Pythagorean Comma

What is the Pythagorean Comma?

There are several ways to explain the Pythagorean comma. In a nut shell you cannot tune a circle of perfect 5ths and end up where you started.

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Start from C and tune perfect 5ths all the way around to B#. You will find that C and B# are not in tune.

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A perfect 5th is 702 cents.

702+702+702+702+702+702+702+702+702+702+702+702= 8424 cents

An octave is 1200 cents.

1200+1200+1200+1200+1200+1200+1200= 8400 cents

8424 - 8400 = 24 cents = Pythagorean Comma

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The frequency for the lowest C on the piano is around 32.7 Hz.

A perfect 5th is a 3:2 ratio or 1.5

32.7 * 1.5 = 49.05 (G)
49.05 * 1.5 = 73.575 (D)
73.575 * 1.5 = 110.362 (A)

Carry this all the way out to B# and you get 4242.705 Hz

An octave is a 2:1 ratio or 2

32.7 * 2 = 65.4 (C)
65.4 * 2 = 130.8 (C)
130.8 * 2 = 261.6 (C)

Carry this all the way out to C and you get 4185.6
Obviously 4185.6 and 4242.7 are not the same frequency.

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Start at middle C and tune perfect 4ths and 5ths in both directions. Stay in the FF octave around Middle C. Leave the pythagorean Comma between G# and Eb.

C - G - D - A - E - B - F# - C# - G# * Eb - Bb - F - C

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Tuning and Temperaments Early Music Institute School of Music IU Bloomington

I welcome any comments or suggestions, email me at psloffer@indiana.edu