temper d-a so that it beats exactly the same as g-d1.
>(7) Check that a-e1 beats ever so slightly faster than twice c-g (actual
>rate is 2.1/sec).

OK, but accurately geometric 5ths are even easier than that, by ear, and 
without counting any specific beat rates for anything, at any specific 
starting pitch.  And all these steps together take less than a minute.

- c2 (octave above middle c) from the fork.
- c1 (middle c) pure to it; c pure to it.
- e1 pure to c1 (and check it with c as a 10th).
- g temporarily pure as a 4th under c1
- a temporarily pure as a 5th under e1
- d1 averaged out, comparing as a 5th above g and as a 4th above a, so 
the beat rate of the 4th is exactly triplets against the beat rate of 
the 5th, as duplets.  (And what that beat rate is numerically, we don't 
care at all!)  Just listen for the 3-to-2 relationship.  It will be fast 
and easy to hear, as both of these (at this point) are 1/2 syntonic 
comma...pretty big.
- g1 averaged out likewise, as a 5th above c1 and a 4th above d1.  Triplets 
of the 4th against duplets of the 5th.
- a1 averaged out likewise, as a 5th above d1 and a 4th above e1.  Triplets 
of the 4th against duplets of the 5th.
- Correct the g and a to be pure octaves from g1 and a1.
- We now have a geometrically accurate C-G-D-A-E with identical tempering 
in all of those 5ths/4ths.

This procedure also works for 1/4 comma meantone, or for any other regular 
system (any size of "meantone" generated by a reasonably-sized starting 
major 3rd somewhat bigger than pure) for the reasons I have described at

That same trick of 4ths triplets vs 5ths duplets (under fingers 5-4-1 of 
the left hand) can be used all the way up the treble of the keyboard, in 
any *regular* temperament, to check the veracity of octaves.  Eventually 
it all gets too fast to hear, but the equal *quality* of the intervals 
is easy to sense.

Starting from an A fork instead of a C fork?  Same basic trick (assuming 
our goal is to set up regular naturals).  Establish the correctly-positioned 
F under A; use the C and D below (temporarily pure to those) to erect 
a correctly averaged G between them, and then correct the C and D as 
averages from F/G and G/A.

Brad Lehman


Date:    Tue, 24 Jan 2006 11:28:01 -0500
From:    Brad Lehman 
Subject: Werckmeister 3 by ear

And Werckmeister 3 from a C fork, geometrically, with no beat-counting?  Easy.

[Assuming that one wants this non-harpsichord temperament on a harpsichord, 
and horrible sounds when playing in all the flat keys beyond two flats in 
the signature (sorry, I hate W3 but we might as well set it up accurately!)....]

- C from fork.
- C-F-Bb-Eb-Ab-Db-Gb-Cb-Fb all pure 5ths.  (Yes, we'll move the last several 
- This gives a C-E that is very slightly narrow; one schisma narrow.
- From this C-E do the 5ths construction trick as described below: temporary 
G below the C and temporary A below the E, each pure.  Use those to construct 
the accurate mean D within that particular C-E (brutally 1/2 Pythagorean comma 
from each of the G and A, a very rough sound).  Use that D with the C and E 
to construct the accurate G and A above them, and then correct our temporary 
G and A as octaves.
- That gives us an accurate C-G-D-A-E cycle of 1/4 Pythagorean comma each.
- Retune E upward to be pure from A, and then B pure from this new E.
- Our last chunk of 1/4 comma has therefore been moved up to B-F# (Gb); same 
quality as our other C-G-D-A.  That accounts for all four of these 
equally-sized chunks.

Brad Lehman


Date:    Tue, 24 Jan 2006 11:40:52 -0500
From:    Brad Lehman 
Subject: arithmetically faked roots

>>Incidentally I ought to stick up for K's mathematics - an arithmetic
division of the comma is for all musical purposes the same as a
'correct' geometric one.<<

I'll second that.  The general principle is that we can fake a fourth root 
by using four neighboring superparticular fractions.

Viz: if we want to whack 81/80 (the syntonic comma) into four geometrically 
almost-equal pieces (i.e. a fourth root) we multiply the whole thing top and 
bottom by 4, and then deal it out thus:

81/80 = 321/320 x 322/321 x 323/322 x 324/323 = 324/320

Starting from a fixed middle C, compare the geometrically correct G above 
it (a tempered 5th of 1/4 syntonic comma) against the arithmetically 
approximated 5th where a pure 3/2 has been flattened by 321/320.  The 
difference is less than 0.005 Hz.  That's closer than any acoustic instrument 
is going to stay in tune for an hour....  Ergo, such fake roots are good 
enough for practical music.

Likewise, the fake square roots of 81/80 are 161/160 and 162/161.

Brad Lehman
(personally not interested in monochord construction anyway)


Date:    Tue, 24 Jan 2006 16:12:48 -0500
From:    Brad Lehman 
Subject: B major

>>I am reminded that on one occasion I was stuck for a temperament due to the
variety of keys (including B major) being used in an orchestral concert, and
I chickened out and used ET. Upon being asked, I told the leader, a
violinist, that I had indeed set equal temperament. He looked shocked and
exclaimed, "But I can't play in equal temperament!"<<

Recently (i.e. today over lunch) I had a good Baroque violinist in for a 
reading session, and we played through a rack of mid-17th-C German stuff.  
I of course had my Bach temp set up on the main hpsi, and I had put 
Werckmeister 3 onto the second one just in case we'd like to try it; and 
there was the remainder of meantone from a couple weeks ago on the virginal 

We got near the end of one piece in E minor and I found that I had a whole 
page of pedal point, reiterating the dominant B major for at least a good 
45 seconds.  It all went fine on the Bach and I wouldn't want it to be any 
spicier than that.  At the end of the page I leapt up, sat back down at 
the Werckmeister, and said, "Let's do that page again to hear how it sounds 
in this."  It sounded pretty much like garbage, with that excessively strong 
B-D# that Werckmeister has.  Then for nut's sake I played the B major harmony 
a couple of times on the meantone, and it was even worse than that because 
the D# was tuned as Eb.

Brad Lehman


Date:    Tue, 24 Jan 2006 16:46:27 -0500
From:    Brad Lehman 
Subject: Re: B major

> We got near the end of one piece in E minor and I found that I had a whole 
page of pedal point, reiterating the dominant B major for at least a good 45 
seconds.  It all went fine on the Bach and I wouldn't want it to be any spicier 
than that.  At the end of the page I leapt up, sat back down at the Werckmeister, 
and said, "Let's do that page again to hear how it sounds in this."  It 
sounded pretty much like garbage, with that excessively strong B-D# that 
Werckmeister has.

I should add: the trouble wasn't so much from the B-D# interval *in isolation*.  
(The size of B-D# is almost the same in both of these, and Werckmeister's is 
actually a smidge smaller.)  To my ear at least, it was the difference of 
having a heavily tempered (1/4 PC) 5th there on the B-F# in the Werckmeister, 
vs having a pure one.  45 seconds of a B major chord, it jangles incessantly 
with those several beats in there vying for supremacy.  If I leave the 5th 
out of the B major chord it calms down quite a bit, but who wants to remember 
to do that while playing?

That plus the way the leading A# in Werckmeister is so bloomin' high on the 
way into B minor/major, either melodically or as part of its dominant.  I 
was playing through the B minor Ouverture/partita (BWV 831) last night, in 
this Werckmeister to remind myself what used to seem normal and unavoidable, 
and I found the notes A#, E#, and B# to be way too high when they come up 
in their various melodic and harmonic contexts.

A few more solo pieces to try, and then I'm switching this hpsi back out of 
W3 (ick!) for at least a good month.  Probably putting back the Young #1 
that I mentioned this morning.

Brad Lehman 


Date:    Wed, 25 Jan 2006 12:33:44 -0500
From:    Brad Lehman 
Subject: Re: 1 step forward and 2 steps back; was B major

Poletti wrote:
>>[basically a diatribe Brad-is-a-moron-for-even-setting-up-W3-at-all, deleted...]<<

As I mentioned yesterday, and I thought I explained clearly enough (but maybe 
not), the main reason I had W3 on my 2nd harpsichord at all this week was to 
play back through old stuff, to remind myself what seemed "normal" some years 
ago when I first learned those pieces.  The key-colors of the harmonies and 
some of the melodic intervals: features that seemed somewhat interestingly 
expressive at the time, but that now sound to me like clunky nonsense.

I was also checking out somebody's comment elsewhere that he actually *likes* 
playing Clavieruebung II (Italian Concerto and Ouverture 831) in Werckmeister 
3 better than in the temperament I've proposed, because W3 is more 
interestingly spicy.  So, I wanted to remind myself how 831 actually sounds 
in W3.  The Italian Concerto is almost indestructible as it uses only a 
small handful of Ab occurrences, while the entire rest of the piece sticks 
to the twelve classic meantone notes.  Almost any reasonable temperament 
will sound OK in that Italian Concerto.  But in 831, a temperament has to 
handle D#, A#, E#, B#, and Fx in various contexts...and W3 is (in my opinion) 
horrible at this.  The other gentleman playing 831 with W3 still fancies it 
anyway; OK, whatever.

I would never dream of using W3 willingly in any concert on harpsichord, 
or even any serious practice session whether solo or ensemble.  I dislike 
it *that* much, musically, along with believing it fundamentally 
inappropriate to harpsichords.  I even said as much (or at least half that 
much) yesterday, when providing instructions to set it up accurately:

Brad Lehman


Date:    Wed, 25 Jan 2006 15:53:23 -0500
From:    Brad Lehman 
Subject: steps forward

>> I did a survey of the 10 or so
harpsi students I will be teaching tuning to next semester to find out what
their level is and how much they know. Temperaments:

meantone, WIII, KIII, Young/Vallotti

One did know "French tuning", by which I assume he meant some kind of 

So I got me work cut out for me.<<


Suggested lab exercise: have them set up an ax in the way I recommend, 
and then play through Figure 179 in the CPE Bach _Versuch_ (English 
translation by Mitchell, p164), along with carefully working out what 
CPE's paragraph 29 thereon is about.  Gist: bring out dissonances more 
prominently than their consonant resolutions, and especially emphasize 
any notes that are coming in from outside the prevailing scale.

Same figured bass exercise that is reproduced in the middle of the page at
.  If you're working with the Breitkopf facsimile of the 1753/62 editions 
(Lother Hoffmann-Erbrecht, 1958/92), it's Figure XIV at the bottom of the 
"Tab VI" pullout page of musical examples, and it's pages 129-31 in the 

If they have ears to hear, playing through these harmonic progressions, 
they will notice that the spots CPE Bach marked with "f" are already 
being emphasized somewhat, as a naturally-occurring result of that particular 
temperament.  (Hypothetically,) CPE here has simply written into his dynamic 
indications a reinforcement of the normal musical priorities that are 
*already* in the harpsichord's sound.  In every one of his harmonies marked 
"f" there is at least one note that is especially strong or noticeable, 
due to a slight frustration of intonational expectations...i.e. whatever 
note is creeping in from outside the prevailing scale.  And it draws 
attention to itself, making the harmony as a whole seem louder or more 
restless: the thing that a dissonance is supposed to accomplish.

And yes, I've checked this in Vallotti, Young, K3, W3, various meantones, 
and various Neidhardts...where this special resultant effect does not 
happen with such a startling clarity (or much of anything).  The point 
is seriously lost, unless one actually listens to it by playing through 
the exercise in the temperament I suggest that CPE was probably using.

Brad Lehman


Date:    Wed, 25 Jan 2006 15:14:50 -0500
From:    Brad Lehman 
Subject: E minor

>>As to what a harpsichord owner did in the 17th century when faced with 
E minor, I should imagine it was obvious.<<

Right -- "do something intelligent" or else pass out suppositories for 
the ears.  If said hypothetical harpsichord owner is playing straightforward 
stuff, like the extant Reincken or Buxtehude pieces in E minor, it's easy 
enough to flick all the Eb keys on the keyboard down to a better D# for 
at least the duration of the piece.

But if said protagonist is essaying the Froberger toccata in this key, 
from 1656, that simply doesn't work.  Eb and D# are both in there, both 
prominently.  It happens to work in this piece, and it's maybe a fluke, 
that one can leave all the Eb intact except switching the one nearest 
middle C down to a D#.  Who needs anything resembling an octave there 
if the octave is not actually played in the piece, right?

Or else--whether as general habit or exceptionally for such pieces--pick 
some compromised positions *between* D# and Eb, and likewise for some or 
all of the other raised keys...at least.  And maybe also F and C, so 
they can act like E# and B# some of the time.

Or else play different music.

Or else sit there wondering why the harpsichord just sounds suddenly 
rotten and then fine again, throughout some compositions in some otherwise 
innocuous-looking keys, without really bothering to figure out why...or 
to do anything about it.

Or get some split-key accidentals built into the instrument, but then 
still sit there wondering why E# and B# in otherwise normal-looking 
music still sound rotten.


I meanwhile have a hard time believing any historical conjectures that 
Buxtehude had Werckmeister 3 on the organ...at least at the point when 
he wrote the F# minor Praeludium.  The piece sounds like wounded dog (@@) 
in W3, whenever the A# or E# or B# come up.  The Cx and the Fx (in the 
G# minor section) aren't so much of a problem here, because we're flipped 
so far around the other side it starts to sound decent again.

Whatever circumstantial evidence argues for W3 in Buxtehude 
(names/dates/places/whatnot), the existence of *this* piece has to be 
circumstantial evidence on the other side of that argument...assuming 
of course that the piece was ever intended to be played at all, as written, 
and assuming that any musicians within earshot didn't really relish a 
serving of wounded dog.  That's also assuming that extant musical 
compositions are allowed to *be* evidence of practical matters: evidence 
with some value approaching that of words and dates and monochord calculations.

@@ "Wounded dog" is of course both a technical term and a euphemism for 
unprintables.  I probably should have made some synaesthetic reference 
to olfactory trauma, instead.

Brad Lehman


Date:    Thu, 26 Jan 2006 11:01:16 -0500
From:    Brad Lehman 
Subject: African or European swallow?  (really Vallotti/Young)

Poletti ribbed Jensen:
>>Oh - reminding myself - I tune Vallotti&Young with an electronic meter.
>How do you decide which day to tune Vallotti and which day to tune Young? Or
>do you tune V. on one harpsichord and Young on the next? Or alternate
>between the two each succesive octave? Or . . .
> ;-)
>But seriously, I don't think we professionals should propogate the error
>created by the Korg company, which I think was the first to conflate the
>two. At least in my experience, the first time I ever saw them "joined at
>the HIP" was in the MT-1200 instruction manual.

Yeah, and what do they *mean* by the conflation "Vallotti&Young", anyway, 
to make a preset on the electronic device?  Which of Young's?

Young's first one is considerably closer to Vallotti than his second one 
is, even though it looks horrendously more complicated on the page.  
Starting from Vallotti, move two notes (F slightly lower and B slightly 
higher: that's all there is to it in practice!).  It alters four of the 
major 3rds slightly by 1/11 syntonic comma each.  Db-F and B-D# both get 
a bit gentler than Pythagorean; F-A and G-B both get a little more vibrant 
than they were.  The other eight major 3rds stay where they were in Vallotti.

Young's second one ("nearly the same effect") transposes the whole 
temperament of Vallotti up a 5th, and *none* of the sharps/flats or the 
F are the same as they were in Vallotti.  [If we keep a common C and A 
to all of these comparisons.]  Also, eight of the twelve major 3rds have 
changed size, some of them by as much as 2/11 of a syntonic comma.  I 
guess that's "the same effect" for rather large values of "nearly".  :)

 9 11 11 11       Ab-C, Db-F, F#-A#, B-D#
 9  7  5  3       E-G#, A-C#, D-F#,  G-B
 3  3  5  7       C-E,  F-A,  Bb-D,  Eb-G

Young #1 (i.e. Vallotti with slightly altered B and F):
 9 10 11 10
 9  7  5  4
 3  4  5  7

Young #2 (i.e. Vallotti transposed up a 5th):
11 11 11  9
 7  5  3  3
 3  5  7  9

Further details about Young 1:
http://harpsichords.pbwiki.com/f/Young.html  (thanks, Gordon)

Theoretical and practical versions:
Explanation (most of it posted in November; new link added):

Vallotti and Young 1 are both symmetrical on the keyboard, in the same 
way the keyboard layout itself is: across D or across G#.  Young 2 isn't.  
It's symmetrical across G or Db.

Brad Lehman


Date:    Thu, 26 Jan 2006 11:16:55 -0500
From:    Brad Lehman 
Subject: Re: African or European swallow?  (really Vallotti/Young)

At 11:01 AM 1/26/2006, Brad Lehman wrote:
> Vallotti and Young 1 are both symmetrical on the keyboard, in the same 
way the keyboard layout itself is: across D or across G#.  Young 2 isn't.  
It's symmetrical across G or Db.

Sorry, that last sentence about Young 2 should be: "It's symmetrical 
across A or Eb."

Brad Lehman 


Date:    Fri, 27 Jan 2006 09:54:15 -0500
From:    Brad Lehman 
Subject: 'tis V

> C-F is definitely tempered, so normally I would say, yes it's V.  But (even
> taking into account the variability inherent in the MT-1200, about which I
> readily agree), something's still not quite right.
> One of the list members sent me this "deviation from equal temperament" list
> for the Mt-1200 V/Y:
> MT1200, Vallotti & Young, ¢ deviation from =temp:
> C - +6
> # - 0
> D - +2
> # - +4
> E - -2
> F - +8
> # - -2
> G - +4
> # - +2
> A - 0
> # - +6
> B - -4

It's Vallotti with everything rounded off to the nearest cent, and lined 
up with a start on A (if we were using a tuning fork).

For a version from a C standard, subtract 6 from all twelve numbers.

There's nothing compellingly "& Young" about it.  Young's two are as I 
described here yesterday, where four or six of the notes are different 
from the above recipe, and four or eight of the major 3rds are different sizes:
(But strike/correct the last sentence: Young's #2 is symmetric across A 
and Eb, not G/Db.)

When can we get around to talking about temperament structures as something 
other than cent "deviation from equal temperament" measurements?  I blame 
Barbour.  :)

When I hear "Vallotti" I immediately think, "Ah yes, all seven naturals 
are spot-on to their positions in regular 1/6 comma, and all five accidentals 
are at compromised positions rising by pure 5ths."  That temperament is 
so simple and elegant, that way.  There doesn't need to be a "Vallotti 
[with or without Young]" setting on a device, if the device already has 
regular 1/6 comma on it.  Just lay down all the naturals from 1/6, turn 
off the device, and then stick five pure 5ths on the accidentals, coming 
off either end from F or B.  It then meets a residual pure 5th on the 
other one, as checkpoint.

Want Barnes?  Tune "Vallotti" and then knock the B upward so it's pure 
from E.  That's all there is to it, that single note difference.

Brad Lehman 


Date:    Fri, 10 Feb 2006 12:14:21 -0500
From:    Brad Lehman 
Subject: mozart tuning for todays modern a'=442 Hz

> It would be more useful to describe this or any other temperament in
> terms independent of reference pitch. Best for some of us is simply
> to express the circle of fifths in terms of either purity or
> fractions of a comma (whichever kind you want) wide or narrow.
> Some will want beat-rates, some won't bother. Some will wants
> cents-deviations from ET, some won't bother.
> But tying the thing to absolute frequencies seems not only cumbersome
> and inflexible, but a good way to make it particularly hard for
> beginners to see the underlying logic of the pattern.

I agree.  It makes no sense to me to tie any temperament to any specific 
starting frequency, as a rigid prescription.  Temperaments are shapes, not 
strings of numbers!  The whole thing slides up or down together--as for any 
specific frequencies and beat rates--if we've started from some lower or 
higher pitch.

If it's a regular temperament ("meantone"), all the notes are in _regular_ 
positions, i.e. consistently spaced from one another, by 5ths/4ths.  All 
eleven 5ths are the same constant size as one another.  The leftover 
diminished 6th, wherever it may be, is some different leftover size (unless 
we're using equal temperament, where it comes out the same as the eleven 5ths).

If it's an irregular temperament (a "temperament ordinaire" or a "well 
temperament" or what-have-you), some of the notes have been nudged off 
their regular positions, more sharpward or flatward.  And the general 
strategy is to get the sharps and flats to work more tolerably as one 
another, when the wrongly-spelled enharmonic note gets played.  (For 
example, playing a D# in the music while the keyboard was tuned originally 
to favor Eb.)

Beat rates are dependent on starting frequency, but a temperament's 
identity IS NOT.  Therefore, beat rates in a book are not a particularly 
useful way to learn a temperament, because they all have to be adjusted 
together if we'd start from some different frequency.  And beat rates 
don't clarify a shape; they're just a bunch of numbers measuring *results* 
instead of the concept.

Vallotti at 440, Vallotti at 442, Vallotti at 396, Vallotti at 420 -- 
they're all equally Vallotti, even though they all have absolutely 
different beat rates from one another, on all the corresponding notes.  
The whole Vallotti shape just moves up or down a little bit in pitch, 
as a constant set of relationships among its notes.  Likewise for any 
other temperament.

Vallotti has all seven naturals in the same position they are for a 
regular 1/6 comma temperament.  F-C-G-D-A-E-B.  And then the sharps are 
nudged a little higher, and the flats a little lower, until all of these 
happen to meet each other in a chain of pure 5ths B-F#-C#-G#-D#-A#-E#, 
or Cb-Gb-Db-Ab-Eb-Bb-F.  Name that chain whichever one you want: sharps 
or flats.  It just works out conveniently that we can make a chain of 
pure 5ths here.

Other related irregular temperaments treat this similarly: over the 
core of regular naturals, nudge the sharps upward a little bit (and 
maybe also B or E), and/or nudge the flats downward a little bit (and 
maybe also F), until it connects with itself somewhere, working from 
one or both ends of the naturals.

That is, the sharps generated by 5ths (being nudged upward some each) 
get some amount of tempering *less* than the regular naturals.  The 
flats generated by 5ths (being nudged downward some each) get some amount 
of tempering *less* than the regular naturals.  Some of these 5ths, 
here and there, might end up pure or maybe slightly wide: but the 
perspective is that they're being narrowed *less* than regularity would 
say.  They're some bit wider than the regular amount, on average.

That distance from B through the sharps/flats back around to F has to 
get spanned somehow.  And the average size in there is pure 5ths, 
unless we've also let the B/E drift up a little bit or the F down a little 
bit.  The main point is that these average 5ths across this region are 
bigger than our regular 5ths on the naturals.  And, the simplest (least 
sophisticated) way to do it is to make them all pure...yielding the 
Vallotti layout.

A hazard is: if we use four or more pure 5ths in succession, the resulting 
major 3rd across them gets wider/rougher than we might want.  So, the 
simple chain of six 5ths from B across the accidentals to F might not 
be best done as all pure.  That's why it's useful to diddle the endpoints 
F and/or B a little bit outward, so the stuff in between doesn't have 
to average as wide as a pure 5th...it lets those extreme major 3rds 
be softened a bit.

Various Neidhardt, Sorge, Bach, and Young #1 layouts are differently 
sophisticated ways of spanning that region, giving the thing more 
interesting musical nuances (or smoother transitions) than the Vallotti 
layout does, and less harsh major 3rds as well.  Some of the 5ths get 
tempered only half as much as the normal amount would be.  It therefore 
makes that note tastefully a little higher (if a sharp) or lower 
(if a flat) as we go along, working through B-F#-C#-G#-D# etc, or 
from F-Bb-Eb-Ab etc.


Getting back round to the original question about tuning for Mozart's 
music: we don't know how Mozart tuned keyboards.  But, the extant 
music by him makes it obvious that the sharps and flats have to work 
very well as one another, regardless of their spelling; we can't stick 
with strictly regular 1/6 comma all the way through 12 notes, or 
we'd be retuning the instrument for every composition...and some go 
beyond 12 notes anyway.

So--start from that basic Vallotti shape mentioned above, with 1/6 on 
all the naturals, and then experiment a bit with the way you're moving 
the sharps up (going upward by 5ths off the B end) or flats down (going 
flatward by 5ths off the F end).  Play with the ideas of taking F 
itself down a bit, or B and/or E up a bit, before continuing that 
tasteful chain in either direction.

On our church's piano I have the layout F-C-G-D-A-E 1/6, E-B-F#-C# pure, 
C#-G#-D#-A# 1/12 each.  (That is, the basic Bach layout I've been 
writing about....)  And it works marvelously for Mozart/Haydn/Beethoven 
and on into the 19th century.  My setup instructions:

This Sunday as part of the service I'm planning to play the first movement 
of Mozart's sonata K282 (E-flat major, with lots of Ab-Db-Gb-Cb in it), 
and the Rondeau en Polonaise second movement of sonata K284 (A major, 
with G#-D#-A#-E#-B# in it).  Noticeably different colors, these two 
pieces, with everything working smoothly.

Brad Lehman


Date:    Tue, 14 Feb 2006 16:59:34 -0500
From:    Brad Lehman 
Subject: naked major 17ths in Bach

> Well we had a big Italian, and our friend JSB, as I've been noticing
> more and more and more in recent months, has an almost intentionally
> perverse habit of exposing raw two-note intervals of two-octaves and
> a third. So that the upper third is measuring its FUNDAMENTAL against
> the lower note's fifth partial. Which is just a complicated way of
> saying that NOTHING will make those moments of this exposed interval
> tolerable. It is bloody AWFUL.
> I've been thinking of getting disciplined and, just in the pieces
> I've been working on, tallying-up all the places where Bach plays
> this very mean trick. It's as though he's just rubbing our noses in
> it.

I'd like to see such a list, too.

By my quickish tally in the four Duetti BWV 802-5 -- where a main point 
might have been to rub organbuilders' noses in it with a set of test 
pieces for tuning -- the major 17th comes up 35, 19, 33, and 6 times 
respectively.  That first Duetto has the remarkable feature in bar 48 
of hitting seven of these in succession, parallel major 17ths on 32nd 

The downbeat of bar 74 in Duetto 2 really nails it on an accented 
entrance of the inverted subject, with Ab in the bass and C in treble.  
Yow.  Elsewhere, this piece contents itself with leaping parallel 
major 10ths....

In Duetto 3, most of the 33 occurrences hit in weak parts of the beat.  
Intrigued by this, I did an experimental alternate take for the 
recording, registering it with the tierce riding the bass line to hear 
these guys duke it out.  The bass's tierce therefore creates some tight 
beats against the treble's fundamental.  Both versions are in this set:
and the more sanely registered version is also on this disc:
What surprised me was that these look plenty awful on paper and sound 
plenty awful on harpsichords that have strong upper partials...but 
the tierce registration somehow worked anyway in the context of real 
music, on organ.

The mother of these, C#-E# at the beginning of the C# prelude in WTC 1 
-- I think it's interesting that Bach's earlier drafts of that piece 
did not have it hitting on the downbeat, but the right hand went 

What I'm wondering is: did Bach perhaps use this bold sound deliberately 
to keep pieces moving forward melodically, instead of stagnating 
harmonically?  There's just enough irritation to keep the listener 
aroused and alert, which doesn't happen so much if all of these major 
17ths are the same as one another (i.e. equal temperament), or if they're 
too well in tune (in the small number of Bach pieces that happen to 
work OK in meantone).  It's irritation that gets an oyster to hork up 
a pearl.

The effect of battling major 17ths also falls off rapidly, 
exponentially(?), as distance increases from the instrument.  The 
harpsichordist hears these strongly while playing, but they sound 
less gnarly to the audience or microphones.  Is this related to a 
string player's and singer's fondness for vibrato, both to warm up 
the sound and to help it project presence at a distance?

Brad Lehman

Date:    Thu, 16 Feb 2006 13:38:40 -0500
From:    Brad Lehman 
Subject: Re: "Far out keys"

>(1) Is the aesthetic of "the farther out you go, the more dissonant it gets"
>documented in the historical record for well temperament, or is this a
>modern-day interpretation we have unconsciously applied?

See Rita Steblin's dissertation, now in its 2nd edition as a regular book:
(Easy to get by Interlibrary Loan...)

Brad Lehman


Date:    Thu, 16 Feb 2006 13:47:12 -0500
From:    Brad Lehman 
Subject: Tuned Bach really an ugly wolf A#-->F?

>> (...) the "upside-down" attempt with an wolf 5th:  A#-->F  broadley
claims that Bach had been so incompetent, that he would had
  tuned such an ugly 704 cents wide sharp wolf (...) <<

Is that intended as a joke at my expense?

An "ugly 704 cents wide sharp wolf"?!?  Ever heard one?  In musical practice 
on acoustic keyboard instruments, its sound is scarcely distinguishable 
from a 700-cent 5th of equal temperament.  Both are the same distance on 
either side of the pure 5th, ~702.  They both sound like a nearly pure 5th 
with a very gentle vibrato.  The beats on 704 are simply wobbling in the 
opposite direction, subtly.

IMO, anyone who objects to the straightforward sound of a 704 cent "5th" 
(or diminished 6th) in musical practice is just stirring up the proverbial 
tempest in a teapot...stocked with red herring.

Bradley Lehman


Date:    Thu, 16 Feb 2006 14:10:08 -0500
From:    Brad Lehman 
Subject: "Far out keys" - dominants of minor keys

[RE: Very strong dominants of simple minor keys, e.g. B major cadencing 
into E minor, or F# major cadencing into B minor...]

>> PPS. Brad's ... temperament addresses this, doesn't it?
>ANY circulating temperament will, even those with Pythagorian thirds.

Not if we at least halfway accept the retorts of Marpurg to Kirnberger 
during their protracted debate.  (Bach-Dokumente III, #815; or page 449 
in the old _Bach Reader_ (1966), or Steblin's chapter about these guys.)  
Marpurg asserted that "The late Capellmeister Joh Seb Bach, who did not 
have an ear spoiled by bad calculation, must have felt that a major 
third enlarged by 81:80 is an execrable interval."

Granted, Marpurg's reasoning is flawed elsewhere in the passage, where 
he asserts that it is not even possible to have such Pythagorean major 
3rds in existence, as long as _all_ major 3rds are at least somewhat 
sharp of pure.  He's taking what looks like a hot-headed poke at 
Kirnberger here, the pure-interval-boy, and he's crossed the edge of 
reason.  But still, at least Marpurg is an 18th century writer with 
general tuning expertise (proven elsewhere) making a remark about 
Bach's taste.

Why indeed should an 18th century temperament connoisseur sit still 
for a performance stuffed with Pythagorean major 3rds, in this common 
situation of dominant-tonic?  (Obviously, some such connoisseurs did, 
and some did not!)  People wrote and improvised B-minor music, and 
presumably the listeners liked it.

Want to see a simple temperament that has all 12 major 3rds somewhat 
sharp of pure, but *three* of those sharpened by 81:80?  Famous old 
Werckmeister 3.
And its B-D#-F# is very lively, too, due to having a 1/4 Pythagorean 
comma narrowing of the 5th....

Brad Lehman

Date:    Thu, 16 Feb 2006 18:38:20 -0500
From:    Brad Lehman 
Subject: Re: Tuned Bach really an ugly wolf A#-->F?

>>Which brings me to an unanswered question of mine regarding said
A#/Bb-F disputed 5th: Why would one refer to it as a diminished 6th in
the context of a circulating temperament? I mean, the lower tone is an
A# or a Bb depending on musical context, is it not? In setting the WTC
Bach temperament (whether your interpretation or my variation of your
interpretation) it is not a leftover, remainder, or any other kind of
different-than-any-other such interval. That's the theoretical and
practical point of a circulating system: Every tone can be either a
sharp or a flat depending on the musical context. The fact that one
may, while tuning, arrive at the A#/Bb tone from a "D#" below doesn't
make it an A#, because of course that "D#" is also an Eb in a
circulating temperament. The use of sharps and flats going off in
opposite directions from C is a convention that has historical and
theoretical and musical import in non-circulating temperaments, but to
carry over the key-specific identity of certain tones from those
non-circulating systems makes no sense.<<

The main point about calling that note "A#", and about having the 
whole diagram start on F, is (I believe) an elegant and straightforward 
concept.  One does all seven naturals, and then one does all five sharps, 
and the work is done.  A start on F is the only way to do all seven 
naturals in succession as ascending 5ths, without crossing into some 
of the accidentals first!  (And one may "start on F" by getting *any* 
of the nearby naturals from a fork; I use either A or C, set my F from 
there with appropriate quality, and away we go, simply being careful 
not to move the one that came from the fork!)  By "5ths" here I mean 
as pitch-classes; one can break it back by an octave here or there, 
or do some of them as 4ths, keeping the same sequence of all the note 

One does not tune the A#-F interval directly; it really *is* a residual 
coming off both ends of the diagram.  So, why not call it the diminished 
6th?  Every temperament has to have one, if we want to stay really 
technical about things.  So does Pythagorean.  Run 11 5ths in one 
direction or the other, either ascending or descending, and whatever 
is left over -- not tuned directly -- *is* the diminished 6th of 
whatever size.

I've had an FAQ section about this very point of the A#-F for at least 
half a year:
with a couple of additional remarks at

Perhaps I did not explain that point well enough in the paper, but 
I have endeavored to do so on the web site to help people over that 
particular point of confusion!  It also addresses the people who 
would bomb the "little C" hook-stroke (above the big C of "Clavier") 
all the way out of the diagram as meaningful.  I believe the diagram 
*still* starts on F, with or without that stroke, for the same reason 
as explained above.  Do all the naturals, and then do all the accidentals.

And the thingy still looks like a very nicely rounded little C to 
me, whether any bean-counters vote it in or out of the diagram!  
The "little C" business has also been fully addressed on the web site, 
and in the paper (a footnote within the second half).  Anyone is of 
course free to disagree with this, and claim variable mileage.

Myself, I'd rather go play music than niggle about nomenclature too 
much.  If we want to call the same key-lever "E#" or "F" on two 
different occasions, in different musical contexts, who *really* 
cares that much as long as the sound coming out sounds good in both 
contexts?  Likewise, whatever "B" is set up on the keyboard, it 
had better sound decent played as a Cb wherever those come up in 
the music.  That is a point of my "enharmonics" analyses.

>>By the way Brad, I am using my slightly altered version of your WTC
"Bach" temperament in my upcoming concert this weekend. I find that it
just works so well (no pun intended) on much 18th century literature.

Best wishes on the concert!  What are you playing?

Brad Lehman


Date:    Thu, 16 Feb 2006 18:50:23 -0500
From:    Brad Lehman 
Subject: circulating temp with a wide 5th in it...

>>I say this as someone
who also is politely troubled by a wide 5th in your interpretation on
the grounds that it seems to be unusual for circulating "well"
temperaments of the 18th century (...) <<

Is the wide interval in Neidhardt's third-circle #4 (1732) disturbing?
Eb to Bb (or D# to Bb, or call it whatever) is 1/12 comma wide of a 
pure 5th; 704 cents.

Here's my quick and easy way to set that temp up on a harpsichord:

NEIDHARDT 1732 "Third-Circle #4"
Ab -60 Eb +60 Bb -120 F 0 C -120 G -120 D -120 A 0 E -120 B 0 F# -60 C# -60 G#

1. F-C-G-D-A-E in regular 1/6 Pyth.
2. C-F-Bb-Eb-Ab pure.
3. E-B-F# pure.
4. C# equally tempered from F# and G#.
5. Retune F down so it's now pure from C.
6. Retune E up so it's now pure from A.
7. Nick D# downward the slightest bit (1/12) which improves it with B,
and makes it slightly wide from Bb.


Likewise, some of the lesser-known Werckmeisters also have wide 5ths, 
considerably wider than 1/12, back in the daddy era of this topic....  
Granted, not the 18th century on those.

Brad Lehman


Date:    Fri, 17 Feb 2006 14:13:27 -0500
From:    Brad Lehman 
Subject: Re: circulating temp with a wide 5th in it...

> Is the wide interval in Neidhardt's third-circle #4 (1732) disturbing?
> http://www-personal.umich.edu/~bpl/larips/00NeidhardtThirdCircle4.jpg
> Eb to Bb (or D# to Bb, or call it whatever) is 1/12 comma wide of a
> pure 5th; 704 cents.
> NEIDHARDT 1732 "Third-Circle #4"
> Ab -60 Eb +60 Bb -120 F 0 C -120 G -120 D -120 A 0 E -120 B 0 F# -60
> C# -60 G#

Das ist:

Ab -1/12 Eb +1/12 Bb -1/6 F 0 C -1/6 G -1/6 D -1/6 A 0 E -1/6 B 0 F# -1/12 C# -1/12 G#


Ab -1 Eb +1 Bb -2 F 0 C -2 G -2 D -2 A 0 E -2 B 0 F# -1 C# -1 G#.


And his third-circle #3, right next to it on his chart, has *two* wide 
5ths of 1/6 comma each!  Also remarkably, this temperament has zero pure 
5ths in it.

Ab +2 Eb -2 Bb -1 F -1 C -2 G -2 D -1 A -2 E -2 B +2 F# -2 C# -1 G#.

Note that feature with the note F#.  It's first tuned pure with both B 
and C# on either side, and then it's knocked upward 1/6 comma...making 
it serve as a brighter sharp, smoother flat, and coincidentally a wide 
5th (706 cents) on one side.  Likewise with Eb: originally pure between 
both Ab and Bb, and then knocked upward 1/6 comma: it too becoming a 
smoother flat and brighter sharp.

Neidhardt *could have* had four pure 5ths in this temperament (B-F#-C# 
and Ab-Eb-Bb), but he chose not to have any, focusing instead on the 
qualities of his major 3rds by nudging some of the notes off the 
expected position.

Knocking the F# upward takes plenty of the edge off an excessively 
bright F#-A#, as a trade-off of a less calm D-F# (which incidentally 
becomes wider than A-C#!).  Knocking the Eb upward softens a very wide 
Eb-G, and widens B-D# so the latter becomes the same size as F#-A#, 
and so Eb-G isn't wider than Eb-G.

So again, not such an odd thing to do.  One just has to stare at these 
temperaments for a while, and better yet try them out directly at the 
harpsichord, to figure out what the strategy is in them.  Nudge a 
single note up or down by some amount, and two 5ths change accordingly, 
and two major 3rds change accordingly.  If the temperament ends up 
with one or more slightly wide 5ths, so be it...as long as the 
temperament as a whole sounds good for the music to be played!

And isn't this the sort of thing referred to by the Sorge teacher/student 
quote, a dozen years later?  One *can* make up interesting and useful 
temperaments that have some wide 5ths in them [via just this sort of 
dinking around with individual notes, thinking at the keyboard], but 
"it's unnecessary"....

Barbour's chart of this Third-Circle #3, "table 158" at the top of page 
171, has both the notes Bb and B transcribed incorrectly, along with the 
# missing from C#.  Yet more of so many typos in this otherwise fine book.  
Barbour has mis-transcribed all three of the Neidhardt 1724 temperaments 
(tables 151/155/156), and this one from 1732 (table 158), among others; 
I haven't found the patience yet to go through everything in the book 
looking specifically for corrections.  His MGG article is riddled with 
a similar batch of typos.

(Obviously we should cut him some slack, though: imagine performing 
this incredibly detailed book on a tracker typewriter, through more 
than one draft!)

Reference on these two Neidhardt third-circles: section VII of
...thx again Gordon for making this widely available....

Brad Lehman

Date:    Fri, 17 Feb 2006 23:57:51 -0500
From:    Brad Lehman 
Subject: irritating vs active

Tom Dent wrote:
> ... one man's 'irritating and unusable' is very close to another man's
> 'active'. Just to be clear, Brad's tuning makes the F# major triad
> *better* than A major, E major and B major, and also B major better
> than A major, so you have to consider what does make musical sense to
> you. Would you extend the bathroom to be larger than the kitchen?

That is an excellent point, Tom.  And, I spent plenty of time in that 
skepticism/incredulity myself.  Then, I gradually realized that this 
temperament fosters a primarily melodic and contrapuntal manner of 
playing, ahead of any triad-thwacking that would over-emphasize ANY 
of the simple harmonies or their relative qualities.

It has transformed my approach to the instruments, as a player 
(harpsichord/organ/clavichord).  Music is full of all manner of 
interesting dissonant/consonant contrasts and horizontal tensions, 
which I sort of knew before, but which has been hammered home to me 
by playing in this temperament for a long time now.

And, I feel that I have learned more about excellent practical keyboard 
musicianship in the past 22 months than in the preceding ten years 
(I stopped formal study 12 years ago...), and my independent fingerwork 
has improved quite a bit as well.  The thing makes me *want* to play 
in the manner that Bach said was desirable, on the title page of the 

If I were an autoharp player, I would do it in meantone -- and have 
done, experimentally.  That instrument fosters chord-thwacking as its 
basic sound.  Strong and sweet harmony are paramount, and no complex 
harmonies are even possible.  Therefore, meantone sounds tremendous 
on autoharp.  But, the classic keyboards playing Bach's complex music 
are not anything resembling an autoharp.

Brad Lehman


Date:    Thu, 2 Mar 2006 17:24:28 -0500
From:    Brad Lehman 
Subject: discoveries:  some relevant squiggles ....

> > As many here know I don't favor Vallotti or Young, of course, but as
> > scholar/scientist I have to accept the authority of the autograph, which
> > clearly shows Bach's intention to everyone.
> In case you plan to publish your discovery, might I suggest inverting the
> diagram to secure popular appeal?

Enough of such caustic smirking (the latter sentence) that both 
misrepresents and belittles my work.  More than enough.

What our resident cynics and rhetoric-spinners fail to grasp, or even 
perhaps to know, is that I had my formula installed and in use on my 
harpsichord for at least TWO MONTHS, before the day I realized that 
*another* convenient way of explaining it in the paper would be to rotate 
the page 180 degrees on a tabletop.

I simply read the thing from right to left, before that, during the 
course of exploring various possibilities in both directions (March 
into April 2004), and being especially drawn to the way the right-to-left 
sounds on real harpsichords in my manner of interpreting it.

My argument doesn't stand *or* fall on flipping the book.  It never did.  
Nor does it stand or fall on accepting or rejecting the "little C", 
which I have also explained in many ways on my web site.  Those who 
have been picking and plucking at my work, for their own (dis)satisfactions 
on these and other little points of argumentation, are apparently unable 
or unwilling to grasp the larger musical picture...at least as I see 
it.  Let me try to explain.


"Popular appeal" is, I believe in largest part, because my resulting 
layout sounds musically attractive and useful on the types of acoustic 
keyboard instruments that Bach knew and worked with.

Whatever anybody thinks of my argumentation or my writing style is 
really neither here nor there.  The thing works so well for Bach's 
keyboard music, and indeed for all tonal music, ...that's the appeal.

The gist of my spring 2004 discovery is this, boiled down to a single 
sentence about the musical phenomenon:

>>>Keyboard instruments playing tonal music sound extraordinarily 
rich when the interval C-E is 3/11 syntonic comma (or environs) sharp 
in size, E-G# is noticeably wider than Ab-C, and all the other major 
3rds are arranged in a smooth progression through these three fixed 

Through some complex confluence of acoustics, human musical perception, 
and the normally rule-based behaviors of tonal music: layouts with 
those specific parameters make tonal music sound strong, colorful, 
emotionally involving, interesting, intellectually satisfying, and 
in some way seeming perfectly natural.

The Bach diagram can be read in at least one plausible and simple 
way that delivers exactly that result, with these C-E-G# checkpoints 
and a smooth enough arrangement around them.  Actually, I know of two, 
and I am rather fond of both of them in my actual practice playing 
the WTC on harpsichord: the right-to-left layout that I have published, 
and a remarkably different left-to-right layout that I have not 
(because the first one on balance makes what I feel is a better and 
more elegant argument in public).  I have put most of my energy into 
that first one, because it to me looks the more intuitively obvious 
reading.  The two sound scarcely distinguishable from one another, 
anyway, having the same pattern and relative strengths of the same 
twelve major 3rds, and comparably smooth 5ths all around.  Up one side 
steeply from C major (calmest) to E major (brightest), and then 
gradually transforming back around the other side until we reach C 
again.  All the scales have their own distinctive "signatures" in 
their melodic/harmonic sounds.

Here's a broader point, though:

Any competent harpsichord tuner-by-ear should be able to set up a whole 
manual with no fuss at all, in 10 minutes or considerably less, by this 
simple expedient and counting *no* beats anywhere:
- Establish C-E at the right size (at or near 3/11 syntonic comma) by taste 
and experience.
- Crank the G# up until it's just at the breaking point of being tolerable: 
just barely inside where it goes as Pythagorean result if we'd do 
E-B-F#-C#-G# all pure.  The sharp keys around E major have a noticeable 
brightness and hardness ("dur"...) to them.
- Ab-C is then, by residue, noticeably calmer and "warmer" than E-G#.  
This gives a rich and mellow sound to the flat-side keys around Ab major, 
as plenty of contrast against sharps-music or naturals-music.  There are 
these three distinctive characters, all gradually blending into one another.
- Average out all four of the 5ths C-G-D-A-E the same as one another 
within the fixed endpoints C and E.  (An aside: this evenness is also 
important if we're going to have any string players tuning all their 
open strings to the result...and we shouldn't forget that Bach himself 
was at least a highly competent string player.)
- Average out all four of the 5ths E-B-F#-C#-G# the same as one another 
within the fixed endpoints E and G#.
- Average out all four of the 5ths Ab-Eb-Bb-F-C the same as one another 
within the fixed endpoints Ab and C.
- Either leave it at this point as finished; or do some tasteful 
micro-adjustments of the same type common to "ordinaire" practice going 
back into the 17th century: most notably, making each of Bb and Eb a 
wee bit lower so the F#-A# and B-D# will have a slightly more friendly 
quality in their musical contexts.

That's the theoretically optimal(?) object that I believe Bach was 
describing by example: very closely approximating this particular shape 
of the 12 major 3rds in their relative qualities.  Its fundamental shape 
has nothing to do, in particular, with exact measurements of 1/6 comma or 
beats or cents or any other modern scientific voodoo; it's harmonic and 
melodic relationships.  1/6 comma is merely a convenient way to measure 
and describe the workable range of "good taste" where the shape works 
itself out with internal consistency, and exhibits the right balance.  
It has nothing to do with any electronic devices of any kind, which 
Bach (obviously) didn't have anyway.  The main point of 
memory/taste/experience is to know the workable quality of C-E as a 
direct interval, at whatever pitch they would happen to be in modern 
Hz measurement.

Sit down and just *do* the thing, instead of arguing about any numbers 
or any scientifically measurable refinements, or anybody's written 
rhetoric.  Just do the music, and the result makes its own case for 
existence.  Those who have ears to hear the music, let them hear.

On some days, if I feel like it, I just do that procedure above, straight 
up and not caring about any of my current derivations from the Bach 
diagram.  The musical result is essentially the same, whether we're 
dealing with the diagram or not, as long as we've got that particular 
set of parameters on the resulting keyboard layout.

And then beyond that: who gives a &%#& if one 5th turns out to be some 
micro-smidgeon wide; or if some people who don't tune harpsichords at 
all (regularly or ever?) can't get their speculative minds around it;
or if the most thorough and overly cautious historians (which is generally 
a good thing!) fail to accept anything as even *possibly* true, until 
it's been demonstrated beyond any shadow of their *own* doubt (and 
preferably in written documentation that's seemingly unambiguous words 
and/or numbers)?  No, the broader point is to play practical music, 
both composed and improvised, to hear what works well.

My practical/theoretical connection then has been: I believe the Bach 
diagram demonstrates (in at least one compelling way) evidence of that 
strategic thought-process, setting up the C/E/G#/C to appropriate 
spacing.  One works directly at a harpsichord with the tuning lever 
in hand, making such tiny adjustments until the music sounds optimally 
good.  And that is a major point of the published paper, and my later 
web site clarifying it.  I believe that Bach found a way to write down 
that particular sound that happens to give outstanding results in music.

I'm trying to be a responsible historian about that, to the best of 
my ability: bringing in as much plausible corroborating evidence as 
I can find.  Some might say that my attempt shows some decent ability, 
or some might say it's near zero.  Whatever anybody thinks of me 
personally is less than relevant; the music still comes first.  My 
construction of any theoretical/historical/musical argument takes a 
back seat to letting the extant music speak for itself, and the 
diagram itself as evidence of same.

I personally wouldn't have seen or seriously tried this point about 
that specific C-E tasteful size, plus the E-G# > Ab-C, because it 
seems so crazy and far off the normally beaten paths; most of the 
unquestionably documented temperaments have Ab-C > E-G#, or at a 
limit they're equally sized.  For many years until 2004 I had accepted 
that situation as simply the only (or normal) way things work for 
unequal temperaments, case pretty much closed, even though some of 
the music sounds raunchy that way....  But, some things in the Bach 
diagram sparked me to experiment with this other set of relationships 
directly at the instrument, E-G# > Ab-C, to hear how it works in 
his real music with the sharps turned up so high and the flats 
correspondingly so gentle as tonics.

Everything else has been merely a writing exercise, along with a huge 
amount of (continuing) research and experimentation.  Anomalies grab 
my attention, and they don't let go, and I try to puzzle them out to 
the best of my ability.  They wake me up at night, fairly often.  
The anomaly of an irregular and asymmetric diagram has done so, and 
continues to do: why would Bach write out such an irregular doodad 
unless it is meaningful in some important way?

And *if* Bach had some excellent-sounding method in hand, as this 
one happens to be, why would he care one way or another if everybody's 
*published* temperaments around him couldn't see the efficacy of 
E-G# > Ab-C?  Excellent musicians, after all, are allowed to think 
and do things that other people have not published or documented to 
their own satisfaction; and part of the job of modern historians is 
to find extant clues into practices that might not have been 
documented to the satisfaction of positivists.

Historiography involves leaps of faith all over the place, and it's 
just a question of where.  I believe that expert musicians of the past 
presumably had ears that were at least as sensitive to nuances, as 
modern sensitive musicians claim to have--allowing that every individual 
is somewhat different from every other, and that the cultures have 
shifted a bit as well.  But that's why they were experts in *their* 
milieus: their ability to hear closely, and to control their materials!  
Any positivistic argument has to cut both ways, to be 
fair/meaningful/useful: one can't prove that the dead experts *did not* 
do things that happened to work beautifully, but didn't trouble to write 
down in some way sufficient to modern satisfaction.  I personally am 
quick to grant that the dead experts really did know what they were 
doing, perhaps more than they're usually given credit for in modern 
skeptical historiography.  Those dead guys (the Bachs and their best 
colleagues and contemporaries) are after all the witnesses to their 
own, and one another's, expert practices.  Any clues to those practices 
are bound to show up, somewhere/somehow, in their extant music plus 
any other written documentation they might have left...not necessarily 
in words.

Documentation doesn't always have to look exactly like anyone later 
might expect it to do.  It only needs to have been sufficient for any 
who used it originally.  Would anybody seriously make a case, anymore, 
that the existence of only unmarked orchestral parts to some composition 
constitutes some kind of proof that any of those musicians played 
it without any nuances, since those nuances aren't notated (or restricted) 
to the satisfaction of the most cautious positivism?  Nay, good musicians 
living or dead will always try to use their best 
instincts/training/sensitivity to make a compelling musical product 
out of whatever sketchy piece of paper they happen to be looking at.  
That's a difference between musical expertise, and mechanistic 
sight-reading by some restrictive set of rules.

Bach knew how to make music sound good, and especially his own music 
that he composed or improvised.  Has that premise ever been in question 
by anybody?  Bach was a genius and a musical expert.  Likewise, does 
anybody seriously dispute that?  Well then, does this particular 
pattern about C/E/G#/C present some plausible view into his historically 
"knowable" practice(s) on at least one occasion?  That's what the 
debate--on its good days--is about.  The reports said that Bach was 
brilliant in melody/harmony, and in thorough mastery of his materials, 
and that he really didn't care about any numbers, and that he disliked 
any unnecessary roughness within his musical practices.  Those points, 
to me, are some awfully important ones.  Bach's task, as with any 
excellent musician, was/is to do what all of one's instincts and 
training and experience show to be simultaneously practical, effective,
and intense enough for the musical content.

And, I personally put exactly zero stock in any notion that any of the 
flourishes in Werckmeister's book are in any musical way meaningful.  

Bach's WTC title page scrawl still appears to me to be a one-off 
notation of his own devising, as far as I have seen to date.  But I 
do still suspect that others in his family before him knew how to tune 
similarly or identically, at least in the most important specifics, 
to get their own music to work.  They too were known as harmonic 
innovators, especially uncle Christoph the organist in Eisenach around 
1685.  And obviously JSB's capriccio in E major (BWV 993) dedicated 
to the other Christoph, big brother in Ohrdruf, had *some* way of 
sounding musically compelling...which it happens to do in my layout 
very nicely, all the way through its double-sharps.


On another topic: what the author of a July 2005 Internet paper fails 
to point out to anybody is: his own readings of the drawing right-to-left 
amount to EXACTLY THE SAME RESULT he'd obtain if he'd flip the book 
around 180 degrees himself.  It's nothing more than rhetorical spin 
by him as a red herring: one can read it from right-to-left, or flip 
the book.  The author just uses that point against flipping the book, 
in more than a dozen occasions I've seen, to try to make Bradley 
Lehman look less than credible in public.  And, that's nothing more 
than cynically-barbed verbal abuse by him.  It says nothing about Bach, 
or music, or scientific inquiry, or historical responsibility.  
It's not even any manner of argument about harpsichord tuning, one 
way or another.

I guess we're just supposed to overlook the fact that this same author's 
June 2004 Internet paper proposed four seriously bizarre 1/3 comma 
temperaments, claiming to stem directly from an esoteric interpretation 
of a wholly different Bach document.  That is, while he was making up 
such things and trying to grab some limelight as some manner of 
researcher, I and several other musicians had already been using my 
now-published layout for several months already in real musical practice.

Likewise, I object to his incessant prattle/scoffing against any layout 
where "the tuning scheme starts on F", while his own allegedly preferred 
"Cornet-Ton" temperament does EXACTLY THE SAME THING putting F-C-G-etc 
into exactly the same position, reading "right to left" as opposed to 
flipping the whole page around.  What is the *&@% difference, other 
than his semantics to make his thing look somehow superior, in some 
allegedly measurable way?  Some public anti-Lehman joke or abuse by 
him that he has now sustained for more than a year, for his own amusement?  
Or is he simply unable to comprehend his own work?  Has the man ever 
actually tuned a harpsichord, by ear or otherwise?  I asked him once, 
some time ago, and he said that his work was done entirely by numerical 

Enough about that person in particular.  His usual red herrings continue 
to be dead fish argumentation, no matter how many times he presents them.

A strong and convincing musical sound always comes first...at least for me.

Bradley Lehman


Date:    Fri, 3 Mar 2006 12:19:27 -0500
From:    Brad Lehman 
Subject: meantone 101

>>>I think what is needed is some way of making tuning easier for the
beginner and amateur.   I try to get my students to learn how tuning
works by doing it, but most are happier with a machine.  I suppose it's
an issue of confidence.  I don't use a machine and find the needle
movement more distracting than my ear, which I trust much more.
When I read much of our list's postings, the tuning 'recipes' do make
sense, but to the novice they may appear difficult. If anyone has found
a way to 'crack the code' in making this easy to amateurs and students I
would love to know.  It's a bit like baking bread or making pasta- to
the unitiated I seems mysterious, yet others say how easy and simple it
is. Any ideas?<<<

Here's a rather quick and informal go at explaining "meantone" (i.e. 
the class of all _regular_ temperaments where 11 of the 5ths are all 
the same size as one another, and then there's one leftover diminished 
6th that is not tuned directly).  A No-Fear general method of regular 
tempering, without using any specific beat rates anywhere.

- 1. It is important to understand that the concept of "same size" means 
interval *quality*, and has nothing to do with beat rates or any other 
type of numeration.  It's the ability to hear a consistent amount of 
wooziness in an interval, no matter where that interval may be located 
on the keyboard.  The interval is typically tuned pure first, and then 
one of the notes is moved slightly so a little bit of deliberate dirt 
is introduced into that clean sound.  Our goal is to introduce the *same* 
amount of dirt into all the places where it is appropriate, and to 
recognize by ear when we have done so correctly.

- 2. To start, get one note (typically C) from a suitably reliable pitch 
source.  Establish middle C.

- 3. Set the major 3rd above it to be pure.  C to E.

- 4. If our goal is to set something that's not exactly 1/4 comma (and 
that *is* our goal for this exercise): move that upper note (E) upward 
a little bit to taste.  Introduce some vibration in that major 3rd, so 
that it still sounds good to you without becoming obtrusive.  That 
vibrato in there is called "beats".  The C to E major 3rd is widened.

- 5. The exact measurement of that amount only matters if we're trying 
to be too scientific about it.  But, for this exercise, we're focusing 
on taste and the ability to hear.  Just crank it up some little amount, 
so it still sounds resonant but gets a little bit of activity to it.  
The beats should be fast enough that they give the sound some nice 
character, but not so fast that they turn into a machine-gun effect or 
a blur, at middle C to E.

- 6. We really don't care for now if we're hitting "1/5 comma" or 
"1/6 comma" exactly, or which of the two "commas" we're trying to split, 
to please any theorist who might care about numbers.  We're just 
focusing on the ability to hear some tasteful amount by experience.  
Anywhere in that continuum of "1/4 comma" up to about "1/7 comma" is 
OK for this.  (The nomenclature such as "1/6 comma" refers to the *specific* 
amount of dirt we are introducing into each 5th/4th; it's not necessary 
for this exercise, beyond that brief mention.  And, "1/4 comma" is a 
shorthand way of saying that our major 3rd is pure.)  Importantly, we 
want to stay shy of the place around 1/7 or 1/8 comma where the interval 
of middle C up to E merely turns into an undifferentiated and ugly blur.  
That would negate a main goal of trying to do "meantone" in the first 
place: which is to have as many excellent *and consistent* major 3rds 
available as possible, and each one considerably smaller than they 
would be in equal temperament.

- 7. OK, we've got our nice major 3rd as our boundary.  Now we have to 
fit four equal 5ths inside that boundary: C-G-D-A-E.  This is most easily 
done by alternating 4ths and 5ths.  This is where we get our most basic 
techniques of listening to the 5ths/4ths for consistency.  A 4th is 
merely a 5th in which we have jumped one of the notes over to the next 
octave, on the other side.  We are dealing with slightly *narrow* 5ths, 
and correspondingly we need to have slightly *wide* 4ths.  Our octaves 
must always be pure.  (Conceptually: start with a pure octave somewhere, 
and put a pure 5th into it, up from the bottom note of the octave.  
Now knock that middle note slightly lower, making the 5th narrow.  
The 4th above it automatically becomes the same amount wide.)

- 8. Tune downward a 4th from C to G, making it pure, and then make the 
G slightly flatter from that so a gentle wobble develops.  The general 
principle is to make 4ths wider than pure, by a tastefully small amount.  
Since we are tuning the lower of the two notes here in C-G, the G, 
make it flatter so the interval becomes wide.

- 9. Tune downward a 5th from E to A, making it pure, and then make the 
A slightly sharper so a gentle wobble develops.  The general principle 
here with 5ths is to make them narrower than pure, by a tastefully small 
amount.  Since we are tuning the lower of the two notes here in E-A, 
the A, make it sharper so the interval becomes narrow.  Confirm that 
the E-A 5th has a similar amount of wobbliness that we gave to the G-C 
4th: not by counting their beats numerically, but by quality.

- 10. Tune upward a 5th from G to D, making it slightly narrow.  In 
this case, since we are tuning the upper of the two notes, it needs 
to be *flattened* from the pure point.  Check that this D makes a 
similarly slightly wobbly sound as a 4th from the A already tuned: wider 
than a pure 4th, i.e. sharper.  An important concept is here: we are 
putting our D at the geometric midpoint between D and A; we are also 
putting it at the melodic midpoint between our original C and E.  
This is the reason such temperaments are called "meantone": the "tone" 
(whole step) within the major third is exactly "mean" or average, as 
the midpoint.

- 11. Check all these 4ths and 5ths C-G-D-A-E again, listening closely 
to make sure all of these have the same quality: the same amount of 
wooziness or dirt away from purity.  C-G, G-D, D-A, A-E.  If some of 
them are too nearly pure, at the expense of others wobbling too fast, 
make the adjustment until they sound the same.  The general principle, 
again, is that all the 5ths have to be narrow and all the 4ths have to 
be wide.

- 12. From this point forward, the rest of the temperament is easier.  
All we have to do is to reproduce this same amount of dirt everywhere, 
consistently, by testing both the 5ths/4ths and any major 3rds that 
become available as we go along.

- 13. Tune a 5th downward from middle C to F, and make it slightly 
narrower than pure, i.e. the F a bit raised.  Test that this F-C has 
the same amount of wooze/dirt to it as the G-D 5th next to it, already 
completed.  Also test that the major 3rd F-A has the same amount of 
brightness to it as our original C-E major 3rd.  We are trying to get 
all of the major 3rds to have this same consistent sound, all equally 
sharp of the point where they would have been pure.

- 14. Tune a 4th up from F to Bb, making it slightly wide.  Test that 
the F-Bb 4th has the same quality as the G-C 4th nearby; and that the 
Bb-D major 3rd has the same quality as our original C-E, and as the F-A 
that we just finished setting up.

- 15. Tune a 5th down from Bb to Eb, making it slightly narrow; test 
its quality to be like F-C, and test the Eb-G quality to be like F-A.

- 16. Tune a pure octave Eb up to Eb, just so we don't forget to do it 
later.  Likewise, tune a pure octave from tenor F up to the F above 
middle C.

- 17. That is as far as the flats go: Eb is the end of the road.  Now 
we must finish the B and all of the sharps.  The notes such as Ab, Db, 
Gb, etc *do not exist* in this temperament!  But, don't worry about 
that for now.

- 18. Tune a 4th down from E to B, wide; test its quality to be like 
the A-D 4th, and test the G-B quality to be like both F-A and C-E.

- 19. Tune a 5th up from B to F#, narrow; test it to be like A-E, and 
test D-F# to be like C-E.

- 20. Tune a pure octave F# down to F#.  Also tune a pure octave down 
from our original E to the tenor E.  Test that the E-B 5th is still like 
F-A and G-D, in quality.  We are just doing a consistency check here.

- 21. Tune a 5th from tenor F# up to middle C#, narrow: same quality 5th 
as G-D and F-C around it, and same quality major 3rd A-C# as the G-B and 
the Bb-D already done.

- 22. Tune a 4th from that C# down to G#, wide: same quality 4th as G-C 
and the A-D around it; same quality E-G# major 3rd as the F-A.

- 23. We now have all 12 notes: Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#.  All 11 of 
those 5ths or 4ths have the same quality as one another, the same geometric 
size.  All eight of the major 3rds Eb-G, E-G#, F-A, G-B, A-C#, Bb-D, C-E, 
and D-F# have the same quality as one another.  You might notice that 
the vibrato gently increases in speed all the way up that test play, 
but that the interval quality still sounds the same.  That is the 
confirmation that we have done it correctly.  Beat speeds (the vibrato) 
increase as we move up the keyboard, because all the strings are vibrating 
more rapidly themselves to make the higher pitches.

- 24. Confirm that the following intervals are *not* particularly good!  
G#-Eb will be quite a bit wide, and maybe unlistenably so, to the point 
of raw ugliness.  That is the "wolf" 5th, really a diminished 6th by its 
spelling of the note-names!  F#-Bb, G#-C, B-Eb, and C#-F are our four 
diminished 4ths: they look like major 3rds on the keyboard, but the 
spelling makes them diminished 4ths instead.  They will all be much 
wider than the eight major 3rds we set and tested earlier.  The notes 
named D#, A#, E#, B# *do not exist*, but are only simulated (roughly) by 
a flat or a natural that was tuned from the other side.  Likewise, Ab, 
Db, Gb, Cb, Fb are not available but are only simulated (roughly) by a 
sharp or a natural.  Our strategy here is to set up the 
sharps/flats/naturals that get used most often in music, making them 
sound especially good by the major 3rds around them, and to let all the 
other incorrectly-spelled notes be relatively rotten.

- 25. You have just met the five "wolf" intervals that are common to the 
"meantone" or "regular" style of temperament.  Most of the music that 
would be appropriate to "meantone" style does not use these five wolf 
intervals, except very carefully as special effects occasionally.  The 
emphasis, instead, is on the sonorous quality of the eight correctly 
spelled major 3rds, all having the same gently tempered quality as one 

- 26. Play the chromatic scale all the way up from Eb, E, F, F#, etc. 
listening closely to the sizes of these semitones.  There are two different 
sizes.  Whenever we go between two notes of the same *name*, such as Eb 
to E, or G to G#, that is a small-sized semitone and is called "chromatic".  
Chroma: color change.  When we go between two notes of *different* name, 
such as E to F or from B to C, that is the large size and is called 
"diatonic".  All of the regular meantone layouts have two and only two 
different sizes of semitones.

- 27. Test also the effects of playing the minor 3rd D-F immediately 
followed by the major 3rd D-F#; the minor 3rd C-Eb followed by major 3rd C-E; 
and the minor 3rd G-Bb followed by G-B.  Try it also with the entire 
triads G-Bb-D and G-B-D.  This is another aspect of the color-change, 
the chromatic phenomenon: these alternations give an effect that we are 
staying somehow on the same harmony but merely changing its tone color.  
That is one of the beauties of regular (meantone) temperament.

- 28. Look all the way back to steps 3 and 4, where we originally had 
our C to E pure.  If we had left it that way, instead of sharpening the 
E a little bit, we could also go through all the rest of the instructions 
using only pure major 3rds.  (Try this on a different occasion.)  With 
the same principle of having all eight of the "good" major 3rds the same 
as one another, all of our test points would give us pure major 3rds on 
those eight: Eb-G, E-G#, F-A, G-B, A-C#, Bb-D, C-E, and D-F#.  There are 
several trade-offs to this otherwise lovely situation.  All of our 
5ths/4ths have to be more aggressively wobbly; the wolf intervals F#-Bb, 
G#-C, B-Eb, C#-F, and G#-Eb end up sounding even more caustic; and our 
two sizes of semitones become even more different from one another.  
When we are listening closely to melodies played in meantone, these 
different sizes of steps can start to become obtrusive...all being a 
matter of taste, of course.

- 29. A general point is to be able to *control* all this, and to realize 
that it all follows as a consequence of the original major 3rd size we 
had chosen, way back in step 4.  That single decision determines all 
the sizes of all the 5ths, 4ths, semitones, major 3rds, and the wolves: 
and the overall melodic character of the results, too!

- 30. Having a chromatic scale that we're happy with and understand, 
all the way up from tenor Eb to the F# above middle C: it is now time 
to finish the rest of the octaves.  Everything from here forward is 
simply pure octaves.  From the tenor D, get a pure octave from the D 
above it, and test that our new D-A 5th still has our consistent 
quality.  Go next to the C#, setting and testing it similarly...and 
continue all the way down to the bottom of the keyboard, chromatically.  
The only interval we cannot test as a 5th, remember, is G# to Eb.  
And, you might notice that the vibrato speed continues to decrease, 
smoothly, all the way down until it's imperceptible: but all our 
5ths still have the same *quality* as one another.

- 31. Similarly, to do all the treble, chromatically upward from 
the G above middle C.  Get the pure octave from below.  Going up 
the treble range, it is a useful exercise and checkpoint to test 
each note not only as a pure octave, but also with *both* the 5th 
and 4th intervening.  Those two need to have the same quality as 
one another, and it is an excellent way to prove the purity of the 
octave.  Once again, remember that G#-Eb has to be skipped over: but 
all the rest of the 5ths/4ths need to have the same quality as one 
another.  Also notice that they have a steadily increasing beat 
speed, all the way up, until they turn into a blur each: and all 
we can here then is the consistently woozy *quality*, not able to 
count the beats in any useful way.

- 32. That crossover point from distinguishable beats into blur 
happens somewhat differently on each individual instrument; and it 
also depends on the size of major 3rd chosen all the way back at step #4.

- 33. Reviewing the basic concepts we have learned here: we have 
learned how to install a basic temperament *shape* all the way up 
and down the keyboard.  That regular shape varies only by intensity, 
all dependent on the size of major 3rd we chose at the beginning.  
The narrower (or closer to pure, or maybe entirely pure) we have made 
our original major 3rd, the more intense all the contrasts are 
everywhere else as a result.  Consistently we end up with five "wolf" 
intervals in the same places; and they vary only in nastiness, 
determined by the size of our original major 3rd.

- 34. We have also learned and heard how we may verify pure octaves 
anywhere on the keyboard: by testing the 5th and the 4th within that 
octave to be sure those two intervals have the same quality as one another.  
This is useful during the setup of the whole temperament, or if we are 
simply correcting a few weather-drifted notes without re-doing the whole 
thing.  Our 5ths and 4ths can quickly tell us which of the two notes 
in our octave is the one that needs to be fixed.  (Well, it is usually 
the one that's farther away from middle C...but not always!  The middle 
register of a harpsichord tends to stay better in tune than the extreme 
treble or bass do, through weather shifts.  That is why we do it first: 
plus the ability to hear the qualities most clearly in that region.)

- 35. This exercise, in part, is to train our ability to hear consistent 
major 3rds; and to recognize the way their size depends on the consistency 
of the four intervening 5ths/4ths.  All major 3rds are generated by 
a cycle of four 5ths/4ths.  This knowledge can be used as checkpoints, 
with the major 3rds proving the 5ths/4ths, and vice versa.


All (or let's say "virtually all") of the other "modified meantone", 
"ordinaire", and "well temperaments" are little more, conceptually, 
than starting from a framework of some _regular_ layout...and then 
making some of those 5ths *less* tempered than we would do if trying 
to set all 11 consistently.  By "less tempered" we mean that some of 
the 5ths are less wobbly, i.e. wider than they would be in their regular 
positions: and maybe even so wide that they become pure, or go slightly 
beyond pure to become a little bit wider yet, and start wobbling again 
in the opposite direction!

The larger goal there is to make the cycles of 5ths and major 3rds meet 
somewhere around the back alley, the musical keys with plenty of sharps 
or flats in the signature: so more of these major and minor scales/harmonies 
sound reasonably playable, and so the wolf leftovers either go away 
entirely or at least become less offensive to the ear.  In those layouts, 
it does become more important to get the starting major 3rd to some 
known and controllably specific size, within reasonable tolerance.

But again, the first thing to learn is the ability to do a complete 
_regular_ layout of all 11 of the 5ths/4ths--of various tasteful yet 
consistent sizes--before starting to mess around with modifications.

Some of my other remarks about various examples of regular systems:

There exist techniques -- easy ones by ear -- to make sure that all the 
intervening 5ths/4ths within the major 3rd are really the same size as 
one another, and not merely a guess.  But, that's not the purpose of 
this initial exercise.  For completeness, I've described my favorite 
one here with an analogy to paper-folding:

Bradley Lehman


Date:    Fri, 3 Mar 2006 16:04:41 -0500
From:    Brad Lehman 
Subject: Re: squiggles on the web

For those that prefer to take their squiggles without hypotheses.<<

Heidelberg?  I thought you were a researcher/professor of theoretical physics 
in Greece, and then recently moved to England?

I think that at least the title page and several other pages of Fischer's 
_Ariadne musica_ should also deserve to be there at your collection.  We 
know for sure that Bach was acquainted with that published book, while we 
don't know one way or the other if he ever encountered Suppig's manuscript.  
Several of the spirals in Fischer's book look virtually identical to Bach's 
individual loops: especially the one at Fischer's name.  (Designed by Fischer?  
Or someone on his publisher's staff?  Variously through the book some of them 
look pre-fab, and some are especially ornate....)

As for Suppig, and the dilettantism displayed by both his handwritten 
document and his interminable piece of music: what appeal would there be 
for expert Bach to parody a dilettante?  It's not as if Suppig's work had 
anything to offer Bach's interest, but mainly some dull pages of numerical 
calculations, accompanied by a thoroughly formulaic composition.

That's why I have suggested that *if* anyone was parodying anyone else 
with their drawings (whether meaningfully or not) and/or their compositions, 
probably Suppig and Bach were *both* imitating the Fischer work, separately.  
Fischer's music offers an inspirational musical labyrinth, years before 
either Suppig or Bach.  See also Craig Wright's book for remarks about this:
...and his articles about the labyrinth BWV 591.

The Fischer book and BWV 591 are both recorded together here....
including a free web sample of all of 591.

I couldn't stomach the thought of wasting session time on the Suppig, 
but I did play through it at home.  It could only be done real justice 
with a split-key keyboard (far more than 12 per octave), anyway, for what 
that's worth.

Brad Lehman


Date:    Sat, 4 Mar 2006 17:35:46 -0500
From:    Brad Lehman 
Subject: Re: discoveries: some relevant squiggles ....

> Subject: Re: discoveries: some relevant squiggles ....
> And, never to forget in the whole: Bach wasn't fond of mathematical
> approaches, which would be necessary to find, refine and apply beat
> rates and beat ratios. Doesn't work intuitively.

Not to ignore a few other technical points, as well (which I'm offering 
here as my own skepticism, to test the theories) :

- 18th century musicians had no reliable way to measure Hz (and why would 
they care?), or to recognize when they were hitting integers exactly.

- No clear source has been presented that *anyone* 200+ years ago cared 
about (or recognized) integer beats--or any manner of synchronized beating 
in the intervals/chords--within musical texture; what difference if any 
does it make to a listener?  Can they even be perceived at all, more than 
a meter or two away from the harpsichord, in anything but the most exposed 
musical textures?  My own empirical checking of this latter point suggests: 

- Why should any integer 1 Hz (one beat per second) be subtracted from the 
*fundamental* pitch, as opposed to subtracting it somewhere in the 
third/fourth/fifth overtones which is where tuners actually listen to beats?  
By subtracting it from the fundamental, any allegedly meaningful beats of 
1 or 2 or 3 per second are actually turning up in three or four different 
octaves of the keyboard...and some of those are too high to hear.  So, 
what's the point?  [Check an easily produced beat-rate chart from the 
newest set of prescribed frequencies.]
Where is a tuner-by-ear to begin setting this up in practice, presuming 
that some hypothetically exact A=410 fork is available, and using only 
18th-century tools?  What's an appropriate sequence to deliver those notes 
onto the keyboard, with no electronics involved?

- Where are the exactly A=410  (or in a former formulation, A=420) tuning 
forks of the 18th century, and did Bach for sure have one?  Where is it now?

- In the newest A=410 scheme, why are D-F# and Eb-G the best major 3rds, 
among some others, all noticeably better in harmoniousness ahead of C-E?

- Why would any musician restrict himself/herself to a scheme that works 
at *only* one specific pitch, where it happens to work out that a bunch of 
integers turn up?  What happens during weather changes, or at any venue 
where the diapason cannot be controlled that closely?


> And it is true that Bradley's published statements, in which he showed
> no doubt whatsoever about, that his solution could possibly not be
> Bach's method, belong to the unscholarly part of the whole thing.
> Bradley says not once in connection with the 1722-squiggles from the WTC
> 1 autograph that it is a hypothesis.
> I can only guess from the one or other I have read from him more
> recently, that he has become more aware of the simple fact, that his
> ideas are a hypothesis and not a discovery of a historical fact.

Point well said (among many others).

Perhaps this is mostly a matter of personality and expression style.  I 
have of course known all along that anything I come up with for any written 
piece is, by definition, a hypothesis.  Likewise for my public lectures 
and interviews about this.  ALL historiography (by anyone!) is a string of 
hypotheses strung together, refined, and as well-constructed as the author's 
ability and resources allow.  I should think that that basic fact about 
historiography is implicit in any careful reader's mind, whether the topic 
is music history or floral arrangement at the court of Louis XIV.  No 
presently living writers or readers were there, and all reconstructions of 
past principles/practices are to some extent guesswork.

Maybe my writing does not come across with a sufficiently scholarly tone 
of tentativeness, to some readers: but that's presentation *style*, not 
substance.  I believe it goes back principally to my training/experience 
and roles as a musical performer: when one walks onstage (or directs 
weekly church services...), all musical decisions and textual choices 
must be delivered with utter conviction and forcefulness, to make an 
effective performance of the music.  Any waffling about background 
decisions must be left offstage, and omitted from one's performance demeanor, 
in the main presentation of the work.

I believe I have found some things that are historically plausible (all 
being a hypothesis, of course, and focused most tightly on 
practical/theoretical matters!), and obviously very widely useful in 
practice on harpsichords, clavichords, organs, fortepianos, and modern 
pianos.  So, in writing the EM piece, I took quite seriously EM's standard 
editorial guidelines to all their authors: to write forcefully, and to 
make a strong case for whatever is being presented.

Also, in case anybody's missed it: for months, *every* page of my web 
site has had a clear explication and disclaimer:
"LaripS.com, © Bradley Lehman, 2005-6, all rights reserved.
All material here on the LaripS.com web site is the personal opinion of 
the author, as a researcher of historical temperaments and a performer of 
Bach's music."

Brad Lehman


Date:    Mon, 6 Mar 2006 09:55:33 -0500
From:    Brad Lehman 
Subject: circulating temps with wide E-G#

> Since there is only one single temperament (of many others) of the time
> and place which comes in some way close to Bradley's suggestion, namely
> Sorge's temperament from 1758 for an organ in Chorton (a very specific
> case, not related to the WTC, and published obviously many years past
> Bach's death), one can really not speak from a "garden variety
> circulating temperament", if Brad's suggested scheme should really
> represent Bach's way of tempering.

That Sorge temp is not alone.  More recently I've been working more 
carefully and systematically through the 1732 temperaments of Neidhardt, 
setting them up for playing tests to hear their effects.

It turns out that two of them have the important (to me) feature of E-G# 
larger than Ab-C, and no other major 3rd larger than E-G#.  One of those--#11 
from his first set there--has E-G# all the way up to the same size (10/11 
syntonic comma sharp) as in my scheme, similarly as the single widest major 
3rd.  C-E is of course still the smallest, with a size shared also by F-A; 
the same relationship as in mine.

The other one--#5 from his second set--is so moderate all-round that most 
of this becomes moot.  That one favors all four of C-E, F-A, G-B, and D-F# 
as smallest, and E-G#, B-D#, F#-A#, and Db-F as largest...but the spread is 
so small it's hardly distinguishable from equal temperament.

Notably to me, at least these two demonstrate a willingness to abandon the 
old meantone-world result that the four widest major 3rds in a circulating 
temperament should necessarily be Ab-C, F#-A#, Db-F, and/or B-D#.  That's 
the musical result that is so familiar from Werckmeister, Vallotti, and 
their ilk...and from the Neidhardt layouts that have numerical *misprints* 
variously in Barbour's and Lindley's pieces, including the current _New 

But, loosen up the familiar old type of layouts that have E-G# noticeably 
narrower than Ab-C...and suddenly the behavior of tonal music starts to 
sound smoother and more natural!

And many others among these 1732 temps have E-G# and Ab-C equally sized.  
Put the C-E to some suitably sharp point, and then the G#/Ab is mean within 
whatever's left over in the octave.

That is, the situation favoring a wider Ab-C is nowhere near as conclusive 
as the typical tuning literature connecting Bach and Neidhardt would suggest.  
(Most notably in Lindley's writing; and he and I have been discussing 
this for nearly a year now, on account of my remarks in part 1 of the 
article: speaking my disagreement with the way he has typically minimized 
temperaments that don't have a wide Ab-C and environs.)

This business of E-G# > Ab-C is what's behind my selection of the term 
"extraordinary" to describe temperaments that have it.  I explained it at 
this page some months ago:
(Not updated yet with these Neidhardt examples....)

Details about the Barbour and _New Grove_ numerical errors on Neidhardt:

Brad Lehman

Date:    Mon, 6 Mar 2006 11:42:14 -0500
From:    Brad Lehman 
Subject: what's the "squiggle" squabble about?

> Date:    Sun, 5 Mar 2006 17:35:54 -0500
> From:    Zach 
> Subject: Re: squiggles on the web
> Can someone explain this whole matter of the squiggles? It seems to
> have been assumed everyone has intimate knowledge of this debate.

Dishing the hot potato onto three big plates:

The piece of arguable evidence most hotly in debate is this one, Bach's 
autograph title page for his own copy of WTC (book 1):

A  - There are some who believe that this page's drawings and layout are 
inadmissible evidence in historiography, for one or more of the following reasons:
(1) it is probably meaningless about matters of tuning, always and forever;
(2) it is too vague to be taken as reliable/credible evidence *today* of 
any such thing (no matter what it may have meant to Bach--or not--or the 
people around him), because it offers too many plausible interpretations;
(3) it is not sufficiently precedented to be taken as relevant to tuning 
(such folks need a firm second piece of evidence demonstrating that *anybody* 
ever did this reliably, writing down a temperament as some line drawing);
(4) it is too heavily precedented as something else, namely a bunch of mere 
calligraphic decoration, *instead of* being somehow meaningful to Bach in 
matters of tuning; or
(5) it is not sufficiently credited to Bach himself yet, in the line-drawing 

B  - There are some (including me) who believe that this page shows evidence 
of a keyboard temperament design, where visual elements of similar size/shape 
are taken to be equivalently sized *geometric* events within the temperament 
(e.g. comma splits of some consistent size(s)).  By this hypothesis, the 
drawing shows in some way(s) a temperament or temperaments related to 
regular-comma practice that Bach and other competent musicians/tuners knew: 
consistent amounts of audible dirt introduced to the intervals, as to 
deliberate (im)purity of certain intervals.  (The *speed* of such dirt 
changes smoothly and regularly as we move up or down the keyboard, being a 
geometric phenomenon: everything doubles or halves speed at each octave, 
and by other appropriately proportional amounts in between.)  And, the 
disagreements within this particular "camp" are on matters of interpretation: 
what specifically might the visual elements mean, as to interval sizes and 
placements?  The temperament models presented deal with the perceivable 
*quality* of the tuned or carefully mistuned intervals, and their relative 
placement; not about counting particular beat-rates at any particular pitch.  
That is, any temperament retains its consistent identity whether we start 
the setup work from C or A or some other note, and irrespective of any 
measurable pitch in Hz.  That identity is geometric; and if we describe any 
of it in numbers for modern scientific consumption, we have to use some 
manner(s) of logarithms.

C  - There are some who believe that this page describes an *arithmetic* 
temperament design, or several: where the visual elements describe constant 
beat-rates of integers (one per second, two per second, three per second, 
etc...within some particular octave(s) of the keyboard where it happens 
to work out, and based on some specific pitch standard measured in Hz).  
In this hypothesis, it is somehow important to have the beats on the 
keyboard synchronized with one another when certain intervals are played: 
during the tuning process, or in musical performance, or both.  The 
adherents of this "camp" do not agree amongst themselves if the integer 
business should be between the *fundamental* frequencies (measured in Hz), 
or among the beat-rates of the overtones that are used in setting up 
fifths/fourths/thirds.  But, they apparently agree that Bach was somehow 
into counting something, or doing other numerical calculations, 
whenever tuning his keyboards in practice.

Inevitably, all of this vigorous debate also leads out to bigger issues 
of epistemology:

- What are appropriate manners of doing historiography, musical theory, 
and practice?

- What are the roles of taste?

- What types of inferences are admissible to argumentation, in any direction?

- What types of "tone" or manner (or projected confidence...) are 
appropriate in the written presentation of anyone's hypotheses?  Who 
gets to decide that, and how many opinions of disagreement/contempt are 
appropriate from people who think the work was far off the mark, in 
judging matters of style for the author and/or publishers?  (Kick 
something in the head hard enough or often enough, and will it go away?)

- If a writer proposes a hypothesis, how rigidly should the readers adhere 
to the specific measurements presented, as to their understanding of 
what was actually being said?  (For example, can a "1/6 comma" temperament 
vary anywhere in intensity between approximately 2/11 or 1/7 comma, or 
use the other comma, to taste, and still "be" the same thing in musical 
practice; or does the lockdown of measurements rule out everything else, 
such that the writer is somehow wrong for suggesting a precise example?  
If Bach himself ever tried to use some arguably precise scheme of his 
own or someone else's invention, but got it too high or too low in some 
features on some particular day, would he be wrong?  How much leeway 
exists in temperament setup-by-ear, by 18th century standards?  And 
by modern standards?)

- What scientific processes are appropriate to any musical or historical 

- Can measurable shapes have existed among long-dead people who didn't 
have any restrictive scientific apparatus (by modern standards) to 
measure/describe them?  If so, what might their representation of such 
shapes have looked like, with a more intuitive or personal approach to 
writing down what they knew?  (And then, how tightly or loosely can 
such a shape be interpreted until it turns into something else recognizably?)

- Are any bits/methods of esoterica admissible to any such debate, 
when dealing with a complex individual such as Bach who had varied 
extra-musical interests and influences...or should we focus *only* or 
mainly on the sound of the music?

- Who are the most important or credible arbiters of other people's 
work, to decide what's historically or theoretically true, and to 
evaluate it for public consumption?  Other practitioners of same, or 
presumably disinterested observers from other fields of expertise, or both?

- How do we decide what "sounds better" than what else, in musical 
practice?  What types of consensus or opinion are important?

- Does "sounds better" have any place in deciding historical hypotheses?

- Does an elegant intellectual argument necessarily have to sound 
outstandingly good in musical practice; or conversely, does something 
that sounds good have to yield a clean and obvious (and controllable) 
intellectual model as well?

- Would Bach have cared about any of this, and why or why not?

- usw....

And, I personally consider that *Bach's extant music* is just as important 
to the analysis as any derivations from this hand-drawn page; but lots 
of people would whittle it down to be only about the page or its calligraphy....

I myself am radical enough (or belligerent enough) to suggest that any 
would-be critic of the work owes to himself/herself to play through 
*at least* the whole WTC book 1, and tuning the instrument only by ear, 
to give a fair test to practical hypotheses of temperament!  (That's 
what Bach himself would have had to do, if testing any proposed layout.)  
Does the musical work make sense in a proposed tuning, or not?....  
The point of "Bach" temperament research is to deliver some workable 
layout that sounds plausible in the music; and I believe this calls 
for empirical testing, actually playing that music on appropriate 
instruments, under the same type of conditions Bach would have had to 
use if testing it himself.

Brad Lehman
(personally not fond of the word "squiggle", & never have been....) 

Date:    Mon, 6 Mar 2006 14:46:27 -0500
From:    Brad Lehman 
Subject: doodle

>>(personally not fond of the word "squiggle", & never have been....)
>Doodle might be more appropriate: "An aimless scrawl made by a person while
>his mind is more or less otherwise applied." (OED)

Boy-oh, talk about biasing the reader with nomenclature!  Bach couldn't 
possibly have been concentrating or *trying* to do something particularly 
meaningful, if all he was making there was a "doodle."

That's like letting baronet conductors call our instrument nothing more than 
a device to generate sound-effects for skeleton porn.

Brad Lehman

Date:    Mon, 6 Mar 2006 16:37:28 -0500
From:    Brad Lehman 
Subject: Re: doodle

>Brad Lehman écrit:
>>Boy-oh, talk about biasing the reader with nomenclature!  Bach
>>couldn't possibly have been concentrating or *trying* to do something
>>particularly meaningful, if all he was making there was a "doodle."
>Well, what about Kellner's "divination" and "esoteric numerology", as
>opposed to your "discovery"?

My assessment of Kellner's method comes from reading a stack of his papers 
that describe his arbitrary assignments of meaning, and from trying to 
play through large chunks of the Bach repertoire using Kellner's temperament, 
and from listening to organ recordings using his.  The thing just sounds 
too ugly in ordinary-looking spots of the music, and the 
numerological/spiritual reasoning makes such arbitrary points, that I 
remain with little respect for the results.  Example: we're really supposed 
to search for the number 369 in Bach's four Duetti and other music, and 
take that to signify *only* Kellner's layout?  How?!  Especially so, 
when the musical result is plenty harsh in those particular pieces, 
tuned Kellner's way?

>What about "Sparschuh's temperament" as opposed to "Bach's own temperament"
>(instead of "Lehman's temperament"), when they both are two different
>interpretations of the same doodles?
>Are these terms of yours not biased?

Point taken, but the phrase "interpretation of the same doodles" shows your 
own bias just as strongly....and, I suspect, deliberately--to make sure I 
get this whack-Lehman-o'er-the-head-yet-again point.  You've already 
dismissed ALL of it as meaningless (and/or at least unknowable on this 
particular evidence), with the assumption that Bach didn't/wouldn't do 
something beyond an idle doodle, on the title page of a composition that 
explicitly says it's about playing in all keys.

The observation that the "doodle" is insufficiently credible evidence 
*to you* is not equivalent to saying it was merely an 
idle/thoughtless/disinterested diversion to Bach.

Brad Lehman


Date:    Wed, 8 Mar 2006 18:29:52 -0500
From:    Brad Lehman 
Subject: another easy circulating temp, emphasizing tasteful adjustment

Here's a simple temp I've formulated and have been messing about with for 
a while, as an all-purpose circulating scheme that takes about two to four 
minutes to set up in the bearings area.  It emphasizes listening to interval 
quality, rather than counting anything, and it all has basically one size 
of 5th.  No notes need to be dinked around (like neighboring "half-tempered" 
1/12 comma 5ths) after they've been put into place.

What should I call this, "Lehman 2"?

- C from fork, and over to middle C.

- Set E from C, somewhere near the quality of regular 1/6 comma (i.e. with 
the familiar sound from Vallotti, or slightly sharper than in Werckmeister 
3).  If it happens to be a little high or low to taste, that's OK.  (*)  
We just want to stay out of the range where it's so high that C-E turn 
into a blur, or so low that there's no room left to work with in the 
following steps.

- Fit the G, D, and A into this so C-G-D-A-E all have the same quality as 
5ths or 4ths.  (**) Whatever regular size we're setting here, this is our 
basic unit.

- From E, pure B, pure F#.  G-B should sound slightly "harder" or brighter 
than C-E, and then D-F# even more so.

- From C, pure F.  F-A has the same character as G-B.  The C major, F major, 
and G major triads are our three best.

- From F, temper Bb as a narrow 5th (or wide 4th) of approximately the same 
quality as the others above.  Listen also that Bb-D is similar quality to 
the F-A already available; and that F#-A# is high but acceptable-sounding.

- From Bb, pure Eb.  Confirm that Eb-G and D-F# have the same quality as 
one another, both sounding much like they do in equal temperament.  Confirm 
also that B-D# has approximately the same character as F#-A#, and the whole 
triad B-D#-F# is quite good, owing in part to the pure 5th.

- From Eb, make Ab a narrow 5th or wide 4th of approximately the same 
quality again, or perhaps a little bit gentler to taste.  Test that E-G# 
is high and bright, but not quite as wide as Pythagorean.  Ab-C should 
sound very slightly wider in character than Eb-G does, but still a good 
complete triad Ab-C-Eb.

- From F#, make C# a narrow 5th or wide 4th similarly.  Check that A-C# 
makes a nice transitional character between D-F# and E-G#.  Also check 
that Db-F has a character resembling that of B-D#.  Finally, note that 
our leftover interval C# to G# is probably slightly *wide* as a 5th (or 
narrow as a 4th), but that the character of C#-G# sounds the same as Eb-G#.  
They simply happen to be tempered in opposite directions, but a similar 
amount, not that anyone would notice during the playing of music.

Play suitable music in a variety of keys, to test that everything works 
nicely. (***)


There's an odd set of properties that come up if the initial C-E is too 
wide and venturing into the blur-range.  (Value judgments, of course; 
others' mileage may vary....)

- The natural keys sound as if they're reluctant to relax, noticeable 
especially in F major; and the Bb major triad gets difficult to set 
with any reasonable quality.  C, F, G, and Bb are still the best four, 
but we've lost a good bit of their resonance.  (And if we're going to 
do so, why not just use equal or something pseudo-equal, instead?)  
The D major triad also emerges sounding rather hectic.

- Around the service entrance at the back of the building, the leftover 
C# to G# 5th (or C# to Ab) turns out narrow.

- The triads of Db major, B major, and F# major turn out much too 
consonant, ahead of the qualities of D/Eb, A/Ab, and E.  It's as if 
the shape flips itself inside-out: making the key of C# major sound 
fantastic, but at the expense of too much "nervous tension" everywhere 


(*) Somewhere in the quality range of 1/5 syntonic, 1/6 Pythagorean, 
1/6 syntonic, or about 1/7 Pythagorean.  Somewhere in there, perhaps 
somewhat differently from instrument to instrument, there is a happy 
medium where we'll end up with a quite good C major, F major, and G 
major.  Taste and experience!  For those who "must" count beats: put 
C-E somewhere between 3.5 and 6 beats per second, and this is easier 
to hear if playing a major 10th from middle E down to the tenor C.

(**) A method to do this accurately is at
...or just guess it out so it's close enough, with those four 5ths/4ths 
all sounding similar.

(***) One of my favorite half-hour test sets is as follows...play as 
much of this as necessary to be satisfied with temperament quality 
and usability -- melodic/harmonic integrity, and enough character:

- Bach's four Duetti (for various examples of enharmonic swappery, and 
chromatic motion within the basic characters of common keys);

- The following major-key preludes of WTC book 2, all spaced a minor 
3rd apart: F major (for smoothness and big resonance in an "overheld" 
harpsichord texture), Ab major (for chromatic adventures and the 
character of flat keys), B major (for a cycle through the sharpest keys, 
with some fairly wide spacings), and D major (for a strong and vigorous 
sound that projects well);

- Flip through the last three _Ordres_ 25-27 of Couperin, for 
spot-checking into these remote areas similarly.  Anything that sticks 
rather closely to meantone doesn't do so well, in these pieces.

Brad Lehman


Date:    Thu, 9 Mar 2006 17:51:24 -0500
From:    Brad Lehman 
Subject: Re: Ordinaire meets Bach

>>Is it credible that Bach 'had the fretted clavichord in mind' when
writing each and every piece in WTC I? Even if not, fretting and
tuning are inseparable.<<

A reading of Richard Loucks's fine article is part of the way to 
explore such a question.

As his article is so thorough on the issue of any fretting conflicts, 
I didn't bother to cover that same material again, in writing the one 
for CI that is outlined at the bottom of this page:

Whatever instrument Bach "had in mind" as primary medium for those 
compositions -- if for some unexplained reason that would be only one 
instrument -- he could surely expect it to played on whatever else was 
handy, additionally.  Similar points come up in Robert Marshall's 
article "Organ or Klavier?" in his argument that the toccatas BWV 910-916 
were (perhaps) foremost organ music...even though they could also be 
played on harpsichord or clavichord.  This article:
(both in his own book and in Stauffer/May's anthology "Bach as Organist")

I'm trying to figure out what a primary instrument (if any) might be for 
the early E major capriccio, BWV 993.  Several passages are downright 
unplayable as written, if lacking a pedalboard.  Did JSB in his early 
20s have access to a pedal clavichord or hpsi?  (Evidence?)  Or if 
it's primarily organ, it had to be one whose temperament handled the 
excursions into the ueber-sharps.  Or did JSB recruit a third hand 
whenever venturing this composition anywhere, on stringed keyboard 
without pedal?

Brad Lehman


Date:    Fri, 10 Mar 2006 09:39:16 -0500
From:    Brad Lehman 
Subject: fretting conflicts in WTC (was Ordinaire meets Bach)

>>My point is that some of the
apparently "impossible" notes in a fretted clavichord piece are also a
convention.  It's quite possible to play through them without thinking
about them - after you get familiar with the instrument and its fretting
pattern, you just adjust automatically. I do, at least. I have no reason
to think that a Baroque musician, much more highly trained than I ever
will be, would have had greater problems. Again, there *are* such things
as truly troublesome fretting conflicts. I just haven't found any in WTC.<<

I've adjusted to them without thinking, too (on a cc with D & A free), 
but one that always trips me up and sounds lousy is in book 2, bar 98 
of the B major fugue, left hand.  The bass comes up to A#, and then 
the tenor has to go B-C#-B-C# while the A# is held.  Of course, as soon 
as we touch the B, the A# goes away...and then as soon as the B is 
released, the A# comes back, sort of.  Awfully hard not to hit the B 
too loudly each time, given that the string is already moving.  And 
then, simultaneously, the right hand has to handle the articulated 
alternation of E and D#, all on a single string....

And the right hand of book 2's F minor fugue, bar 12, where we're supposed 
to hold the top Db as the other voice goes C, Bb, Ab under it.  
Analogous spot, bar 41, where we have to hold Ab and immediately play 
the G under it for another voice.  The C and the G *cannot* be played 
without releasing the higher note first...and then it's gone.  This makes 
the thing sound like merely a single voice playing a scale, instead of 
two voices.

Just some of the "Achtung!" spots I've pencilled into my copy....

Brad Lehman


Date:    Fri, 10 Mar 2006 16:14:01 -0500
From:    Brad Lehman 
Subject: Re: another easy circulating temp, emphasizing tasteful adjustment


Gee.  All of that pedantic/patronizing diatribe flows from a misunderstanding 
of the temperament I posted, plus some presumption that I'm stupid, 
plus an obvious unwillingness to try the thing by ear.  Um, "thanks!"

One who actually tries that temperament, with my instructions here:
...might perceive that the resulting major 3rds vary smoothly from 
approximately size 3 (C-E) to 10.5 (E-G#): a variety that I believe 
is appropriate for the musical examples I listed.

That's on an 11-point scale where 0 is a pure major 3rd, 11 is Pythagorean, 
and 7 is equal temperament.  Same way I've been measuring all the others.

My instructions even included such explicit phrases as: "The C major, 
F major, and G major triads are our three best." "Test that E-G# is 
high and bright, but not quite as wide as Pythagorean."  How could 
any of this be confused with equal temperament?


For reference on those measurements: 1/4 syntonic comma meantone has 
(0, 0, 21) in each set of three major 3rds.  And regular 1/6 Pythagorean 
comma has (3, 3, 15).  Equal temperament obviously has (7, 7, 7).  
Each set of three has to add up to 21.  [Neidhardt believed it was 24, 
but he was off by a schisma in each; Sorge corrected him later.]

There also exist temperaments whose major 3rds are all sized (7, 7, 7) 
but that are not equal temperament.  Neidhardt gave two such examples 
in 1732, next to equal temperament itself.  Just slide all three 
notes of a set up or down together, preserving the spacing among them.  
That is, these notations of four sets of three numbers are not 
*unique* ways to describe a temperament; one also has to look at the 
5ths, and/or the minor 3rds, and/or semitones.

Brad Lehman


Date:    Fri, 10 Mar 2006 09:39:16 -0500
From:    Brad Lehman 
Subject: fretting conflicts in WTC (was Ordinaire meets Bach)

>>My point is that some of the
apparently "impossible" notes in a fretted clavichord piece are also a
convention.  It's quite possible to play through them without thinking
about them - after you get familiar with the instrument and its fretting
pattern, you just adjust automatically. I do, at least. I have no reason
to think that a Baroque musician, much more highly trained than I ever
will be, would have had greater problems. Again, there *are* such things
as truly troublesome fretting conflicts. I just haven't found any in WTC.<<

I've adjusted to them without thinking, too (on a cc with D & A free), 
but one that always trips me up and sounds lousy is in book 2, bar 98 
of the B major fugue, left hand.  The bass comes up to A#, and then 
the tenor has to go B-C#-B-C# while the A# is held.  Of course, as soon 
as we touch the B, the A# goes away...and then as soon as the B is 
released, the A# comes back, sort of.  Awfully hard not to hit the B 
too loudly each time, given that the string is already moving.  And then, 
simultaneously, the right hand has to handle the articulated alternation 
of E and D#, all on a single string....

And the right hand of book 2's F minor fugue, bar 12, where we're 
supposed to hold the top Db as the other voice goes C, Bb, Ab under it.  
Analogous spot, bar 41, where we have to hold Ab and immediately play 
the G under it for another voice.  The C and the G *cannot* be played 
without releasing the higher note first...and then it's gone.  This 
makes the thing sound like merely a single voice playing a scale, 
instead of two voices.

Just some of the "Achtung!" spots I've pencilled into my copy....

Brad Lehman


Date:    Fri, 31 Mar 2006 16:49:43 -0500
From:    Brad Lehman 
Subject: Re: Bach Tuning Website

and in addition to Daniel Jencka's fine set of questions....

I'll add my own following questions, which (for now) are only about 
the technical and practical details of the offered temperaments.

1. Ummm...it is remarkably difficult to temper 4ths directly upward, 
accurately by ear, in the octave(s) above middle C.  Even more so, 
if the allegedly prescribed beat rate for them is as slow as 1.0.

And pipe organs, more so again, but we'll come back to that below.

Focusing on the harpsichord version:
why would Bach prescribe such a difficult-to-hear method, especially 
right at the *start* of the tempering C-F-Bb-etc. where it is most 
crucial not to get going with cumulative errors?

Has the author actually tried it on harpsichords?

2. One of the author's stated premises at
is: "Tuning is performed for some twelve semitone contiguous range."  
What is the basis of this assumption, instead of allowing Bach to 
work across (say) an octave and a half, or some other loosely-defined 
practical range, to get the job done?  Can the author supply a 
historical precedent for tuners restricting themselves to work within 
only a twelve semitone contiguous range, wherever that range might 
happen to be on the keyboard?

If so, is there any precedent or corroboration for that range to be 
entirely above middle C?

(As I noted above, in my question #1, that octave is a particularly 
difficult place to temper 4ths directly...but 4ths are the bread and 
butter of this hypothetical method!  It appears to me that the author 
is adopting 4ths instead of 5ths, mainly to avoid any accusation of 
reading Bach's diagram right-to-left and/or "upside down"...but this 
to me is not so much a sticking point, as the impracticality of 
setting 4ths in that particular octave on harpsichords.  The listening 
point for beats in a 4th is two octaves above the lower note.)

3. Comparing the separate temperament readings offered for "Cammerton" 
harpsichord and "Chorton" or "Cornet-ton" organ: they're not simply 
transpositions of one another.  The premise of constant arithmetic 
beat-rates (with the drawing allegedly encoding two entirely separate 
layouts!) compels them to be absolutely different from one another 
in the disposition of the resulting intervals.

Why would Bach prefer/prescribe such a situation, instead of a 
straightforward transposition (where Bb major of one sounds identical 
to Ab major of the other)...or, instead of preserving the *same* 
relative disposition of major 3rds etc despite the change of overall 
pitch level (keeping some consistent Ab major quality/character to 
the Ab major of both)?

4. Where might one obtain an F 373 tuning fork, to commence the 
advertised "Cornet-ton" layout?
Now and/or in 18th century Germany?

5. Where might one obtain an A 458 tuning fork, to try the putative 
"Labyrinthus musicus" temperament of Suppig's?

6. If Suppig were into esoteric "winding index" calculations of 
secret irregular temperaments, instead of the handwritten prose 
and ratios his manuscript actually offered (in extant format, 
anyway) about theoretically equal divisions of the octave, why didn't 
he just say so?

Brad Lehman 


Date:    Sat, 1 Apr 2006 10:54:37 -0500
From:    Brad Lehman 
Subject: Re: Bach Tuning Website

Charles Francis wrote:
>>It's more than a year now since I read the "Rosetta Stone" article, which
appeared shortly after my own paper on the WTC spiral.<<

...Which is absurd in at least several ways.

1. The "Rosetta Stone" article (by me) was not yet available in its 
entirety, at the time (February 2005) when Mr Francis took it upon 
himself to write a "review" of it, and distribute same on the internet.  
The prose portion of my article was not all released until the May 
2005 issue of the journal, including the web portion of a dozen pages 
that were too long for the print.  The audio-examples portion is 
*still* not yet available from Oxford, when I checked this week 
(March 2006).

2. As for my work "appearing shortly after" Mr Francis's, that is only 
a red herring technicality by him to be able to make such a claim for 
himself.  He deposited a self-produced PDF file onto the internet, on 
a web site that prides itself openly on having no peer review process 
(Eunomios.org), and made sure to do so three or four days before my 
first printed section at Oxford was offered for web download.  This, 
to Mr Francis, gives his work prior place in some odd sense of publication.


>>I know Jorgensen argues in his red book (certainly, not the Little 
Red Book) that because of a
flawed theory predicting beats of fifths at the fundamental, the 
unfortunate people of the past were unable to hear them. But this is 
readily shown to be incorrect: my wife doesn't know that the beating 
of fifths occurs between second and third harmonics, but she still 
hears the Bach beats perfectly

On what real harpsichord?  Beats in synthesizer-produced audio files 
are not the same thing.  Please supply details of the model and stringing 
materials of harpsichord(s) that Charles Francis has ever personally 
tuned by ear using his proposed system(s)--since the hypothesis as 
presented is that Bach allegedly did so.

Brad Lehman

Date:    Wed, 5 Apr 2006 15:04:23 -0400
From:    Brad Lehman 
Subject: If temperaments were SUVs

If temperaments were SUVs.

[Warnings: has sound; requires Flash player...]

Brad Lehman

Date:    Fri, 7 Apr 2006 11:05:37 -0400
From:    Brad Lehman 
Subject: Bach Tuning Website

>>Incidently, I'm curious what you
>>make of the following:
>>(BTW, if anyone here plays violin, they can tune it: a'-2-d', d'-1-g,
>I play violin, and I have no idea what the sentence above is supposed to mean.

He's asking, instead of using your musical training and experience, 
for you to set up three *different* tempered 5ths on your violin: having 
the G to D strings beat once per second, the D to A twice per second, 
and the A to E twice per second.

And "he" isn't Mattheson or Bach or Suppig or any other 18th century 
person, but a 21st century user of Mathematica (tm) speculating how 
acoustic instruments should be tuned.  It is up to those who play and 
tune acoustic instruments to test the speculations of those who just 
play with numbers and electronics.

Mattheson's excerpts suggest that some 18th century musicians, including 
violinists, knew how to do their job using normally-tempered 5ths, and 
recognizing the correct interval quality in a suitable way "INSTEAD OF 
[using] a clock".

And, as anyone who's ever worked with any *regular* type of 5ths (any 
strain of "meantone") should know: if trying to make the D-A and A-E 
5ths the same quality as one another on this violin, and if the D-A 
is beating "2" of something, the A-E has to beat "3" of that same 
something (triplets vs the duplets of whatever speed it is).  This is 
true because the notes are 3/2 higher in frequency than their counterparts.  
Musicians don't have to know that technical reason, necessarily; but 
we do have to be able to recognize when two tempered 5ths have the same 
musical *quality* (not beat rate) as one another.


As for what "the mathematics" demonstrate about beat rates....

demonstrates by example:

By running the Briggs type of beat-rate algorithm backwards (having 
adjusted some coefficients) to find any interesting coincidences with 
starting frequencies and the 12th unknown 5th (one variable being 
independent and the other dependent), people who play with Mathematica (tm) 
instead of harpsichords may find interesting coincidences.  Then they 
may convince themselves that they've picked the best of all possible 
ordered sets of coefficients, using whatever arbitrary esoterica can be 
roped into the task.

[By the way, there's nothing wrong with Briggs's algorithm.  It simply 
confirms the truism that if one assumes eleven specific beat rates on 
particular intervals in a particular octave, and then also assigns an 
assumed beat rate as well for the missing twelfth 5th, the whole thing 
constrains the layout to start from some particular frequency on some 
reference note.  That is, such temperaments cannot be transposed up or 
down in pitch, overall, because the beat rates would change.  This truism 
is useful, for example, in assessing Owen Jorgensen's books: all the 
prescribed beat rates in his tuning-recipe explanations constrain A=440, 
and conversely, A=440 has generated all of Jorgensen's beat rates.]

It doesn't prove that Suppig or Bach used *any* of this method themselves, 
however.  Indeed, historical evidence suggests otherwise (and not only 
the observation that electronic calculation was unavailable to 18th 
century musicians).

Suppig attempted to calculate something entirely different (rational 
approximations to equal-temperament systems of multiple division), 
explicitly and mathematically.  A quip at
calls this his "exoteric" meaning, implying that some other type of 
meaning is running as some allegedly more important undercurrent?

And Bach was--according to his son--"like all true musicians" unconcerned 
with such mathematical matters.  That is, he did his acoustical and 
musical work without needing to calculate numerical coincidences.  As 
was remarked in his obituary, he could discern the interesting acoustical 
properties of a room just by taking a quick look [, no measurements].  
The man dealt with shapes and relationships, not needing to pause for 
the intermediate steps of assigning numerical calculations to his observations.

Going beyond the absurdity of having three different sizes of tempered 
5ths installed among the violin's open strings:

Furthermore, why would Bach be constrained to use some keyboard temperament 
with *differently* related intervals in it, whenever confronted with 
an occasion where he'd need to start from some pitch other than a 
favorite?  If some dude walks in with an oboe or recorder built to 
some in-the-cracks pitch, and Bach wants to tune a harpsichord to play 
along with the guy, does Bach make some *differently* shaped temperament 
dependent on the guy's starting pitch, letting all the harmony chips 
fall where they may?  Absurd.  Why should the *musical* quality of a 
key like B-flat major, in a temperament, vary as a dependency of starting 
pitch?  Absurd.  But, that's what these beat-rate algorithms force 
to happen, trying to preserve 11 or 12 specific beat rates by seconds; 
as if Bach wouldn't know any better, about recognizing interval quality 
in any tempered 5ths.

Bach, by age 37, had already been tuning harpsichords by ear for more 
than zero occasions.  Why would he strap his understanding to a clock, 
pendulum, or any other such crutch of temporal measurement, instead of 
actually just doing his job musically?  Taste and experience show how
much "twang" any particular tempered 5th has, to be correct, without 
having to quantify this by an external device.  In any temperament 
that's even remotely related to meantone principles, such a "twang" 
can be heard consistently as the intervals are listened to up or down 
the keyboard, as a constant quality, even though the beat rates are 
doubling per octave etc., or in triplets-to-duplets ratio when the 5ths 
are a 5th apart.

Brad Lehman

Date:    Sat, 8 Apr 2006 09:21:01 -0400
From:    Brad Lehman 
Subject: Re: Bach Tuning Website

>>Any statement about the equality of fifths is meaningless without specifying
a metric.<<

Preposterous.  I have here a collection of tennis balls that are equal 
for all practical purposes.  Anybody can perceive that they're the same 
size, without actually measuring their diameter, mass, or any other 
numerical "metric".

>>For example, if we take the twelfth root of the Pythagorean Comma and 
repeatedly multiply some starting frequency, we generate 'Equal Temperament'. 
(...) Perhaps some here have used logarithm tables? Over time, they 
were replaced by slide-rules, calculators and computers.<<

Logarithms, twelfth roots, and the like are merely mathematical models 
to describe phenomena.  Those phenomena can exist without such mathematical 
"metrics" that turn basically analog processes into digital [mis]understandings.

Tune a real harpsichord, entirely by ear, in 10 minutes without doing 
a single calculation of anything, and without measuring anything 
whatsoever against any external reference (not even a tuning fork).  
Understanding *might* follow.

And, now I think I'll go throw a few tennis balls for my dog to chase; 
she enjoys that game.

Brad Lehman

Date:    Sat, 8 Apr 2006 17:53:57 -0400
From:    Brad Lehman 
Subject: "roots of commas don't exist"

>>Phenomena do indeed exist without mathematics and beat rates are a 
measurable phenomenon. Roots of commas are not phenomena, however.<<

Well: since a comma-adjustment to an interval (such as a 5th) is really 
a multiplication (if we care a whit about numbers) adjusting the 
relationship between two frequencies, conceptually we're indeed working 
with a logarithmic scale, and a concept such as "1/6 comma" really is 
a root, even though it looks like simple division instead.

But consider this (and I presume that the mathematical concept of 
"continuous function" is understandable): starting from a pure 5th C-G 
and sounding both notes together, lower the G gradually and listen to 
the interval get increasingly sour.  At some moment during this process 
of continuing to lower that G, you'll meet or pass through the point 
where that 5th has been tempered by the amount "1/6 comma" because this 
process of lowering the note's pitch is a continuous function.  That 
point exists.  Phenomenal.

How do musicians recognize when that point has been reached?  Taste, 
and/or experience, and/or hands-on training, and/or counting beats, 
and/or comparing it with some other interval that was already set up.  
How do you know when you've put too much tabasco sauce onto your omelet?  
Stop and measure everything in Scoville Units, then consult some chart 
that says what numbers are decent?  Or just taste a bite to see if it's 

Brad Lehman
(B.A. in mathematics, along the way to more interesting things)

Date:    Sun, 9 Apr 2006 16:57:18 -0400
From:    Brad Lehman 
Subject: Re: Tuning Challenge !

>>Sorry, the correct link is:

As I recall, an important question was into the author's use of *any* 
real harpsichords to test *any* of his own work, before disseminating 
it to the public (since the project purports to be about JSB's tuning 
of at least one real harpsichord).  That's not about beats, per se; 
it's about research methodology.

Not in question was the author's ability to spin pages upon pages of 
web site, with synthesizer samples and games to play.  Nor was the 
question about the perceptions or amusements of web visitors, playing 
a "Tuning Challenge!" with sustained sawtooth and sine waves.

What is to be gained, in the expectation that *other* people should 
perform the vetting of one's "research" instead of oneself?  And in 
the expectation that sawtooth waves from a synthesizer have anything 
whatsoever to do with Bach or Suppig?

Also questionable is the notion that "worst to best " is a function 
of *only* the mathematical impurities; and that "Maybe you also found 
the best (no beats)" should agree with everybody's notion of "best".  
What if some people like vibrato/tremolo in their 5ths, to liven up 
the sound a bit, as being more pleasant/interesting than beatless 5ths?

Brad Lehman

Date:    Tue, 11 Apr 2006 09:57:34 -0400
From:    Brad Lehman 
Subject: Passio DNJC secundum Joannem

> > >>Krähete der Hahn zum achzigsten [after all the
> >rehearsals] Mal. Doodeedaadee squorch.<<
> >
> >Did you change from hpsi to organ, or vice versa,
> >*during* this recit between that rooster squorch
> >and the chromatic bass where Petrus goes out to bawl?
> Nah. I reserved the to-and-fro gymnastics to "es ist vollbracht," outer
> parts and middle part, and there were some very quick changes between
> arias.
> Had a problem with confined space and very different keyboard levels.

Sounds exciting!

I was supposed to play a Johannes earlier this season, but the gig 
fell through.  This was a harpsichord-only hire (with orchestra), 
IIRC--no organist, and a remarkably small stage given the size of 
chorus they planned to use.  I had practiced it into playing shape, 
but then I made the tactical error of confirming to the conductor and 
impresario that I planned to tune the harpsichord myself.

The conductor refused to entertain any talk of any unequal temperament, 
alleging that his musicians would all be confused "having to adjust" 
to it; even more, he refused to hear a half-hour demonstration of the 
specific temperament, when I assured him there would be no musical 
problems with it and I offered to play some of the bits directly.  
I pointed them toward some of the examples at
I also mentioned that this temperament had already been used 
successfully for the Johannes, by another ensemble.

As the dialogue continued I stuck to my convictions and said that I 
couldn't in good conscience play the piece in equal.  I pointed out 
(among other things) that equal makes it unnecessarily difficult to 
interpret the piece's shifting Affekt, making it harder work for 
everybody--having to project natural musical gestures more artificially, 
and so on, instead of just listening closely to the music and reacting 
naturally to its tonal tensions.  The point of the unequal temperament 
is to make the music easier and more beautiful, more naturally flowing; 
not harder or confusing.

I also proposed the slightly gentler compromise of using Sorge's 1758 
(similarly transposed to offset the modern transposition of the written 
part), saying that I'd be willing to play it thus; but likewise the 
response was Ni.  Unequal temperaments are right out.

So, we mutually and politely agreed that they could go hire somebody 
else to play it their way.  (And, no quoting of Wanda at this point!)  
I forgot to go hear their performance; maybe I should have made greater 


I'm listening to the Fasolis recording of Johannes this morning.  
Pretty interesting interpretive choice in the first movement already: 
treating the repeated bass strokes strongly, rather like an anticipation 
of nails into wood.

Brad Lehman

Date:    Tue, 11 Apr 2006 11:50:40 -0400
From:    Brad Lehman 
Subject: Passio DNJC secundum Joannem

>>I wonder...just wonder...if you had tuned Bach/Lehman or Sorge or
>>something similar, and called it Equal, would he or anyone else even
>>noticed? (Not that you would do anything so disingenuous.)

Indeed, probably not.  In retrospect, maybe I should have just shown 
up and done my thing without saying a word; and if someone asked why 
the harpsichord seems to sound more beautiful and present than they're 
accustomed to hearing, I could say something like: "Just trying to do 
my job of playing beautifully, and listening closely to the acoustics 
here in this hall...."  If they *really* pressed further and seemed 
interested, I could hand them a copy of the printed explanation, 
erster & zweiter Teil.

Or, my other repertoire remark of "This is equal temperament with just 
a dab of MSG in it."

Poletti schrieb:
>>Winds, if they are
>>good, will more or less ignore the keyboard temperament and play as close to
>>pure as they can push it (as a number of old texts say melody instruments
>>should). If they aren't good, they will do whatever. Strings ditto, though
>>they are much less prone to playing pure than winds, mostly because they gig
>>around too much and play too much classical stuff. Winds, too, but there
>>ain't nothing purer than a good Harmonie, even playing Beethoven. Once
>>again, I think you are projecting the keyboard tonal universe into a world
>>where it simply doesn't apply.

Obviously Paul, I know all this.  I don't expect anybody choral or 
orchestral to conform to any keyboard (and please don't confuse any 
straw-Lehman in your mind with speculators who might think that way).  
The points of having a good keyboard temperament are zwei: (1) Have 
the instrument sound as good as possible under the circumstances, for 
the repertoire to be played; and (2) to have a plausible/inspiring/beautiful 
backdrop against which everything happens, whether anybody else is 
trying to match pitch or not.  (Both points *you* obviously know!)

Brad Lehman


Date:    Wed, 12 Apr 2006 14:23:55 -0400
From:    Brad Lehman 
Subject: Re: Bach Partita in b minor

>>I don't know why one would want to play the c minor version; the 
lower pitch and the pungency of the temperament adds so much to the 
music. But anyway...<<

The pungency of *Werckmeister 3* adds some things to the music.....

(I *do* like your CD of it, David--have it on right now, in fact--but 
those F# and B and C# major chords are just way too pungent for me!  
Kudos to Owen on the instrument, too.)

Brad Lehman


Date:    Thu, 13 Apr 2006 08:03:51 -0400
From:    Brad Lehman 
Subject: St Matthew on Netherlands Radio, channel 4.

A broadcast in progress at the moment: Bach's "St Matthew".  The 
performers are the Netherlands Bach Society directed by Jos van Veldhoven.

Continuo temperament in use: Bach/Lehman 1722, D version from pages 
17-18 of the article....

The internet streaming link is here:
"luister live" at the left side of the page.  Netherlands Radio, 
channel 4.

Brad Lehman

Date:    Thu, 13 Apr 2006 08:56:33 -0400
From:    Brad Lehman 
Subject: St Matthew on Netherlands Radio, channel 4

Jede Mutter lobt ihre Butter!
And the radio-link doesn't work because it is for win-doof-s.<<

Umm.....please explain: what is the basis for criticizing/scorning 
this sound in actual musical practice, in Bach's dramatically flowing 
composition, without actually being able to listen to it in that practice?

Brad Lehman

Date:    Thu, 13 Apr 2006 08:53:43 -0400
From:    Brad Lehman 
Subject: Re: St Matthew on Netherlands Radio, channel 4.

>>I like the continuo tuning, but aren´t the other musicians mostly more 
or less in equal temperament?<<

Yes, pretty close.  My intro about that, where somebody asked me to write 
a version that his teenaged daughter should be able to grok:

Brad Lehman


Date:    Thu, 13 Apr 2006 10:34:39 -0400
From:    Brad Lehman 
Subject: Mersenne 10/sec

>>"And if one can recognize these beats or throbbing without the ear, it will
not be necessary in tuning the organ, for if, for example, the pipes must
beat ten times in a second to make the tempered fifth, one will be assured
that the organ will be in tune, when all the fifths beat ten times and the
ear is not necessary for this, if the hands which hold them and touch the
pipes can tell the said number of beats through the sense of touch."<<

Interesting.  To demonstrate that Mersenne the writer is not merely 
using rhetorical hyperbole, please present a human specimen who can 
count 10 beats/second in an organ 5th by that method.  10 beats/second 
is very fast, for such direct and accurate tempering; not to mention the 
confusion of feeling the pipe vibrations individually, sorting out 
what's coming separately from beats.  (Granted, I don't have the touch 
sensitivity of a Helen Keller.)  *And* discounting any drawing of two 
pipes together, with proximity on the same chest....

It looks even more like hyperbole that "all the fifths" should beat thus.  
Mersenne was obviously smart enough to know that beats double per octave, 
usw.  Isn't Mersenne simply making the non-technical rhetorical point 
that methods of measurement exist *in addition to* listening?  "Ten" 
is such a round-off number, to catch the broad gesture of "some".  (My 
wife likes to use "fifteen" regularly in her own hyperbole, while 
negotiating traffic turns: "Oh man, I could've made it out there 15 times 
in front of *that* guy!")

Similarly, 1 beat/second is very slow for direct and accurate tempering, 
if it's attempted upward by 4ths (raising the top note of the 4th 
sharper than pure) above middle C on harpsichords.  Difficult enough to 
be impractical; but that's just my opinion from attempting this particular 
1-beat-per-second-Cammerton-scheme multiple times over half a year, on 
only three different harpsichords.

I agree with Poletti's recent anecdotal comments about the difficulty 
of tempering 8' or 4' flute registers directly, because the tone is 
too close to sine waves to generate many useful beats.  When I tried 
it again most recently (February), experimenting on a continuo organ 
with a different temperament I hadn't had opportunity to try yet on 
harpsichords (was traveling at the time), I had to set some of the 
Principal 4' pipes first as reference points even though I wouldn't be 
using them further in the music I wanted to test.  The stopped flutes by 
themselves were too congenial and beatless, for any reasonable accuracy.  
On the other hand, last year for a concert I did the B/L 1722 directly 
onto 8' Gedeckt in under 20 minutes, without any problems or references 
(other than a single A fork to start): listening for quality instead 
of trying to do anything with beats.  Get the F-A major 3rd going with 
its right quality, get the right character into F major and C major 
triads, and everything else flows from there without effort.

Brad Lehman 

Date:    Thu, 13 Apr 2006 11:00:15 -0400
From:    Brad Lehman 
Subject: Re: temperament propaganda directed to young people


>>Brad's young person's guide to his thinking and tuning ("Informal
lecture (tribute to the style of Feynman)"
is mainly only propaganda for the author's modern ideas.<<

What is unclear about my disclaimer that appears at the top of every 
single web page?
"LaripS.com, © Bradley Lehman, 2005-6, all rights reserved.
All material here on the LaripS.com web site is the personal opinion 
of the author, as a researcher of historical temperaments and a performer 
of Bach's music."


>>How does the teenager, you are adressing, get to know your recently
awaked (obviously not very developped) self critical attitude?
Don't you think that especially young people should be raised to be
critical thinkers instead of getting shiploads of unsubstantiated claims
presented in factual way?
Especially young people should learn to make the distinction between
facts, hypothesises, and beliefs. In this webpage you not presenting
anything as hypothesis. Do you think that that is a responsible way to
educate young people?<<

The personal ad-hominem quip against my character aside, I agree with 
you that I should add a couple of more-prominent notes to help a teenager 
understand differences between hypothesis and fact.  (But I disagree 
with the phrase "shiploads of unsubstantiated claims", obviously.)

Point taken; I'll make a few changes to that informal essay to try to 
clear that up.


>>And, Richard Feynman, the physicist, to whom the website seems to be a
tribute (wonder why), was not a propagandist but a scientist. (But he
can't defend himslef anymore, since he died 1988)<<

I've listened (so far) to more than 30 of Feynman's lectures on tape, 
from the early 1960s; and studied the more interesting of those next 
to the published (and cleaned-up) transcript of same, where they turned 
these improvised lectures into books.  I've also read several biographies 
of him, and the books of his personal anecdotes about all sorts of 
things--scientific and otherwise.  I like the way the man taught, 
presenting difficult ideas with directness and concrete examples, 
and the way he was able to incorporate *play* into both his 
experimentation and presentation.  It's inspiring to me as a pedagogical 

I can only attempt to explain something I feel I understand, there at 
my own web page, borrowing Feynman's *style* for the moment.  Nothing 
here about propaganda; it's an attempt to teach a complicated subject 
in simple and obvious terms.  Since the topic is music, it involves 
directly playing and testing and demonstrating music.

Brad Lehman

Date:    Thu, 13 Apr 2006 17:49:28 -0400
From:    Brad Lehman 
Subject: Re: Bach Partita in B / C minor

>>Well, maybe someone who had an instrument with nothing below C and 
no short octave would like to keep the C minor version to kick around... 
one might try retuning bottom C downwards but it might not be very 

My main hpsi only goes down to BB, but the B minor version of this 
BWV 831 goes down to AA a handful of times.

It's been too long since I last borrowed a score of the C minor 
version, as I usually just transpose ad hoc from the B version if I 
want to check something on the music's character.  Is the C version 
different enough in the bass that it really does not go down to any 
occurrences of BBb?

I tune my BB down to BBb frequently for other things; it doesn't seem 
to mind.

>>C minor tuning might have a certain pungency of its own depending on 
how one
does the flats.<<

The way I tune, "pungency" isn't the word for it...in *either* B minor 
or C minor.  The piece is a sweet thang, both ways.  Different both 
ways, different characters, but equally usable in either one.  Right 
away, if playing in C minor (and if using some temperament that too 
much resembles either Werck3 or Vallotti), one would have to watch out 
for the naked Ab-C on the downbeat of bar 3.

Key changes: likewise the F major fughetta BWV 901 turning into the A-flat 
fugue of WTC 2.  Or the C# major fugue of WTC 2 coming from a couple of 
C major ancestors.  Or the Eb fugue there existing also in D.  This last 
one is less of a switch, as my D major and Eb major sound rather similar 
to one another.  But those others, a definite change of character when 
the music gets transposed...and yet equally usable either way.

Brad Lehman

Date:    Thu, 13 Apr 2006 18:03:15 -0400
From:    Brad Lehman 
Subject: L'Epineuse

>>The rejigging of the little C/G major Passepied into B/F# major 
(with excursions via C# major) is really quite bizarre for any 'French 
style', (...)<<

What about the pointy one, in Couperin's 26th ordre, 1730 some years 
ahead of Bach?  F# minor rondeau, with a nice little couplet in F# major.

When Bach flips from B minor to major in that 831 passepied, who says 
it has to sound bizarre?  Just because *some* temps have a cruddy B-D# 
and/or F#-A# and/or C#-E#, doesn't mean the bizarrity they impart is 
native to the piece.

Brad Lehman

Date:    Thu, 13 Apr 2006 18:25:13 -0400
From:    Brad Lehman 
Subject: pure Bb-D-G#

>>The chords 4:5:7 (roughly Bb D G#) 20:25:28 (C E F#) and 25:28:35 
(Bb C E) should certainly be heard at least once with absolute purity 
on a harpsichord, by anyone who likes meantone. (...)
Perhaps some people prefer the slightly out-of-tune 5:7's in ordinary 
1/4 comma meantone. Views?<<

Agreed, those are excellent sounds.

On a whim I stuck an E on top of Bb-D-G#; the E being pure from the G#.  
Then I had an uncontrollable urge to resolve it up to F anyway, and then 
pop on a CD of Wagner's Liebestod.

Count me as one who likes to have a *little* bit of motion in that 
Bb-D-G#, i.e. ordinary 1/4 comma's sound, instead of that D-G# tritone 
pure.  If the chord is just gonna sit there with no zot to it at all, 
we've swung too far toward basking in a puddle instead of moving forward.  
(Everyone's mileage may vary on that point....)

The thing that makes Ligeti's Hungarian passacaglia cool, for me, is 
not that series of stagnant pure M3/m6 all the way through the ostinato,
but rather the beating and forward motion created by the free voice, 
contradicting all that blissful purity.

Brad Lehman


Date:    Thu, 13 Apr 2006 18:34:50 -0400
From:    Brad Lehman 
Subject: Re: Mersenne 10/sec

>>You are doubtless correct that Mersenne understood that the beat rate
doubles per octave. Accordingly, when he refers to 'all fifths', it would be
reasonable to assume he references the fifths used for setting the bearings.
If such fifths are equal as described in his example, the octave is the Four
Line octave and the organ would be pitched at A = 460.801 Hz. For
comparison, at 9 beats per second, the organ pitch would be A = 414.721 Hz,
while at 11 beats per second, its pitch would be A = 506.881 Hz.<<

Someone should set an organ temperament by listening way up there as 
the bearing octave?  Please pull the other one.  Oh, wait, the other 
one is the forcing of these three pitches as result.  Both have now been 

>>By ear, I am able to distinguish fifths in the Four-Line octave beating at
10 Hz, from those at 9 Hz and 11 Hz. Not everyone can do this, however. An
experiment would determine whether such differentiation can be made in a
tactile manner as Mersenne intimates.<<

The question wasn't about "distinguishing" them by ear.  It was about 
*counting* them by ear, like one-two-three-four-five-six-seven-eight-nine-ten 
(or in French or whatever, or without words) within one second, accurately 
enough to rely on them as gospel.  I should hope that anybody present who 
actually tunes by ear (plus some who don't!) can "distinguish" that, 
without necessarily using it to set bearings as the principal section of 
the keyboard.

I don't trust any speeds above about 7 beats/sec, to set any intervals; 
but that's just me.

>>My best wishes to all for the Easter season.<<

Something I can agree with!  Cheers!

Brad Lehman

Date:    Fri, 14 Apr 2006 11:53:04 -0400
From:    Brad Lehman 
Subject: Re: 'temperament proganda' - veering OT

>>Musicians, organ builders and others have been convinced by Brad's
matter-of-factly style that he "discovered" Bach's tuning recipe, and
that the alleged "Bach-temperament" should be used. And all repeat
Brad's uncritically proposed wording of "Bach's" temperament or tuning.<<

What organ builders and musicians allegedly repeat such an "uncritically 
proposed wording"?  Please, who are these straw-men?

Anyone who invests the time to read the scholarly articles (and not 
merely surfing the web as pseudo-library) should be formulating his/her 
own critical response to the evidence as presented, evaluating the 
hypotheses, and not simply taking anyone's word for it.
Taylor & Boody call it "Lehman-Bach temperament" in their printed literature 
that I've seen.  Their web page about that particular Opus 41 organ says 
currently: "This organ is tuned in an unusual temperament, as described 
by Bradley Lehman and derived from the title page of J.S. Bach's 
"Das Wohltemperierte Klavier"."

In their newer Opus 46, which incidentally is being dedicated in recital 
on Sunday May 7th coming up soon:
...they simply call it "Lehman-Bach temperament":

Orgelbau Koegler says this about theirs, in Helsinki:
"Neue Orgel in Finnland Helsinki Laajasalo, Die erste Orgel in Europa 
mit der neu entdeckten Bach Temperatur von Bradley Lehmann."

The printed booklet in Richard Egarr's CD says: "Temperament: 
Lehman-'Bach' temperament (2004)".  And in his booklet essay he hedges 
to: "A timely coincidence, musicological discovery allowed me to use 
what may be Bach's own tuning system for this recording. (...) I am 
sure that Lehman's idea will not receive universal acceptance, but 
I find it utterly convincing."

The printed booklet in my organ set (on the page of organ specs) simply 
gives the recipe by 5ths without giving the temperament *any* name.  
Describe the animal by its genome, not by anybody's name for it.  
The essay leaves things open by saying: "what I believe to be" Bach's 
(so does my biography in there).

My harpsichord CD's essay makes it even clearer than that, without 
naming it "Bach's temperament" or any such:
"The present album and its companion set for organ (LaripS 1002, 
"A Joy Forever") explore the specific temperament I have proposed 
as reconstruction of his practice, the discovery of his documented 
expectations. This tuning method is deduced from historical context 
and Bach's written evidence: his drawing at the top of the 
Well-Tempered Clavier's title page (1722) plus an analysis of his 
music--in that book and elsewhere."

Granted, some readers might not be fond of the phrase "his documented 
expectations".  But again, it's an enthusiastic statement of personal 

If musicians are using this temperament elsewhere, for music not by 
the Bachs, I hope the #1 reason to do so is that they fancy the 
musical/plausible sounds it creates in practice, for the music at hand; 
and not because of any names or allegations of its parentage.  I've 
been using it here myself quite a bit in Couperin and Mozart, for the 
simple reasons that it makes both the instrument and the music sound 
terrific, and I don't ever have to stop and retemper the instrument 
between compositions, and it inspires me to give [what I believe to be] 
well-committed and beautiful performances.


On the same website, there is a small article by Brad in which he writes:
   "This was Bach's preferred system, and it solves all the practical
problems in his music and the music of his sons. Indeed, it turns out to
be an excellent tuning solution to play all music, both before and after
Bach's."  I don't know how one could present something more as a matter of
fact. And no disclaimer there makes a reader aware, that the author is
presenting just his own invention.<<

I will send them a suggested revision for that web page, today, to make 
it clearer that it's a hypothesis (and that it's my *opinion*) instead 
of universally accepted fact.  Those several paragraphs there were 
written more than a year ago, before we started inserting the hedge-words 
to make such things more palatable in the writing style.


All modern historiography, on any topic, is hypothesis stacked upon 
hypothesis in multiple layers -- is it not?

To get even this far, back in 2004, I had to peel off and re-question 
pretty much everything I had ever encountered in the standard 
historical-tuning literature, to identify what's somebody else's 
long-standing hypothesis that I don't necessarily agree with as fact.  
Among the biggest ones that were all in the way of my initial 
understanding: (1) Ab-C must be smaller than or equal to E-G#; 
(2) Db-F should be big and ugly if anything is; (3) Vallotti is "late" 
(1750s+) rather than a 17th century commonplace; (4) Bach gave even a 
fig about any of Werckmeister's tuning schemes; (5) measurement schemes 
of deviation from equal temperament (and evaluations of quality thus) 
are of any use whatsoever in understanding 17th/18th century work.

Those premises (other people's hypotheses as ersatz common-knowledge 
facts!) were all blocking my thinking, before that.  Whatever Murray 
Barbour and other 20th-century writers thought was highest priority 
in analysis of the historical record, wasn't necessarily so.

For a different example along this line: the notion (seeming silly to 
me now) that Young's #1 is somehow harder in practice than #2.  
Young's #1, if we look at its genome *in practice actually on a keyboard*, 
is nothing more than dinking two notes very slightly off their Vallotti 
positions, making them nicely transitional between the 5ths on either 
side....  The difference between paperwork historiography/speculation 
and musical practice, setting it up and playing music.

In Barbour's defense, though: I should point out that the Vallotti 
pattern wasn't available to Barbour to recognize that connection.

Anyway, on Barbour's writing about Young #1, plus Young's own comments, 
generations have been saddled with the wrong impression that (1) the 
math is too hard to understand, and (2) #1 couldn't be a practical 
temperament.  Unfortunate, because the thing is so easy to set up...*if* 
we don't jack around with ratios or beat rates or cents or any other 
calculations first, but just sit down and do it at the keyboard.

Brad Lehman

Date:    Fri, 14 Apr 2006 17:25:07 -0400
From:    Brad Lehman 
Subject: Re: 'temperament propaganda'

>>I think I have pretty much seen and read everything from Werckmeister's
published writings on temperament (also those books in which temperament
forms just a minor part).  In what I read there is no reference to
Italian temperament practice at all, and none of his schemes comes even
close to Vallottis still 100 years later unpublished scheme.<<

1. Run six pure 5ths consecutively around the flat side.
2. We therefore still have a whole Pythagorean comma to deal with, among 
the remaining notes which all happen to be naturals, to meet where we 
ended up from step 1.
3. Temper some or all of those natural-5ths until the result sounds good.

Is that a description of Werckmeister 3, or Vallotti, or Young 2?  Or all 
three of them, in their basic tuning strategy?

Nobody ever thought of--or tried--such a tempering strategy on their 
keyboards, until these three guys happened separately to write down a 
favored result of such a process?  (Especially any nameless, straightforward 
tuners finding it expeditious to run as many pure 5ths as they can bear 
to listen to, and never troubling to write it down?)

Things don't exist until a pedigreed authenticated extant document says 
they do?

This isn't proof that such a common practice existed; I'm merely questioning 
the defaults here in the methods of reasoning.  Can things be *even possibly* 
true about a historical situation, although the documentation is lost or 
perhaps never existed on paper?  Or are such things automatically false 
until proven true?

I assume that every member of this list ties his/her shoes.  But how do 
we know?  Have any of us ever written down the method to do so?  Somebody 
trying to figure out our culture, looking back 200 years from now, will 
they find *any* printed documentation about tying shoelaces?

Brad Lehman
Date:    Fri, 14 Apr 2006 22:06:15 -0400
From:    Brad Lehman 
Subject: Re: simple historical question

>>Had anyone published *any* theoretical irregular tuning schemes involving
1/6 PC and 1/12 PC fractions, before Neidhardt in 1724? Before that, the
only known use for 1/12 comma was to tune equal.<<

Like the Metius temperament pre-1650 that Barbour presented as an anomaly?  
Page 177ff and table 177.

What do you think of Barbour's hypothetical temperament in table 125, where 
one takes a fretted clavichord and retunes the naturals to 1/6 comma "as 
in Bach's day", and lets the sharps fall where they happen to fall?  Barbour 
gives those as approximately 100 cents from their higher neighbors.  Page 

Brad Lehman
Date:    Sat, 15 Apr 2006 08:24:47 -0400
From:    Brad Lehman 
Subject: Werckmeister's proto-Vallotti, etc

>>In my previous mail I asked Brad, referring to
   "You have claimed earlier that the "Vallotti"-scheme  (published
    from the unpublished mid-18th century ms., 2oo years later in
    1950) was of venetian origin (you called it "Proto-Vallotti")
    and you claim that Werckmeister knew it.
       The substance for such claims?  You refer to Devie as a
    secondary source, who according to your mail refers himself
    to a ternary source (Jean Bosquet).<<

Yes; see my footnote #45 of part 2, where I provided the same information 
in a link to Devie's book.  See also the paragraph to which it is attached, 
on page 217, where the whole thing is in passing.  My paper is not primarily 
about Werckmeister (and, as I have said elsewhere within it, I do not 
put much faith in Werckmeister/Bach tuning connections anyway from the 
standpoint of direct keyboard practice).  Could my writing style in that 
paragraph on page 217 be improved?  Yes, probably.

And, did I make a reasonable attempt to find out more about this 
proto-Vallotti/Werckmeister, following up the lead of Bosquet?  YES.  
I requested Bosquet's papers through our interlibrary loan, but only 
one of them showed up: the 40-page survey article "Sons musicaux, 
intervalles et tempéraments".  It didn't have this proto-Vallotti/Werckmeister 
point in it, unfortunately.  I tried to locate contact information for 
Bosquet; no luck.  I would appreciate some, if anyone here has it.

So, I failed to follow one of my offhand remarks (page 217 bottom left) 
out to cross every T and dot every I, in the prioritization of getting 
the main substance of the paper done.  Does this omission make me a bad 
scholar, by your standards?  Perhaps.  How much does this affect/degrade 
the main substance of my paper, that one of the ancillary points off the 
track of my argument isn't followed up to your satisfaction?

This comes back to my remarks yesterday, again, about how we take defaults 
when information is missing.  Is any possible knowledge (to Werckmeister) 
of a 6+6 system of pure 5ths and tempered 5ths automatically 
false/impossible, until demonstrated true with absolutely pedigreed 
extant documentation?  Or--my approach--is it reasonably and possibly 
true, even though Bosquet's reference is only a ternary source to me?  
Are people from 300 years ago allowed to have known things, even if they 
didn't write them down in a format that modern researchers would allow?

This issue, indeed, needs to be (in part) about epistemology.  Reasonable 
doubt to one researcher isn't necessarily the same reasonable doubt to 
another.  It's a matter of style in epistemology.  Mine is not logical 
positivism.  Any readers will have to take that into account, evaluating 
the argument and evidence that *are* presented, by whatever rules suit 
their own fancy and own epistemology.  I suspect that the argument can't 
be proven at all, *within* a requirement of logical positivism; but that 
doesn't mean that the argument is necessarily wrong.

>> The scholar in you then
    should not just rely on such information, but seek the
    original information: Did you check that source (Bosquet) and
    where he got it from? Did you check finally in Werckmeister's
    writing, in he facsimiles - which are anyway basics for an expert
    in temperament?
    Why didn't you peel at this part of the fruit?<<

I have Werckmeister's 1691, but unfortunately not the 1681.  (Plus, of 
course, Rasch's excellent introduction to the 1691.)

Why didn't I peel at this part of the fruit?  Because, again, the paper 
is not primarily about Werckmeister, and this (to me) is all a rather minor 
point that I have brought up in passing.


>>You seem to propose that a structural similarity  already allows to
construct a meaningful link beyond the structural aspect.<<

Well, yes, at least to me: I think in structures when I'm sitting at the 
harpsichords, figuring out what it would take to convert temperament X 
into temperament Y.  How many notes to move, how they're related, how the 
temperament shapes are put together.  If I find a very easy concordance 
between two methods, I take them as reasonably related.  (Like the Vallotti 
and the Young #1 I have pointed out: move the F and the B.)

For another example, a biggie: John Barnes's "Bach" temperament is 
identical to Vallotti except for moving the single note, B...and even 
though Barnes didn't say so in his article about this.

Did Barnes *know* this?  We don't know; he didn't say, one way or the 
other.  The fact remains that we can convert these two temperaments to one 
another, merely by dinking the B.

This is how I think.  Maybe it makes me a bad historian?  But, the evidence 
is sitting right there installed into the keyboard: 11 of the 12 notes are 
the same.  I think as a practitioner, in my epistemology.  Does the thing 
work, and is it reasonably easy to deliver?  Does it make the music sound 
good, and therefore be worth bothering with?  And if so, mightn't other 
practitioners have done rather likewise, without bothering to say so?

Similarly, my own proposed "Bach" temp has 7 of the 12 notes identical 
with Vallotti.  It comes back to the same observation that the Vallotti 
layout is very easy, easy enough and symmetrical enough to be 
common-practice, whether anybody before Vallotti wrote it down (and 
distributed it) or not.  Same core of regular 1/6 through the naturals, 
and then we adjust stuff.  This is not hard; and I see that same core 
of regular 1/6 naturals all over the place, even in the most arcane 
Neidhardts that have very few 1/6 comma 5ths remaining, and yes, even 
in Werckmeister 3 and 4: as I said in the paper.  (Werckmeister 4's 
naturals: start with regular 1/6 comma 5ths and then knock half of them 
sideways, making them pure against a note on one side, leaving the other 
1/3 comma residue into the other side.)

1/6 comma regular layout.  We can then square up the whole flat side by 
making it a bunch of pure 5ths, if we want to.  Or we can knock some of 
the sharps increasingly sharp, making them 1/12 or pure or whatever to 
do so, spiraling upward, as far as they sound good, and until they meet 
something reasonable coming around the back side.  Straightforward 
"modified meantone" here; that's the type of structure I see.  Turn a 
regular (i.e. "meantone") 1/6 into something more circulating, to be 
able to play in more or all keys.  That's raising some sharps, and/or 
lowering some flats, until we meet ourselves somewhere that everything 
sounds decent.

Did Bach know that Vallotti layout in practice, for sure?  I don't know.  
But the thing is so blessedly simple, my default (in my mind) says: 
why would he *not* know it, as by age 37 (1722/3) he'd already tuned 
harpsichords for 20+ years.  Wouldn't he have tried, even once, the 
effect of six pure 5ths in succession around the flats, and then bashing 
through some music with it to hear what happens?  And then, found a 
few ways to make it sound better to him?

No logical positivism here.  But, it is *logical*.


1) What Venetian temperament of a "proto-Vallotti"-type (I stick here
    to that chronologically upside-down label you invented) is known
    to have existed in or before Werckmeister's time.

2) How is Werckmeister likely to have known it? Is there evidence
    for Werckmeister to have known Italian temperament sources? Does
    he refer to any Italian source or mention Italy, Italian culture,
    in connection with his writings about temperament?

I don't know.  On a hunch I have been following up Poglietti, on and 
off, but it hasn't panned out yet working from illegible printouts 
from the microfilm.  Any assistance in this would be welcome.

Brad Lehman
Date:    Thu, 20 Apr 2006 10:12:36 -0400
From:    Brad Lehman 
Subject: Re: 'Harmonic' bells and polytone bells and...

>>The traditional European bell, the result of the efforts to get a
definite pitch, produces the fundamental and well-tuned (though how
exactly I'm not sure) 2nd, 3rd and 4th harmonics - along with a strong
and persistent overtone at a minor tenth, 2.4 times the fundamental.
So even harmony using major triads becomes problematic. Temperament
for such bells is a nonsensical concept and any attempt to apply
temperaments designed for keyboard instruments with a true 5th
harmonic is utterly pointless. One might as well have 'well-tempered'

I'm sorry, but the assertion "temperament for such bells is a 
nonsensical concept (...) utterly pointless" appears to me to be 
absurd.  Here's why:

The set of bells, as a whole, needs to be able to play music written 
in most or all of the 24 major/minor keys.  That is, all the *scales* 
must be usable, whether melodically or harmonically or both.  It's 
not merely about sounding three bells of a major triad together.  
It's about handling all the possible two-note intervals (dyads) and 
having smooth-sounding melodies as well, within all the scales that 
the set of bells is expected to play.

If the owners/players of the bells agree that they'll only play in 
the easy scales of few sharps/flats, a meantone-based system makes 
some sense (not withstanding the semitones of vastly different sizes, 
melodically....).  If not, there has to be a smoother circulating 
system and/or equal temperament, for the fundamentals.

Will there be some clashes in bells, among various overtones, no 
matter what we do with centering the fundamental pitches?  And 
especially so, due to a prominent minor 10th?  Of course.  But the 
point is to provide overall utility for the instrument, is it not?  
One has to temper the scale *somehow* to be able to use more than a 
few isolated scales.  (As you know!)

For five years I lived within easy earshot of a carillon.  I heard 
it all the time, playing all manner of music in all sorts of keys.  
That one happened to be in equal temp.  It wouldn't have had to be; 
only something close to equal.  A slightly irregular configuration, 
if done carefully, would have made the instrument as a whole 
somewhat more interesting to listen to: more variety of musical 
resources, stemming from subtleties of intonation.

Brad Lehman
Date:    Fri, 21 Apr 2006 18:40:22 -0400
From:    Brad Lehman 
Subject: JSB's and Friedemann's math skills

>> (Der seelige war, wie ich...s.o.) In diesem Zusammenhang darf 
das J.S. Bach zugesprochene Urteil
über C.Ph.E. Bach nicht fehlen: " 's is Berliner Blau! 's verschiest!"
 (The blessed was, as I am......)
 In that context one may not miss JSB's conviction about C.P.E:
 ""it's misplayed Berlin blue!""

 C.Ph.E. Bach wird als Zeuge für eine adäquate J.S.-Bachforschung 
 Hence C.P.E. becomes doubious as witness for adequate JSBach research.

Granting for the moment that JSB personally didn't fancy *some* of 
CPEB's compositions:

...How does that (of itself) make CPEB *in any way* unreliable/dubious 
in his reportage of his father's work, where we can say with reasonable 
certainty that any particular statement by CPEB about JSB is false?  
Especially, *because of* such criticism offered in the opposite 
direction by JSB?  [What if CPEB actually took his father's criticism 
to heart and revised his own practice, thereafter, benefitting from 
it?]  Where, in any of CPE Bach's writings, is there evidence that he 
was in any way petty against his father or his father's work, as 
retribution for the "Prussian blue" quip?

Throw out the witness of CPEB entirely, as allegedly too unreliable, 
on something like that?

Let's try another example, just to exhibit how absurd the construction 
is.  A police officer stops me and gives me a reprimand, because one 
of my headlights is burnt out.  Does that event (of itself) make me an 
unreliable witness, forever thereafter, in any commentary I could ever 
possibly offer about the local police force and the way they conduct 
their business?  Whether I go and fix my headlight, or not?  What if 
I go about saying especially nice/appreciative things about the police 
force, in my thankfulness that it was only a reprimand and not a 
formal citation for breaking the law?  The guy was really polite and 
everything, and I was indeed in the wrong: the light was burnt out 
(less bright than Prussian blue, even!), creating a public hazard.  
But the event itself somehow makes me unable to report fairly about 
any item of local law enforcement?  If anything, the event makes me 
even *more* appreciative of their practice, because at the time I did 
not know that the light was out, or that my driving that evening was 
offensive in any way.

Brad Lehman

Date:    Fri, 28 Apr 2006 15:35:48 -0400
From:    Brad Lehman 
Subject: Re: Lecture of Mark Lindley with (live) music examples, Heidelberg

> This will take place in Heidelberg next week at the Musicology
> department. If all goes well it should include live harpsichord music,
> with tuning supervised / realized by Prof. L. I
> "Was war für Johann Sebastian Bach ein wohltemperiertes Klavier?"

Wish I could be there!  Both to hear the presentation (of course), and 
to try out especially BWV 823, 906, 993, 910, 998, 590, 831, 802, 803, 
805, 795, 777, 848, 857, 859, 872, 880, 881, 886, 892, 903, 846, 
808(Sarabande), 591, 622, 702, 949, 547, and 1079 on whatever he's set 
up for the demonstration.  Longer list here:

Lindley brought up *some* of those in his 1985 and 1997 articles, but 
it's these other pieces where some of the most challenging stuff happens.

Norrback's book makes excellent points about BWV 544, too....

Brad Lehman

Date:    Mon, 1 May 2006 09:05:52 -0400
From:    Brad Lehman 
Subject: Egarr's BBC 3 broadcast yesterday

Yesterday's live broadcast of BBC 3's "Early Music Show" featured 
harpsichordist Richard Egarr playing music by Frescobaldi, Louis Couperin, 
Handel and Bach.  He used my temperament as a convenient tuning for 
that program, and talked about it briefly between the pieces.

It is available through this week on the "Listen to the latest 
programmes - Sunday" link at this page:

The Taylor & Boody organ Opus 46 is being dedicated next Sunday, also 
using this tuning.  That one is a two-manual, 13 stop instrument at a 
Presbyterian church in the Shenandoah Valley, Virginia.

Other recent and upcoming usage (but I usually don't know about it 
very far in advance):


Brad Lehman

p.s. Thanks to another HPSCHD-L member for tipping me the news about 
Egarr's program yesterday; I hadn't known ahead.

Date:    Mon, 1 May 2006 10:28:40 -0400
From:    Brad Lehman 
Subject: test pieces for Bach tuning

I wrote:

>Wish I could be there!  Both to hear the
>presentation (of course), and to try out
>especially BWV 823, 906, 993, 910, 998, 590, 831,
>802, 803, 805, 795, 777, 848, 857, 859, 872, 880,
>881, 886, 892, 903, 846, 808(Sarabande), 591,
>622, 702, 949, 547, and 1079 on whatever he's set
>up for the demonstration.  Longer list here:
> http://www-personal.umich.edu/~bpl/larips/testpieces.html

...and Tom D replied:

>How do I hate BWV numbers... let me count the ways. I suppose they are
> quick to write, but I make no progress in remembering any of them
> aside from 565.

OK then, mapping those numbers onto titles:

823 - F minor suite
906 - C minor fantasia
993 - E major capriccio
910 - F# minor toccata
998 - Eb prelude/fugue/allegro
590 - F major pastorella, esp. "aria" 3rd mvt
831 - B minor partita/ouverture
802-805 - Duetti
795 - F minor sinfonia
777 - E major invention
848 - C# major WTC 1
857 - F minor WTC 1
859 - F# minor WTC 1
872 - C# major WTC 2
880 - F major WTC 2
881 - F minor WTC 2
886 - Ab major WTC 2  (Play also the earlier F major version of its fugue!)
892 - B major WTC 2
903 - Chromatic F&F
846 - C major WTC 1
808 - G minor English suite (Sarabande)
591 - Harmonic Labyrinth
622 - Eb "O Mensch"
702 - Bb "Das Jesulein soll doch mein Trost"
949 - A major fugue
547 - C major praeludium/fugue
1079 - two ricercars of the Musical Offering, plus the modulating 
canon using all 12 minor scales

Brad Lehman
Date:    Tue, 2 May 2006 12:42:49 -0400
From:    Brad Lehman 
Subject: Re: ... but some temperaments are more equal than others

Tom Dent wrote:
>>I find it a bit depressing that someone playing both Frescobaldi and
Handel G major + a few major-key Goldbergs would choose a near-equal
temperament, with or without historical sources, over something purer
& more unequal. Why - unless you like ET for its own sake?<<

Tom -- what temperament do you recommend personally for Frescobaldi's 
"Cento partite"?  (With the premise of playing on a harpsichord that 
has only 12 keys per octave....)  And please play all the way through 
the piece before answering...noting the swapping of G# to Ab, Eb to D#, 
C# to Db, partway through.....

And no, that's not the Frescobaldi piece that Egarr played on the 
program, but I think it's an interesting question about Frescobaldi 

What you're dismissing as "near-equal" and "depressing", some people 
might hear as "subtly nuanced" (or similar), and as such it might be 
more interesting to listen to than a regular temperament is...in 
addition to its flexibility for modulations.  The music is arguably 
*more* colorful here, rather than less (as it would be in real equal, 
or in *regular* 1/4 or 1/5 or whatever).

Food for thought.....along with the notion that "purer" is in the ear 
of the beholder.  Egarr's broadcast sounded exceptionally pure to me: 
"pure" not to major 3rds sitting there as stagnant sonorities, but pure 
to accuracy and smooth melody.  The pieces all had what seemed to me to 
be a lovely and easy flow, not only in the intonation but also in the 
phrasing/delivery.  And the melodies were considerably less bumpy (i.e. 
"purer" in a different sense) than they would have sounded in any regular 
temp, or a more slightly modified 1/5 or 1/4 as you suggested.

By the way, I agree with you that F's toccata settima *also* works fine 
and sounds terrific in more regular temperaments.  That doesn't mean it 
*must* or *should* necessarily be played in regular temperaments in all 
situations, however.  (Which isn't what you said, either, but I'm merely 
pointing it out.)


Brad Lehman

Date:    Tue, 2 May 2006 14:30:41 -0400
From:    Brad Lehman 
Subject: an hour to set up meantone?!

>>Unless they were all in ET or 'Vallotti-Young' before, I can't help
but think this is a huge backwards step. So no-one can teach or
perform (let alone practice) in meantone or ordinaire without
occasioning at least a solid hour of retuning?<<

What "solid hour of retuning" is this?  The regular meantone layouts are 
trivially easy in under 15 minutes.  Just get the first few major 3rds 
going correctly, and then each time you set a 4th or 5th after that, 
check that the newly created major 3rd has the same quality as those 
first ones.  The middle region of the keyboard takes under 5 minutes, 
and then everything after that is merely copying octaves/unisons.

I've already posted at least five or six sets of practical instructions 
to do *any* regular temperament....

June 2004:

September 2004:

April 2005:

September 2005:

March 2006:

Because my virginal had drifted sharp enough to notice (from recent 
weather shifts), I gave it a completely fresh go just now with a regular 
1/5 comma, and timing the work.  9 minutes.  It usually takes less, 
but a couple of my pins have gone slippy.

Brad Lehman

Date:    Fri, 5 May 2006 10:39:50 -0400
From:    Brad Lehman 
Subject: Re: On the fly

>>But the solution here - a small piano - is no solution at
all. The sound of a piano is so radically different from any historical
instrument that it makes the whole exercise pointless. Too much
inharmonicity and not enuf overtone development.<<

I agree; pianos don't tell us anything subtle enough about any 
temperament, especially in the treatment of 3rds and 10ths.  They just 
more generally sound decent or indecent.

>>The best system for demonstrating temperaments is a good (and I mean GOOD -
there are lots of bad ones) synthesizer. (...)
Furthermore, with a patch to Sibelius or other sequencer, you can demo
whatever music you want, regardless of your personal playing skills.<<

I disagree with this part, as to "demo whatever music you want, regardless 
of your personal playing skills".  Sequencers, no matter how good they 
are at supplying notes with whatever detail is programmed into their 
files, still don't react to the temperament *itself* as part of the 
musical interpretation, if the temperament changes from the time when 
the sequence was programmed.  Sure, they're fine to demo a bunch of notes 
assembled end-to-end, but such a demo is not musical interpretation.  :)

As a human player, whether it's simple or complex music, my articulations 
and timings are profoundly/directly affected by the sounds I'm getting 
back from the temperament: adjustments a pre-recorded computer file 
can't do.  If the intonation is giving me an extra bit of resonance in 
a chord, or extra poignancy in the melodic inflection of an interval 
within a phrase, my articulation/timing adjust to bring it out.  My 
dynamics do, too, on clavichord or other keyboards that are more 
touch-sensitive in that regard than harpsichords and organs.  This is 
"on the fly" adjustment without much conscious effort of cognition.

I should think that any sufficiently sensitive players do likewise; 
there's nothing terribly special about my own practice of reacting to 
the musical sounds as they go along.  Hit all the right notes at the 
right times...informed by intonation, room acoustics, and more, not 
only by the score!  But sequencers can't/don't/won't do that.  They'll 
send the same timing/articulation whether the synthesizer is tuned in 
1/3 comma meantone or equal.  If the sequence was prepared by somebody 
really playing a keyboard, it becomes important what temperament was 
onboard when he/she did so, as to the nuances that make it into the 

Take a (speculative) example of early-20th-century sequencing: 
Rachmaninoff making rolls for reproducing pianos.  I dare say that 
*if* he had made those recordings on a more unequally-inflected piano, 
he might have done some of his phrasing or timing or tempos differently: 
reacting to the sound he was getting, different from equal.  To 
reproduce his rolls accurately now, whether computer intervention is 
in there or not, one would have to reproduce not only the touch 
sensitivity of the keyboard and pedals from the piano he used, but also 
its temperament.  (Which Wayne Stahnke's reproductions have done, 

Count me old-fashioned.  IMO, temperament demos are most startling and 
clear when done on Italian or Flemish-styled harpsichords, and on 
real pipe organs (including availability of purely-tuned mutations 
and mixtures).  Granted, the latter involves considerably more work 
than the former.


Anybody else on-list coming up for the Taylor & Boody #46 dedication 
this Sunday afternoon?  The player is Mark Brombaugh.

They had a nice little recital there last Saturday afternoon, too.  
The organists (neither of which was me) brought out repertoire ranging 
from Clerambault and Buxtehude to eight 20th-century compositions.  
Yes, even Alain's "Litanies" with its chromatic extremities.  A piece 
by Isaac Barton from 1917 brought out some surprising colorful effects, 
switching suddenly from naturals to a rack of flats, and it was like 
listening to an entirely different instrument...even though the man 
hadn't touched his registration, but merely played the sudden modulations.

Then I had about half an hour afterward to try out some Pachelbel and 
some improvisations, for fun.  A very nice Nasat and Cornet on there.

The other organist on Saturday tried out the odd effect of using the 
Cornet III all by itself as a solo stop, during the concert!  I'm not
sure Mr Taylor intended it to be used in that way, but it *did* make 
a convincing-sounding (if ethereal) musical effect, IMO.

Brad Lehman

Date:    Fri, 5 May 2006 10:57:52 -0400
From:    Brad Lehman 
Subject: Re: On the fly

>The usual method of tuning requires listening to the beats of the 3rd
>harmonic of the A and the second harmonic of the E which gives a beat
>frequency of 2.8 beats per second which is a value you can use.
>Fortunately harpsichords, clavichords and many organ pipes are rich
>in second and third harmonics.]
>I suspect that even when tuning unisons in real life, one homes in on
>the beats between harmonics rather than fundamentals.

Yes indeed.  That's part of the reason, also, why one can't substitute 
a string of the wrong material or gauge when replacing one: the unison 
with the old string won't work out properly.

Beats between fundamentals are too slow, and low-pitched, to be of much 
use...unless we're on soprano recorder or the next two smaller sizes, 
already way up there in pitch.

Brad Lehman

Date:    Fri, 5 May 2006 17:47:21 -0400
From:    Brad Lehman 
Subject: BWV 622 and 671, etc

Thanks for the report, Tom.

I'm especially intrigued by this part:

>>On the flat side we had some fine excerpts from O Mensch Bewein.
[BWV 622] (...) The extremely chromatic end of the big Kyrie BWV 671
was played, to somewhat startling effect. That was the one point at
which I thought the Lindley style might be too unequal - indeed it is
more unequal than the Neidhardts.<<

Did he then play my recordings of those two pieces, from the CD set?  
Those recordings exist, in part, because those several compositions 
illustrate some of the thorniest spots in temperaments that (in my 
considered opinion) are unbalanced.  I chose to play/record them because 
they sound not rough, but spectacularly beautiful in the spots where 
other temps have problems.  They would be excellent illustrations for
this type of lecture.

Has Lindley mentioned that he's even *heard* any of my recordings yet, 
since the early January release?  These two:


>>After the formal lecture Lindley outlined to me the crux of his
musical case against Brad by first saying -  'His E major is like my
F# major' [i.e. too hard & impure] - playing the E major prelude
suitably transposed - then 'His Ab major is too relaxed' - playing
that prelude in a lively and detached manner. We also noted the open
fifth in b.5 of that piece, as opposed to the full Eb major chord when
the analogous passage occurs in the dominant.<<

Why should one have to play the Ab prelude in a "detached manner"?  
Bach knew how to write quavers followed by rests when it's supposed to 
be a crisper sound, for example in the Ab fugue that is this prelude's 
partner.  One shouldn't have to make things especially detached due 
mainly to temperament considerations (i.e. avoiding lousy-sounding chords).

And why should the open 5th in bar 5 say anything, one way or another, 
about the need of that particular 5th to be pure?  The sound is (arguably) 
more interesting/lively if that 5th is slightly *impure*...which point 
I've made also in regard to the D# minor fugue of WTC 1.  (If pure 5ths 
suddenly intrude into a sparse texture, the music can sound suddenly 
dead...nothing vibrating.)  I hope the argument wasn't about that 
downbeat chord of bar 5 *itself* avoiding having a C in it; Bach has 
already used one of those in bar 3, with the C included!

And what's "too hard and impure" in my E major, for the WTC (either 
book) or for the invention/sinfonia in E; or for any of Bach's E major 
or A major suites?  Especially so, as I've found from lots of playing, 
that the E major and A major can foster a predominantly *melodic* manner 
of playing, with smooth lines moving forward?

Brad Lehman

Date:    Fri, 5 May 2006 22:30:52 -0400
From:    Brad Lehman 
Subject: BWV 552

>>Then Lindley connected the Klavieruebung III (with its three-flats
dominance) with the visit to Naumburg which housed a new Hildebrandt
organ with a 'Neidhardt' tuning. Lindley's opinion is that the big Eb
major emphasis in this publication was a signal to any would-be player
that meantone or simple modifications were out. Staccato root position
chords of Ab major and Db major in the big Prelude were mentioned as a
sign of 'tailoring' to a tuning in which these chords are still not so
pure as others. <<

If the tuning argument really wants to go into this piece, BWV 552 
prelude at the front of ClUb III, fine....

Let's just not have those staccato chords be red-herrings arguing for 
dodges of rotten tuning of A-flat and D-flat major chords!  (Get off 
the chord quickly *because it sounds bad*....)  In that very same piece, 
the argument should have to deal with the obviously sustained examples 
of those same harmonies: in bars 44-50, 97-130, and especially the 
downbeat of 165.

Brad Lehman

Date:    Sun, 7 May 2006 07:16:19 -0400
From:    Brad Lehman 
Subject: Re: popular misinformation

> From:    Thomas Dent 
> Subject: popular misinformation
> In my righteous wrath yesterday I left out the worst piece of
> pseudomusicology from Egarr. Here:
> "Lehman's 'Bach' tuning is rooted in the seventeenth century (a
> sixth-comma meantone tuning)..."
> I think someone responsible should tell him just how nonsensical this
> is. The historical record for 1/6 comma meantone is a big fat zero up
> until practically 1700.

Sorry, this self-righteous ranting against Lehman's theory is getting 
really tiresome.

It's "a big fat zero" perhaps to people who haven't read J Murray 
Barbour's book closely enough: specifically pages 124-128.  And it 
should be obvious enough that the publication date of Kircher's big 
book was 1650.

Also cited there is Johann Beer's _Schola phonologica_ (1695) cited 
by Mattheson (in 1722-25).  Here's a bit about Beer (1655-1700):

As I had cited in the article, there's also the vocal tutor of Tosi 
(1723), writing near the end of a long and illustrious career as 
performer/pedagogue, and advocating conservative taste.  His system, 
too, was this 55-division (i.e. 1/6 comma)...this same widespread 
and intermediate manner of tempering, inherited from the 17th century.

I'd already mentioned on-list, more than a year ago I believe, the 
corroboration that Poglietti's tritones (given as ratios) were the 
same size as those in 1/6 comma.  But since I hadn't followed that 
part up systematically enough, I didn't make a huge point about it.  
I didn't, and don't, think there needs to be one.  The people who 
have already decided for themselves that I'm wrong aren't going to 
listen to such historical evidence anyway, it appears.

> Does this mean that we are due for a slew of recordings of 17th
> century music in 1/6 comma meantone (let alone 'Vallotti'-style)? I
> hope not. But can it be prevented if supposedly-knowledgeable people
> swallow whole what they read in the papers?

That perhaps depends (in part) if they read self-righteous rants on 
the internet, rants by people who still can't fathom or swallow the 
idea that my theory *could possibly* be correct, or that Egarr and 
other fine musicians who use it actually know what they're talking about.

Just because ranters don't know about the 1/6 comma division, aka 
55-division, as used from the 17th century forward doesn't mean 
that ordinary and better musicians didn't use it!

This tie to the 55-division was the strongest and most important 
part that was given to me in response, by Oxford's anonymous 
peer-reviewers, when I submitted the first draft in summer 2004.  
They advised me to make more (than I had been doing so far) of this 
55-division connection back into the 17th C.  Very good advice.

And now I'm going to go spend the day enjoying my family, and a 
fine organ concert, and the beautiful weather, and getting ready 
for tomorrow's lecture/demo--to a group of piano tuners who are 
interested in the merit and usefulness of my work.

Brad Lehman

Date:    Fri, 19 May 2006 12:54:38 -0400
From:    Brad Lehman 
Subject: de-obfuscation and insignificant figures

>>an absurd number of (in)significant figures

An excellent sensible perspective is in Grove's Dictionary, 5th 
edition (1954) edited by Eric Blom, the "Temperaments" article by Lloyd.

After using a logarithm-based discussion that describes the diesis 
as [effectively] 21/11 of a syntonic comma, he remarks about people 
who get too wrapped up in too many (in)significant digits, out there 
where it makes no audible difference.

"Using enough decimal places in this way we could also verify that 
there is no acoustical difference between a schisma and one eleventh 
of a comma.  Arithmetically we should find that the difference 
represented a very small interval which, as a mistuned unison at the 
pitch of G of the treble clef, would produce a beat not more often 
than once every three-quarters of an hour.  Musicians, interested 
only in the differences the ear can detect, will ignore the 
discrepancy.  'Interminable decimals' unrolled by theorists should 
always be brought down to earth by translating them, in this way, 
into beats of mistuned unisons."


21 schismas in a diesis:


And I agree with Anne A's remark: what is to be gained, anywhere, by 
belittling those of us who use spreadsheets as one tool (among others) 
to analyze temperaments?  "As to spreadsheets, they are a convenient 
tool for testing theories and observing relationships."  From about 
1983 through 1988 I used my BASIC and Pascal programs instead of 
spreadsheets, for this type of thing, and then I started over with a 
spreadsheet because they were more convenient and flexible and portable.

My old freebie from 1990s is still sitting out there for anyone who 
cares to give it a try:
...but my personal copy of it has evolved much beyond that, since then, 
and it generates a dozen different views into temperament characteristics, 
including my summary chart.  An example printout for Young's #1, with 
the 1/11ths of syntonic comma, 1/21sts of the diesis, cent deviation 
from other regular layouts, etc etc:

And not a beat rate in sight, there, because beat rates are dependent 
on starting frequency.  The spreadsheet generates beat rates, too, on 
demand...but they're scarcely interesting to me anymore, so I use them 
only when there's some directly relevant point to be made about a beat 
rate.  All the rest of the time I consider them to be irrelevant and 
obfuscatory.  Temperaments are proportions and relationships...not 
collections of beat rates.  Beat rates down to a precision of 1/10 of 
a second are valid for about half an hour, on harpsichords; therefore 
not terribly useful or interesting to me.


And if anybody gives a flying #*&^ what the calculated frequencies are 
for Young's #1, at A=440 Hz, to a numbingly irrelevant 15 digits of 
precision, according to my spreadsheet it's this:

A               220.000000000000000
Bb             233.872844886736000
B               246.662948474423000
Middle C    262.513392515506000
C#             277.182630976872000
D               293.996576986312000
Eb              311.830459848981000
E               329.255534170749000
F                350.413338583079000
F#              369.576841302496000
G               392.881760362129000
G#             415.773946465308000
A               440.000000000000000

...although any number of inaudible round-off errors may have crept 
into there, anywhere after the first several digits.  Obviously the 
decimals should not all end in 000, since we're dealing with irrational 
numbers and logarithms and such; but so what, in musical practice?  
Frequency accuracy of one digit after the decimal (like 311.8 Hz) is 
more than sufficient for anybody's perception of music played in 
such a temperament, in this octave.

Sharpshooters will of course recognize that this presentation of 
Young's #1 is itself already a round-off to my 1/12 Pythagorean comma 
model of it, as against Young's own calculated machinations of 3/16 
syntonic comma.  So?  The 3/16 comma version comes out like this, 
to three digits:

A     220.000
Bb   233.889
B     246.663
Middle C 262.569
C#    277.202
D      294.017
Eb    311.852
E      329.232
F      350.463
F#    369.603
G     392.937
G#   415.803
A     440.000

That is, it's "the same" frequencies all the way across, to within 
less than 0.1 Hz.  That is, my 1/12 PC model and Young's 3/16 SC model 
sound the same in musical practice, unless one gets picky down to the 
level tighter than 0.1 Hz.  When one's playing real music, is that 
going to be noticed?  Let's see, line up the two middle C examples as 
a unison.  They beat 56 times in 1000 seconds.  That's one beat per 
about 18 seconds.  That's pretty slow.  Who's gonna hold a middle C 
for 18 seconds to hear it, and be bothered by the inaccuracy?

Brad Lehman

Date:    Fri, 19 May 2006 17:33:27 -0400
From:    Brad Lehman 
Subject: Re: In place of spreadsheets

>>The one and only thing you need to know to do calculations (if you
find it useful to have a theoretical model) is that one Pythagorean
comma is about 12/11 syntonic comma, or if you use 13/12 you are not
far out.<<

Yup.  And each of these 1/12 PC or 1/11 SC units is, for practical 
purposes, the same as the official schisma.  So, there's one easy 
and simple unit that unifies most of the typical reckoning that we'd 
have to do.

Then, in each batch of four consecutive 5ths, add up the number of 
these units that have been burned off, and subtract from 11 to get 
the size of the resulting major 3rd, expressed in this same unit.

Demonstrative example, the Neidhardt 5th-circle #11 from 1732, the 
one that happens to have *three* slightly wide 5ths in it, but also 
no major 3rds as wide as Pythagorean 11:

Eb -1 Bb +1 F -1 C -1 G -2 D -3 A -1 E +1 B -1 F# -2 C# +1 G# -3 D#

The 5ths across the line always have to add up to -12 units, the 
whole Pythagorean Comma.

Eb to G: 11 -1 +1 -1 -2 = 9
Bb to D: 11 +1 -1 -1 -2 = 8
F to A: 11 -1 -1 -2 -3 = 4
C to E: 4
G to B: 6
D to F#: 7
A to C#: 8
E to G#: 10
B to D#: 6
F# to A#: 6
Db to F: 9
Ab to C: 7

The whole diesis is 21 units to be burned off, in each stack of 
major 3rds such as C-E-G#/Ab-C.  Why is it 21?  Because it's two 
syntonic commas (11+11=22) less one schisma that we pick up from 
going round the circle the other way, in the enharmonic shift.

And 1/4 syntonic comma meantone has each of its four stacks as 
0, 0, 21.  Each of the two pure major 3rds owes a syntonic comma of 11, 
and then those two (less the schisma) get schlocked into the 
diminished 4th that we don't care to play.

All of which is fine and easy to do on paper, *in addition to* 
anything interesting that spreadsheets and direct harpsichord application 
can tell us.

>>What you don't see from any spreadsheet is that tuning by ear involves
an inevitable approximation, so it hardly matters if your arithmetic
has a fudge factor too, and you can throw away anything smaller than
1/25 comma if it makes things look neater. By working with fractions
of a comma you are already dealing with very tiny differences.<<

Why is it alleged that spreadsheets can't/don't show such 
approximation?  A spreadsheet designer can do exactly that, easily, 
by choosing how many significant digits to preserve in the display 
of any given result.  Don't blame the tool for a presumption of more 
accuracy than is really practical.  The tool can do round-offs to 
whatever reasonable precision we want.  Blame the people who misread 
the tool's results, the wrong expectations of a too-mathematical accuracy.

Even if we have achieved a nice convenient set of 12 integers for 
all of the major 3rds, as calculated above (whether we've used 
paper or a spreadsheet), we should still understand that *in practice* 
they're all going to have a tolerance of maybe 0.5 or as much as 1, 
on either side of that integer.

As for anyone who can't stomach the 1/6 comma or 55-division 
theoretical calculations in my work, I suspect that they too are 
misreading my presentation: in overestimating the accuracy *I'm* 
actually expecting.  (As if Bach or his students--or any 17th century 
people!--weren't allowed to have used "about that much" major 3rds 
resembling those of 1/6 comma, or whatever, before some theorist took 
to publishing such divisions more formally?)  I have a section in 
that regard, in my FAQ page:
I haven't asserted that anybody living or dead ever has to hit those 
55edo marks (or 1/6 comma points) exactly, in practice.  That's just 
a convenient theoretical framework, uniting the "major thirds a little 
sharp" "about *that* much" with the 18th century pedagogy where there 
are 9 commas within a tone: 4 commas for the chromatic semitone and 
the other 5 for the diatonic semitone.

And what are the loop thingies in Bach's drawing, in practice?  One 
way to think of them (IMO) is little wrist flicks, knocking the 
slightest rotation onto the tuning pin to impurify the otherwise 
pure 5th or 4th.  It doesn't have to be more mathematically precise 
than that.  Taste, experience, and a trip through all 12 notes together 
show how much is the right amount, in each such flick or double-flick, 
such that it all works out.  It happens to work out that it's the 
same as those 1/11 SC or 1/12 PC or the schisma, that same unit.  
The mathematics of all this are nothing more than a convenient way 
of explaining, and reproducing, that practical musical result.  Do 
the whole tuning by trial and error, to get the right shapes and 
relationships; and then work out all the maths later if that's even 

Brad Lehman

Date:    Mon, 22 May 2006 17:59:47 -0400
From:    Brad Lehman 
Subject: the audible schisma

>>It is unscientific to use numbers that are more precise
>>than what you can hear or tune.
>Many people are not able to discriminate even the Schisma
> PC/SC=32805/32768=5*38/215 ~2Cents,
> and is hence for them "unscientific"?

In what musical contexts, and in what tuning contexts?

It's crucial to be able to hear schisma discrepancies within unisons 
or octaves, for any reasonably skilled musicians in playing or 
singing...on notes that are sustained for more than a few seconds.  
Notes within violin tessitura especially.

And crucial to be able to hear and control schisma discrepancies 
within 5ths, 4ths, and nearly-pure major 3rds or 10ths, for harpsichord 
tuners....recognizing that it's easier to do on the front 8 than the 
back 8, and easier on some harpsichords than others.

But beyond those situations (any others?), the usefulness diminishes 

Brad Lehman

Date:    Tue, 23 May 2006 17:41:39 -0400
From:    Brad Lehman 
Subject: Re: Fess up you blokes!

>>Brad, Andreas, Paul, Thomas (Tilman?).
>>How often do you tune your harpsichord(s)??

1. Italian virginal, which is left almost all the time in a regular 
meantone and has a short bass octave: average about once every three 
weeks, plus I check/fix all the treble octaves quickly each time I 
play it, if I have more than a few minutes to do so.  Some of this 
depends how much stuff is piled on top of the instrument, because I 
have to open the lid and set up the music desk to do any tuning.  If 
I'm just having a quick bash at a piece (usually from Fitzwilliam or 
contemporary of it) and the octaves haven't drifted much, I don't bother.  
As for *which* regular temperament to use on it, it could be anywhere 
from 1/3 comma up to 1/6; I usually do it differently each time I go 
from scratch, for variety's sake.  Whatever size major 3rd I feel like 
having for the F-A or the C-E, I start with that, and everything else 
automatically follows as derivation.  On the rare occasions when I 
want to try something circulating on this virginal, I've used 
Bendeler 3 a number of times, or Kirnberger 3.  It seems to respond 
well to those two.

2. Flemish single, my main hpsi for serious work: I check all the 
octaves and 5ths each time I play, i.e. almost daily.  I scrap it and 
start the tempering over whenever the weather has changed seriously, 
or whenever a spot check of a dozen notes (or less) doesn't take care 
of it, or whenever the entire pitch level is noticeably off the fork.  
All round, starting over on average about five times a month.  This 
one's usually in my Bach or in other modified meantone variants of 
similar interest; but sometimes I set it in equal or some of the 
Neidhardts, if I want to check out some really fine points in the 
repertoire.  Even the slightest out-of-tune drift on the octaves or 
5ths bothers me intensely on this hpsi.  I have to stop and fix them 
immediately.  And, "as much as ye ear can bear" on the 5ths/4ths never
goes tighter than 1/5 comma, on this hpsi; more intensely tempered 
5ths sound nasty on this instrument, with its bright 8.  Let one of 
the treble natural notes drift as much as a schisma in either direction, 
and I catch the awful-sounding 1/4 comma 5th or 12th immediately.

I'm lazy about the 4-foot, unfortunately.  I know that if I kept it 
in tune more often, the whole instrument would probably stay more 
stable.  Sometimes it goes a couple of months between the times that 
I use the 4-foot.  This instrument with just a single 8-foot often 
seems too loud for the room it's in, even with the lid closed, and 
my wife starts to gripe if I open it up or if I put on the 4....

3. Other more generic single (a playable but not great beater 
inherited from a relative), about once a week on average.  This 
is the hpsi on which I'll try just about anything, and especially 
for comparison with the main one sitting next to it.  That's 
usually one or another of the recipes I have at
, or trying more of the Neidhardts.  Sometimes when I have people 
over for a demo, I put a regular 1/6 or tighter onto this one all 
the way.  But if I'm not experimenting at the moment, I usually 
leave this one close to the "vocal version" of my Bach temp (see 
part 1 of the paper, and/or
) so I can play through Bach vocal works.  That's much easier than 
transposing all the music at sight.

I stuck this particular hpsi back into Werckmeister 3 again over 
the weekend, just to remind myself again of W3's limited range.  
Sounds good for Bb, F, C, G, D majors and for D and A minors, but 
that's all the praise I can muster for it.  If we're gonna stick to 
those keys anyway, meantone works better....  [/soapbox]  

Time to go put on another Neidhardt or something.  Well, before doing 
so, I think I'll leave all the flats/sharps where they are and dink 
around with the naturals, to convert this to Kellner or Young 2.  
It takes more time to say so than just to go do it.

4. Fretted clavichord double-strung: about once every three or 
four months from scratch.  More often if I'm doing serious practice 
on it (like I did Saturday) for an upcoming gig, or in situations 
where wife and child are both taking a nap and must not be 
disturbed...on the other side of the same wall.  Even if it's going 
to be a harpsichord or organ gig, if time allows I always try to 
get in some clavichord practice to *make sure* I've thought seriously 
about my fingering and my phrase control.  But the instrument as a 
whole doesn't need to be tuned very often; just the occasional check 
of octaves and unisons.  And whenever I do so, I find that I have to 
play some of the octave notes *separately* (like melodic octave leaps) 
to be sure if they're too flat or too sharp.  Beat counting in any 
4ths/5ths isn't really much of an option, since the notes don't 
sustain long enough or loudly enough to generate them.  Interval 
quality is just a recognizable quality in itself, for the purpose of 
tuning this clavichord.  "Yeah, that sounds like about 1/6 comma...and 
this other 5th is too good, the octave must be wrong...."

5. Piano at our church: volunteer contribution whenever it needs it, 
which I have to go do sometime soon; recent weather has hosed too many 
of the octaves, and banging kids have hosed too many of the unisons.  
(And yes, I have it set in my Bach all the time...not that anyone 
notices.  The other pianists just go ahead and play anything, as if 
it's equal, and it works fine for everything.)  I usually touch it 
up every few months.

6. Germanic double at a local university: I go in whenever I want to 
try something on a double, or do adjunct/freelance stuff for them.  
A couple of months between visits, sometimes.  But on that one the 
back 8 is muddy enough, it's hard to do a tuning on it straight up.  
I have to do its front 8 first and then copy unisons.  Somebody on 
staff keeps that instrument in or close to equal, with their Korg, 
all the time...and whenever I go in and encounter it, the C/G/F majors 
sound awfully dissonant to me.  Equal sounds inside-out to me.  What's 
the point of having different keys if they're all going to sound 
average-ugly?  [/soapbox]

Always by ear, all of these, from a single A or C fork.  Zero assistance 
from spreadsheets, beat rate charts, or any electronic device.  I generally 
don't allow myself to set up a temperament unless I've first memorized 
its theoretical shape, and therefore know what I should be listening for.  
(And that's why I don't consider the huge batch of Neidhardts to be either 
"merely theoretical" or "impractical" as I work my way through them; but 
I'm surely in a minority on this point.  The prejudice against them goes 
way back to Murray Barbour, at least, in those speculations that they're 
paper theory.  But, they work nicely *for music*.)

Among these various instruments, I'm tuning at least half a dozen notes 
on *something* almost every day.  I'm really picky, and can't stand 
needlessly out-of-tune notes, or the sound of wrong enharmonics.  If 
the music says G-flat, then confound it, the note that comes out had 
better sound within a comma of G-flat, relative to its melodic or 
harmonic neighbors.  So, meantone really yanks my chain unless I'm 
playing correctly spelled music with it.

I leave my slide rule in the car to calculate gas mileage.  It does 
beautifully at that.  I like analog devices better than digital ones, 
for that sort of thing where one doesn't need too many digits of 
precision.  I'm just barely old enough, and curmudgeonly enough, to 
think that slide rules should be taught in public school the way we were 
shown how to use them in junior high.  It's a really good tool!  How 
can kids who punch calculator keys get the *concepts* of a slide rule?  
How can they be allowed to work cash registers if they can't even 
calculate the simplest subtractions and percentages in their head, to 
give change or recognize a discount?  And how hard is a 15% tip to 
calculate in a restaurant, mentally, in two seconds?  10% is trivial and 
half of that again is trivial; sesquialtera.  Grumble grumble grumble.  

Brad Lehman
Date:    Tue, 23 May 2006 18:11:59 -0400
From:    Brad Lehman 
Subject: Neidhardt ignoring the schisma

>> he [Neidhardt] ignored the schisma, as you do here.
>No I don't. The sum of deviations from Pythagorean intonation over
>three major (or four minor) thirds is one Pythagorean comma, with no

Stack three pure major 3rds.  Each of those pure major 3rds is a 
syntonic comma shy of a Pythagorean major 3rd.  So, our octave accumulates 
three of those...less the Pythagorean comma in the opposite direction 
that we get from renaming one of the outside notes.

Example: C - E - G# - B#.

Falling short of an octave, the diesis is three syntonic commas (11 
schisma-sized units each) less one PC (12), total 21.

Neidhardt apparently reckoned this diesis to have 24, rather than 21, 
portions...losing a schisma somewhere in each major 3rd, and/or in some 
other muddle.  His equal and quasi-equal examples come out as triples of 
8, 8, 8 on the average major 3rd sizes, instead of 7, 7, 7.  Sorge 
checked this out and fixed it.


Different issue: if we stack three Pythagorean major 3rds instead of
three pure major 3rds, that's the same as doing 12 pure 5ths in 
succession...generating one PC overshoot, no fudging.  That's what the 
PC *is*.  Tautology.

Brad Lehman
Date:    Wed, 24 May 2006 09:50:03 -0400
From:    Brad Lehman 
Subject: solfege with "be" syllable in minor scales...

>>The hexachord system is ambiguous because Ffaut, Gsolreut, Alamire and
Elami (not to mention various types of B) can exist either in the
tenor or the middle octave. Also in the classic Guidonian system there
is no name for the lowest F.<<

This reminds me of a question that occurred to me recently.  When I 
got out the volume from 1954 Grove to quote from its temperament article, 
I also took a look at its article about solfege.

My question is: for those of you who learned your solfege minor scales 
starting on "la" instead of "do"....the way you learned it, does there 
exist a distinguishing syllable "be" ("bay") to substitute for "fi", 
raised 6th degree melodically, when "fi" isn't going to "sol" on the 
lowered 7th degree?

La ti do re mi be si la, for the rising melodic minor.  The argument 
in that 1954 Grove article is that "fi" has such a strong tendency to 
go only to "sol" (not to the raised "si"), there needs to be something 
other than "fi" there.  The situation where it's five whole steps in 
a row, do re mi be si.

Anybody here ever hear of this?

We learned our minor scales as "do re me fa sol le te do" and "do ti 
la sol fa me re do" up and down; "may" "lay" "tay" as the minor scale's 
difference from "mi" "la" "ti" of major.  For melodic minor rising, 
it was simply "do re me fa sol la ti do."  But, the theory class the 
year after me had a different instructor, and he taught his minors 
starting on "la".  Just a matter of style.

[I've regularized the spelling here, for consistency; that Grove article 
gives "ba" but makes it clear it's pronounced "bay", so I've rendered 
it here as "be" with consistently Italianate vowels....]


Another funny related topic: in the CD booklet notes of the 1998 Broadway 
cast of "The Sound of Music", the annotator remarks in passing about 
the theorist who made up Italian syllables for the notes.  But it's not 
Italian; it comes from Latin plainchant!  Ut queant laxis.

I think it's amusing that immediately after the "Doe, a deer" on that 
CD comes the song that is probably the most difficult of the whole show, 
if done in solfege: "16 going on 17".  What is that young Nazi Rolf 
teaching tender innocent Liesl about chromaticism, in their clandestine

I also think "My favorite things" is a brilliant melody: harmonizable 
without melodic change, in either minor or major.  And that's what 
happens in the next stanza of it.

In that 1998 production, during the solfege song where Maria teaches 
the kids that we use one note per word, "When you know the notes to sing, 
you can sing most anything"...Brigitta pipes up and complains that 
"anything" had not one but three notes.  Maria's explanation is simply 
to fluff it off cheerily, "Sometimes we do that!"  Uppity kids nowadays, 
arguing immediately with things they're learning by rote.

Brad Lehman

Date:    Wed, 24 May 2006 14:10:34 -0400
From:    Brad Lehman 
Subject: Re: solfege with "be" syllable in minor scales...

> My question is: for those of you who learned your solfege minor 
scales starting on "la" instead of "do"....the way you learned it, 
does there exist a distinguishing syllable "be" ("bay") to substitute 
for "fi", raised 6th degree melodically, when "fi" isn't going to "sol" 
on the lowered 7th degree?

On closer inspection, it's possible that this substitute "bay" business 
*only* applies specifically to the mid-19th-century pedagogical system 
by John Curwen, "Tonic Sol-Fa".  That feature of solfege in general 
may have died out since then; hence my question (which wasn't about 
Kodaly, specifically, but rather about solfege methods more widely).

The 1954 Grove article is by an H W Shaw.

My sight-singing textbook was by Berkowitz et al, 1960/75; and 
emphasizing moveable Do instead of fixed Do.

Mi sol la, re fa re sol, do mi do fa re sol sol!  - The Nairobi 

Brad Lehman
Date:    Thu, 25 May 2006 12:27:32 -0400
From:    Brad Lehman 
Subject: Re: Poll on Personal Temperament Preferences (Tom Dent's ordinaire)

Re: Dent's ordinaire today at

A nice temperament (IMO) for its stated purpose.  It's easy to set up 
in a couple of minutes, musically effective as stated, and its 
process encourages the "right" types of thinking/listening by ear 
with the tuning lever in hand....

I set it up and then played through some selections from Couperin's 
last three ordres.  Pantomime (f#, 26), Epineuse (f# and F#, 26), 
Exquise (b, 27), Pavots (b, 27), Ombres errantes (c, 25), Monflambert 
(c, 25), Misterieuse (C and a, 25).  All of those are from 1730 publication.

Having done so, the D# and the A# were too high for me, too hot both 
melodically and harmonically when they're the 3rds of major 
triads.  So is the E# playing within b and f# minors, but not worth 
messing with.  As soon as I went back and knocked down the D# and A# 
the slightest bit, each, all those pieces sounded better to me.  The 
low Ab and the high E# still bug me, but the process of converting 
the D# and A# already improves the melodic contexts around Ab and E# 
and make it less of an issue.  So, I left it alone at that point, 
instead of trying to nudge down the F slightly or the G# up.  Good 
points in Dent's final paragraph.

Then I went back and played the first three pieces in ordre 18 
(1722): two in f minor and one in F.  All the moments where Db and Ab 
come up really bug me, sorry--especially where C is used melodically 
a major 10th or 17th above Ab.  It just makes the melody sound like a 
soprano singing too sharp....  But, my unhappiness with f minor is a 
small price, with everything else sounding so nice otherwise.  I'll 
probably leave this on for at least a week for more play.  And I've 
already griped enough that I'll probably go knock the F down slightly 
and the G# up slightly, anyway, eventually....

Time to go play through some Haydn in this, including the f minor 
variations and some pieces in D and A.


Technical remarks:

 From the description of the way the notes C and B are set up 
relative to one another, it's obvious that this is a 1/5 syntonic 
comma temperament.  (C-G-B pure, before moving the G later.)

That whole comma is burned off within C-G-D-A-E-B.  All that's left 
over is to fudge out a schisma elsewhere.  As stated, B-F#  takes 
care of that ("about as in ET") by itself, or optionally it's ironed 
out with some in F#-C# as well.

If we're manufacturing an *additional* schisma-sized bit of tempering 
there in F#-C# by making it also "about as in ET", we're making our 
sharp keys sound good at the expense of the major 3rds Db-F and 
Ab-C...but that's a pretty normal thing to do, especially for those 
who have closely studied lots of Lindley....

If the C# is made extra low like that, it also gives us some 
overshoot wiggle-room for placing the final Eb, as stated.  [If we 
had simply burned off the schisma and nothing more within B-F#-C#, 
there is nothing left; and all of F-Bb-Eb-Ab are effectively 
pure.]  In this wiggle-room, one or both of Ab-Eb-Bb are slightly 
*wide*...not that that's any problem in itself.  Such slightly wide 
5ths have an interesting bit of motion to them, instead of sitting 
there sounding dead.  (And it was plenty good for Neidhardt to have 
two or even three of those wideys in a temp, 1732....)

When I took the D# and A# down, it's just moving this wideness 
wiggle-room around a bit.  Even if we'd had F-Bb-Eb-Ab initially 
pure, in the context of this temperament as it works out, I'd still 
want to tweak the D# and A# down the slightest bit each.  I don't 
consider it a liability to impurify the 5ths of that region making 
slight wideness...because it sounds good, and makes such a huge 
improvement in the sharps' behavior with such a small cost to the 
flats or those 5ths.  IMO, of course.

*Pure 5ths are not automatically "better"--musically--than slightly 
tempered 5ths!!*

These are slight adjustments.  1/5 comma core doesn't leave a huge 
amount of fudging room to take the D# downward, without making E-flat 
major music start to sound crunchety....


All of this is of course the normal "ordinaire" strategy.

- Set up some core of regularly circulating naturals, all the same 
size as one another and yielding good-sounding major 3rds in the easy keys.

- Jimmy the sharps progressively sharpward (maybe as much as pure 
5ths, or nearly so), and the flats progressively flatward (ditto, or 
even slightly wider than pure 5ths) until they meet one another at 
some tasteful and decent point coming around the back.  Dink the 
notes around slightly, until happy with the overall sound for the 
music to be played.

- Consider also F (flatward) and B (sharpward) as candidates for 
adjustment off the line of regular naturals, to get it all smoothed 
out enough...to taste.  If those original naturals are pretty tight, 
like 1/5 or 1/4 comma, the F at least has to start moving: toward 
less or no tempering in the C-F, and getting to sound like a decent 
E# when playing in keys as simple as b minor and e minor....

- And the faster the flats fall off, the better results we get in the 
important dominants B major and F# major...but at the expense of 
playing music in f minor and the other deepish flatty keys, three or 
more in the signature.  The payback has to be in there somewhere, and 
more so the tighter our original naturals were.

Brad Lehman

Date: Thu, 25 May 2006 16:44:59 -0400 
From: Bradley Lehman  
Subject: Haydn on harpsichord in Dent

>> >> But. Mature Haydn in ordinaire - however watered-down? I wouldn't do
it myself. And the F minor variations... in the same spirit as the
Beethoven sonatas 'for piano or harpsichord'?!? There is a work that
cries out for the fortepiano, if ever there was one. I will be
extremely surprised if Brad likes what he hears in the coda, with its
rococo right-hand decorations above massive chords of Db and Ab major,
just made for an instrument with lots of resonance and a sustaining
pedal (Broadwood?).<<

What's wrong with it?  The piece sounds terrific on harpsichord, and 
doesn't need a sustaining pedal.  Obviously one has to approach some of 
it with normal "finger pedal" technique of holding stuff as long as 
sounds good beyond the notated values (like in those arpeggios on the 
penultimate page), but that's just basic harpsichord-playing and taste. 
  Sure, it would sound good *additionally* on any type of piano, with 
or without pedaling.  It's just a well-written piece of music.

And, for that matter, who's more rococo than Francois Couperin?

>> >> I tend to think Haydn learnt something quite close to ET early on -
but we've had this discussion before. Porting tuning habits from
clavecinistes across the cultural gulf is strictly at your own risk.<<

Ummm....what "cultural gulf" is that?

>> >> My synthetic tuning is just one point on the continuum between
Rameau's modified meantone and Rameau's cod-equal. The point of having
a few adjustable notes is that they *should* be adjusted, particularly
when you jump from three sharps to four flats. Haydn does this in the
middle of a movement sometimes. If one doesn't hop about between
extreme keys all the time (does Brad ever use C major or D minor?) the
adjustments should be minimal.<<

Y'mean like those C major and F major pieces by Couperin that I 
explicitly said I used to test this temperament?  Look again.

And I do play in easy keys quite a bit, especially in play-and-sing 
sessions with my 3 year old.  The songs in her books are in really easy 
keys and use only a few different harmonies.  If I weren't too lazy to 
take the five minutes, I'd maybe switch the instrument to some nearly 
regular meantone where those keys sound even better (knock each black 
key one way or the other), but then I'd have to switch it back for 
other stuff *I* want to play.  She enjoys the games of playing slow 
even trills, and gently pressing the keys instead of hitting them, and 
carefully playing every single note up and down the keyboard in a 
steady rhythm.  These lessons even paid off recently when I was playing 
Elizabeth Farr's CD of Jacquet de la Guerre, in the car: from the back 
seat came the comment, "Hey, that music on the radio is playing 
trills!" *instead of* demanding that I switch to a kid's music CD.

I like the idea of having a couple of designated "adjustable notes" 
within a temperament, and it works, if the user gives a dang.  But, 
human nature being what it is, adjustable things really don't get 
adjusted all that often in real practice.  Give the users an option and 
they'll still leave it set the way they generally like it most of the 
time, instead of trying to use flexible options--whether we're talking 
a temperament or a home-stereo equalizer or software or the cutting 
height of a lawnmower.

>> >> I imagine the Frenchies making up an initial prelude to check the
tuning, playing a chunk of pieces in that key, then making up a
prelude in a different key, then wielding the tuning hammer for a
minute or two, before settling down to the next Ordre.<<

More and more, over the past couple of years, I'm suspecting that that 
idea is pretty much something that Murray Barbour made up, in the 
several articles and the book.  Free preluding yes (and I have done); 
but getting out the hammer to change an optionally flexible batch of 
notes *that* often between suites, I doubt it.  At least for 18th 
century dudes like F Couperin, whose music moves around enough _en 
route_ that it hardly works to leave the ax set in *any* regular 
meantone for a whole ordre (i.e. no single adjustment would work for 
the whole thing--since enharmonic respellings come up).  Or for any 
full-length Bach suite, except maybe the D minor French Suite that 
stays in the classic set of meantone notes the whole time (which still 
doesn't mean it *should be* or *was* played in strict regular 
meantone).  Or as I've opined in print, the Bach inventions/sinfonias 
that refuse to stick to only twelve notes per two-page piece.

A couple of weeks ago I played through the book of extant Marchand, 
which is in D minor and then G minor.  I used a regular temp for the 
whole thing and it was beautiful that way...and yes, I then tweaked one 
note per octave before playing the G minor music.  But, Marchand seems 
a world apart from Couperin or any of the Bachs, or Froberger, in the 
enharmonic notes that have to be available and sounding decent, and the 
way the harmonies are just the basic several (albeit souped-up in 
interesting ways).

I was surprised to read yesterday that Alexander Brailowsky in recitals 
would improvise little chord-modulation connections between the 
Bach-Beethoven-Chopin-French-Russian-whatever pieces on his programs.  
Transitions of mood for the audience?  I like those little amorphous 
improvs by Chick Corea and Bobby McFerrin on their disc of Mozart piano 
concertos, setting up the mood like (as they say in the booklet, IIRC) 
a butler at the door greeting the guests.

Brad Lehman

Date:    Tue, 30 May 2006 16:34:38 -0400
From:    Brad Lehman 
Subject: Re: I am not a temperament... I am a free man!

 >>Hm, I go off-list for a few days and find my name mysteriously
transmuted into a random mixture of French and English mid-18th
century tuning ideas, after having undergone the rigorous but
eccentric Lehman testing procedure. Is it worth it, I wonder?
... I suppose it will now be called Dent No.6?<<

Dude, what's "eccentric" about setting up a well-designed temperament on 
a decent instrument, and then trying out whatever music the user wants 
to play?  Your temp is still on my harpsichord here and I'm still 
enjoying it.  It makes the instrument sound strong and clear.

Has anybody else here tried Tom's fine ordinaire?  Use is better than no 
use.  Anybody else tempted to move any of his four optional notes 
slightly, as I reported to have done on two of them?

Over the weekend I corrected our church's piano (it's in "Bach/Lehman 
1722" or whatever anybody wants to call it).  There were a few slightly 
wobbly unisons in the treble clef area, but the main problem was that 
the bass had been drifting upward since the winter, which had made the 
octaves and 5ths sound noticeably wrong.  Fixed all those as octaves 
from the two-string notes on down, and it was good to go; less work than 
I expected/feared.  No need to start any of the temperament over, on 
there.  This piano has been very stable since last summer, with only two 
or three of these little half-hour sessions of correcting a few 

Then I hauled out whatever sorry remnants still exist of my high-school 
piano technique (octaves and big chords, ouch!), and had a play through 
the following pieces in a Chopin book:

- Mazurka in Bb, Op 7 #1
- Mazurka in A minor, Op 7 #2
- Mazurka in G minor, Op posth 67 #2
- Mazurka in A minor, Op posth 68 #2
- Prelude in B minor, Op 28 #6
- Prelude in A, Op 28 #7
- Prelude in Db ("Raindrop"), Op 28 #15
- Waltz in C# minor, Op 64 #2
- Nocturne in F minor, Op 55 #1
- Polonaise in A, Op 40 #1

For what it's worth, on modern piano (a decent Yamaha upright), this 
temperament sounds terrific in that repertoire.  It doesn't call undue 
attention to itself but merely highlights what the music is already 
doing...and clarifying Affekt(en).  It gives plenty of poignancy without 
sticking out, and music in the simple keys relaxes more than it does in 
equal.  Music in A major seems to project itself especially easily (even 
the quietest stuff with a shimmer to it--Mozart and Haydn being 
exquisite), and the F minor has an attractively somber darkness to it.

Usually I play Bach, Scarlatti, Haydn, Mozart, Beethoven, Grieg, and 
Brahms on there instead of Chopin.  Chopin has some starker key 
contrasts and more exotic effects...rivaling some of the quick tonal 
shifts in Grieg.

I came home and put on somebody's good CD of all the mazurkas...and 
within 20 minutes or so I felt that the whole thing was rather dull in 
equal temperament, not having enough natural contrast to it.  (Granted, 
it's disingenuous to expect *anybody* to listen to more than a dozen of 
those mazurkas in a row, if even that many, in whatever temperament.) 
What's the point of having different keys if they're all going to seem 
the same?  Even if the composer uses some exotic-sounding scales 
melodically, the sameness sets in quickly.

Somebody sent me a recent concert CD of a professional pianist using the 
B/L'22 on a Boesendorfer model 225.  Repertoire was some of Schumann's 
late chamber music: the folk-style cello pieces Op 102, the D minor 
violin sonata Op 120, and the third piano trio Op 110.  The thing just 
sounds so natural and unremarkable to me, I don't notice it as a 
listener.  But when I listen to other modern-piano recordings (of 
whatever), or play on pianos myself in equal temp, the C major and F 
major seem grossly dissonant to me compared with what I'm now accustomed 
to.  I wish they'd resolve!

Brad Lehman

Date:    Wed, 31 May 2006 09:25:05 -0400
From:    Brad Lehman 
Subject: Re: I am not a temperament... I am a free man!

 >>>(it's in "Bach/Lehman
 >>>1722" or whatever anybody wants to call it)
 >How about Squiggly #99?

That would be meaningless inflation.  In May and June 2004, when this 
particular temperament was already in professional concert use, it was 
still less than the 10th temperament derived from that diagram (since 
the 18th century, of course): other people hadn't got around to filling 
up the duck pond with their 100+ decoys yet.

It's based on straightforward principles (among the additional 
historical material as presented):

- Axiomatically (& historically), no major 3rd should be smaller than 
C:E.  Some others might be similar or the same, but none smaller.

- Axiomatically (& historically), at least the natural 5ths from 
C-G-D-A-E should all be of consistent size geometrically...i.e. from the 
core of "regular" (aka meantone) tempering practice, with _ordinaire_ 
types of adjustments outside that core.  With those regular 5ths, what's 
good for violins/violas/cellos/violas da gamba on the open strings is 
good for music: gentle and consistent tempering of the naturals.  (And 
Bach himself was a string player; what might he do as normal practice on 
those instruments?)

- Axiomatically (& historically), the C major scale is the natural 
center of harmony, and the one that should be most regular 
melodically...again from meantone practice.

- Axiomatically (& historically), there cannot be any noticeably bad 
5ths/4ths anywhere; all major and minor triads have to be usable.

- Axiomatically (& historically), if the major 3rds in a temperament are 
changing sizes, it has to be gradual and sound steady when we modulate 
normally around the circle of 5ths.  The easiest test is to play major 
triads all the way around in both directions, like dominant or 
subdominant progressions.  No major 3rd should be grossly different from 
the ones immediately on either side of it, in root motions by 5ths.

- Premise: the whole WTC is playable (and to be played) in a single 
temperament without stopping to retune any notes between pieces.  A good 
solution makes everything playable and sufficiently interesting as well. 
  On fretted clavichords and organs, retempering between pieces is out 
of the question anyway.

- Practical observation (from experimentation and from historical 
models): it works well to have E:G# smaller than or equal to Ab:C, not 
vice versa, because Ab is closer to C than G# is (C-F-Bb-Eb-Ab, vs 
C-G-D-A-E-B-F#-C#-G#).  The closer we are to the home key of C major, 
the less tempered out of regularity we should be.

- Practical observation: it works well to have F to Bb slightly wider 
than a pure 5th (as in Italian/French _ordinaire_ practice), yielding a 
decent major 3rd on F#:A#; the cost to Bb situations is much less than 
the gain in A# situations, both melodically and in dominant harmonies.

- Practical observation: the major 3rd C#:E# must be rather good, as 
Bach audaciously started the C# major prelude with an open exposed 
occurrence; likewise plenty of G#:B#.  Also on the subject of Db:F, this 
interval is very important to music in the frequently used keys of C 
minor, F minor, and Eb major, among others; it just doesn't do to have 
this interval be nasty or obtrusive.  Music (such as Bach's F minor 
prelude/fugue of WTC 1, or the Eb major or the Bb minor p/f, or the much 
later F major Duetto BWV 803) develops suddenly intrusive bumpiness on 
the occurrences of Ab:C and Db:F, in temperaments like Werckmeister 3 
where those major 3rds are the widest.

- Practical observation: the major 3rd B:D# must also be very good, for 
straightforward use in E minor and A minor.

- Practical observation: if an organ is tempered with the WTC's 
temperament (in at least one or more accompanimental ranks, if not the 
whole instrument), it also has to handle the Chorton/Cammerton 
transposing continuo parts for the compositions that were written that 
way, playing the continuo in its originally notated keys; and this 
affects the overall sound of the ensemble.

- Premise: Bach was clever enough (and musically enterprising enough) to 
have understood all this and made full use of it *before* writing his 
music, treating temperament issues as a musical virtue rather than an 
unwelcome liability.  The tuning style perhaps affected his creative 
imagination, symbiotically, as to the types of themes and harmonic 
adventures that made their way into his music; and they only pop back 
out most clearly if we can re-create the same or similar tuning balance 
to hear those effects directly.  Set up the same conditions he likely 
had at home or in his office, during the compositional process, to hear 
how its sound can influence improvisation and composition.

- Practical observation: the best way to test all this is to play the 
music directly.  The compositions themselves tell us more than any paper 
argumentation does, in their sound.

Brad Lehman

Date:    Thu, 1 Jun 2006 14:01:50 -0400
From:    Brad Lehman 
Subject: Re: I am not a temperament... I am a free man!

 >>- Axiomatically (& historically), at least the natural 5ths from
 >>C-G-D-A-E should all be of consistent size geometrically...
 >Funny how the "historical" figures Sorge, Neidhardt, and Werckmeister 
 >all seem to have been unaware of this axiom. Oh well, nuthin' but a 
 >bunch of 2nd-rates, that lot!

Funny how Paul P's own professed favorite among the Neidhardt 
temperaments, namely the 1724 Village, has absolutely regular 5ths 
C-G-D-A-E as its core.  This one:

And did Sorge, Neidhardt, and Werckmeister themselves know that the 
regular meantones--and equal temperament--have consistently-sized 
C-G-D-A-E and more?  Yeah, probably, but I guess that's one of the funny 
things we can't prove if cynicism trumps all.


 >>- Axiomatically (& historically), the C major scale is the natural
 >>center of harmony, and the one that should be most regular
 >>melodically...again from meantone practice.
 >Uh, gee, I was under the impression that the scales in the good keys 
 >all had the same melodic structure in strict meantone, beings as all
 >the whole and half steps are of consistent size. Must be missin' 
 >sumthin' somewheres.

Of course they do, and the thing that "must be missin' somewheres" is 
the picking/plucking at the way I stated my point, instead of *getting* 
the point (which perhaps I should have said better).  Das ist: if any 
scale is going to be regular or most nearly so, it needs to be C major. 
  Can anybody name a historical temperament of impeccable pedigree 
wherein all seven diatonic notes of some other major scale *instead of* 
C major happen to be regularly spaced, while C major is not?


 >Let's not even mention the fact that as yet Brad has provided us with
 >absolutely no evidence that the slight MELODIC differences created by
 >circulating temperaments are even perceivable, when there is plenty of
 >research to the contrary.

"Research to the contrary" involving what?  Repeated playing of the most 
directly relevant music (namely the WTC) on suitable harpsichords, 
clavichords, and organs, to test what's "perceivable" or perceptible in 
various temps?

I guess I keep forgetting that not everybody (except me) has spent 
months listening to Watchorn's WTC recording (in his draft edit) more 
than 20 times, and my own published recording of excerpts from WTC et al 
(at least 30 times per piece during the production process), and other 
people's recordings in various temps...and playing the pieces myself in 
other temps to listen closely to the music, hands-on.

Such empirical inquiry into what's perceptible apparently doesn't matter 
much, next to speculative assertions that melodic variations of size 
can't be perceived.

Please supply some good hard research that proves melodic variations 
among the keys aren't perceptible, within circulating temperaments. 
Especially in the behaviors of differently sized semitones, if it's not 
too much trouble.

Funnily enough, whenever I put on somebody's recording where the 
temperament isn't identified, and listen closely to whatever repertoire 
in whatever circulating temps, I *can* pick out various melodic 
inflections among the keys.  Especially in the ways semitones move, and 
most nakedly the way they behave during suspensions.  (Like in Bach's F# 
minor fugue of book 1, for instance.)  That's without knowing ahead of 
time which ones might be expected to be smaller, since the temp isn't 
identified.  And it's much clearer on harpsichords than it is on pianos, 
clavichords, organs, or electronic simulations.  But hey, my claimed 
ability to be moved and pleased by expressive semitones doesn't really 
speak to anyone else's musical perceptions, does it?  I think I get your 

Try some.  Put on Christiane Jaccottet's recording of WTC that is 
available for dirt cheap just about anywhere.  No temp is specified, and 
it's obviously not equal, but moderate.  Put on the C# minor prelude and 
fugue, for a decent example.  Those variously sized semitones in there 
somehow don't have differently expressive quality...and aren't 
perceptible at all?  Try the same piece with any other recordings of 
your choice...and better yet, by directly playing it on harpsichord 
yourself, in any and all temperaments of your choice.  Not perceptible?


 >Of course, what happens inside Brad's head, in the
 >full knowledge of what the differences are and where they lie, may be
 >something quite different from what happens inside the mind of the 
 >listener.  But then, I'm not sure such distinctions are important for 

Oh, the citation of a universal listener: to trump whatever I claim to 
hear when I play my harpsichords, and whatever I allow to affect my 
timing and phrasing as I play.  I see.

Samples of the F# minor fugue, et al:

Sorry about the tone of this posting; but I'm pretty much just 
reflecting back the tone of the criticism to which I'm responding.  It's 
hard for me to fathom how the things I hear in music are allegedly not 
perceptible.  Even harder for me to fathom how they allegedly couldn't 
have occurred to a much better and more experienced musician than I, 
namely JSB.  I'm humble enough to realize and admit that JSB knew his 
perceptions (whatever they might have been) at least as well as I know 
my perceptions; and probably he did better at all this because he worked 
at it more diligently.  I mean, he's the guy who wrote a whole piece on 
the motto that fa-mi and mi-fa are a foundation of all the music.  This 
leads me to suspect that he could hear expressive semitones quite well, 
and knew how to handle his modulations or hexachord mutations.

Brad Lehman

Date:    Thu, 1 Jun 2006 15:41:16 -0400
From:    Brad Lehman 
Subject: hearing melodically

> > Please supply some good hard research that proves melodic variations 
> > among the keys aren't perceptible, within circulating temperaments. 
> > Especially in the behaviors of differently sized semitones, if it's not 
> > too much trouble.
> > 
> > Funnily enough, whenever I put on somebody's recording where the 
> > temperament isn't identified, and listen closely to whatever repertoire 
> > in whatever circulating temps, I *can* pick out various melodic 
> > inflections among the keys.  Especially in the ways semitones move, and 
> > most nakedly the way they behave during suspensions.

In addition to semitones going up or down, with or without any context 
above a bass, the other two intervals that often make huge melodic 
impressions on me are the leaps of major 6ths or minor 6ths.  If they're 
even the slightest bit larger or smaller than the melodic/harmonic 
context around them has led me to expect, those leaps draw attention to 
themselves expressively.

Good pieces to test that: the A major prelude of WTC 1, or the various 
fugues of both books that have leaps of 6ths in their subjects.

The leap of a 6th in the Eb prelude (Ab up to F) often sticks out to me, 
too, in temperaments where Db-F is especially wide.  The Db isn't even 
being played there immediately, but the Ab and F seem so far apart in 
this melodic context.

As for hearing whole tones, I find it more difficult, but if I'm at a 
harpsichord or an organ and play Do-Re-Mi-Re-Do melodically a few times 
(in whatever key) I can usually pick out accurately if the Re is higher 
or lower than the midpoint, or if it's a reasonably accurate mean tone 
within that key.  More difficult on recordings, and harder still with 
other synthesized electronic tones that don't resemble acoustic instruments.

A decent test piece for overall melodic smoothness through modulations: 
D major prelude of WTC 1.

And the Eb prelude, already mentioned.  Temperaments that have a low C# 
or a low G# sound really bumpy (to me) in that opening section, whenever 
there is melodic motion from Eb to Db or from Bb to Ab: a whole step 
that's too big.

Not only keyboard.  When I hear solo singers or choirs or oboes doing 
some melody and they put the 6th degree of the major scale uncommonly 
high, it bugs me.  (This phenomenon seems to happen with boy trebles 
more than other voice types, it seems....)  If La is farther above Fa 
than Mi is above Do, ouch.  Even in something as simple as monophonic 
chant.  It doesn't seem to matter if I know the piece or not, as to 
having any expectations where the La "should" be; it's all from melodic 
context set up by the lower degrees in the scale.

Everyone's mileage in this may of course vary.  I'm simply reporting 
what I hear.  I'm curious if anybody else hears this stuff similarly.

Brad Lehman

Date:    Thu, 1 Jun 2006 18:17:51 -0400
From:    Brad Lehman 
Subject: wide fifths

 >>an ahistorical inacceptable faulty overbrode-beating wolf-5th: Bb>F 
of ~704Cents,
  wrongly labeled by it's caned author as an "diminished-6th",
  in order to hide the evidently inherent shortfalling drawback
  inbred from misleading "French-ordinaire" confused taste,
  appearently undue inept compared versus the elaborate tuning 
standards, in Thuringia & Saxonia at the early 17th century, but
  all of them without the slightest indication of an overbroade wolf 5th 
on Bb>F,
  however tiny small debased, because no good futile in Bach's region 
since Werckmeister 1681, hence:
  The "French"-taste must be considered
  musicologically completely inacceptable!<<


Werckmeister IV and V (from 1691 as you know) both contain 5ths that are 
even bigger than 704 cents, described by him as good temperaments, and 
he wasn't French.

To quote my favorite fictitious doctor from the television: "Please 
state the nature of the medical emergency."

Is your objection that a widey exists at all, or that it's specifically 
on Bb-F?  (Werckmeister involved the note E-flat in his.)  Or something 

What do you make of Mark Lindley's temperament developed for Grosvenor 
Chapel, having 704-cent wideys on both of Eb-Bb-F?  The man certainly 
knows more about historical temperaments than I do.

Similarly, Harald Vogel's circulating temperaments frequently have one 
or more wideys in them.

And what about the several exemplary circulating temperaments among 
Neidhardt's 1732 set that have one, two, or three wideys?  Also not French.

Finally, what do you make of the fact that Bach regularly incorporated 
French and Italianate compositional elements into his music?  Where, 
then, is the step that a French/Italianate (*) practical idea about wide 
5ths should be verboten from hypotheses into Bach's tuning practice(s)?

(*) Not just French/Italianate, but also used by some notably skilled 
Germans and Americans (whether living or long-dead)...

Just curious....

If there is a musical or historical objection to 704-cent (or even 
slightly wider!) 5ths, within circulating temperaments, please state it 
and support it.  Thank you.

Brad Lehman

Date:    Fri, 2 Jun 2006 10:15:34 -0400
From:    Brad Lehman 
Subject: scientific evidence, etc

 >>   Can anybody name a historical temperament of impeccable pedigree
 >> wherein all seven diatonic notes of some other major scale *instead 
 >>of* C major happen to be regularly spaced, while C major is not?
 >Since this would require semitones and tones to be of equal size, the 
 >answer is that no scale of any system has ever had "all seven diatonic 
 >notes ... regularly spaced".
 >Or perhaps Brad meant that the tones and semitones were each uniform, 
 >that the six consecutive fifths which define the relative spacing of a 
 >diatonic scale's notes are all the same size?

Yes; sorry I didn't state that more clearly.  Regular diatonic scales, 
like the type that are treated in Easley Blackwood's book about scale 
structure.  Sequences of regular-sized 5ths, collecting notes until we 
have seven of them (generated by six 5ths), and then rearranging those 
pitch classes melodically to form a diatonic scale.

 >In the following temperaments the indicated major scales are thus 
 >perfectly regular:
 >K-II: G#, Eb
 >K-III: G#
 >Stanhope: B, F#, C#
 >W-III: C#.
 >All have irregular C major scales.

Of course this is correct.  These temperaments that have a long sequence 
of pure 5ths around the back side yield some Pythagorean diatonic 
scales, and those meet the criterion.

These Pythagorean scales also contradict my other premise about the 
major 3rd C:E being the smallest.

The temperaments that fit *both* these premises are (obviously) all the 
regular meantones, and the modified-meantones where the seven naturals 
are regular while the five sharps/flats are at some compromised 
positions.  Set all the natural notes to some regular size, and then 
fudge the sharps/flats until things sound decent enough for the music to 
be played.  And--although this probably isn't admissible evidence to 
positivism--anybody who happened to use the easy "Vallotti" layout in 
their musical practice before it was written down in extant sources.  A 
layout that is symmetrical in the same way the keyboard's positioning of 
key levers is.

 >> Such empirical inquiry into what's perceptible ....
 >Look up the term "controlled experiment".

OK, let's have a controlled experiment, as to musicians' ability to 
perceive some differently-sized melodic intervals while going about 
their usual musical practice.  I suggest that the sample set of people 
should be musicians who are personally able to perform Bach's tasks: 
temper an entire harpsichord in under 15 minutes by ear, play all of 
Bach's extant keyboard music, and improvise from his basso continuo 
parts.  This might tell us something about what's plausible for Bach to 
have done, in playing/improvising/composing with some expressive 
shadings from temperament...and the parameters of such a temperament 
that would work reasonably well for at least the whole of WTC book 1.

What do you do with all the 17th/18th century people cited in Rita 
Steblin's book, claiming to associate different moods or colors or 
expressions to the various major and minor keys?  Were they all just 
deluded and imagining things, or were they (at least in part) influenced 
by irregular tempering manners, which Steblin treats in a chapter?  Is 
all such opinionated evidence by musicians and listeners merely 
anecdotal, and therefore to be fluffed away?  (Steblin's book is one of 
the first places I looked for supporting evidence, myself, but there's 
unfortunately not enough from people who were firmly in Bach's milieu, 
as to which keys did what.)  Generally, *what* were such people hearing, 
and what caused them to form their aesthetic judgments?

 >While I'm on this subject I'll respond to one item of Brad's from 
 >> the best way to test all this is to play the
 >> music directly.
 >Google "anecdote" and "pseudoscience" together.

OK, I've done that.  But, how is googling better than reading Michael 
Shermer's book three times (as I've also done) and agreeing with it? 
This one:
And his chapter of 25 common fallacies is reproduced here:

 >It also would be most helpful if Brad could supply references for all 
 >those historical sources in which authors remarked on the importance 
of >*melodic* considerations in devising a temperament that is specific 
to >keyboard instruments.

OK, I'll look.  But if it takes me too long to provide those to your 
satisfaction, does that itself prove that good keyboard tuners *did not* 
ever consider melody to be important in fashioning temperaments?

Can musical practices ever have existed, apart from the selective types 
of evidence that are the only things allowed to the table by a 
positivistic method of inquiry?  How would we "know" that, for example, 
Froberger or Kuhnau never set up any circulating temperament for 
themselves because there happens to exist no firmly pedigreed evidence 
stating that they did?  (Shouldn't the same argument leave just as wide 
open the possibility that they *did*, and just didn't happen to write it 
down, being such a commonplace or taste-based practice that doesn't 
agree with modern scientific methods?  Bolstered by any evidence of 
their extant music that uses more than twelve enharmonic notes within 
the same composition?)

There we go schwinging off into epistemology again, but it's important. 
  Who gets to decide what manner of evidence is admissible to reasonable 
arguments?  Several here keep pointing out, repeatedly and correctly, 
that my theory doesn't satisfy a positivistic epistemology.  I realize 
that.  But why does the game of discerning hypothetical historical truth 
(and plausible practices) have to be a purely positivistic one?  Some of 
us would really rather spend the time playing the extant music, and 
trying to create beautiful performances, for whatever that's worth.

Maybe I'm unduly influenced by a favorite quip from Buckminster Fuller. 
"When working on a problem, I never think about beauty. I think only of 
how to solve the problem. But when I have finished, if the solution is 
not beautiful, I know that it is wrong."

Brad Lehman

Date:    Fri, 2 Jun 2006 10:52:13 -0400
From:    Brad Lehman 
Subject: Axiomatic

 >> Funny how Paul P's own professed favorite among the Neidhardt
 >> temperaments, namely the 1724 Village, has absolutely regular 5ths
 >> C-G-D-A-E as its core.
 >That's one temperament - ONE - among the dozens offered by 
 >Werckmeister, Neidhardt, Sorge, and Marpurg.  Nearly every other 
 >unequal temperament offered by these authors has irregular fifths here.
 >Look up the definition of "axiomatic".

Axiom, n.  "A statement accepted as true as the basis for argument or 
inference."  > POSTULATE.

Postulate, v.  "To assume or claim as true, existent, or necessary : 
depend upon or start from the postulate of.  To assume as a postulate or 
axiom (as in logic or mathematics)."

Postulate, n.  "A hypothesis advanced as an essential presupposition, 
condition, or premise of a train of reasoning. AXIOM."

Premise, n.  "A proposition antecedently supposed or proved as a basis 
of argument or inference; specifically : either of the first two 
propositions of a syllogism from which the conclusion is drawn. 
Something assumed or taken for granted : PRESUPPOSITION."

I've stated my axioms/postulates/premises clearly in public,
...which started this quibble about "axiomatic."  Maybe I should have 
said "Premise" on all the ones where I said "Axiomatically"?

The same list is here:

Can we please play music now?

Brad Lehman

Date:    Tue, 6 Jun 2006 12:55:38 -0400
From:    Brad Lehman 
Subject: temps with ordinaire-type adjustments


 >>The schisma then falls at C#-Ab, by taking F-Bb-Eb-Ab pure. Then go
backwards from C# by making C#-F#-B each about as narrow as C#-Ab,
then B-E pure, then you're almost done... yes, it is a bit odd to be
tuning the naturals last, but it automatically ensures circularity.<<

Yyyyeahhhh...that sequence of installing the notes might be a little 
odd.  But, at least, the most important thing is that the *resulting* 
layout is reasonable/reliable, whatever sequence may have been used to 
install the individual notes.

I like Quantz's practical suggestions to flautists and string players, 
to get themselves best in tune with the keyboard (and therefore the 
whole ensemble, congregating around whatever pitch standard the keyboard 
happens to offer):

- Tune your flute to sound good with the keyboard's F, even though the 
most obvious scale for the flute is D major.  Or if you're going to be 
playing a piece in E-flat or A-flat major, put the flute a little low on 
purpose, so things will work out best.  Or on an especially warm day, 
ditto.  (All of this is about putting the *center* of the flute's 
intonation in some place where it will do the most practical good, and 
be easiest to perform in tune by tastefully adjusting some individual 
notes as the music goes along.)

- Tune the string players' open strings to the tempered 5ths on the 
keyboard, individually, instead of letting those string players set pure 
5ths or even a little bit wide 5ths (!!) for themselves...as they might 
do if playing alone.

And presumably, the same types of ideas obtain in reverse, if the 
keyboard has to be tuned from scratch against some flute whose pitch (at 
some rogue amount too high or low) would set some _de facto_ standard 
for some particular gig.  The center of the keyboard's intonation scheme 
has to be where the flute can hit reasonably accurate notes, most 

...Perhaps best to lay down the keyboard's naturals (from whatever 
external pitch standard) before the keyboard's accidentals, for 
practical purposes, if it's for non-solo work.  And that's usually gonna 
be agreeing with somebody's naturals somewhere, not their sharps or flats.


Hence part of the reason why I believe, for practical purposes, 
keyboards for ensemble work especially well when at least the natural 
5ths C-G-D-A-E (and maybe more, on one side or both) have some regular 
tempering, no willy-nilly pure 5ths in that particular cycle of 
naturals.  Some regularity, downtown in those naturals: so the string 
players and maybe also the winds have some reasonable expectation of 
consistency.  Diatonic melodic themes can be played on a different 
string, modulating by a 5th at a time, without having to adjust the 
fingering unduly (as one would have to do if the 5ths varied too much 
among the strings).

That suggested syntonic comma temperament of
"... F -2 C -2 G -2 D -2 A -2 E 0 B -1 F# -1 C# -1 G# 0 D# 0 Bb 0 F "
is a good example for practical use.  All of those naturals are 
regularly spaced, except B.  And some "-1" in there is really a schisma 
wearing a mask, taking care of the enharmonic shift from coming round 
the other side.  All the syntonic comma temperaments have to stick that 
schisma in somewhere, like a freebie extra 1/12 comma.

A next hypothetical step is that temperaments that sound good in 
ensemble music also sound good in solo music...for the same reasons. 
The intonation of their notes falls at reasonably expectable places, 
making tonal music sound smoothish, and nothing startlingly far out of tune.

Brad Lehman

Date:    Tue, 6 Jun 2006 13:32:09 -0400
From:    Brad Lehman 
Subject: Miracle cure or placebo effect?

 >>Than I suggest and reccomend to choice in future merely musicians,
that able to perform the overwide overwide wolf 5th Bb>F precisely at 
~704 Cents exact, deliberately aware by intension in order to meet the
"Rosetta-stoned" smack also by all other members of the ensemble too.
Over-tempering 5ths in tuning means, alike
-oversalting dishes in cooking-
both a sign of questionable taste.<<

That line of gratuitous Lehman-bashing is really getting tedious.  Long 
ago, already.

Do those who object to 704-cent 5ths actually set them up and PLAY MUSIC 
IN THEM, to find out how it sounds in practice?  Or is that all a 
speculative argument of complaint, like "any port in a storm" to say 
something that looks important?

As to any beating, a 704-cent 5th and a 700-cent 5th are quite difficult 
to tell apart (played as an open 5th in the music, when sounded 
together), since the beating is comparable: the same amount, on either 
side of a 702ish pure.  It's merely a slight beating in the opposite 

The D# minor fugue of WTC 1 is a good example here.  Three voices and no 
clutter.  It has some exposed 5ths A# to E#, struck together, at several 
places in the piece...in subject entrances, no less.  Similarly there 
are some open 5ths of D#-A# and G#-D#.  Play the daggone piece.  Do any 
of these open 5ths sound noticeably bad in the temperament I 
recommended, if you've done a careful job of tuning the harpsichord? 
Especially, do you even notice that 704-cent 5th AT ALL, as any 
perceptible difference from those several 1/12 comma 5ths that are 700s? 
  If not, well then, please let this cherry-picking argument rest, and 
stop spitting these cherry pits in my direction.


N.B., especially to Daniel and any others who would prefer a pure 5th on 
their Bb-F: a pure 5th on Bb-F *does* stick out as different from the 
others, in the context of this composition.  Its inactivity makes it 
sound different.  Whether that's a good or a bad difference, 
musically/aesthetically as to value, is a separate question and there 
will surely be some reasonable opinions on either side.  PERSONALLY, I 
happen to believe the slightly active 5ths are more interesting to 
listen to, and a suddenly non-vibrato 5th (especially at a subject 
entrance or these stretti) startles me a bit too much.  The king-daddyo 
of those is in bar 77.  The entrance is already drawing attention to 
itself by *being* an entrance, soon revealed to be an augmentation, and 
a stretto of all three voices; does the thing have to be over-emphasized 
by being non-vibrato as well?  Why should the music suddenly stop 
shimmering for a brief moment?

This particular point about the A#-F or A#-E# 5th has also been available at
for at least half a year.  Among much else.

Brad Lehman

Date:    Wed, 7 Jun 2006 19:08:25 -0400
From:    Brad Lehman 
Subject: a practical response to Poletti's objections

> > Other things, like the irregularities of gut, the amount of moisture the
> > string has absorbed, the bowing point, the temperature, the humidity, and so
> > on, add to these factors to make string playing a constant activity of
> > aural-motor feedback. You’re supposed “expectation of regularity among the
> > strings” is nothing but the speculative musings of someone who obviously
> > knows nothing about the instrument, neither the acoustical theory nor the
> > practice.

"Nothing" about the instrument?  "Speculative musings?"  I used to be a 
violist, years ago.  I gave it up and focused on keyboards, although I 
do remember a thing or two.

But more to the point, and much more relevant: today I played (in 
concert) a Corelli solo sonata--Opus 5 #1--, with a professional Baroque 
violinist using gut strings.  His regular practice, years before he met 
me, has been to tune each of his four strings to the harpsichord's 
tempered A-D-G and E; he can do it himself, with pinky of the left hand 
playing the harpsichord key while the rest of his left hand is turning 
his pegs.

The man's dissertation was in mid-17th-century Austro-German violin 
music, and he prepares editions of 17th-century music for publication; 
he certainly knows how to do his job as musician and scholar, and it's a 
delight to play with him.  We *do* treat it as friendly play, listening 
and reacting to one another, and that's what makes it both easy and fun.

He's quite happy with my 1/6 comma naturals.

Our Corelli sonata today was in D major, but it went all over the place 
harmonically, and most notably it had a hefty rack of F# major and C# 
major dominant harmonies.  We played it directly from a facsimile of the 
Walsh edition, and both of us improvised a scheissload of 
Geminiani-style embellishment and harmonic-crunching into it, the better 
to try to catch the full measure of his good-taste treatise saying how 
it should be done.  It works because we're both listening and reacting 
all the time.

His double-stops and his arpeggiated harmonies in the Corelli were 
excellent, too, three or four notes going at a time.  There's obviously 
no question of trying to match any of that exactly to any harpsichord, 
but just trying to make it sound good through all its wild-ass 
modulations, including the F# and C# major chords he was playing. 
Corelli sure knew how to write exciting music.

I'm pleased that my tempered sharps agree nicely and sound good with 
where this excellent violinist was already putting them.  Sure makes the 
music easier to play.  Dare I suggest that this anecdotal evidence makes 
*some* manner of oblique argument for Bach (a better musician than I) 
possibly doing likewise?  Aw crap, now the positivists are gonna jump on 
me again for using anecdotes from the heat of real-practice musical 

Other pieces on the program (both of us together accompanying a singer) 
were in Bb, F, E, D.  And the singer and I did some other things in C, 
F, G, A, E, B, Bb, F minor, C# minor, and G minor.  All of those keys 
had to work, to do this program.  Some of these were transpositions 
expressly requested by the singer, the better to fit his most effective 

Obviously these fine musicians were adjusting their singing and playing 
*all the time* instinctively, and listening carefully, as the music goes 
along.  Every key has to make sense both melodically and harmonically, 
all the way through the gig.  I've known this from more than 20 years of 
playing professionally, and listening to what my colleagues do.  It's 
called musicianship and experience, and I don't need to be lectured 
about this.  Just pointing out the obvious, here.

Over the past couple of weeks of rehearsal, with this particular group 
to prepare this gig, there were only rare times I ever had to make any 
comments about intonation.  These were when the occasional G# was being 
done even higher than its place in my already-bright E major, or when 
the high D was occasionally sharper than its position within my B-flat 
major.  I mentioned these points once each, it was fixed immediately by 
simply drawing 30 seconds of attention to it, and all those passages 
went fine thereafter.  Good musicianship and experience by my colleagues.

Paul, please don't conflate/confuse me with any people who would insist 
on everybody matching the keyboard's pitch/temperament all the time, on 
every single little note; that's silly, and we've already addressed this 
point before.  It's also not doing your argument any good to make up a 
straw-Lehman who is allegedly totally stupid about such things, and then 
ranting in public about it.

I hate to burst *your* dissing bubble, Paul, but I'm not a moron or any 
mere speculator about musical practices.  When opportunities arise I 
play real concerts, and I especially enjoy making up the whole 
accompaniment from figured bass, like today's gig.  Both because that's 
a historically plausible way to render such music, and because it's 
supposed to sound really good.  The people who commented to me or asked 
questions after the concert all said it was (good-sounding, I mean). 
They also said they'd never heard such dynamic or resonant 
harpsichord-playing before.  That's what counts, IMO: that the listeners 
have a great time and that the music comes across clearly/strongly.  Not 
that *you* need to be lectured about *this*, but I'm responding 
defensively to your personal attack, so hey.

And not a word was said to anybody at the concert that I personally 
believe this was probably Bach's own temp.  It wouldn't be relevant, nor 
did we play any Bach today.  The relevant thing is to prepare the 
instrument to sound as good as it can, in the rack of various music to 
be played, and then get up there and try to play the heck out of it: to 
give a good time to the people who showed up to hear it.  Congregate at 
the appointed time and try to do some good musicianship.

And now it's time to go schlep the instrument out of the van and back 
into the house.  Looks like y'all have been busy here today.  I come 
back from a gig and there are 25 postings to catch up with.

 >>Now, Brad is coming along as a roadway engineer and saying, “It would 
be much easier to steer your car around the turn if the turns are made 
with irregularities of no more than 1 cm every 50 meters.” His precision 
is too far below the normal corrections to be of any real advantage.<<

What straw-Brad is that?

A longtime friend of mine *is* a roadway engineer, and he has a T-shirt 
that proclaims: "Civil engineers guarantee their erections."  But I digress.

I dare to suggest that today's gig would not have sounded quite as good 
if we'd merely used Vallotti, or Workmaster, or something too close to 
pure meantones.  Too many forays into F#, C#, B, and Ab major played 
directly.  Some of the Neidhardts I've been fussing around with *might* 
have worked OK, but those are touted as organ temperaments; how would 
they have anything to do (one way or another) with what might sound good 
in practice on an Italian-styled virginal?

Brad Lehman

Date:    Thu, 8 Jun 2006 11:15:32 -0400
From:    Brad Lehman 
Subject: Re: temps with ordinaire-type adjustments (but about fiddles)

Bill J wrote:
 >>>Point is, you don't just clamp down the finger, fingering was in
 >particular key and passage dependent and you have to listen and above
 >all practice. Always. But that doesn't deter from the fact that having
 >the open strings tuned to regular fifths of any size is an advantage.
 >That is how I understood Brad. Not about playing in those keys.

Paul P responded:
 >this is what Brad said:
 >"Diatonic melodic themes can be played on a different
 >string, modulating by a 5th at a time, without having to adjust the
 >fingering unduly (as one would have to do if the 5ths varied too much
 >among the strings)."
 >If this doesn't mean playing in those keys, then I don't know what it 
does mean.<<

Well then, Paul, you didn't catch my drift.  Let me try once more.

Try some easy first-position exercise using diatonic intervals.  "Ut mi, 
re fa, mi sol, fa (new string...) la, sol ti la ti ut" will do nicely, 
with no chromatic notes to worry about.  Play it on pairs of strings: 
G&D, D&A, A&E.

All those notes in first position should be in the same corresponding 
spot on every pair of strings (give or take a few microns to adjust for 
string thickness/material, which was *your* argument!) if you've got 
regular 5ths set up on that fiddle's open strings.  Put the first 
(pinky-fingered) "sol" in the place where it's going to agree with the 
open-string second "sol".

The exercise comes out the same on all three pairs of strings, because 
all three pairs have the same relationship there.  And, the "la" is 
fingered at the same place on the second string that the "re" was 
fingered on the first string.  That "fa" to "la" transition has to sound 
as decent as an "ut" to "mi" would be, and that's the point of playing a 
smooth (meantonish/regular) major scale to locate all the diatonic 
intervals.  Whatever size 5th is installed from string to string, that 
determines where everything else goes for smoothest result.

But, if you've got some screwball scheme like Werckmeister 3 or 
Kirnberger 2, with the violin's open strings matched to some keyboard in 
such, where some of these string-crossings are pure 5ths and others are 
tempered 5ths, the exercise suddenly gets harder.  Same exercise calls 
for appreciably different fingering points, to get the transition 
between the strings to be smooth.

And this is a melodic point to make reasonable-sounding music; that 
keyboard doesn't even need to be playing at all.  The violinist isn't 
trying to match dingdong points in whatever keyboard temperament; s/he 
is trying to play a nice simple diatonic melody reasonably well in tune 
with itself.  This is not about deferring to any keyboard's crappy 
intonation in whatever keys.

And yes, of course, the finer players will be making various adjustments 
within such an exercise or melody to get the thing to sound even better; 
but that wasn't the point.  Never mind trying deliberately to put the 
"mi" and "la" and "ti" a little lower, each, in their contexts as major 
3rds above a different degree--that's a refinement for closer to just 
intonation.  The point at a more rudimentary level was that the regular 
5ths set up a basically sane position, where one doesn't have to 
remember that (for example) A-E open strings have a noticeably bigger 
5th than D-A, so every note on the E string would have to be fingered a 
little lower.  Yecch.

Bill J understood me correctly, I believe.

Brad Lehman


Date:    Tue, 20 Jun 2006 16:58:27 -0400
From:    Brad Lehman 
Subject: "regular"

 >>Hmm, where do you get this "regular" definition of "regular"? I call 
anything which uses only one size of tempered fifth "regular".<<

Yeah, me too, and I'd think that "regular" applies to the whole range of 
temps that I described back here in March:

Eleven 5ths with the same size, and one leftover diminished 6th of 
whatever size.  That last guy is the same size as the 5ths if we're in 
equal; or he's larger if the 5ths were tighter than that; or he's 
smaller if we're getting toward a Pythagorean tuning of pure 5ths.  But 
all these are "regular".

Back from three days of business traveling where I was staying at a 
Holiday Inn Express.  I still haven't seen the commercial and still 
don't get the joke.

Brad Lehman


Date:    Fri, 23 Jun 2006 09:06:31 -0400
From:    Brad Lehman 
Subject: "regular"

 >>That questionable "dim-6th" concept as closing-interval the circle of 
5ths corrupts all Such private use of "regular" produces evidently 
deviations inbetween theory-labeling versus the really obtained interval 
seizes in practice.<<

As I pointed out, in some systems (such as Pythagorean) that diminished 
6th "closing-interval" is smaller than the 5ths; and in some other 
regular systems it's bigger than them.  That's not news.  Whatever its 
size, it's still a break.  The only regular temperament in which it 
coincidentally *is* the same size as the 5ths, is 12-note equal temperament.

All the 12-note "regular" (or "meantone") temperaments have 11 5ths of 
one size, plus a closing diminished 6th where the spiral breaks.  They 
are a subset of 12 notes selected from an enharmonic spiral that keeps 
going outward in both directions.  The diminished 6th is the gap, across 
which the notes get renamed to a different enharmonic name (such as Ab 
to G#).

Any two-note intervals that cross that spelling gap are *not* the same 
as other intervals that would look to be the same distance, on the 
keyboard.  For example, C# to F is not a major third, but a diminished 
4th, due both to the spelling and to the different size.  For example, 
C# to F in Pythagorean happens to be *smaller* than the major 3rds 
available in correct spelling, within that system.

To understand all these "regular" tuning systems (and "regular" meaning 
"not irregular"), I suggest reading the following book of mathematical 
theorems about this topic: Easley Blackwood's _The Structure of 
Recognizable Tunings_

Brad Lehman


Date:    Mon, 26 Jun 2006 18:22:16 -0400
From:    Brad Lehman 
Subject: Re: tuning woes

 >>The computer for me would just be an extremely flexible set of tuning 
forks, pretty much, for setting temperaments. Even then, I
usually just use a fork (or some other tone) to get the A and do the
rest by ear, but I'm not so accurate that I can do two harpsichords that
way and be sure that they'll match, and it can be risky on unfamiliar
instruments, or in noisy environments. Or when the tuner is 

I'd rather match the several harpsichords to each other, rather than 
matching each in turn to an electronic device (or even to a couple of 
forks).  Easier to hear harpsichord tone blending with other harpsichord 
tone, for unisons, than to other types of tone.

Last year for the Bach quad-concerto I tuned for all the rehearsals and 
the perf the same way:

- set up the loudest hpsi first, completely, as the standard;

- with an assistant, tune the tenor region's F-G-A and C-D-E as unisons 
from the main hpsi, onto each of the other three hpsis...good clean 

- everything else is merely pure 5ths or slightly tempered 5ths from 
there, so each other hpsi got a fresh set of the remaining notes on 
itself working from the already-fixed placement of C, D, E, F, G, and A; 
establish about an octave and a half, center of the keyboard;

- go back to #2 and finish all its bass, then treble;

- go back to #3 and finish all its bass, then treble;

- go back to #4 and finish all its bass, then treble.

The rotation of the last three hpsis was to give each one a few minutes 
while the next one was being done, in case we were going to get any 
drifting.  It would be caught when coming back to that one, re-testing 
the C, D, E, F, G, A just in case.

Worked fine, and the whole tuning task took about 35 to 40 minutes to 
get all four instruments done...only one 8' on each, which was all we 
needed.  No electronics.  The assistant had to be present for only about 
three minutes.

Brad Lehman


Date:    Tue, 27 Jun 2006 14:40:05 -0400
From:    Brad Lehman 
Subject: Re: tuning woes, or whatever else

 >for crying out loud! just learn to do it. It ain't that hard.<

Nice side effect: develop awareness of and complete control over 
intonation, which is one of the strongest expressive resources available 
on the harpsichord.  Whatever intervals are set up, whether they're pure 
or tempered, the harpsichord brings them out with stark clarity...and 
that's part of the musical delivery both melodically and harmonically. 
(Assuming that one actually listens to oneself while playing!)

I find it hard to imagine (anymore) playing harpsichord without having 
awareness of whatever temperament's expressive shape, as I play.  It's 
an essential part of the musical interpretation.  Tune the ax in some 
appropriate way, as part of deciding what music to play for the session, 
and then go play it.

At the start of any practice session, before playing, I have to check 
the relationships in whatever temperament I've set up (or do a fresh one 
from scratch if I feel like it), along with checking all the outer 
octaves in both directions.  If I don't take the minute or two to do 
that, to combat whatever the weather has done in an intervening day, I 
know I'm going to waste time and frustration *during* the playing 
stopping to fix gamey notes.  I'd rather get it out of the way before 
playing, so there just won't be any.  And no little gizmo is going to 
tell me what sounds right, better than carefully trained ears will do.

Violinists tune before playing.  Why should it be any different for 
harpsichordists?  Just part of the required routine, like tying the 
shoes before going for a walk in them.  Sure, it takes some time and 
practice to learn, and skills can *always* be improved further; but so 
do most things worth doing.

Learn it early (teen years or before)...like with languages, and typing 
100 words per minute, and driving stick shift...it ain't gonna get 
easier later, after there are inefficient bad habits to break first, or 
mental blocks about how hard it allegedly is.


One unfortunate side effect: it can become difficult to listen to 
anybody's recordings (or live perf) where they didn't take sufficient 
care with their octaves or unisons, and especially with a gamey 4-foot. 
  Excruciating, knowing that the *&^# thing would have taken only 30 
seconds to fix if they'd care enough to stop and fix it.


A question from me, which might sound disingenuous but which is an 
earnest one: what is the hard evidence (if any!) to perform even a 
single piece of Froberger's harpsichord music in regular 1/4 comma 
meantone?  (Some of the eminent players *do* it, of course, and have 
done for years, but what's the firm musicological justification for it?)

And I'm not, for the moment, accepting the circular argument that some 
wildly wrong enharmonic moments are definitely part of the musical 
texture, which somehow only regular 1/4 comma is able to bring out 
sufficiently.  What is the argument that *any* pure major 3rds--along 
with the strongly dissonant wolf diminished 4ths, elsewhere--should 
exist anywhere within Froberger's harpsichord music?

Brad Lehman


Date:    Tue, 27 Jun 2006 17:08:19 -0400
From:    Brad Lehman 
Subject: Re: tuning woes, or whatever else

 >> I tend to agree with Bill about just quitting making such heavy
 >> weather of it and practicing how to tune.
 >Couldn't agree more. My question is how you practice?
 >Obviously the most important thing is just to tune a
 >lot! But I was wondering if anyone, when trying to improve
 >their tuning skills, set up any sort of exercise for themselves?

Yes.  I set myself the exercise/practice of tuning *any* arbitrary
meantone (i.e. "regular"), with a time limit of 15 minutes for a
complete 8-foot register.  Or less.

- Get either C or A from a fork.

- Put an E over the C, or an F under the A, with only the criterion
that it's some bit wide or narrow (!) as a major 3rd.  But not any 
pre-determined numerical amount, in beats.  Just *some* amount
(which might even be almost as wide as equal temperament's major
3rd).  That puts us somewhere in the range between 1/3 syntonic
comma and 1/11, but we don't care about any of those numbers (for
the purpose of this exercise, which is a *listening* exercise).
It's all a continuum.  Our exercise is to get all 11 5ths the same
as one another, in quality, generated from whatever size of major

- Generate all 12 notes accurately, choosing some reasonable
disposition of 12 notes: from Eb-Bb-F up to F#-C#-G#, or from Ab
up to C#, or from Bb up to D#, or from F up to A#.  The diminished
6th wolf ends up at any of the four places G#-Eb, C#-Ab, D#-Bb,
or A#-F.  The size of that wolf is determined wholly by the size
of the initial major 3rd we chose.

- By "accurately" I mean using the triplets-to-duplets method I
have described:
* http://www-personal.umich.edu/~bpl/larips/tetrasect.html
...but the exercise can also be done by simply listening for
equal *quality* within the 5ths and the resulting major 3rds,
and not counting any beats anywhere.  As I described at a different
time here:
* http://tinyurl.com/oca2y
Practice it both ways on different occasions: either the hardcore 
beat-counting of 3 to 2 speeds, or the quality-matching without
counting anything.

- Set all the octaves outward in both directions, checking every one
of them (except the wolf) with the intervening 4th and 5th to be
sure the octave is true, with that same triplet-to-duplet rule, or
the sensing of equal quality.

- Play music.

With enough runs of this exercise, at different sizes of starting
major 3rd, the process itself becomes very quick and 
accurate...especially the part about accurate octaves, which also
serve as a good test for the regular 5ths/4ths.  If any of those
5ths or 4ths have already drifted, they get corrected immediately.
My hands just automatically do the tests and corrections, without
my mind needing to be very consciously engaged; could even be
thinking of something else at the time.  Sometimes I'll find that
I've done at least half an octave up or down, chromatically,
while my thoughts were entirely elsewhere.

Making the thing that much "second nature," the basic shape and
behavior of meantone also become crystal clear.  Just keep track
in the mind which enharmonic note-names exist in the chosen
disposition, and which other ones are going to be way out (and
therefore making the music sound spicy).

1/4 comma is only one special case of this general shape,
where the starting major 3rd happened to be beating 0.

And the tuning chore hardly takes any time or conscious effort,
eventually.  The instrument is going to start to drift within the
hour anyway, so get the tuning part done quickly and spend most of
the time playing!


"Modified meantone" is a cinch, as to lowering flats
progressively and/or raising sharps progressively, if the skill
of reliably regular 5ths is already in place.  Different exercise:
whenever we get into smelling range of the place we're going to
put the wolf, deliberately start to temper the regular 5ths less
(maybe even as wide as pure, or--gasp!--the slightest bit of
overshoot into wide 5ths).  The temperament becomes irregular,
and usable in more keys, as soon as we start to relax some of
those regular 5ths into something wider, and testing all the
resulting major 3rds as they come up in the sequence.


If only I were this diligent about sit-ups or jogging or weights!
But it's a lifestyle choice to be decent at hearing regular
tempering, and to have really well-tuned octaves.

Brad Lehman


Date:    Wed, 28 Jun 2006 14:47:18 -0400
From:    Brad Lehman 
Subject: tuning 2x8

 >>I completely concur with Owen's suggestion here about time limits. 
(...) However I am not fast enough to do the whole thing in 20 minutes, 
because these days I tend to do octaves and check the new interval with 
both of the octave pitches.<<

Sounds to me as if that process is making it unduly complex, to get both 
8-foot registers done at the same time.  (Is that what "check the new 
interval with both of the octave pitches" means?)

If setting up a 2x8, here's what I do--and I'm curious what others do:

- Set the temperament on the *front* 8...where any beating is easier to 
hear because of the brighter tone.

- Establish the bearings from approximately tenor c up to the a' above 
middle C.  [Elapsed time: usually under 4 minutes]

- Go chromatically all the way down to the bottom of the instrument, by 

- Engage the back 8, and set unisons chromatically all the way *up* from 
the bottom of the instrument until reaching the top of the region where 
we had the bearings.  [Elapsed time: usually under 8 minutes]

- All the tension adjustment to big strings is now already done, and we 
haven't touched the treble of either register yet.

- Disengage the back 8.

- Finish all the front 8 chromatically upward, by octaves, from the 
place where we abandoned it earlier.

- Engage the back 8, and take it up with unisons from that same point. 
Both 8s are now done.  [Elapsed time: usually under 12 minutes]

- If there's a 4-foot: engage it, turn off the back 8 (!), and do the 
whole thing chromatically up from the bottom of the bass with the 4-foot 
plus the front 8.  Again the brightness of the front 8 tone helps us 
out.  [Elapsed time: usually under 15 minutes]

- Turn the back 8 back on so we've got everything, and do a small bit of 
playing to find/fix any obvious errors anywhere.

- If the 4-foot hadn't been tuned for a month or more, do it again 
because it's probably already out.  The point is not to fuss about it 
too much or too long on first pass, above, if some of those notes are 
going to take two stages anyway.

Brad Lehman


Date:    Thu, 29 Jun 2006 11:35:17 -0400
From:    Brad Lehman 
Subject: Re: tuning 2x8

 >>Your suggestion of doing the lower notes of both ranks before the 
 >>trebles is an interesting one- I'll give it a try next time.
 >Brad's method is really the best, since the highest tension is always
 >in the low tenor on every instrument, both harpsi and fortepiano. Thus
 >one does the lower half of both registers first, and any moving of the
 >structure caused by tension changes will be pretty much over with. I
 >find that if the instrument is reasonably out of tune, doing one 8'
 >complete and then doing the second inevitably results in an out-of-tune
 >treble. I'm not talking about a touch-up tuning, of course.

To be clear, it's not "my" method doing that chromatic upward bass thing 
before touching the treble; I learned it from Parmentier.

Before that, I would do one whole 8 first, then engage the second one 
and go from middle C all the way down, then up, copying unisons.  And 
when I'd check the end result by retesting octaves, I'd run into 
problems especially in the treble: that my first 8 had already moved to 
give lousy octaves before I got back to it, and I'd have to do all that 
treble again.  Having already done the unisons from the other 8, I had 
two notes to fix for each drifted problem.  Or three if I'd already done 
the 4-foot.  Ugh.

Since switching to Parmentier's sequence, I could do the whole 
instrument about twice as fast as before *and* have it stay in tune 
better...so I kept it, obviously.  No wasting time doing the treble half 
of the instrument twice.  There are better things to do, like play music.

Here's my posting about this from October 1995, saying essentially the 
same things:

Brad Lehman


Date:    Thu, 29 Jun 2006 19:08:19 -0400
From:    Brad Lehman 
Subject: Re: Fretting over frets yet again

 >>Not to pour salt on an open wound, but quit by accident I’ve 
uncovered more evidence that Brad is off on a wild goose chase with his 
idea that having similar fifths among violin strings somehow makes the 
job of playing the violin easier because you can just slap the fingers 
down in the same places to play the same pattern on all four strings. 
More important, the fact that one CANNOT do so has been recognized for 
quite some time. (...)<<

The thickness/materials thing is all well and good, and thank you for 
the information.  But, that wasn't the point of the issue.

It's this: all other considerations being equal, is it better for the 
string player to have regularly spaced 5ths (of whatever *consistent* 
size) at all three places, or have differently weirdo 5ths to keep track 
of at different places?

Which conceptual model helps the player imagine the note and hit a 
nearly-correct spot *before* playing it, best?  Regularity, or having 
some mixture of impure and pure 5ths?

Putting this another way: if the player is *additionally* dealing with 
the intonational anomalies of the string materials, weather, etc among 
the problems...why make things yet harder by inducing uneven 5ths?

Brad Lehman


Date:    Wed, 5 Jul 2006 11:47:54 -0400
From:    Brad Lehman 
Subject: Re: non-scientific tunings

 >>Most of these instructions are scientifically "problematic" because 
they are based on temperament instructions of the following kind:
"Tune C. Tune E and G above it so that they sound good. Tune F and A
below C so that they sound good. Tune B and D above G, so that they
sound good. Check the fifths E-B and A-E to see that they sound good.
If not, go back and check what you've done so far. (...)"  There are so 
many of these non-scientific sets of instructions that we should at 
least ask ourselves the fundamental question of how ordinary players 
tuned at the time.<<

OK...how about the following set, in that style, as oral instruction at 
a lesson:

- Tune the home key, the C major scale, first.  We do that by putting 
the Ut-Re-Mi-Fa-Sol-La into order by 5ths or 4ths.  Fa-Ut-Sol-Re-La-Mi. 
  That is F-C-G-D-A-E.  Tune all of those in the normal way, which is to 
make the 5th or 4th pure and then give the tuning lever the slightest 
couple of nudges to make the 5th narrow (or the 4th wide).  When you get 
to E-B, just leave it as a pure 5th.  Play a few things in C major to 
check that it all sounds good.  Yes, the major 3rds are a little wide.

- All our natural notes are done.  We just need the accidentals now, 
doing the B major scale for convenience.  Fa-Ut-Sol-Re-La-Mi is 
E-B-F#-C#-G#-D#.  We already have E-B pure.  Make B-F#-C# pure also. 
This gets the sharps up good and high so they can also be used as flats. 
  For the C#-G#-D#-A#, when you do the G#, D#, A# in turn give them a 
single nudge a little flatter, each, so they don't all get too high 
where they would sound bad.  Test the B major scale with some simple 
music, and improvise around to some other keys if you feel like it. 
Check the C# major, F# major, and other spots that would sound rough in 
other methods, and make sure they sound decent to you here.

- That's the whole setup.  Now play through this book that demonstrates 
all the scales sounding marvelous, in some worked-out music.  *All* the 
major scales Ut, Re, Mi, and *all* the minor scales Re, Mi, Fa.

- Here's a simple little picture showing that same nudging process as a 
reminder which notes get what, but you've probably already memorized it. 
  The C major scale is this half with the double nudges, and the B major 
scale is that other half.  They overlap with the E-B, there in the middle.

Brad Lehman
(Works for me, easily in under 5 minutes and counting no beats or commas 
or anything)


Date:    Thu, 6 Jul 2006 09:28:06 -0400
From:    Brad Lehman 
Subject: Re: non-scientific tunings; enharmonics vs meantone

 >>In view of temperament for example, this kind of source evaluation is
important for my research, as I see it:
– If organ-related sources, printed matter (intentional tradition) and a
multitude of area-wide information preserved in manuscript (in various
archives, or documented in source editions and published archival 
research, like examination reports, letters etc.), i.e. unintentional 
tradition,  so far have shown only, that meantone temperament was seen 
and used as the (intended) standard for organ temperament in Northern 
Germany the 17th century and until quite long after the turn of the 
century, (as represented in my research, which I hope to publish in the 
near future), and that well-tempered systems and equal temperament are 
curiously absent from all documents (of the practice of organ 
temeprament in that area), then the conclusion is appropriate, that 
meantone actually was the intended standard. That we have just a handful 
cases of unintentional modifications, and (about five) known both 
unsuccessfull and criticized cases of unspecified modifications around 
1700, seems to corroborate my present view.  Now, I would never claim, 
that I have proven that claim as close as the holocaust has been proven 
to be a fact of history, and I will follow any better evaluation of the 
multitude of evidence, but the burden of proof (i.e. disproving muy 
claims or hypothesis) is now at those who claim something else.<<

In response, I'd point out some things that you have surely heard before.

Crossing against the meantone-as-standard hypothesis: there exists a 
body of composed MUSIC for organ, having lots of notes in it such as E#, 
B#, Cb, Gb, Db, Ab, A#, D#--notes that DO NOT EXIST within the typical 
regular meantone schemes, 12 notes in the octave.  This isn't purely an 
aesthetic problem; the notes are misspelled within the music (from the 
perspective of regular temperament) AS IF those notes should sound 
plenty normal, not unduly odd.  The assumption of regularity has to be 
questioned, due to the presence of those notes and the way they're 
spelled within tonal context.

If it is necessary to cling to the assumption of regularity anyway, one 
would have to demonstrate a series of dodgy excuses to get past that 
problem of enharmonics:

- perhaps that the music in manuscripts and publications wasn't actually 
*played* (but was only theoretical/speculative) with any expectation of 
sounding unproblematic;

- perhaps the players only transposed it to keys that happened to work 
without enharmonic error (which is absurd for pieces such as BWV 547 
that has more than 12 chromatic notes functionally named within it...and 
there are plenty more compositions where that came from, not only by Bach);

- perhaps the music, even if it's organ music, was played mainly on 
pedal harpsichords or pedal clavichords that the players could retune at 
a whim, diddling around to differently tuned notes so the enharmonic 
misspellings wouldn't sound as noticeable;

- perhaps the musicians and lay listeners simply couldn't hear, or 
didn't care, that the enharmonically misspelled notes sound wrong within 
their tonal contexts;

- perhaps those pesky compositions are only supposed to be played on 
split-key instruments?...but then again we're still lacking B#, E#, and 
Cb at least;

- perhaps the use of "special effects" in the form of suddenly-rough 
intonation is an expressive and integral part of the music...bringing 
some aesthetic decision into the picture, how rough is too rough for 
satisfying *musical* effect?...and how do we "know" that the composer 
deployed wolf intervals purposefully for such a strong shock effect?;

- perhaps some of the organs were tuned more moderately, such as 1/6 
comma rather than 1/4 comma, offering moderated wolves and some 
additionally useful chords (such as B major and F# major in first 
inversion) due to some attractive acoustic phenomena;

- perhaps "modified meantone" was more prevalent than the hypothesis of 
hardcore meantone would suggest, especially around the notes G#/Ab and 
D#/Eb (builders and organists "cheating" by taking those notes off the 
normal meantone positions, and neglecting to document it satisfactorily);

- others....

Shouldn't at least some of the burden of proof be on those who would 
assert that the music "should" sound enharmonically rough (to satisfy an 
antiquarian approach or whatever)?  That the music "should" draw 
attention to itself with sour sounds, whenever we encounter an 
misspelled note, one that doesn't exist in the temperament?


I used to be a big fan of meantone, myself: playing in it almost all the 
time, for years, either in its pure/regular form or with slight 
modifications.  But, eventually I got really tired of hearing ugly 
sounds whenever playing music that looks perfectly normal, and the 
virtues (extra euphony in *good* keys) didn't add up enough, to me, 
anymore.  Meantone worked in *some* music, but it also took a big part 
of my joy out of practicing any other music.  But this is just a 
personal comment by me.  And, I'd emphasize, I *still* keep one of my 
home instruments in various meantone shades at least 95% of the time, to 
play earlier music for which meantone *is* obviously appropriate.

On the other hand, I have a really hard time believing that Froberger, 
Buxtehude, Bach, Handel, et al would have written the marvelous music 
they did, if they had been stuck firmly with *only* 1/4 comma meantone 
and its resources, on all or most organs.  Why would they bother writing 
music that sounds harsh, within such a standard, in the keys they did?

Lots of Bach's music that raises enharmonic problems in meantone:


I'd also point out that any connections with holocaust-denial (on either 
side of any argument!) contribute a not very helpful spin to the 
argument, by association/suggestion.  There's a vast difference between 
killing people (institutionally or otherwise) and playing music. 
Mention "holocaust" or "holocaust denial" and people's emotional hackles 
rise.  Emotional hackles rise with music, too, obviously; but again 
there's a vast difference between trying to produce satisfying music 
(with reasonable historical/antiquarian accuracy as well), vs 
exterminating human beings.

Brad Lehman


Date:    Thu, 6 Jul 2006 13:53:26 -0400
From:    Brad Lehman 
Subject: Re: Historical method

I wrote:

 > - perhaps the players only transposed it to keys that happened to work
 > without enharmonic error (which is absurd for pieces such as BWV 547
 > that has more than 12 chromatic notes functionally named within
 > it...and
 > there are plenty more compositions where that came from, not only by
 > Bach);

Ibo apparently missed my point, and replied:

 >>We have a multitude of evidence that transposition occurred frequently
(actually one of the tasks, like complex improvisation, constantly 
examined thoroughly, in organist's examinations), both for ensemble 
music and chorale playing. Transposing keyboard pieces is described 
within clavier pedagogy, even very complex pieces.  In that context the 
existing written down transpositions of Bach's pieces are interesting 
and would contradict your view that a Bach-piece with "more than 12 
chromatic notes functionally named within it" would be "absurd": One
of Bach's most harmonically most challenging pieces the fantasia g-minor
542a) exists in a (later) f-minor version. <<

Let me try my point once again, to make this clear.  I didn't say 
anything (or mean anything) about transposition *itself* being absurd.

Rather, I pointed out: some compositions (such as BWV 547 that I cited) 
DO have more than 12 differently named chromatic notes used within 
functional harmony, directly, as the piece goes along.  And, the kicker 
part: no amount of transposition is going to solve that, playing on a 
regular meantone keyboard without split keys.  *Some* notes are always 
going to be wrong enharmonics, not existing in the temperament, no 
matter how much the whole piece gets transposed around.

Similarly, on my CDs, I recorded an organ piece by Froberger--a 
liturgical toccata for the elevation--that goes beyond 12 notes....

Brad Lehman


Date:    Thu, 27 Jul 2006 11:00:32 -0400
From:    Brad Lehman 
Subject: on the air today

Peter Watchorn and I are the scheduled guests on the "Global Village" 
radio program today.  It's a production of KPFK-FM in Los Angeles, and 
the Thursday host is John Schneider (composer/guitarist and a specialist 
in unequal tunings).

Details, and the webcast link:

The time of this segment on today's program is the hour 10:00a.m. to 
11:00, Pacific Daylight time.  Adjust accordingly later into the day, 
for your own time zone....

The discussion topic is my Bach tuning research, focusing especially on 
some musical results of my published method as it sounds on harpsichords 
and organs.

We will hear and discuss excerpts from Watchorn's newly-released 
complete WTC book 1:

We will similarly hear and discuss some examples from my several recent 
recordings, as linked below.  I have a single disc of harpsichord 
selections and a 3-CD set on organ, all chosen to demonstrate the 
various characters of the keys where this temperament remains the constant.

I believe Schneider will be bringing in some comparative examples of 
other people's recordings, in different tunings.  I don't have details 
on that part; the whole program will probably be very informal and 
improvisatory by all of us.  That's similar to the program that 
Schneider did with Richard Egarr a few months ago, during his US tour 
with the Goldberg Variations in this tuning.

Brad Lehman

CDs and free downloads:
harpsichord...     http://tinyurl.com/gd5up
organ...           http://tinyurl.com/zhcse
trumpet/organ...   http://tinyurl.com/jrn4g

personal web site: http://tinyurl.com/jnjo4


Date:    Fri, 28 Jul 2006 13:33:13 -0400
From:    Brad Lehman 
Subject: "these sciences died with him"

 >>C.P.E. in 1774, wrote about his father in Dokumente III, 1972 Kassel, 
#801, p.285
   "Das reine simmen seiner Instrumente so wohl, als des Orchestres war sein
   vornehmstes Augenmerck. Niemand konnte ihm seine Instrumente zu 
Dancke stimmen und bekielen. Er that alles selbst....."
   ..."The pure tuning of his instruments as well, as of the orchestre 
was his
   exclusive attention.  Nobody was able to tune and quill with
   pinfeahters his instruments to his satisfaction. He did everything 
   also there on p. 284 C.P.E. declares finally:
   "Diese Wißenschaften sind mit ihm abgestorben"
   "That sciences died togehter with him."<<

I guess you'll need to reconcile that "died with him" quip (feeding 
Forkel for the biography) with the fact that CPE Bach himself was the 
regular deputy harpsichord tuner for the main Leipzig church, with JSB 
as music director.  1731-33, which would have made CPE about 17 or 18 at 
the time, old enough and skilled enough to be entrusted with that job. 
If meticulous tuning and regulation were that important to JSB, as 
claimed here, the deputy job doesn't go to just any old hack but to 
somebody worthy.  Some details:

Should CPE have gone on to promote himself in that letter by saying, "By 
the way, the exception to my general statement here is that Dad also 
approved *my* work, after teaching me how to do it..."?

Brad Lehman


Date:    Fri, 28 Jul 2006 13:50:14 -0400
From:    Brad Lehman 
Subject: Re: On the air today

 >>I got Brad Lehman's e-mail about the show at around ten minutes to 1, 
so was able to listen to it.  I think that Brad Lehman and Peter 
Watchorn were ill served by the announcer's constant reference to the 
Lehman tuning system as "rediscovering the Holy Grail," hype that is 
really unhelpful for any discussion.  As someone who has been sort of 
following the debate on this listserv, I can certainly see what is at 
stake but think that that kind of hype only obscures the issue.  The 
comparison of chords and scales in the Lehman temperament and in 
meantone provided a very clear contrast--but, of course, meantone is 
unlikely to have been the alternative; some other kind of 
well-temperament is, and, there, the contrast and the benefits are not 
as immediately obvious to the listener.<<

Glad that some members here could listen in, and I appreciate the other 
comments I've received off-list.

John Schneider's program was of course for a popular daily series and a 
general audience, not for tuning specialists.  The show is regularly 
very eclectic.  The audience might not even be into "classical" music at 
all, or know anything about *any* unequal temperaments.  We had to pick 
the clear contrasts that might be made within that particular game.  I 
thought John did his job well to hold general interest.

Within the context of a single one-hour program, and for listeners who 
might never have read *anything* about the topic, contrasts have to be 
as clear and direct as possible.  If those listeners want to follow-up, 
John made clear reference to places where they can read much more about 
it, and purchase the recordings for the leisure of listening again more 

Peter Watchorn did say quite a bit, in his segments, about the 
experience of trying out the music in other subtle "well-temperaments", 
and his reasons to choose this one in musical practice.  He said all the 
things I didn't need to say, and well, from the perspective of playing 
the whole Bach solo repertoire.  His practical trek through this is 
paralleling mine, independently, reacting to the music and sometimes 
bringing out different observations.  He does likewise in the booklet 
notes of his WTC set, offering his personal response to the music.

I wasn't comfortable with John's repeated hype about "holy grail" 
either, and as you might have noticed I made several comments during the 
program try to bring the thing much more down to earth than that.  It's 
a straightforward method to alter meantone for better practical 
circulation through all keys, by changing six of the twelve notes on the 
keyboard according to simple rules illustrated in Bach's diagram (taken 
to *be* a diagram!).  As I said directly in the program, countering a 
question about grail, I wasn't *seeking* a grail!  Just continuing my 
college and doctoral research of the 1980s/90s, into aspects of practice 
and history.

But once again, it's a popular radio show and the announcer/producer can 
play to his audience as he sees fit.  He and his staff initiated the 
program, and Peter and I were his invited guests for the day, by phone. 
  He was following up to his own program from the spring, where he had 
Richard Egarr live in the studio to talk similarly about practical 

As I've kept saying for more than a year now (here), but didn't bother 
to say pedantically on-air, anybody who really wants to compare one 
"well temperament" vs others needs to do so HANDS-ON by playing the 
repertoire directly.  Do the walking through Bach's toes and fingertips 
to find out what the existing music he produced really requires.  This 
checklist, especially:
And, my survey of putative "Bach" temperaments continues to be here:

Anybody wishing to do a truly thorough pass through all that stuff, it 
takes more than a year, and the commitment/ability to tune the 
instrument by ear and play the music.  Not material for a popular 
one-hour radio show for beginners.

My FAQ pages continue to grow at

Yesterday's show was entirely improvised; Peter and I didn't know any of 
the questions ahead of time, and we just went with the flow of whatever 
might be good to listen to (and talk about) next.  A jam session among 
three musicians who have all grappled with PRACTICAL issues of tuning. 
It was a lot of fun and a highlight of my summer, for enjoyment.

I liked John's kudo to the memory of Joseph Spencer at the end: a 
reference to the chutzpah of tuning a real harpsichord directly on the 
air during a live program.  I never got to hear Joe's Berkeley radio 
shows where he did that type of thing; wish I could have.

Incidentally, my demo instrument yesterday was an Italian virginal built 
by this list's own Anne Acker.  Cheers!  I don't know how well it 
sounded on-air, played here about a meter away from a $30 speakerphone 
as the microphone, but hey.

The after-the-fact playlist pops up now at
My unannounced little ditty in C major that they've documented as 
"Renaissance Virginal piece" was the first strain of a Byrd alman, from 
the Fitzwilliam, #63.  I picked that at random by flipping open the 
book, 5 minutes before the show; it selected itself.  The "E major 
Sinfonia #6(out of tune!)" was the first half of the Bach 3-part 
invention in E, *in* tune according to meantone, and running into all 
manner of misspelled notes as it does.  D#, A#, E#, B# while the 
instrument is tuned to Eb, Bb, F, C!

I see that John had the Blandine Verlet recording of WTC available in 
his studio, and played unannounced, earlier in the show.  I wish I'd 
known that, because I would have stepped in and said "Yes, play that one 
as it sounds like Werckmeister 3", when Peter suggested a hearing of the 
C# major prelude comparatively.  That's indeed a good illustration of a 
piece that sounds dicey in other unequal temperaments such as W3. 
Opportunity missed.


A new article by me hits the sale stands starting next week, in BBC 
Music Magazine's August issue.  The longer original version of that one 
will be added to my web site shortly, as well.  Commissioned piece in 
spring 2006 to bring the basic issues to the attention of *general* 
non-specialist music lovers; and no hype about holy grails in it.  I 
don't know what title the BBC's staff have given the article.

The point is to listen to (and hopefully also to play directly) the 
music, not just read or speculate about it.

Brad Lehman

CDs and free downloads:
harpsichord...     http://tinyurl.com/gd5up
organ...           http://tinyurl.com/zhcse
trumpet/organ...   http://tinyurl.com/jrn4g

personal web site: http://tinyurl.com/jnjo4


Date:    Fri, 28 Jul 2006 17:55:12 -0400
From:    Brad Lehman 
Subject: T-lever: perpendicular or parallel?

Here's an item I don't remember ever seeing discussed before.

If you have flat undrilled pins, do you prefer the T-shaped tuning lever 
that has the slot parallel to the T-top, or perpendicular?  I almost 
always use the perpendicular one.  But, I pulled the other one out of 
the toolbox by mistake today and then gave it a try for old time's sake; 
it seemed harder to control for really fine adjustment, or maybe it's 
only acclimatization or habit?  The practical difference comes in the 
effort never to bend the pin sideways in any direction, while turning it.

Preferences?  All things being equal, as to a snug fit on the pin, 
depth, etc.

Brad Lehman


Date:    Wed, 30 Aug 2006 17:27:14 -0400
From:    Brad Lehman 
Subject: BBC Music Magazine - Sept 2006

The newest issue of BBC Music Magazine has an article by me on pages 
42-44.  It's a much-shortened version of the article that's here:
and retitled "In Good Temper".


Bradley Lehman

CDs and free downloads:
Harpsichord...     http://tinyurl.com/gd5up
Organ...           http://tinyurl.com/zhcse
Trumpet/organ...   http://tinyurl.com/jrn4g

Personal web site: http://tinyurl.com/jnjo4

Article "Bach's art of temperament": http://tinyurl.com/ztvnu


Date:    Wed, 30 Aug 2006 17:34:49 -0400
From:    Brad Lehman 
Subject: Re: BBC Music Magazine - *August* 2006 (not September)

Sorry, correction -- their August issue (the current one) Volume 14 
Number 13, not September!  It's been available since about August 7th. 
This one:

The article was originally scheduled for print a few months earlier into 
the spring.  Anyway, it's out there now.



Date:    Mon, 4 Sep 2006 14:49:20 -0400
From:    Brad Lehman 
Subject: Re: Which WTC I?

 >>Please pardon me if this has been discussed to death in the past, but
I have decided that I want a new recording of the Well Tempered
Clavier, Book 1, and I need some recommendations. Amazon.com is not
extremely helpful, particularly as the reviews rarely mention the
instrument(s) used.  I'd prefer an appropriate Germanic instrument as
opposed to a French Double, but ultimately the interpretation is the
important thing. Whose recording would you recommend and why? And
don't be shy if you have recorded it yourself!<<

I'm very happy with the new one, released a few months ago, by Peter 
Big Germanic harpsichord, and he uses 16' and 8' pedal on some of those 
preludes and fugues as well; a thrilling sound all around.  This is also 
the first complete recording to use what I believe is an optimal tuning 
for the music...Bach's own...as explained in its booklet notes (some 
written by me, some by Watchorn).

See also the feature review of this recording on pages 28-29 of the 
September/October 2006 issue of _American Record Guide_, which just 
arrived this weekend.  It was written by Peter Catalano.  A few 
excerpts: "Progressing through Bach's monumental score, it's really 
noticeable that chords and imitated phrases each have a different 
timbre, virtually an entire orchestral palette that can be reedy, 
stentorian, limpid, or bosky.  The distinctions are palpably audible. 
Measure for measure, chord by chord, this recording is a serendipitous 
adventure.  Adding to the opulence is Watchorn's sweeping vision and 
masterly playing.  No bolting through this score!  This is one musician 
who so savors every note on this two-disc recording that Book 1 is a 
good half-hour longer than rival sets.  (...) Watchorn's WTC sounds 
spontaneously improvised.  The aching fantasie-like E-minor and 
E-flat-minor Preludes are spellbinding.  Likewise, the recitative-like 
sections, especially the C-minor and B-flat-minor Preludes, really call 
to mind the emotional drama of French and Italian baroque opera. 
Watchorn gives every prelude and fugue a distinct character without 
sounding fussy or contrived.  All in all, this MusicaOmnia recording is 
a 'must have', a seminal interpretation."


Another favorite of mine, but tuned differently, is Edward Parmentier's 
on Wildboar.  Wonderful clarity and spontaneity there too.
That was recorded a long time ago, but only released as recently as 2004.


I've also released several recordings of my own, January 2006, playing 
selections from this book and using the same temperament: the C major, f 
minor, f# minor, bb minor, and B major on harpsichord, and the Eb major 
on organ.  Coincidentally those two sets are reviewed in this same issue 
of _American Record Guide_ on pages 70-71.  The harpsichord I played on 
there was a very bright-sounding Franco/Flemish.

The links to those discs are below.


Bradley Lehman

CDs and free downloads:
Harpsichord...     http://www-personal.umich.edu/~bpl/larips/cd1003.html
Organ...           http://www-personal.umich.edu/~bpl/larips/cd1002.html
Trumpet/organ...   http://www-personal.umich.edu/~bpl/larips/cd1001.html

Personal web site: http://www-personal.umich.edu/~bpl

Article "Bach's art of temperament":


Date:    Tue, 5 Sep 2006 13:01:11 -0400
From:    Brad Lehman 
Subject: Re: Which WTC I?

 >>Peter Watchorn's set:
Big Germanic harpsichord, and he uses 16' and 8' pedal on some of those
preludes and fugues as well; a thrilling sound all around.<<

I should clarify one small bit of that: in book 1 it's the *pedal* 
harpsichord that's "Germanic", styled after Hass principles.  The manual 
instrument on this recording is a non-Germanic double: copy by 
Hubbard/Broekman of Ruckers/Blanchet/Taskin.

It's book 2 where Watchorn will have a Harrass copy, as was also heard 
in his set of the toccatas BWV 910-916 a few years ago.

To Ray's short list of other people recording book 1 on a German 
instrument, don't miss Glen Wilson's 1989 recording on a copy 
(Sassmann/Kramer) of the 1728 Zell.

Brad Lehman


Date:    Wed, 27 Sep 2006 17:40:00 -0400
From:    Brad Lehman 
Subject: Re: Brilliant Classics Bach Edition

 >>The double and triple concertos are played by Anton and Erna Heiller
plus Kurt Rapf in the triple, with I Solisti di Zagreb conducted by
Antonio Janigro. Nice performances though the stereo separation between
the 2 harpsichords is rather wide for my taste - listening with
headphones that is.
So there are clearly differences between the two issues. I suspect that
the 160 CD version is the more recent, but no way of checking.<<

No--the 155-disc version is definitely the newer one.  It has 
performances as recent as "May/June 2006" on the disc credits: the 
Belder/Henstra/vanDelft set of doubles/triples (replacing Heillers), and 
the Belder set of Brandenburgs.

I was able to confirm my suspicions from the sound, yesterday, by 
checking with Henstra: yes, they used my Bach temp on those concertos. 
The tip-off, to me, was hearing the smoothness in the C minor 1062, and 
the way the C major concertos have a noticeably stronger 
brilliance/crispness when the music moves to sections of more sharps. 
Noticeable at least to me!  A very lively set of performances all 
around, on that disc of concertos 1061-1064.

In the double 1060 and the quad 1065, this Brilliant set uses a 1992 
recording by Schornsheim/Thalheim/Stark/Liebsch.  I'm especially moved 
by Schornsheim's uncommonly slow and pensive performance of the F minor 
concerto 1056.  Most of the difference comes in the first movement, 
taking 4.5 minutes rather than the more usual 3'15" or so.

Brad Lehman


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