Interval Cost Function

Here is a new piece: 53edo nxd.

This uses the same scale that I showed in yesterday's post.

For this new piece, I changed the interval cost function. This table shows a large part of the cost function. The composition algorithm prioritizes intervals with low cost. Intervals of the same number of half steps follow diagonals from upper left to lower right. The uppermost such diagonal shows the cost for the half step interval. In an equal tempered scale, these half step intervals would all have the same cost. In this scale, however, the interval from C to Db is not exactly the same size as the interval from Db to D, and so their costs differ.

Marvel Piano

Here's a new piece: 53edo 7x7x7

This is in a 12 note per octave scale that would work quite nicely on a piano!

53edo has much more accurate major thirds than those of conventional 12edo. Moving from C to A# in this tuning moves through two major thirds. This makes the A# quite a bit flatter than conventional tuning. The interval from C to A# is a very sharp approximation of 7:4 in conventional tuning. With this 53edo scale, the 7:4 is just 5 cents off!

Traversing Marvel

Here is a new piece of music: 53edo scale 10

This piece is in a ten note per octave scale in the 53edo tuning system. I designed this scale to support traversing the marvel comma, 225:224. This scale allows using conventional note names because it doesn't distinguish any commas that conventional 12edo tempers out. It would be straightforward to extend this scale to the full twelve notes of a conventional scale: D and A would fit naturally into this structure. My goal here was more about traversing the marvel comma; matching a conventional scale was not on my mind.

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.

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.