The Other Side of 612edo

When a tuning system tempers out a set of commas, it will temper out any linear combination of that set... assuming that the tuning system includes linear combinations of any subset of its intervals. Here "linear" means integer multiples. So e.g. two perfect fifths makes an octave plus a major second, etc. Half of a perfect fifth doesn't really make so much sense musically. So the natural mathematical structure for these sorts of tuning systems is a module. Someday maybe I will learn more about modules. So far I have just got the name straight! The set of commas tempered out by these tuning systems form a submodule. One can find basis sets for these submodules, i.e. a set of commas where every comma tempered out by the tuning system is an integer combination of the commas in the basis set. A couple days ago I posted here one comma tempered out by 612edo. Here is another comma tempered out by 612edo; with that earlier comma, these form a basis set for the (5-limit) commas tempered out by 612edo.

9010162353515625:9007199254740992 = 3^10 * 5^16 : 2^53

This comma is about 0.57 cents, i.e. extremely small. This is a reflection of the precision of the 612edo tuning system.

Here is a new piece of algorithmic music that traverses this comma 36 times: 621edo scale 52.

I used a scale for this that has 52 notes per octave. This scale has a period of 306 steps of 612edo, i.e. the scale pattern repeats twice in an octave, with 26 notes in each repetition. The scale was generated by the interval 83\612, which corresponds to 1125:1024. The scale has step sizes of 5\612 and 21\612. I came up with this scale just by staring at the tonnetz diagram for 612edo to see what might work!

Each cell represents a pitch class of 612edo. Moving one cell to the right is moving up a perfect fifth, e.g. from pitch class 0 to pitch class 358. Moving up a cell is moving up a major third, e.g. from 0 to 197. Moving up three major thirds does not return one to the starting pitch class, e.g. from 0 one moves to 197, then 394, then 591. 591 is 21\612 flatter than the starting 0. This reflects the fact that 612edo does not temper out the diesis.

The notes of the scale are highlighted in this diagram. It's easy to see that they form a path from one occurrence of the pitch class 0 to another instance of the same pitch class. A more accurate geometrical representation of this tuning system would be a torus, wrapping this diagram around in two directions so the pitch class occurences would fold back on themselves... more accurate, but less easy to see!

A graphical score for the piece shows the structure of the traversal:

This score folds the actual score in two ways. All the pitches are folded into a single octave, so the pitches along the vertical axis run from 0 to 611; and all the traversals are folded into a single traversal, so time along the horizontal axis runs from 0 to 78 seconds, the length of each traversal.