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.

# Mathematical analyses of Bach's temperament

Full explanation of these two versions is in my Early Music article, February and May 2005.

See also my practical instructions [Beginner] [Intermediate] [Expert] to set up this first layout easily and quickly by ear.

Users of electronic tuning devices will find the magic numbers in the rows "d ET (A) c" (deviations from equal temperament keeping a fixed A, in cents), and "d ET (C) c" (deviations from equal temperament keeping a fixed C, in cents). For example, using the fixed-A numbers for the first version, the notes are:
A (0), Bb (+3.9), B (0), C (+5.9), C# (+3.9), D (+2), Eb (+3.9), E (-2), F (+7.8), F# (+2), G (+3.9), G# (+3.9).

But, I strongly encourage the use of the by-ear methods (or the other explanations of its formulation given in the Early Music article). That hands-on experience is part of Bach's tuning lesson, to understand the relationships among the intervals and to heighten the ear's sensitivity to pitch!

### Solo

The normal layout of Bach's temperament, for solo and instrumental-ensemble work:

The tetrachord discussions are in the second half of the Early Music article. The diesis discussion is at the tuning maths page of this web site.

#### Syntonic comma (SC)

The lines "Maj3rd % SC" and "Min3rd % SC" give the portion of the syntonic comma that the interval is out of tune, as measured from a pure major third (5:4 frequency ratio) or a pure minor third (6:5).

The smooth sound of a keyboard temperament is related to the way in which it handles this error, which must average 63.6% across all twelve major thirds. The adjoining fifths, as shown here in the chart, need to have similar percentages in these major thirds: no jumps to grossly different portions of this syntonic comma.

For example, when music modulates from E major to B major, those two triads should sound similar enough to one another in the quality of their major thirds that it's not a jarring change (which is a problem in some temperaments, such as regular meantone). Likewise, the "Eb" and "G#" edges of this chart are connected in practice, needing a smooth transition. The chart here is essentially a flattened-out cylinder or torus.

Furthermore, any major thirds at or exceeding 100% of this SC error sound rough (which, of course, is somewhat subjective and on a continuum of perception--and taking into account the reports of Bach's own preferences in this assessment), and so a basic part of the tempering strategy is to avoid having any of the major thirds that wide.

#### Pythagorean comma (PC)

The lines "Error % PC" and "Enharm err % PC" give the portion of the Pythagorean comma that the named note deviates from its expected position in regular 1/6 comma temperament (i.e. the 18th century standard of "55 equal divisions of the octave"; see the FAQ).

Each time we go twelve positions around the spiral of fifths, in any 12-note keyboard temperament, we either add or subtract a Pythagorean comma to get to the new enharmonic name of the note. Keyboard temperaments are designed such that the notes all have a primary name or function at which they sound best, while they can also function to some extent as the enharmonically named note a PC away in either direction.

To be usable in most or all keys, temperaments need to have some smooth progression as portions of this PC error are distributed gradually, as seen here. Read these two lines of data together, wrapping around the edges, noting the progression of the error from Bbb, Fb, Cb, Gb, Db, Ab, Eb, Bb, F, C, G, D, A, E, B, F#, C#, G#, D#, A#, E#, B#, Fx, Cx. (Of Bach's 1722 music, the book of inventions and sinfonias requires all 24 of these notes; and WTC book 1 requires additionally Gx, Dx, and Ax!) Take the example of the F minor Sinfonia, BWV 795: it uses Cb, Fb, Bbb, B, E, and A all in the same composition. Therefore any temperament to play this piece must have a distribution where none of those notes sound harshly out of place, within their different enharmonic contexts.

This metric of percentages of PC is especially useful when comparing the 40+ other temperaments presented in my supplementary materials from Oxford, showing how the various temperaments handle this either smoothly or badly (and, in the case of "Werckmeister 3" and "Kellner" and regular systems, very badly). The second half of the Early Music article presents this topic of enharmonic equivalences in fuller detail.

#### Circular diagrams

The fractions here are reckoned from the Pythagorean comma.

In Temperament Units that equivalent diagram is here, with the same values around the perimeter. The numbers inside each triangle represent the percentage of the syntonic comma sharp: for example, C-E is 27% sharp, E-G# is 91% sharp, and Ab-C is 73% sharp. This is the same as "Maj3rd % SC" explained above.

#### Cents

For those who are most comfortable with tables of "cents" instead of fragments of commas, here are the fifths and thirds as rounded off to the nearest tenth:

```Note   fifth   minor3rd   major3rd
Eb     700.0   298.0      400.0
Bb     703.9   300.0      398.0
F      698.0   296.1      392.2
C      698.0   298.0      392.2
G      698.0   300.0      396.1
D      698.0   305.9      400.0
A      698.0   305.9      403.9
E      702.0   305.9      405.9
B      702.0   302.0      403.9
F#     702.0   298.0      402.0
C#     700.0   294.1      403.9
G#     700.0   296.1      402.0
```

Yes, the E-flat major triad sounds exactly like its counterpart in equal temperament! Played on an organ tuned this way, Bach's book Clavierübung III (published 1739) begins and ends in equal temperament...but is not in equal temperament for most of the 90+ minutes in between the opening and closing triads....

#### Cent offsets

As noted above, if we start from equal temperament and keep A exactly where it was, all the other notes end up here:
A (0), Bb (+3.9), B (0), C (+5.9), C# (+3.9), D (+2), Eb (+3.9), E (-2), F (+7.8), F# (+2), G (+3.9), G# (+3.9).

But, if there is some overwhelming concern to keep the whole instrument's equilibrium as it was in equal, we can simply add the same constant to all 12 numbers. This lets us adjust it so no single note moves more than 5 cents in either direction.

Subtract 3, and round them all to the nearest cent:
A (-3), Bb (+1), B (-3), C (+3), C# (+1), D (-1), Eb (+1), E (-5), F (+5), F# (-1), G (+1), G# (+1).

The overall tension in the instrument stays the same: within each octave, 5 of the notes move down a total of 13 cents, and the other 7 notes move up a total of 13 cents.

#### Beat rates

The following charts show how absolute beat rates vary, depending on the choice of a starting pitch standard from a tuning fork. (This is a truism in basic physics, not a characteristic of any particular temperament.)

Cent relationships and comma-fragment relationships remain constant, irrespective of starting pitch, as they are geometric (and expressed logarithmically). They are the same from octave to octave, as well. The vibrating frequencies, on the other hand, double at each octave and so do the corresponding beat rates.

The beat rate for any particular interval is found by taking the frequencies of the two notes and finding the least common multiple. For example, if A is 220Hz and the E a fifth above it is (nearly) 330Hz, the multiple is 660Hz. That is the pitch at which beats are audible, subtracting the expected frequency (i.e. 660) from the actual frequency (~658.5) of the upper note, yielding -1.49. It is heard as a slight wavering when both notes are played together on the keyboard.

That is how we harpsichord tuners set tempered intervals accurately by ear: by counting that resulting "wavering" speed of 1.49 beats per second (narrow) for that particular fifth, for this particular temperament (and similar ones), in this particular octave, starting from an "A 440" tuning fork. The notes are all derived from a single starting source, either by counting specific beat rates or by judging the proportions of beat rates accurately, all being a set of geometric relationships. More about this is at my practical instructions page.

### Melodic characteristics in diatonic scales

As described in "Bach's extraordinary temperament: Our Rosetta Stone", every major scale and every minor scale sounds distinct from every other, melodically/harmonically. This gives the temperament a wealth of expressive inflections.

In major scales, as we modulate upward or downward by 5ths to the next scale, the most noticeable changes happen in Mi, La, and Ti. The note is slightly higher or lower than it was in the scale we just left, reckoning from the new Do. As we modulate into sharper keys, the entire diatonic scale sounds "brighter". As we modulate into flat keys, the scale is "more mellow". The E-flat and A-flat major scales are the closest to hitting the spots of equal temperament.

### Vocal accompaniment

The transposed layout of Bach's temperament, for keyboards in the vocal works that were written for a Chorton/Cammerton transposing situation. (For example, all of the Leipzig vocal music: cantatas, masses, passions, oratorios.) Tune the organs and harpsichords using this alternate version, for keyboard players who are reading from modern parts that have been transposed up to the key of the orchestra:

The discussion of the vocal music has its own page explaining why this distinction matters.

A suggested list of Bach repertoire to test these, giving them a play-through to hear how they sound in practice, is here.