Pythagorean comma.html
 
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In music, when ascending from an initial (low) pitch by a cycle of justly tuned perfect fifths (ratio 3:2) (play (help·info)), leapfrogging twelve times, one eventually reaches a pitch approximately seven whole octaves above the starting pitch. If this pitch is then lowered precisely seven octaves, it will be discovered that the resulting pitch is 23.46 cents (a quarter of an equal tempered minor second) higher than the initial pitch (play (help·info)). This microtonal interval

\left(\frac32\right)^{12} \!\!\bigg/\, 2^{7} = \frac{3^{12}}{2^{19}} = \frac{531441}{524288} = 1.0136432647705078125 \!

is called a Pythagorean comma, named after the ancient mathematician and philosopher Pythagoras. It is sometimes called a ditonic comma.

Put more succinctly, twelve perfect fifths are not exactly equal to seven perfect octaves, and the Pythagorean comma is the amount of the discrepancy.

This interval has serious implications for the various tuning schemes of the chromatic scale, because in Western music, 12 perfect fifths and seven octaves are treated as the same interval. Equal temperament, today the most common tuning system used in the West, accomplished this by flattening each fifth by a twelfth of a Pythagorean comma (approximately 2 cents), thus producing perfect octaves.

Chinese mathematicians had been aware of the Pythagorean comma as early as 122 BC (its calculation is detailed in the Huainanzi), and circa 50 BC, Ching 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: history of 53 equal temperament).

Other intervals of similar size are the syntonic comma, and Holdrian comma.

Ascending by perfect fifths
Note Fifth Frequency ratio Equivalent ratio
C 0 1 : 1 =  1 : 1
G 1 2 : 3 =  1 : 1.5
D 2 4 : 9 =  1 : 2.25
A 3 8 : 27 =  1 : 3.375
E 4 16 : 81 =  1 : 5.0625
B 5 32 : 243 =  1 : 7.59375
F♯ 6 64 : 729 =  1 : 11.390625
C♯ 7 128 : 2187 =  1 : 17.0859375
G♯ 8 256 : 6561 =  1 : 25.62890625
D♯ 9 512 : 19683 =  1 : 38.443359375
A♯ 10 1024 : 59049 =  1 : 57.6650390625
E♯ 11 2048 : 177147 =  1 : 86.49755859375
B♯ (≈ C) 12 4096 : 531441 =  1 : 129.746337890625
Ascending by octaves
Note Octave Frequency ratio
C 0 1 : 1
C 1 1 : 2
C 2 1 : 4
C 3 1 : 8
C 4 1 : 16
C 5 1 : 32
C 6 1 : 64
C 7 1 : 128

Circle of fifths and enharmonic change

In the following table of musical scales in the circle of fifths, the Pythagorean comma is visible as the small interval between e.g. F♯ and G♭.

The 6♭ and the 6♯ scales* are not identical - even though they are on the piano keyboard - but the ♭ scales are one Pythagorean comma lower. Disregarding this difference leads to enharmonic change.

C-sharp major.html A-sharp minor.html F-sharp major.html D-sharp minor.html B major.html G-sharp minor.html E major.html C-sharp minor.html A major.html F-sharp minor.html D major.html B minor.html G major.html E minor.html C major.html A minor.html F major.html D minor.html B-flat major.html G minor.html E-flat major.html C minor.html A-flat major.html F minor.html D-flat major.html B-flat minor.html G-flat major.html E-flat minor.html C-flat major.html A-flat minor.html la.html ti.html do.html re.html mi.html fa.html so.html la.html ti.html do.html re.html mi.html fa.html so.html mi.html fa.html so.html la.html ti.html do.html re.html mi.html fa.html so.html la.html ti.html do.html re.html ti.html do.html re.html mi.html fa.html so.html la.html ti.html do.html re.html mi.html fa.html so.html la.html fa.html so.html la.html ti.html do.html re.html mi.html fa.html so.html la.html ti.html do.html re.html mi.html do.html re.html mi.html fa.html so.html la.html ti.html do.html re.html mi.html fa.html so.html la.html ti.html so.html la.html ti.html do.html re.html mi.html fa.html so.html la.html ti.html do.html re.html mi.html fa.html re.html mi.html fa.html so.html la.html ti.html do.html re.html mi.html fa.html so.html la.html ti.html do.html la.html ti.html do.html re.html mi.html fa.html so.html la.html ti.html do.html re.html mi.html fa.html so.html mi.html fa.html so.html la.html ti.html do.html re.html mi.html fa.html so.html la.html ti.html do.html re.html ti.html do.html re.html mi.html fa.html so.html la.html ti.html do.html re.html mi.html fa.html so.html la.html so.html la.html ti.html do.html re.html mi.html fa.html so.html la.html ti.html do.html re.html mi.html fa.html re.html mi.html fa.html so.html la.html ti.html do.html re.html mi.html fa.html so.html la.html ti.html do.html la.html ti.html do.html re.html mi.html fa.html so.html la.html ti.html do.html re.html mi.html fa.html so.html mi.html fa.html so.html la.html ti.html do.html re.html mi.html fa.html so.html la.html ti.html do.html re.html ti.html do.html re.html mi.html fa.html so.html la.html ti.html do.html re.html mi.html fa.html so.html la.html
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* The 7♭ and 5♯, respectively 5♭ and 7♯ scales differ in the same way by one Pythagorean comma. Scales with seven accidentals are seldom used, because the enharmonic scales with five accidentals are treated as equivalent.

See also
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