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Pythagorean Comma

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Although musical notes can be calculated mathematically, doing that creates notes which are slightly off pitch.

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.

 


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