Pythagorean Comma

Pythagoras Picture

The Perfect Fifth is the lowest ratio harmonic you hear and is mathematically calculated by multiplying a frequency the ratio by 3/2 or 1.5. For example, in the note of C, the perfect fifth would be a G. In harmonic terms, the perfect fifth is one octave below the third harmonic.

Pythagorous of Samos (c.582 - c.507 B.C.) discovered that you could make a musical scale by continuing through the Circle of Fifths, and dividing down harmonically with The Law of Octaves to determine the pitch for each note.

There is a distinct problem in this procedure, however. It does not add up correctly but leaves a small residual error that has been the frustration and bane of musicians ever since.

It works out like this. Say we start at C at 256 Hz. G would be 1.5 x C or 384. D would be 1.5 x 384 or 576. This continues as follows:

C > G > D > A > E > B > F# > C# > G# > D# > A# > F > C.

C 256 > G 384 > D 576 > A 864 > E 1,296 > B 1,944 > F# 2,916 > C# 4,374 > G# 6,561 > D# 9,841 > A# 14,762 > F 22,143 > C 33215

Then utilize the Law of Octaves to divide down C at 33,215 Hz by 2 until you obtain 259.5 Hz. The difference between C at 256 Hz, and C at 259.5 Hz is known as the Pythagorean Comma. It works out to the ratio of about 74/73 or 743/733. There is another comma defined in music, the comma diesis which is equal to about 81/80 or 5.4 savarts.

This small excess ratio means that a music scale cannot be completely harmonic with regard to octaves and with regard to the interval of a fifth since this musical error must reside somewhere in the scale. The question is- where?

Thousands of musical scales have been invented to reduce the effects of this error. There is more on the subject on our page about Musical Scales.