Pythagorean Comma - Circle of Fifths and Enharmonic Change

Circle of Fifths and Enharmonic Change

The Pythagorean comma can also be thought of as the discrepancy between twelve justly tuned perfect fifths (ratio 3:2) ( play) and seven octaves (ratio 2:1):

\frac{\hbox{twelve fifths}}{\hbox{seven octaves}} =\left(\tfrac32\right)^{12} \!\!\Big/\, 2^{7} = \frac{3^{12}}{2^{19}} = \frac{531441}{524288} = 1.0136432647705078125 \!
Ascending by perfect fifths
Note Fifth Frequency ratio Decimal ratio
C 0 1 : 1 1
G 1 3 : 2 1.5
D 2 9 : 4 2.25
A 3 27 : 8 3.375
E 4 81 : 16 5.0625
B 5 243 : 32 7.59375
F♯ 6 729 : 64 11.390625
C♯ 7 2187 : 128 17.0859375
G♯ 8 6561 : 256 25.62890625
D♯ 9 19683 : 512 38.443359375
A♯ 10 59049 : 1024 57.6650390625
E♯ 11 177147 : 2048 86.49755859375
B♯ (≈ C) 12 531441 : 4096 129.746337890625
Ascending by octaves
Note Octave Frequency ratio
C 0 1 : 1
C 1 2 : 1
C 2 4 : 1
C 3 8 : 1
C 4 16 : 1
C 5 32 : 1
C 6 64 : 1
C 7 128 : 1

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.


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, reconciled this by flattening each fifth by a twelfth of a Pythagorean comma (approximately 2 cents), thus producing perfect octaves.

Another way to express this is that the just fifth has a frequency ratio (compared to the tonic) of 3:2 or 1.5 to 1, whereas the seventh semitone (based on 12 equal logarithmic divisions of an octave) is the seventh power of the twelfth root of two or 1.4983... to 1, which is not quite the same (out by about 0.1%). Take the just fifth to the twelfth power, then subtract seven octaves, and you get the Pythagorean comma (about 1.4% difference).

Read more about this topic:  Pythagorean Comma

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