About Meantone
Pythagorean 5th degree … about 701.96 cents
The fifth degree of the meantone … about 696.5 cents
The fifth degree of the equal temperament … 700 cents
700✕12 = 8,400… the number of cents per fifth degree of the equal temperament✕12
701.96✕12 = 8,423.52 … the number of cents in the fifth degree of Pythagoras✕12
*The error is 23.52 cents (Pythagorean comma)
701.96✕4 = 2,807.84 … Pythagorean fifth degree✕4
Approximately 696.5 cents✕4 = 2,786 … 5 degrees of the Mean Tone✕4
*The error is 21.84 cents (syntonic comma)
23.52 – 21.84 = 1.68 cents (scisma) *Difference between the Pythagorean comma and the syntonic comma
It is well known that meantone temperament is a temperament method with limited playability of key (however, the meantone here refers to the quarter-comma mintone, or “Aron’s meantone”). In actual performance, however, this tuning method was modified as necessary. In other words, depending on the key of the piece being played, the four basic fifths (C-G, G-D, D-A, A-E) could be changed to (D-A, A-E, E-H, H-F♯) or (E♭-B♭, B♭-F, F-C, C-G). This would allow us to play tunes with many #’s and many flats. However, there is no proof that this was done frequently. There is no doubt that the tunings (C-G, G-D, D-A, A-E), for example, were re-tuned during concert breaks, such as “G♯→A♭” and “D♯→E♭”. Then, what did they do with pieces in which both G♯ and A♭ appear? In that case, it was probably decided according to the frequency with which they appeared. Aaron’s meantone in cents would be as follows.（Figure1）
696.5✕11 = 7661.5
738.5 … Wolf (G♯-E♭)
7661.5+738.5 = 8,400
This tuning method was used predominantly until the middle of the 17th century.
Another meantone, “1/6 comma meantone” is also known as “Silbermann’s tuning method”. This tuning method consist of the all fifths are (Pythagorean fifths – syntonic comma✕ 1/6) and adds the remaining error to the last Wolf (G♯-E♭). The Wolf in “Aron’s meantone” is 738.5, whereas in “Silbermann” it is 718.48, a considerable improvement in sound.（Figure2）
(701.96-21.84✕1/6)✕11 = (701.96-3.64)✕11 = 698.32✕11 = 7,681.52 …(Pythagorean fifth degree – syntonic comma✕1/6)✕11
718.48 … Wolf (G♯-E♭)
7,681.52 + 718.48 = 8,400
#baroque #meantone #temperament #片山俊幸
