Bach's schematic, as it appears on the page, © Bradley Lehman, 2005-22, all rights reserved.
All musical/historical analysis here on the web site is the personal opinion of the author,
as a researcher of historical temperaments and a performer of Bach's music.

Carl Philipp Emanuel Bach

I believe that CPE Bach's career temperament was the same as his father's.

Here are two salient excerpts from CPE Bach's own book, 1753, Essay on the True Art of Playing Keyboard Instruments. They do not describe this particular temperament uniquely, but they describe the general strategy of tempering "most of" the fifths, and they describe its effect in musical practice. This first passage brings out the distinction that this is the new correct way to do things, as opposed to the older regular (i.e. "meantone") schemes.

CPE is describing harpsichords and clavichords, and he remarks:

"§. 14. Beyde Arten von Instrumenten müssen gut temperirt seyn, indem man durch die Stimmung der Quinten, Quarten, Probirung der kleinen und grossen Tertien und gantzer Accorde, den meisten Quinten besonders so viel von ihrer größten Reinigkeit abnimmt, daß es das Gehör kaum mercket und man alle vier und zwantzig Ton=Arten gut bracuhen kan. Durch Probirung der Quarten hat man den Vortheil, daß man die nöthige Schwebung der Quinten deutlicher hören kan, weil die Quarten ihrem Grund=Tone näher liegen als die Quinten. Sind die Claviere so gestimmt, so kan man sie wegen der Ausübung mit Recht für die reinste Instrumente unter allen ausgeben, indem zwar einige reiner gestimmt aber nicht gespielet werden. Auf dem Claviere spielet man aus allen vier und zwantzig Ton=Arten gleich rein und welches wohl zu mercken vollstimmig, ohngeachtet die Harmonie wegen der Verhätnisse die geringste Unreinigkeit sogleich entdecket. Durch diese neue Art zu temperiren sind wir weiter gekommen als vor dem, obschon die alte Temperatur so beschaffen war, daß einige Ton=Arten reiner waren als man noch jetzo bey vielen Instrumenten antrift. Bey manchem andern Musico würde man vielleicht die Unreinigkeit eher vermercken, ohne einen Klang=Messer dabey nöthig zu haben, wenn man die hervorgebrachten melodischen Töne harmonisch hören sollte. Diese Melodie betrügt uns oft und läßt uns nicht eher ihre unreinen Töne verspüren, bis diese Unreinigkeit so groß ist, als kaum bey manchem schlecht gestimmten Claviere.

(From the Einleitung of Versuch über die wahre Art das Clavier zu spielen, Berlin, 1753.)

§. 14. Both types of instrument must be tempered as follows: In tuning the fifths and fourths, testing minor and major thirds and chords, take away from most of the fifths a barely noticeable amount of their absolute purity. All twenty-four tonalities will thus become usable. The beats of fifths can be more easily heard by probing fourths, an advantage that stems from the fact that the tones of the latter lie closer together than fifths. In practice, a keyboard so tuned is the purest of all instruments, for others may be more purely tuned but they cannot be purely played. The keyboard plays equally in tune in all twenty-four tonalities and, mark well, with full chords, notwithstanding that these, because of their ratios, reveal a very slight impurity. The new method of tuning marks a great advance over the old, even though the latter was of such a nature that a few tonalities were purer than those of many present non-keyboard instruments, the impurity of which would be easier to detect (and without a monochord) by listening harmonically to each melodic tone. Their melodies often deceive us and do not expose their impurity until it is greater than that of a badly tuned keyboard.

(This is the standard English translation by William J Mitchell, Essay on the True Art of Playing Keyboard Instruments. New York: Norton, 1949.)

(NB: He does not say if the 5ths are all narrow, or with one or more possibly wide; only that a slight bit of impurity is introduced, which could describe either situation.)

For the hypothesis that CPE was describing the same all-purpose temperament his father used, the stronger corroboration is done here by playing through CPE Bach's own music, and the other music he knew at court. This includes his father's Musical Offering, where the ricercars work unproblematically in this tuning but sound rotten in the other typical unequal temperaments. The music is not merely "unproblematic" in this tuning; those ricercars have phenomenally beautiful tone-color as the music works its way through some of the most extreme flat keys! So does CPE's music, going through all the sharp and flat keys with impunity. The color contrasts are vivid, and there is never any harshness anywhere. I am convinced that CPE's use of those color contrasts in his music was deliberate, and based on this particular tuning method he had learned from his father, his only music teacher.

And, if CPE Bach had meant equal temperament or something aurally indistinguishable from it, wouldn't he have said "all of the fifths" instead of only "most of the fifths" get tempering?


What did CPE mean here by the remark about keyboards being "the purest of all instruments, for others may be more purely tuned but they cannot be purely played"? This comes back to the sound of his father's temperament.

Keyboards have fixed intonation for each note, as opposed to the variable intonation of winds, strings, voice, etc. A good keyboard temperament turns this limitation of fixed pitch into a positive musical advantage. The other instruments and the voice might be "more purely tuned" on occasion for particular intervals, as in being closer to beatless pure intervals (such as 3:2, 4:3, 5:4, or 6:5). But, whenever they do this, other parts of the musical fabric such as linear motion and the tuning of other vocal/instrumental parts come into conflict...or the whole ensemble drifts up or down in pitch, as different parts are adjusted to one another to make the momentary intervals sound more nearly pure. In the bigger picture, tuning is all relative to other things that are happening at the same moment in the music, and the forward motion of that music. The fixed-pitch nature of keyboards provides a stability against all this. That is what I believe he meant by "purely played".

As for "The keyboard plays equally in tune in all twenty-four tonalities...", that is self-evident upon setting up his father's temperament and playing in it, listening closely. It sounds "in tune" all the time! No, the keys do not all sound identical. But, they are all equally usable all the time, equally "pure" in sound and flexibility, "alle 24 Ton-Arten gleich rein".

Flexibility and beauty

Likewise, his father wrote two books of preludes and fugues through all the tonalities: the "Well-Tempered Clavier". Music can be written (or improvised!) with uncommon purity and flexibility, because everything is available all the time.

As CPE wrote to Forkel for the biography of JSB,

"The exact tuning of his instruments as well as of the whole orchestra had his greatest attention. No one could tune and quill his instruments to please him. He did everything himself." (1774, New Bach Reader, #394)
Forkel then presented it thus:
"Nobody could install the quill-plectrums of his harpsichord to his satisfaction; he always did it himself. He also tuned both his harpsichord and his clavichord himself, and was so practised in the operation that it never cost him above a quarter of an hour. But then, when he played from his fancy, all the 24 keys were in his power; he did with them what he pleased. He connected the most remote as easily and as naturally together as the nearest; the hearer believed he had only modulated within the compass of a single key. He knew nothing of harshness in modulation; even his transitions in the chromatic style were as soft and flowing as if he had wholly confined himself to the diatonic scale. His Chromatic Fantasy, which is now published, may prove what I here state. All his extempore fantasies are said to have been of a similar description, but frequently even much more free, brilliant, and expressive." (1802, New Bach Reader, p436. English translation by Kollmann, 1820.)

Dynamics due to harmonic tension and surprise

In part 1 of the Oxford article I wrote:
I have a further hypothesis that the 1722 Bach temperament, or at least something with the same shape (may we call it the "brook" shape?), remained CPE Bach's career temperament: and was also the ordinary temperament at the royal court during his tenure there. In his own treatise he remarked that only "most of" the fifths are tempered in his ordinary practice (and eight or nine of twelve is "most"), and more importantly, his musical examples in the book (and his broader keyboard oeuvre) work beautifully in this tuning.

The flute treatise by Quantz confirms this court hypothesis. Quantz in his Versuch einer Anweisung die Flöte traversiere zu spielen (1752) presented an exemplary Adagio (pp257-8 of Reilly's translation into English): to teach the keyboard accompanist how to play dynamics expressively by the harmonic tendencies of dissonances and consonance, listening to the sound of every moment. Quantz's piece includes 77 dynamic markings in its 45 bars! I have played this piece using Bach's tuning and noticed that the dissonant/consonant content of the harmony matches his dynamic scheme, with regard to the treatment of tension, resolution, and surprise. Readers are invited to explore this same exercise, to hear the dynamic effects as they derive directly from the quality of intonation, the temperament's subtle irregularities. In more equalized temperaments than Bach's, or in regular 1/6 comma, these directional tendencies in the sound are minimized and Quantz's notated dynamics can seem artificial and overdone. Quantz himself worked directly with at least three former pupils of JS Bach (Johann Friedrich Agricola, Christoph Nichelmann, and CPE Bach), giving him plenty of opportunity to learn and play with the Bach temperament.

This point is related to paragraph 29 in CPE Bach's "Performance" chapter. He explains what the piano, mezzo forte, and forte indications mean:
(...) Damit man alle Arten vom pianißimo biß zum fortißimo deutlich zu hören kriege, so muß man das Clavier etwas ernsthaft mit einiger Kraft, nur nicht dreschend angreiffen; man muß gegentheils auch nicht zu heuchlerisch darüber wegfahren. Es ist nicht wohl möglich, die Fälle zu bestimmen, wo forte oder piano statt hat, weil auch die besten Regeln eben so viele Ausnahmen leiden als sie festsetzen; die besondere Würckung dieses Schatten und Lichts hängt von den Gedancken, von der Verbindung der Gedancken, und überhaupt von dem Componisten ab, welcher eben so wohl mit Ursache das Forte da anbringen kan, wo ein andermahl piano gewesen ist, und offt einen Gedancken sammt seinen Con= und Dissonanzen einmahl forte und das andere mahl piano bezeichnet. Deswegen pflegt man gerne die wiederhohlten Gedancken, sie mögen in eben derjenigen Modulation oder einer andern, zumahl wenn sie mit verschiednen Harmonien begleitet werden, wiederum erschienen, durch forte und piano zu unterschieden. Indessen kan man mercken, daß die Dissonanzen insgemein stärcker und die Consonanzen schwächer gespielt werden, weil jene die Leidenschafften mit Nachdruck eheben und diese solche beruhigen (a). Ein besonderer Schwung der Gedancken, welcher einen hefftigen Affeckt erregen soll, muß starck ausgedruckt werden. Die so genannten Betrügereyen spielt man dahero, weil sie offt deswegen angebracht werden, gemeiniglich forte (b). Man kan allenfalls auch diese Regel mercken, welche nicht ohne Grund ist, daß die Töne eines Gesangs, welche ausser der Leiter ihrer Ton-Art sind, gerne das forte vertragen, ohne Absicht, ob es Con- oder Dissonanzen sind, und das gegentheils die Töne, welche in der Leiter ihrer modulirenden Ton-Art stehen, gerne piano gespielt werden, sie mögen consoniren oder dissoniren (c)

(...) Spielt man diese Probe=Stücke auf einem Flügel mit mehr als einem Griffbrette [Tastatur], so bleibt man mit dem forte und piano, welches bey einzeln Noten vorkommt, auf demselben; man wechselt hierinnen nicht eher, als biß gantze Passagien sich durch forte und piano unterscheiden. Auf dem Clavicorde fällt diese Unbequemlichkeit weg, indem man hierauf alle Arten des forte und piano so deutlich und reine heraus bringen kan, als kaum auf manchem andern Instrumente. Bey starcker oder lärmender Begleitung muß man allezeit die Haupt=Melodie durch einen stärckern Anschlag hervorragen lassen.

(From the "performance" chapter Vom Vortrage of Versuch über die wahre Art das Clavier zu spielen, Berlin, 1753.)

(...) In order to control all shades from pianissimo to fortissimo the keys must be gripped firmly and with strength. However, they must not be flogged; but on the other hand there must not be too much restraint. It is not possible to describe the contexts appropriate to the forte or piano because for every case covered by even the best rule there will be an exception. The particular effect of these shadings depends on the passage, its context, and the composer, who may introduce either a forte or a piano at a given place for equally convincing reasons. In fact, composite passages, including their consonances and dissonances, may be marked first forte and, later, piano. This is a customary procedure with both repetitions and sequences, particularly when the accompaniment is modified. But in general it can be said that dissonances are played loudly and consonances softly, since the former rouse our emotions and the latter quiet them (exercise a). An exceptional turn of a melody which is designed to create a violent affect must be played loudly. So-called deceptive progressions are also brought out markedly to complement their function (b). A noteworthy rule which is not without foundation is that all tones of a melody which lie outside the key may well be emphasized regardless of whether they form consonances or dissonances and those which lie within the key may be effectively performed piano, again regardless of their consonance or dissonance (c).

(...) If the Lessons are played on a harpsichord with two manuals, only one manual should be used to play detailed changes of forte and piano. It is only when entire passages are differentiated by contrasting shades that a transfer may be made. This problem does not exist at the clavichord, for on it all varieties of loud and soft can be expressed with an almost unrivaled clarity and purity. A loud, boisterous accompaniment must always be balanced by a stronger melodic touch.

(This is the standard English translation by William J Mitchell, Essay on the True Art of Playing Keyboard Instruments. New York: Norton, 1949. Emphases mine.)

Exercises (a), (b), and (c) for this passage are as follows.

The strength of this passage comes through when those exercises are played on a harpsichord tuned in the Bach temperament. (Print out this page and do so!) Use only one manual, as CPE specified, and listen closely to the sound of the instrument during each chord. As he has described in this paragraph, there is an emphasis whenever the harmonic progression brings in new notes foreign to the key, as a surprise. These are in the chords marked f, forte in the exercises.

That is: the temperament is already reflecting the tensions and resolutions that are natural to a dynamic musical texture. Each major or minor scale is slightly irregular melodically/harmonically, enough that they all sound distinct. When a note from one scale intrudes into the context of another, that irregularity is enough to draw attention to the intrusion: and the whole chord seems louder.


Why did CPE Bach not spell out exactly the 5ths that are tuned pure (outside "most of" the tempered 5ths)? I believe it is due to the practice of vocal-ensemble music, tuning a Cammerton harpsichord for matched use with a Chorton organ, pitched a whole tone higher. (Dual continuo parts for organ and harpsichord, whether for rehearsal or performance or both. The harpsichord continuo part matches the keys of the rest of the orchestra; the organ part is notated a step lower to offset its difference of pitch.)

Quite simply, a different set of "most of the 5ths" gets tempered when doing this. The organ is tuned in the main version of the temperament, with pure 5ths on E-B-F#-C#. When a Cammerton harpsichord is tuned to match the organ, but the player will be reading a part notated a whole step higher, the pure 5ths move to F#-C#-G#-D#.

According to Wanda Landowska, CPE himself as a teenager was the deputy tuner in this situation. "It is known that for several years the tuning of Johann Sebastian Bach's harpsichord at the Thomaskirche was entrusted to Karl Philipp Emanuel. Proof that a harpsichord was there! Johann Sebastian played it himself to conduct his cantatas." (Landowska on Music, 1964, p129) See also part 2 of Peter Williams's article "Basso Continuo on the Organ" (Music and Letters 50 (1969)): CPE Bach was the regular staff tuner for that church harpsichord from 1731-33. Williams (p233) cited Arnold Schering (1936); Landowska didn't give the reference.

This page is the expanded version of footnote #65, and the top paragraph of page 17, from the first half of the Early Music article.


Bach's schematic, rotated for use