About Bach’s Tuning Method
https://excelkobo.net/Bach-WTC1_manuscript.pdf
What does Bach mean by “The Well-Tempered Clavier”? Some people claim that it is implied by the pattern Bach wrote above the title of the autograph in the first volume of “The Well-Tempered Clavier” as if it were a “rebus” (Fig. 1). They say that this is the so-called “five-degree circle” laid out in an open horizontal line. When this pattern is displayed in the form of a circle, it looks like this (Fig. 2). And there are three types of the five degrees, each of which represents the following five degrees.
Example of Bach’s Tuning Method Answer by Emile Jobin
◆Bach’s figure
(1) Triple circle: Five degrees of meantone (five)
(2) Single circle: Five degrees of Pythagoras (three)
(3) Double circle: ? ? ? (three)
5 degrees of Pythagoras ・・・ about 701.96 cents
5 degrees of the meantone ・・・ about 696.5 cents
5 degrees of the equal temperament ・・・ 700 cents
700×12 = 8,400・・・ the number of cents in an octave of the equal temperament
701.96×12 = 8,423.52・・・ the number of cents in the fifth degree of Pythagoras×12
*The error is 23.52 cents.
(1) 8,400 – (696.5×5) = 4,917.5・・・ 1 octave minus meantone 5 degrees ×5
(2) 4,917.5 – (701.96×3) = 2,811.62・・・ “1 octave minus meantone 5 degrees ×5” minus “three Pythagorean 5 degrees”
(3) 2,811.62 ÷ 4 = 702.905・・・ Double circle & 5 degrees of F-C
◆Calculation
(696.5 × 5) + (701.96 × 3) + (702.905 × 4)
= 3,482.5 + 2,105.88 + 2,811.62
= (Total) 8,400 (Fig. 3)
By modern physics it can be explained, in essence, that if all the perfect fifths in a tuning circle are tuned about two cents ( = Pythagorean comma) narrower than the Pythagorean fifth, an equal temperament tuning is automatically produced. However, this unit of “cent” is impossible to measure without using mechanical means. Assuming that Bach knew the “equal-temperament”, then how did Bach achieve the “equal-temperament”? Needless to say, tuning in Bach’s time was 100% done by ear. In other words, the only possible means was to listen to the number of “beats”. But another possibility, though less likely, is that the lute was used to tune the clavier. This is because the lute is unquestionably a “equal-tempered” instrument.
