Bach's schematic, as it appears on the page
LaripS.com, © Bradley Lehman, 2005-13, 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.

Summary of the layout

The circle of 5ths, in Bach's tuning

Tune this; play anything.

Everything else follows from the sound of this, in practice. The music of JS Bach and his sons fits the resources of this expected sound: this particular set of intonational nuances in their complex combinations.

I believe Bach's diagram shows a tuning sequence. Starting at F-C, C-G etc, go clockwise around the circle. Temper the 5ths narrow by 1/6 (2/12) comma, 1/6, 1/6, 1/6, 1/6, 0, 0, 0, 1/12, 1/12, 1/12.

Bach's schematic, rotated for use

That is: F-C-G-D-A-E with "double" tempering; E-B-F#-C# pure; C#-G#-D#-A# with "single" tempering. The last leftover interval, from A# back to F, has "single" tempering in the opposite direction. (One of the video demonstrations shows all of this more clearly in action; so does the article "Bach's art of temperament"....)

The varying sharpness of major 3rds:

C-E 3, Ab-C 8, E-G# 10
F-A 3, A-C# 9, Db-F 9
G-B 5, Eb-G 7, B-D# 9
Bb-D 6, D-F# 7, F#-A# 8

C-E, F-A, G-B, and Bb-D are calmer and more resonant than in equal temperament;
D-F# and Eb-G are the same as their size in equal temperament (size 7);
F#-A#, Ab-C, A-C#, B-D#, Db-F, and E-G# are more active than in equal temperament.
[More about this measurement, where equal temperament's major 3rds have size 7]

Some subjective remarks about key character, when playing with this temperament: Music in sharp keys tends to sound increasingly crisp and brilliant with each added sharp. Music in flat keys tends to sound mellow and warm, with a vibrant glow at each added flat. C major and its nearby keys sound plain-spoken and resonant, similar to 1/6 comma meantone.

Music in the flat minors tends to sound dark, troubled, sorrowful, yet noble. Music in the sharp minors tends to sound intense and incisive, with forceful dominant tensions. Music in major keys tends to sound smooth, having fewer strong contrasts than in minor-key music.

More detailed instructions

Seekers of cent values and other mathematical modeling will find such things here.

Another series of diagrams is in my JMU lecture notes from October 2008, here.


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Bach's schematic, rotated for use