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.


The main historical background is in the Early Music article "Bach's extraordinary temperament...", parts 1 and 2. That article provides the necessary context for the following additional resources.

The historical and practical assumptions within my research

Also posted to HPSCHD-L discussion group, May 31 2006...stating these directly in public, so there should be no confusion from anyone else's allegations of my assumptions!
  • No major 3rd should be smaller than C:E. Some others might be similar or the same, but none smaller.
  • 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?)
  • The C major scale is the natural center of harmony, and the one that should be most regular melodically...again from meantone practice.
  • There cannot be any noticeably bad 5ths/4ths anywhere; all major and minor triads have to be usable.
  • 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. [Test pieces]

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