LaripS.com, © Bradley Lehman, 2005-22, all rights reserved.
All musical/historical analysis 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.
Frequently Asked Questions, part 3
Continued from part 2...
What about the people who in argument would delete the letter "C" from Bach's diagram, as merely a capitalization stroke? (See the diagram above....) I had already brought up and answered this objection myself, in footnote #63 of part 2 of the Oxford paper printed May 2005. Also, my explanation has been available here on the web site since March 2005.
The cycle of naturals has F at the end of the line because that's what keyboards do; F is the end of the naturals before we get into the flats, tuning by 5ths. To tune the entire C major scale (C-D-E-F-G-A-B-C) first, and with the notes all related to one another by 5ths, it is necessary to start with F (F-C-G-D-A-E-B...).
With or without Bach's "C" as corroborating evidence (to those who would make the calligraphy a make-or-break point!), Bach's diagram matches the keyboard layout: all seven naturals are on one side, and all five accidentals on the other side. This point is explained more fully in the 2006 article "Bach's art of temperament".
See also my lecture notes from 10/22/08, where I presented the argument without any necessity of the "small C".
Why didn't Bach write down the tuning process in some other format more familiar to us, if he had something, like a set of step-by-step instructions with a musical staff? Or a set of theoretical mathematical divisions? Well, the short answer is that it's not necessary to think along those lines to get the job done; only to think within some notation that delivers the result, clearly enough!
Let's recall that Bach grew up learning music, writing and improvising it, with tablature notation in addition to any work with staves. He continued to use it, at least into his 30s or 40s (the 1720s); see, for example, Orgelbüchlein with some of the music finished in the margins, in tablature. With the note-names on a blank area of paper (no lines ruled), a large amount of music can be packed into a small space. The note-names give the proper sequence in which the pitches are to be played, relative to one another. See also a photo gallery of music that Bach wrote down at age 15, in 1700 (newly available, August/September 2006).
In this same way, I believe, the WTC title page's drawing is a tablature for tuning: the proper sequence in which the pitches are to be tuned, relative to one another. It is an elegant way to write down a process, with a minimal amount of cogitation or fuss.
In 75 words or less, how does Bach's temperament eliminate the seemingly random harshness we encounter in all the other familiar unequal temperaments? The problem is enharmonically misspelled notes where the intonation and tonal music disagree. The other temperaments sound rough whenever Ab, Db, Gb, or Cb are in the bass; or those or D#, A#, E#, or B# in the treble. This is egregious in all the regular "meantone" layouts, and in the French/Italian "ordinary" temperaments, and Werckmeister and Kellner. Others such as "Vallotti" and the Neidhardts merely ameliorate the problem through compromises, without solving it.
All this math, and additional musical analysis in Bach's repertoire, is covered thoroughly in the May 2005 half of the Early Music article. I ask that any more probing mathematical questions be held until the people asking them have read and thoroughly digested that second part, and (even more importantly) have listened to Bach's and other temperaments hands-on at a harpsichord. There is no substitute for playing through the repertoire--and listening very closely in hands-on thoroughbass improvisation--to experience this comparison directly.
What about other readings of the same Bach diagram, for example the proposed method by Emile Jobin taking the various loops differently? His reading assigns 1/4 syntonic comma narrow to C-G-D-A-E-B, pure B-F#-C#-G#, approximately 1/12 comma wide Ab-Eb-Bb-F each, and the (arbitrary) F-C pure off the ends. This is an excellent example of an ordinary temperament. It is beautifully effective for most of the 17th century French/Italian/English repertoire. It also works nicely for most of the first three books by Francois Couperin (published 1713-1722), although ordre 18 is questionable...and ordres 25-27 from the fourth book (1730) are seriously problematic as well.
But more importantly, as a proposed method for playing Bach's music and specifically the WTC book 1, which should be the most obvious place it needs to work, it gives some remarkably rough sounds. Try, for example, the F minor and B-flat minor preludes and fugues, and the C# major, F# major, and A-flat major. Even the G minor and C minor pieces have some abruptly sour moments. (The reason for these problems is explained thoroughly in the "enharmonics" section of part 2 of the Early Music article, and only briefly here.)
To decide on suitable tunings for Bach's repertoire, individually or as a whole, why can't we simply survey the key signatures to see what notes are needed most often? Tonal music doesn't behave that superficially in practice, as notes are brought in from outside the key signatures. A temperament must be able to handle everything that comes up during the course of a composition, not only the basic scale. The notes should not sound randomly or suddenly out of place within their musical contexts.
For some examples:
How can we know which of several "Bach" temperaments are most plausible to play his repertoire? Play his music. Some of the toughest tests are listed in this "temperament-killer" roster.
But that list is huge, and would take months to play through! One might start most productively where, to hear the most obvious features immediately? Close study of the musical evidence does take time, patience, and fluent playing ability, with no shortcuts around hands-on work at harpsichords/clavichords/organs....
OK...the four Duetti BWV 802-5, the F# minor Toccata BWV 910, the E major French suite BWV 817, the C major Prelude and Fugue for organ BWV 547, both Ricercars of the Musical Offering BWV 1079 (C minor with a chromatic subject), and all of both books of the Well-Tempered Clavier. Play the music in all the keys, not only the easy-reading keys!
This short list of pieces has some of the stiffest examples of equal-temperament promoters, arguing that Bach somehow "required" equal temperament at least sometimes before the end of his life. These compositions at least bring up some of the roughest train-wrecks when played in other unequal temperaments! That can be experienced both directly at a keyboard, and by listening to published recordings in various temperaments.
And to test the vocal version and its features, play through at least the B Minor Mass, St Matthew Passion, St John Passion, and the cantata "Ich geh' und suche mit Verlangen" BWV 49...the one that recycles the E major harpsichord concerto, but puts the organ solo into D major while the orchestra stays in E. Some details about that are in the printed part 1, "Ensemble music" in the outline.
On the hypotheses that Bach used his drawing as teaching material, and/or as audition material sent ahead to Leipzig to get a job: if it is so essential to the book of WTC, why does it appear only on the autograph and not in his students' later handwritten copies of this music? Excellent question, and I don't know. But, some counter-questions of my own:
I don't understand the 1/12 comma wide 5th resulting at Bb-F. It makes some of the intervals less pure than I like them. And, the setup instructions in part 1 of the article perplex me, as to why that step is there at all to tune F-Bb-Eb pure and then move the Bb later. It looks like arbitrary manufacture of mistuned notes. Please explain. OK.... Several points here:
Why do the instructions in the article and the web site give electronic-tuning values of cents, and so on? This bit of math is only to accommodate those who habitually tune with electronic devices. I do not do so myself! The whole harpsichord can be done quickly and accurately (15 minutes with practice) by ear, with no electronics whatsoever. See the "twang" instructions or the better geometric instructions.
To learn these procedures of careful and accurate adjustment at an instrument, turn the machines off and learn how to listen closely for interval quality, testing 5ths and 4ths! (I'm sorry if this sounds crass or off-putting, but Bach obviously did not have any battery-driven device or computer, and it's not necessary.... I don't believe it's helpful even to think in cents anymore, myself.)
What are the best books, for more about all this? The best books I have read about historical and practical matters of tuning are Dominique Devie's Le temperament musical (in French) and Mark Lindley's Stimmung und Temperatur (a book-length article, in German). Additional selections in English are cited inside my article, but (in my opinion) these Devie and Lindley resources are clearer and more thorough. As for the fundamental mathematics behind questions of temperament, seeing the basic structure inside the generation of musical scales, Easley Blackwood's The Structure of Recognizable Diatonic Tunings stands out.
Bear in mind that all three of those resources are at a post-doctoral level of inquiry! For a more basic level of pedagogical presentation: obviously my own Early Music article here, and this www.larips.com web site, are my contributions to a filling a gap I perceive in the English-language tuning literature. That is, I'm trying to write here the set of introductory resources that I myself would most like to encounter as a student of keyboard temperament.
Where are some other clear reproductions of the WTC title page, in print? In addition to those I have cited in the article (books and musical editions): see the encyclopedia MGG (Die Musik in Geschichte und Gegenwart), as an inset within its article about Bach. I have checked the 1966 edition and another from the 1930s-40s; both have a clear reproduction (which might have been copied from the 1911 Grove dictionary!). Andreas Sparschuh has posted a photocopy of this MGG excerpt on his own web site in June 2005; that is apparently the copy he has been working from in his own research.
In addition to the photographic reproduction in various books and editions of the music: I have the 1958 LP set of the complete WTC played by Wanda Landowska, RCA LM-6801. There is a full page given to a photo of this title page, approximately 75% of actual size, in its deluxe booklet of program notes. The page already (by 1958 or earlier) shows the closely cropped margins that have removed part of Bach's drawing, and places where the ink has faded, and the bottom right side of the page looks to have some mildew damage (or similar) with spotting. The photo in the Neue Bach-Ausgabe (NBA) looks almost the same. The photos on page 96 of Wolff's The New Bach Reader, and in the front of Richard Jones's ABRSM edition, each show stronger contrast in these features. Of these, the one in Jones/ABRSM does the best at showing the cropping of the book (the photo is cropped outside the present edges of the page), where obviously some of Bach's drawing has been excised.
What about Kellner's claim to have found important 5s and 7s in Bach's seal, indicating five tempered 5ths and seven pure 5ths? What of it? One could just as easily claim that there are numbers 1, 3, 3, 5 in the crown of it, and that they indicate the one wide 5th (A#-F), the three 1/12 PC 5ths (C#-G#-D#-A#), the three pure 5ths (E-B-F#-C#), and the five 1/6 PC 5ths (F-C-G-D-A-E) of my proposed temperament here. And it's from the same year--1722--as the WTC title page! Wowee!
But esoteric games of this sort are not sufficient evidence of anything, on their own because the interpretation is so arbitrary, against other equally arbitrary readings of the same monogram. Plenty has been made of the various 14s hidden in this same design. What do they prove? What do they disprove against other readings?
Bach's strongest medium of expression, as an expert musician, was musical sound; I believe we should judge evidence and interpretations by the way Bach's sound is handled. Real sound at good instruments. Not esoteric and arbitrary Quatsch!
Is your temperament like Kellner's, musically? There is no resemblance, except in the obvious sense that they're both unequal temperaments!
Kellner's temperament [Chart] is an offshoot of Werckmeister 3 [Chart], made slightly gentler by the alteration from 1/4 to 1/5 comma but keeping the same basic pattern. They share the same feature of sounding increasingly harsh in music that uses more than two flats. I have further remarks about Kellner's temperament and argumentative methods in the main article; suffice it to say that I am not pleased by his results or methods, either musically or intellectually. I concede that his temperament sounds usable in most 17th century music, and is at its best in that repertoire. But, its historic claims in doing so are nil. The stronger effect is made by Werckmeister's original temperament, anyway, or by staying more strictly to the meantone temperaments that are its ancestor.
Kellner's [Chart] and mine [Chart] have no intervals in common, except for the arbitrary coincidence of two pure 5ths. The major 3rds are noticeably different sizes, and arranged in a different pattern: see the section "Handling diesis within the octave" at the bottom of each chart. The tensions and resolutions when playing music are obviously different, as any interested reader should confirm hands-on for oneself.
Whenever I play in Kellner's temperament occasionally, the features I notice most prominently are the egregiously low notes Db, Ab, and Gb in their melodic/harmonic contexts. When those notes are approached melodically by step or leap, it seems to me like singing with poor breath support. Kellner's triads of Ab major, Db major, and F# major strike me as noticeably raucous and ugly...but at least they are not as rough as their counterparts in the regular meantone temperaments.
I am aware that some people are fond of the sound of Kellner's temperament; there will be differences of opinion. I invite these Kellner fans to play WTC 1's prelude and fugues in Bb minor, F minor, F# major, and C# major, and then try that same music in the smoother temperaments presented here.
Did Bach understand that two regularly tempered 5ths in succession beat at 3:2 ratio? Or was he wrapped up in "equal-beating" schemes where 5ths and 4ths are made to coincide exactly with one another 1:1, or with some external time-beating source? The principle in question here is explained here and in the practical instructions: that two 5ths of the same geometric amount of tempering must beat at 3:2 speeds around their common middle note. This remains true when the upper note (for example, G of F-C-G) is brought down an octave to make a 4th within the other 5th: the C-G 4th beating 3/2 as fast as the F-C 5th. The simple reason is that the C, being a 5th above the F, has its fundamental frequency along with all the overtone components of its tone (where beats are heard for harpsichord/organ tuning purposes) beating 3/2 as fast as the F. Everything is multiplied by 3/2 each time we move upward by a 5th on the keyboard, if we are retaining geometric equality as the criterion of size in our 5ths.
So, did Bach understand this? Why would he not understand it, from even a week or two of careful work as a harpsichord tuner? The burden of proof is placed on the wrong side here. (As in, why would anyone argue that JS Bach was not a comprehensive genius in musical arts, as observed in all the other facets of his craftsmanship, and his lifelong care with getting the detail of his work into best reasonable accuracy?) It should be obvious to anyone who does such tuning work hands-on that everything doubles in speed at each octave, and goes at a 3/2 increase at each 5th; this is fundamental knowledge to set up any regular "meantone" layout, before proceeding to do any tasteful adjustments (later) that would give some of the 5ths some irregular size, with an ordinaire strategy.
This is also fundamental knowledge for violinists (as Bach was) or players of the related stringed instruments: stop a string at 1/3 of its length, and the remaining length vibrates 3/2 as fast as the whole string did...producing a note a 5th higher. All the vibrations increase in speed, together, by a factor of 3/2. The same phenomenon determines the tangent-striking points on a clavichord; or how to touch a violin string lightly at its nodes to produce harmonics...this is all simple proportional divisions of the whole. Or, tune the violin's open strings (5ths!) to the regularly tempered 5ths on a keyboard that is based on any meantone-family temperament, and then the violin's strings will beat with one another in 3:2 speed going up, when bowed as open 5ths: A-E is 3/2 as fast as D-A, which in turn is 3/2 as fast as G-D. Yes, A-E beats 9/4 as fast as G-D (since that is 3/2 squared, two regular 5ths in succession), and it's slightly over one octave higher, where it would have beat exactly twice as fast.... (Would not any proponents of "equal-beating" 5ths on keyboards also have to assert that a violin's open strings should beat together at something other than 3:2? Or that violinists 300 years ago didn't bother to tune their open strings to a keyboard when playing with a keyboard?)
Further corroboration that Bach understood regular 5ths properly, and that he listened for beats at all, can be found in the drawing he put at the bottom of the WTC title page. We keyboard tuners by ear listen to these regular beat rates, and gauge them against one another, to know when pure intervals have been detuned by the proper amount...creating appropriate beat rates and relationships. And that regularity for listening only settles into the sustained sound after the noise of the plucking has quickly died away. Bach drew this:
The spiral in the middle of the drawing may represent the regular 5ths (F-C-G-D-A-E as seen in the other drawing), which are geometrically equal to one another; and then the three swishes on one side vs the two on the other side indicate that this is how one may listen to deliver that equality accurately. (Thank you to piano tuner Michael Bensusen--in private e-mail 9 July 2005, cited here by permission on this idea--for this observation that Bach's drawing actively demonstrates an understanding of the 3:2 beat rates, specifically. This goes beyond my more generalized suggestion that this drawing simply illustrates how to listen for beats at all, past the initial noise.)
How about some of the other calligraphy/graphology points on Bach's title page? My main remarks are several paragraphs in the "Major 3rds, minor 3rds, and mean accidentals" section, part 2 of the printed article. [Outline] Those paragraphs (page 222 and its footnote #63, which see) suggest some possible meanings to the shapes and dots, especially in the word "Das", along with the capitalization-stroke issue.
Additionally, notice that Bach consistently used an italicized handwritten script on the page wherever he used Latin words or their derivatives, in contrast with the style of the German words. This is especially in the words of the middle of the page, but also as part of the title: "temperirte Clavier" is Latin(ate) and written in these italics, while "Das Wohl" is in the different style. The technical term for the mixture of languages, itself, is "macaronic".
From at least as early as Weimar documents of 1714, Bach's tendency in correspondence was to use macaronic expressions (whether borrowing from Latin, Italian, or French) and to italicize them, when preparing documents in any official capacity: giving a learned air to the communication. Whenever he needed to ingratiate himself to someone, or to be obsequious, he reached outside German, which also saluted the education of his recipient(s). By contrast, for less formal personal letters and in other notes he simply expressed himself in German and there was nothing to italicize.
Bach's use of italicized script on the WTC title page suggests, at least, that he was painstaking with both the layout and the textual details (visually) of this page, making it look like an official presentation copy to be shown to somebody, or a semi-formal archival copy. If he were writing out all the words here simply for his own amusement or to provide prosaic information, why would he bother making the macaronic distinctions visible, or bother with the funnel-shaped layout? Or provide any "decorative" drawings? Or why would he use such convoluted language that parodies the written titles by Kuhnau, his predecessor at Leipzig? (Kuhnau provided music in only some of the major/minor keys; and Bach, in attempting to surpass him, is providing music in all the keys!)
And given that carefulness by him in wording and italicization: why then would he make his top drawing so irregular and asymmetrical, unless it were somehow also meaningful to the topic at hand, namely the provision of exemplary pedagogical music to be played using all 24 major and minor keys?
Also, returning to the matter of the words' calligraphic appearance: here is an ambiguous and perhaps too speculative point (where I dare not draw any too-firm conclusions from the drawing itself). In the odd-looking "t" of "Wohltemperirte", is there any chance that it doubles as an "s"? Conjecturally, with something such as "semper vi" or "semper la vie" written onto the page first, and then converted into "Das Wohltemperirte Clavier"? I was discussing this briefly in March 2005 with a Latin scholar, but haven't chosen to make anything of it other than mentioning the possibility here.
What do Fischer's spirals and flourishes look like, in Ariadne musica? In part 2 of my article, at page 226 in endnote 3, I wrote: "The 1715 edition of Ariadne musica has various spirals and flourishes on both its title-page and interspersed among the pieces; one on the title-page (attached to the name 'Fischer') is exactly the same as Bach's, when turned upside-down." That is shown here, along with the page from the G minor fugue:
What do you think of the articles in Early Music November 2005, disagreeing with your presentation in various ways?
My detailed responses to each article have their own pages:
How long have you personally been into unequal temperaments? See the bottom half of the discovery page.
One of the most uncommon features here, at close inspection, is that E-G# is wider/brighter than Ab-C is. That's odd, among other contemporary temperaments (Neidhardt, Vallotti, Werckmeister, etc), where Ab-C is larger or where the two have the same size. How is that explainable? The familiar "unequal temperaments" sound that we know, and that some of us love, is so firmly based on having E-G# somewhat sweeter than Ab-C! See the "ordinary/extraordinary" page.
What about the objection from several vociferous critics that the tuning calculations start on F?
1. There aren't any "calculations!" Bach didn't care for calculations (according to the written testimony of CPE Bach, his most loquacious son and expert student: both of these expert teachers and musicians disliking all the "dry mathematical stuff"). Bach's methods were direct practice and demonstration, in music, not in mathematical computations of anything. It has nothing to do with generating string lengths or a set of specific frequencies, from any particular starting length or frequency on any particular note. It has everything to do with sitting down at a harpsichord and tuning it by ear in 10-15 minutes, by listening to the quality of slightly out-of-tune 5ths, and not necessarily using beats to do so (although the use of beat-counting can give a more accurate modern result).
2. Bach demonstrated a picture of the resulting layout as his tuning diagram. One can start to set it up anywhere, in practice, from any convenient reference pitch. I usually do it from an A fork myself, but it is just as easy from C, or from any of the other natural notes (C, D, E, F, G, A, or B) simply working along the diagram in both directions, tempering each 5th in turn as illustrated. I suggest that rather crassly: anyone who is not able to do this in practice, tempering an entire 8-foot register of harpsichord strings by ear in 10 minutes, from any of these starting pitches (with a single tuning fork) does not understand the principle well enough to criticize it! Those of us who do understand it and are able to do so (by testimony from myself and from other harpsichord-tuning colleagues) have had no trouble with it. Practice is reality. Theoretical speculation of critics, without direct practice, is irrelevant.
3. The loudest and most aggressive such critic starts calculations from F in his own papers, while bad-mouthing this principle! And those papers are nothing but belligerent puffery, pseudo-science of arbitrarily selected coincidences, and red-herrings against serious musicology and practical experience at harpsichords. Contact me for details, if interested in the references about this.
What about the standard line that Bach had probably gone to equal temperament, or at least didn't care about nuances anymore, by the time of writing the six-voiced ricercar in the Musical Offering? That's what New Grove says. Hear that ricercar for yourself, with the unequal nuances that contribute to the composition's beauty, played on organ here.
Would people be able to use this tuning on pianos? Absolutely! Its broadcast premiere was on a fortepiano (Robert Hill, May 2004, Swiss Radio). I have tuned several modern pianos to it myself, as have others. It is beautiful for the standard piano literature, all the way from CPE Bach, Mozart, Haydn, Beethoven and into the 20th century and beyond. To me, the sound of Brahms and Grieg in this temperament is a special treat, with the way those composers handled modulations and enharmonic relationships. This temperament also agrees with a general style of Victorian-era piano temperament: nearly equal, but slightly favoring the sweetness of flats ahead of sharps.
One might be able to make a case that atonal music should not be played in anything but equal, since the absolute equality of all twelve notes is a premise of such compositions (and, for example, in some of Debussy with chord-streaming and the like).
But a broader question is, what types of music will be played on a piano most often by the people who own or use it? And in a majority of circumstances, that's typically tonal music. Hence there might be some advantage of tuning pianos in suavely and subtly unequal temperaments, favoring the keys that are played in most often: F, C, G, D, and the others nearby with few sharps or flats. That is a point of existence for circulating or irregular or "well temperaments": the ability to reveal subtle color-changes within the music as a tonally-based texture modulates.
A friend asked me, "If I get my home piano tuned the Bach way, will I have any trouble playing my guitar with it?" I replied, "The piano at your church has already been in this since June and you've played and sung with that piano multiple times, any problems?" "No!"
Several years into all this, having observed all the argumentation for and against this hypothesis, what emerges as the personal pet peeve? Nay-sayers who form a loudly and persistently negative opinion, but having not done three crucial parts of the analysis themselves:
I am especially annoyed by people who focus their negativity on my writing style, or the title of the article, or on hearsay, or anything else not having to do with musical performance.
What temperament(s) do you use yourself in 17th century repertoire (Kuhnau, Froberger, Buxtehude, etc)? It is not necessary to have all 24 keys as tonics, but we'd still like to avoid having any individual notes wildly out of tune. My additional temperaments for this are described in the "bonus" sections of the practical instructions page. See especially #5 there.
How do you respond to complaints and allegations that your work is too mathematical, speculative, "theoretical", or academic? Such complaints usually make it clear that the complainer doesn't understand the work....
My approach is hands-on in the music: playing Bach's music directly on harpsichords, clavichords, organs, and venturing into earlier 17th century Germanic music as well. I tune by ear, not by calculation. I demonstrate that process in video presentations: electronic devices are absolutely unnecessary (and Bach and his predecessors didn't have them, of course).
My tables of mathematical analysis are only a way of describing sounds that have already been installed onto keyboards working by ear. I go through books of music, carefully tabulating the individual notes that are required in each composition (such as the need for all of Bb-F-C-G-D-A-E-B-F#-C#-G#-D#-A#-E# within the same D major Capriccio by Böhm: 14 different notes, with overlap on Bb/A# and F/E#). Then, I seek tuning methods that deliver plausibly smooth results, even (especially!) where the same key-lever on the keyboard must play two differently-named notes within the same composition. Such evidence demonstrates to me directly that regular meantone does not work, and I go on from there! I find carefully moderated solutions to play extant music, using the music itself as primary evidence.
Bach's Das wohltemperirte Clavier (book 1) as a whole needs 27 different notes: Bbb, Fb, Cb, Gb, Db, Ab, Eb, Bb, F, C, G, D, A, E, B, F#, C#, G#, D#, A#, E#, B#, Fx, Cx, Gx, Dx, and Ax. Most of the individual preludes and fugues in that book need 13, 14, 15, or more notes, where enharmonic exchanges must work smoothly. (See my video about that....) These facts themselves constrain any proposed tuning solution to be within a small range of possibilities. Bach's drawing on the title page then confirms that same thing, and shows in sufficient detail how to set it up without any calculations: nudge some of the notes one or two bits off their pure spots, by listening and experience, working hands-on at the instrument. The drawing is a picture of a very easy solution, one that takes only a few minutes to set up.
That's really the kernel of the whole thing. It's a straightforward right-brained concept. Everything else is merely supporting material to satisfy left-brained processes, too: the explanations of historical context, the recorded demonstrations playing the music directly, and the mathematical modeling for people needing to measure/understand things more scientifically (at the risk of losing the music in the process).
What if my piano tuner doesn't want to try this, claiming that it will throw the whole instrument out of balance too much? Show the tuner the math page, where it is demonstrated that we can keep the whole instrument at the same overall tension. Within each octave, five of the notes move down by a grand total of 13 cents. The other seven move up by a total of 13 cents. Like this: A (-3), Bb (+1), B (-3), C (+3), C# (+1), D (-1), Eb (+1), E (-5), F (+5), F# (-1), G (+1), G# (+1).
How about a temperament for Rameau? Rameau's published preference in 1726 was apparently for a system with regular 1/6 comma tempering in Bb-F-C-G-D-A-E-B, and the other four notes tastefully arranged to fill the gap. My presentation of this is in section 6 of the "practical instructions" page.
How do you figure out temperaments for other music? As I presented on the front page, I have a procedure documented by flowcharts to explain how I analyze and set up whatever notes are needed, working by ear. See the simple and more detailed versions.
I also addressed many high-level theoretical principles in my 2009 review of a tuning book by Claudio Di Veroli, along with debunking the absurd methods of John Barnes (1979).
What's newest? There is "The Notes Tell Us How to Tune", written 2021-22. It is my attempt to present everything as direct musical practice (i.e., with as little math as possible), and to explain the fundamental principles to people who did not understand or engage the central enharmonic argument 17 years earlier.
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