The Pythagorean Comma

"Rook di goo, rook di goo!
There's blood in the shoe.
The shoe is too tight,
This bride is not right!"

from Cinderella, by Jakob and Wilhelm Grimm

Like the ugly sisters of Cinderella, through history musicians around the globe have been cutting the heels and toes of natural musical intervals to make them fit the glass (or gold) slipper – the impossible notion of a perfect tonal system.

The Pythagorean Comma has its origin in the Pythagorean Tuning System, which is generated by 12 successive perfect fifth intervals (ratio 2:3). But the span of 12 perfect fifths exceeds the span of 7 octaves (each of a ratio 1:2) by a tiny interval, about 1/4 of a semitone – the Pythagorean Comma – leading to the necessity of tempered tuning systems.

As long as we stay in the world of drones and their harmonic overtones everything is relatively simple: From a mathematical perspective it is all a game of the first 20 integers or so and the proportions between them. But as soon as we also include melody and subsequently harmony, the whole picture becomes more complex, and we have to realize that the most fundamental functions from the harmonic series don't meet accurately.

The most fundamental structure in music is the division of the octave (ratio 1:2) in tonal intervals generated by the perfect fifth (ratio 2:3)..
In the modern equal temperament (12-TET) all semitone steps are mathematically equal (arriving to the following frequency by multiplying by the twelfth root of 2) but in this arrangement the just intervals have been sacrificed – most notably the thirds and sixths, which at modern keyboard instruments are around 1/7 of a semitone off pitch.
The reason for this is that the structure originally derives from the primary elements of the harmonic series, the octave, 1:2, and the perfect fifth, 2:3.

The span of 7 octaves – 2 to the power of 7 – is almost the same as the span of 12 perfect fifths – (3/2) to the power of 12 – but if you calculate the values you will come to 128 and 129.746 respectively.
So something needs to give way when the 12 fifth-generated halftones are arranged within the frames of an octave in a tonal system.

In the video the tonal points have been arranged within a circular frame, which gives an immediate impression of the interval sizes. The 12TET tonal lines are partly hidden underneath the spiral structure. This is an equal division of the circle like the full hours of the clock, but please note, that the ear has other criteria of harmony. The tonal lines of The Pythagorean Tuning is on top of spiral.
One winding, 360 degrees of the spiral equals one octave, ratio 1:2.

The equal tempered semitone equals 100 cent (30 degrees of a circle) and the octave is 1,200 cent (one turn of the circle, 360 degrees). The Pythagorean Comma equals 23.46 cents (7.04 degrees) and is the result of the accumulated inconsistency caused by the 1.96 cent (0.59 degrees) difference between a perfect fifth (701.96 cent, 210.59 degrees) and the fifth of 12TET (700 cent).


Please notice, that the purpose of the keyboard in the video is only to illustrate, as the spiral/circle representation of the octave is unknown to the majority. In reality, it is not – as it is now hopefully clear – impossible that the key of the seventh octave has the same frequency as the twelfth perfect fifth!