The Circle of Fifths and the
Pythagorean Comma

Having seen that the Pythagoreans derived scales by building up stacks of perfect fifths and then octave reducing, there would seem to be a natural connection between this "fifth building" and our contemporary diagram of the circle of fifths:

We have learned to look at this neat and seemingly perfect diagram and appreciate that musical relationships seems to move in a circle and that you can go around the circle and end up back at the starting place: begin at C, move forward by perfect fifths 12 times, and end up back at C.

How wonderful!

Except for one problem: it's a lie! The numbers don't actually work out. Let's see why.

The Pythagorean Comma

According to Pythagorean principles, one could continue to go up by fifths and derive a series of discreet pitches through continuing this process as far as possible.

On the modern piano keyboard, if we start on the lowest C and go up, labeling all of these as sharps, we'd end up on the highest note of the piano, calling it a B#:

We could also start on middle C and go outwards in both directions, ending up at the higher end of the keyboard on an F#, and at the lower part of the keyboard on a Gb.

In modern equal temperament, we think of these notes as enharmonic equivalents, but when we actually do the math on these, using only perfect fifths, we discover that:

B# does not equal C and
Gb does not equal F#!

This is called the Pythagorean Comma.

The Pythagorean Comma is most easily expressed by stating that no power of two can equal any power of 3.

If take a very low tone, represented by the lowest C on the piano and assign that the value of 1, and go up to the top C on the piano, we encompass 7 octaves, and since octaves are expressed as the ratio of 2:1, we find ourselves with the number 128, which is 2 to the seventh. If we go back to that low C and play up by perfect fifths on the keyboard, we of course go up the Cycle of Fifths...

...which for the purposes of Pythagorean thought should be notated as all sharp notes, ending on B#. In equal temperament, of course, the B# becomes the enharmonic equivalent of C natural, but in Pythagorean tuning we need to think in all perfect fifths. Thus, we must go up in perfect fifths 12 times. This would be 3/2 raised to the 12 power (1.5 to the 12th). This results in the number 129.74632.

This ratio, 129.75632 : 128 is the Pythagorean comma. 12 perfect fifths do not equal up to 7 perfect octaves:

(3/2)^12 ≠ (2/1)^7 or you could say (3/2)^12 / (2/1)^7 ≠ 1.

As Gareth Loy says in his great book Musicathics: "Contrary to the wishes of scale builders and musicians from antiquity to the present, the powers of the integer ratios 3/2 and 2/1 do not form a closed system."

This is the basic process in tuning a piano in equal temperament, where each fifth is reduced by a twelve of a semitone, thereby distributing out the comma equally among the twelve pitches of the chromatic scale.

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