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Among quite some number of proposed temperaments, as being the one applied by

J. S. Bach    for his   Well  Tempered  Clavier ,

it is highly probable that

the   Well  Tempered  Meantone    stands among the most plausible ones.

Within the   C–major   tonality, this   Well-Tempered Meantone   contains five fifths and three major thirds holding perfectly equal interval beating rates, when measured in a chromatic scale on   F   , the partition normally used for the initialization of the auditory tuning of a keyboard. This equality of beating rates allows quick and easy tuning of a keyboard.

The characteristics of all the fifths of this   Well-Tempered Meantone   are perfectly at par with those of the loops in the figure above, taken from a   Well-Tempered Clavier   score, handwritten by   J. S. Bach.

A video demonstration of the auditory tuning of this    Well-Tempered Meantone   is made on the website

https://www.youtube.com/watch?v=lwfESoMxd8Y

An explanatory text is available here.

The fifths circle, and the corresponding interval data are shown below.

fifths circle
C C# D Eb E F F# G G# A Bb B
262.83
277.07
293.98
311.90
329.03
351.22
369.42
393.27
415.60
440.00
468.04
492.57

Tuning Table (A3 = 220)
A3 = 220 Equal beating of 5 fifths and 3 major thirds
Partition
F3 - F 4
Beats Major Thirds Beats
A4 - A3 0.00 F3 - A3 1.95
A3 - D4 1.95 Gb3 - Bb3 12.51
D4 - G3 -1.95 G3 - B3 1.95
G3 - C4 1.95 Ab3 - C4 12.32
C4 - F3 -1.17 A3 - C#4 8.27
F3 - F4 0.00 Bb3 - D4 5.84
F4 - Bb3 0.39 B3 - D#4 16.17
Bb3 - Eb4 -0.39 C4 - E4 1.95
Eb4 - Ab3 0.39 Db4 - F4 19.54
G#3 - C#4 0.00 F4 - A4 3.89
C#4 - F#3 0.00

F#3 - B3 0.00

B3 - E4 1.95

E4 - A3 -1.95

E4 - A4 -3.89


Broekaert Johan 2021-01-05   broekaert.devriendt (at) gmail.com

Very sincere thanks to:
       Prof. E. Amiot, A. Calvet, E. Jobin, T. Paintoux: they were kindly open for communications and interchanges of ideas, leading to this disclosure.
      G. Baroin: for the compilation of a magnificent video dealing with the subject
       My daughter Hilde: she drew my attention to investigate on what musicians want (diatonic interval purity) and not on what might be someone's preferred  musical temperament.