Simplicity

Here's a new piece in 31edo: 31edo scale 9.

This nine note scale is intended to channel the composition algorithm into producing a traversal of the comma 126:125. I can't find any structure like that in the score, or in any of the scores I generated in my various trials. Something is not working as anticipated! One possibility is that this traversal is just too simple, i.e. too easy to jostle away. Another detail I suspect might be causing trouble is the new interval scoring formula I have been using. It scores 7:6 as more consonant than 8:7. This may be providing a downhill pathway to breaking apart the traversal.

Traversal or no, this piece sounds nice, so I thought I should share it. Plus, there is almost always something to be learned from failure!

It's Called Blackjack!

Here is a new diagram for the 21 note scale I posted this morning. I learned a lot about the scale from the effort to untangle the graph!

And here is a new piece... less austere! 41edo blackjack

The kind folks in the facebook microtonal community let me know: this scale is known as blackjack!

21 out of 41

Here is a new piece in 41edo: 41edo scale 21.

The diagram above shows the scale used here. Green arrows are perfect fifths, blue arrows are major thirds, and red arrows are 7:4. The scale is generate by the 4 step interval, which corresponds to a semitone 16:15. A curious feature of this tuning is that six semitones make a perfect fifth, instead of the usual seven.

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.