Big-Small

A comma in musical tuning is a ratio made of small primes which is very close to 1. Two classic commas are the syntonic comma, 81:80, and the Pythagorean comma, 531441:524288. These are quite similar in size. The ratio between them is even closer to 1, the comma known as the schisma, 42467328:42515280.

Commas are important in music because consonant intervals have frequency ratios built from simple primes. Combining consonant intervals then generates more complex ratios that are still built from simple primes. Commas thus correspond to combinations of consonant intervals that are very close to unison. This closeness has potential to cause trouble and potential also to cause delight; in any case, managing this closeness is an important musical task. The main tool for this is temperament, adjusting intervals slightly so that when they are combined the result is never awkwardly just slightly different than unison: it is either exactly unison, or distinctly different.

Conventional tuning, twelve tone equal temperament, tempers intervals so that the syntonic comma and the Pythagorean comma both vanish, i.e. the corresponding combinations of tempered consonant intervals results in exact unison. But there are other musical possibilities!

Here is a new piece: 612edo scale 82.

This is in the tuning system that divides octaves into 612 equal steps. Any tuning with such small steps will be extremely precise. 612edo is one of the most precise, for intervals like perfect fifths and major thirds, among other tuning systems with similarly small steps. I am using it here, though, to explore commas. The Pythagorean comma is 12 steps of 612edo; the syntonic comma is 11 steps. Thus the schisma, the difference between these, is 1 step. 612edo is so precise that it does not temper out the usual commas.

I don't know a name for this comma:

450359962373049600:450283905890997363

but 612edo tempers it out! Factored into primes, this is 2^54 * 5^2 : 3^37. It is about 0.3 cents off of unison.

This piece uses a scale with 82 notes per octave. A perfect fifth is 358 steps of 612edo; the octave and the perfect fifth have a greatest common divisor of 2, which means that there are two cycles of fifths. The scale I used here is made of sequences of 41 perfect fifths, one for each cycle of fifths.

This piece traverses this big-small comma 25 times.