Bach's schematic, as it appears on the page
LaripS.com, © Bradley Lehman, 2005-14, 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.

Regular "meantone" strains

The regular temperaments, having eleven consistent fifths of a single size plus a wolf diminished sixth, have a basic sound similar to one another. They differ primarily by degree (intensity) on the way they "lock in" major and minor triads, with harmonic stasis.

That said, they also have little quirks of their own, and hidden features ("Easter eggs").

For the purpose of this discussion, assume that the disposition is held constant at Eb-Bb-F-C-G-D-A-E-B-F#-C#-G# where the wolf happens in the closure of G# back to Eb.

In all of these where the fifths are narrower than equal temperament's fifths, the chromatic semitones are smaller than the diatonic semitones.

The word "meantone" can apply here, because in all the correctly spelled major thirds (such as C-D-E), taken melodically, the middle note is exactly at a mean (average) frequency. It is a mean tone, as opposed to a 9:8 tone or a 10:9 tone from just intonation. The two fifths to generate that tone, from either direction, are equally sized because all the fifths are equally sized. Therefore it is at the center, average, position within the major third.

See also tetrasecting a major third, and practical temperament instructions by ear....

The size of the regular fifth can be derived directly from some other intervals, starting at a fixed C.

Regular 1/3 syntonic comma (aka "19-division")
  • From C, tune E and then A pure; or simply tune A directly to C as a major sixth. (The E will move later; this step is only to establish the proper A from C.)
  • A and C have been established at one syntonic comma apart, since there was one pure major third employed to get there.
  • The fifths C-G-D-A must then each take up 1/3 of that syntonic comma error.
  • Tune all of C-F-Bb-Eb and A-E-B-F#-C#-G# with the same-sized fifths that were established in C-G-D-A.
  • All the correctly spelled minor thirds are pure. Consequently, diminished triads and their sevenths also have a startlingly piquant and static sound.
  • Major thirds are all 3.7 schismas narrow of 5:4 purity.
  • The four diminished fourths are all 28.3 schismas sharp of 5:4 purity.
  • Chromatic semitone: 63.5 cents; diatonic semitone: 126.1 cents. Yes, almost twice as large!
  • This is equivalent in practice to a 19-note equal division of the octave.
Regular 1/4 syntonic comma (aka "Aron meantone" or "31-division")
  • From C, tune E pure.
  • E and C have been established at one syntonic comma apart, since there was one pure major third employed to get there.
  • The fifths C-G-D-A-E must then each take up 1/4 of that syntonic comma error.
  • Tune all of C-F-Bb-Eb and E-B-F#-C#-G# with the same-sized fifths that were established in C-G-D-A-E.
  • All the correctly spelled major thirds are pure as 5:4.
  • The four diminished fourths are all 21 schismas sharp of 5:4 purity.
  • Chromatic semitone: 76.0 cents; diatonic semitone: 117.1 cents. That is, the two semitones are in 3:2 ratio with one another.
  • This is equivalent in practice to a 31-note equal division of the octave.
  • Easter egg: the three augmented seconds F-G#, Bb-C#, and Eb-F# are close to the sound of a pure 7:6 minor third and therefore sound consonant. These contribute to the fairly well-known characteristic of F minor as a specially usable key in meantone.
  • Easter egg: the augmented sixths Bb-G# and Eb-C# are close to the sound of a pure 7:4 minor seventh, and therefore sound remarkably resonant and static.
Regular 1/5 syntonic comma
  • From C, tune E pure; and then B as a pure fifth from the E. (The E will move later; this step is only to establish the proper B from C.)
  • B and C have been established at one syntonic comma apart, since there was one pure major third employed to get there.
  • The fifths C-G-D-A-E-B must then each take up 1/5 of that syntonic comma error.
  • Tune all of C-F-Bb-Eb and B-F#-C#-G# with the same-sized fifths that were established in C-G-D-A-E-B.
  • All the correctly spelled major sevenths (C-B, D-C#, Eb-D, F-E, G-F#, A-G#, and Bb-A) are pure. The ratio is 15:8, and they sound like a pure fifth plus a pure major third. For example, if C-B is played by itself, the ear can fill in a phantom E or G or both, even though the E and G that are really on the keyboard are tempered off those positions.
  • Major thirds are all 2.2 schismas sharp of 5:4 purity.
  • The four diminished fourths are all 16.6 schismas sharp of 5:4 purity.
  • Chromatic semitone: 83.6 cents; diatonic semitone: 111.7 cents.
  • Easter egg: The misspelled minor sixths C-Ab, D#-B, F-Db, and A#-F# are all surprisingly usable, even though they do not work in inversion as major thirds. These four are slightly narrow 11:7 minor sixths, close enough that the ear locks into them.
  • Easter egg: B major, C# major, and F# major work in first inversion for basso continuo due to the 11:7 sixths noted above.
Regular 1/6 syntonic comma
  • From C, tune E pure; and then pure fifths E-B-F#. (The E and B will move later; this step is only to establish the proper F# from C.)
  • F# and C have been established at one syntonic comma apart, since there was one pure major third employed to get there.
  • The fifths C-G-D-A-E-B-F# must then each take up 1/6 of that syntonic comma error.
  • Tune all of C-F-Bb-Eb and F#-C#-G# with the same-sized fifths that were established in C-G-D-A-E-B-F#.
  • All the correctly spelled augmented fourths (such as C-F#) are pure. The ratio is 45:32, and they sound like a pure 9:8 major second plus a pure major third. For example, if C-F# is played by itself, the ear can fill in a phantom D, even though the D that is really on the keyboard is tempered off that position.
  • Major thirds are all 3.7 schismas sharp of 5:4 purity.
  • The four diminished fourths are all 13.7 schismas sharp of 5:4 purity.
  • Chromatic semitone: 88.6 cents; diatonic semitone: 108.1 cents.
  • Easter egg: Those ringing tritones give the diminished triads and diminished seventh chords a strong resonance.
  • Easter egg: The misspelled minor sixths C-Ab, D#-B, F-Db, and A#-F# are all surprisingly usable, even though they do not work in inversion as major thirds. These four are slightly wide 11:7 minor sixths, close enough that the ear locks into them.
  • Easter egg: B major, C# major, and F# major work in first inversion for basso continuo due to the 11:7 sixths noted above.
Regular 1/5 Pythagorean comma
  • From C, tune pure fifths around the flat side: C-F-Bb-Eb-Ab-Db-Gb-Cb. (All of the F-Bb-Eb-Ab-Db-Gb will move later; this step is only to establish the proper Cb from C, and then to reinterpret it as B.)
  • B and C have been established at one Pythagorean comma apart, since we have not done any tempering at all to get there, but have approached it from the opposite direction.
  • The fifths C-G-D-A-E-B must then each take up 1/5 of that Pythagorean comma error.
  • Tune all of C-F-Bb-Eb and B-F#-C#-G# with the same-sized fifths that were established in C-G-D-A-E-B.
  • Major thirds are all 1.4 schismas sharp of 5:4 purity.
  • The four diminished fourths are all 18.2 schismas sharp of 5:4 purity.
  • Chromatic semitone: 80.8 cents; diatonic semitone: 113.7 cents.
  • Easter egg: The Eb seventh chord Eb-G-Bb-Db, and the Bb seventh Bb-D-F-Ab, are surprisingly resonant. The misspelled Eb-C# and Bb-G# are close coincidences to a pure 7:4 interval. (See also 1/4 SC for this same phenomenon.)
Regular 1/6 Pythagorean comma (aka "55-division") - the especially important one for background of the Bach temperament
  • From C, tune pure fifths around the flat side: C-F-Bb-Eb-Ab-Db-Gb. (All of the F-Bb-Eb-Ab-Db will move later; this step is only to establish the proper Gb from C, and then to reinterpret it as F#.)
  • F# and C have been established at one Pythagorean comma apart, since we have not done any tempering at all to get there, but have approached it from the opposite direction.
  • The fifths C-G-D-A-E-B-F# must then each take up 1/6 of that Pythagorean comma error.
  • Tune all of C-F-Bb-Eb and F#-C#-G# with the same-sized fifths that were established in C-G-D-A-E-B-F#.
  • Major thirds are all 3 schismas sharp of 5:4 purity: still sounding nearly pure and with strong "gravity" into cadences, but also with a discreet vibrato.
  • The four diminished fourths are all 15 schismas sharp of 5:4 purity.
  • Chromatic semitone: 86.3 cents; diatonic semitone: 109.8 cents.
  • "Subsemitone" enharmonic pairs of notes such as G# (available on the keyboard) and Ab (not on the keyboard) are exactly one Pythagorean comma apart, with the flat higher than the sharp.
  • The whole octave is comprised of 55 of these equal steps, one comma each. C=0, C#=4, D=9, Eb=14, E=18, F=23, F#=27, G=32, G#=36, A=41, Bb=46, B=50, C=55. Other frequently-used but missing notes include Db=5, D#=13, E#=22, Gb=28, Ab=37, A#=45, Cb=51, B#=54.
  • Easter egg: The misspelled minor sixths C-Ab, D#-B, F-Db, and A#-F# are all surprisingly usable, even though they do not work in inversion as major thirds. These four are slightly wide 11:7 minor sixths, close enough that the ear locks into them.
  • Easter egg: B major, C# major, and F# major work in first inversion for basso continuo due to the 11:7 sixths noted above.
Regular 1/12 Pythagorean comma or 1/11 syntonic comma (i.e. Equal temperament)
  • All the fifths are equally one schisma in size.
  • Major thirds are all 7 schismas sharp of 5:4 purity. (In all temperaments this average across all twelve of the major thirds must be 7 schismas; and within each stack of major thirds such as C-E-G#-C the diesis of 21 schismas must be absorbed somehow into one or more of those major thirds. See also 1/4 SC where all 21 of the schismas are placed into the four diminished fourths.)
  • No intervals are pure, or derived directly from other pure intervals, except the octave.
  • All semitones: 100 cents.
Bach (irregular, but based on regular 1/6 PC)
  • Major thirds range from 3 to 10 schismas sharp of 5:4.
  • Semitones range from 94.1 to 109.8 cents.
  • Easter egg: All intervals everywhere are equally usable, but with a pleasing variety as they are transposed to all twelve positions. Therefore the music has a dynamic effect.


Regular meantone doesn't work for a Bach D minor piece!

Even though D minor (along with C major) is typically the "home" key for regular meantone layouts, here is a simple proof by example that Bach couldn't have intended such a regular layout for the inventions/sinfonias.

This excerpt is from the three-part invention (sinfonia) in D minor, BWV 790. An analytical video about it is here.

In bar 14 a D# and an Eb occur within half a beat of one another, during a rapid modulation from A minor to G minor! The D# must sound good with the G# two bars earlier. The Eb must sound good with the Bb two bars later. Where's the wolf going to be? That D#/Eb lever must be tuned to an interim pitch so either note is passable: both within the 5ths Eb-Bb and G#-D#, and within the major 3rds (or minor 6ths) B-D# and Eb-G.

This handful of bars also argues that the naturals must not be as tightly tempered as 1/4 comma, or 1/5 comma: either B-D# or Eb-G (or both) get uncomfortably wide, wherever the compromised D#/Eb is placed between them. If the naturals F-C-G-D-A-E-B are all in regular 1/6 comma, the B-D# and Eb-G each have to average size 9 (where 11 is a full Pythagorean ditone; [explanation]). Size 9 is already very active and unsettled. The narrower the major 3rd G-B is, the wider both B-D# and Eb-G have to be, to compensate for it.

In Bach's own temperament (as argued here), Eb-G is size 7 and B-D# is size 9. The note B is slightly higher than its position would be if in regular 1/6 comma. G-B is size 5, where it would be size 3 in regular 1/6 comma.

Granted, there exists another D minor piece by Bach that does work fine in any regular meantone: the D minor "French" suite, which happens to have no notes outside Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#. (That's not an argument that the D minor French suite ought to be played in meantone...but only an observation that it may be played that way, without catastrophe.)


Practical instructions to set meantone by ear (27 October 2007)

Try the following sequence. The whole thing takes less than 10 minutes, with practice, to do an entire keyboard of about 50 strings of "8-foot" pitch:
  • Get A from a tuning fork, and make a pure octave down to tenor A.
  • Tune F as a major 3rd below tenor A, and make it 0 to 4 beats wide...to your taste. This will give us anything from 1/4 comma (if we picked 0 beats here) up to about 1/7 comma, automatically generated by the following set of instructions. If you pick 3 beats per second you get 1/6 comma.
  • Build two temporary notes: tenor C below tenor F pure as a 4th, and tenor D below tenor A pure as a 5th. These are being done only so we can put the G exactly where it belongs between F and A, using these two reference points.
  • Working from that C up to tenor G as a 5th, and from that D up to tenor G as a 4th, put G where it has exactly the same quality from both. It will be fairly rough from both. The 4th will be beating faster than the 5th, and if you care to count beats, you have reached the correct point when the 4th is beating as triplets against the 5th as duplets.
  • Working from that tenor G and tenor A up to the D above middle C, similarly put the D into position so the G-D 5th and the A-D 4th have the same quality as one another. Again the 4th will be beating as triplets against the duplets of the 5th. This is always true when doing correctly equalized 5ths/4ths, playing the notes that are under the fingers 5-4-1 of the left hand.
  • From this middle D, correct the lower temporary D to be a pure octave. Check it with both G and A for equal roughness.
  • Working from tenor F and tenor G up to middle C, put C into place so the F-C 5th and G-C 4th are equally rough. Same triplets-against-duplets shtick.
  • From middle C fix the lower C as pure octave, and check its 4th and 5th.
  • Now F-C-G-D-A are all similarly tempered with one another, smoothly. Regular meantone!
  • From tenor A, tune middle C# with the same tasteful quality you used on F-A. It will be beating faster since the notes are higher, but the quality of wideness (or none) will be the same.
  • From tenor A build a temporary pure E 4th under it. From that C# build a temporary tenor F# pure as a 5th. We are going to use both of these to build a correct B next to middle C.
  • Make a B that is equally rough from both that E as a 5th, and F# as a 4th. As usual, the 4th makes triplets against the 5th's duplets.
  • From B and C# build the F# above middle C. Copy it down to tenor F# erasing the temporary one we did a moment ago.
  • From A and B build the E above middle C. Copy it down to tenor E erasing the temporary one.
  • We're almost done. We already have F-C-G-D-A-E-B-F#-C# all as equally sized 5ths/4ths, smoothly. We only lack G#, Bb, and Eb.
  • Copy middle C# down to tenor C# as a pure octave.
  • Build G# from C# as a 5th, sounding in quality like all the other 5ths around it. Check it as a major 3rd above E also, for the same tasteful sound we have already given all the other major 3rds.
  • Copy tenor F up to the F above middle C, if we didn't already do it.
  • Tune Bb as a 5th under F, giving it the same quality as all the other 5ths nearby, and checking it as a major 3rd with middle D.
  • Tune Eb as a 5th under Bb, giving it the same quality as all the other 5ths nearby, and checking it as a major 3rd with tenor G.
  • Copy tenor Eb up to middle Eb.
  • Treat yourself to a nice Eb major chord. And check all the other major triads, too (the correctly-spelled ones, not the B or F# or C# or Ab major that aren't). The eight good major triads should all have the same quality as one another.
  • Finish the instrument by octaves.
  • In the treble especially, a good way to check pure octaves is to test the 4th and the 5th within them...for the similar quality, and the triplets-to-duplets if you care to listen for them.


Beat rates

I do not consider specific beat rates per interval to be especially interesting (anymore!), as they are dependent on the choice of a particular starting frequency.

Music is listened to with harmonic and melodic relationships, i.e. various tensions and relaxations as music progresses...not numbers. The numbers are only a measurement of the physical phenomena.

On harpsichords, at least, these beats can scarcely be heard at more than a meter away from the instrument's soundboard; and then, primarily on long-sustained chords of several seconds when nothing else is moving. This is mostly moot, except for the process of tuning.

However, some readers might find the following charts useful in comparing the different qualities of intervals in these temperaments. Some might also use them to spot-check their work at a keyboard, listening for the proper speeds for the chosen pitches.

I wish to emphasize: it is much easier to set up these temperaments (except equal temperament) by using the principles I present above, and at practical temperament instructions by ear, instead of going mad by trying to count all these beats to tenths of seconds.... The 3-to-2 rule presented in my tetrasecting page is easier to use, judging comparative beat ratios, than trying to count anything against a timekeeping device.

That said, here are the beat-rate tables, based for convenience first on A=440 and then on A=415! Included for comparison are:

  • Regular 1/5 Pythagorean comma
  • Regular 1/6 Pythagorean comma
  • Bach/Lehman (being based on regular 1/6 Pythagorean comma for its naturals)
  • Equal temperament
  • Regular 1/5 syntonic comma
  • Regular 1/6 syntonic comma
These tables are generated by my free Keyboard temperament analyzer/calculator from the 1990s.


A=440


A=415


Video demonstration on YouTube, August 2007: setting meantone accurately [Transcript of the captions]


Video demonstration on YouTube, August 2007: 1/6 comma meantone morphed into Bach


Video demonstration on YouTube, September 2007: setting late 17th century temperament ordinaire, and playing Purcell. This also shows the sequence to get regular 1/5 comma naturals going smoothly, as the first half of the temperament.

Other video demonstrations...

Jean-Philippe 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.


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Bach's schematic, rotated for use