Editing Pythagorean comma (section)

==History==
The first to mention the comma's proportion of 531441:524288 was [[Euclid]], who takes as a basis the whole tone of Pythagorean tuning with the ratio of 9:8, the octave with the ratio of 2:1, and a number A = 262144. He concludes that raising this number by six whole tones yields a value, G, that is larger than that yielded by raising it by an octave (two times A). He gives G to be 531441.<ref>[[Euclid]]: ''Katatome kanonos'' (lat. ''Sectio canonis''). Engl. transl. in: [[Andrew Barker (classicist)|Andrew Barker]] (ed.): ''Greek Musical Writings. Vol. 2: Harmonic and Acoustic Theory'', Cambridge, Massachusetts: Cambridge University Press, 2004, pp. 190–208, here: p. 199.</ref> The necessary calculations read:

Calculation of G:
::<math>262144 \cdot \left(\textstyle{\frac 9 8}\right)^6 = 531441</math>

Calculation of the double of A:
::<math>262144 \cdot \left(\textstyle{\frac 2 1}\right)^1 = 524288</math>

Chinese mathematicians were aware of the Pythagorean comma as early as 122 BC (its calculation is detailed in the ''[[Huainanzi]]''), and ca. 50 BC, [[Jing Fang]] discovered that if the cycle of perfect fifths were continued beyond 12 all the way to 53, the difference between this 53rd pitch and the starting pitch would be much smaller than the Pythagorean comma. This much smaller interval was later named [[Mercator's comma]] (''see: [[53 equal temperament#History_and_use|history of 53 equal temperament]]'').

In George Russell's ''[[Lydian Chromatic Concept of Tonal Organization]]'' (1953), the half step between the Lydian Tonic and {{music|b}}2 in his Altered Major and Minor Auxiliary Diminished Blues scales is theoretically based on the Pythagorean comma.<ref>[[George Russell (composer)|Russell, George]] (2001) [1953]. George Russell's ''[[Lydian Chromatic Concept of Tonal Organization]]''. Volume One: The art and science of tonal gravity (Fourth (Second printing, corrected, 2008) ed.). Brookline, Massachusetts: Concept Publishing Company. pp. 17, 57–59. {{ISBN|0-9703739-0-2}}.</ref>