114edo Tweak

In yesterday's post I mentioned that the scale there could probably get straightened out a bit, so that's what I did. The simpler structure has a simpler description, too! The scale is built of two similar sequences generated by the interval of 5 steps of 114edo. The pitch classes in the scale thus go 0, 5, 10, 15, ..., 55, and then 57, 62, 67, 72, ..., 112. So there are two intervals of 2 steps seperating the two sequences of 5 steps. The tonnetz diagram above looks much the same as yesterday's, but just a little more symmetric.

I made a new piece with this tweaked scale: 114edo 24b. I used a somewhat different process to make this. Both today's and yesterday's piece use my usual thermodynamic algorithm. Yesterday I started the simulation at a very high temperature and gradually cooled in until global order emerged spontaneously. The piece is a snapshot from around the transition temperature. In today's simulation, I just fixed the temperature at the transition temperature that I observed in yesterday's piece. I initialized the system with a traversal of the comma 245:243. This traversal is repeated eight times over the course of the piece.

114edo

I've been exploring diaschismic tunings, tunings that temper out the diaschisma comma, 2048:2025. 2025 = 25 * 81, so the comma gets tempered by making both perfect fifths and major thirds a bit sharp. A few days ago I looked at 90edo, which has a perfect fifth sharp enough to do all the work, leaving the major third very close to just. Today's exploration is 114edo, which has perfect fifths and major thirds equally sharp, sharing the workload.

Looking at the tuning errors of 114edo, in the upper left area one can see that 3:1 and 5:1, corresponding to perfect fifths and major thirds, are not so accurate, being sharp enough to temper out the diaschisma. Since they are equally sharp, minor thirds, 5:3, are quite precise. But what jumped out at me with this table of errors is that 114edo has a quite accurate approximation for 7:1. That should spice things up!

Staring at the tonnetz diagram for a while and contemplating 7:1 which corresponds to 92 steps of 114... well, two steps of 7:1 would be 70 steps, which comes back near the the starting point. This corresponds to the comma 245:243, which I see people call the minor Bohlen-Pierce diesis. I looked for a scale that would support traversals of this and the diaschisma. I came up with the 24 note scale highlighted above. This is the union of two scales generated by the interval of 62 steps of 114edo, the two scales offset by 67 steps. As I have thought more about this, probably offsetting the scales by 5 steps would have been better, but I got pretty far down the road with the 67 step offset, so that's what I have here.

The tonnetz diagram layout shows perfect fifths, moving from a cell to its neighbor on the right, and major thirds, moving from a cell to its neighbor above. I used colors to show movement by 7:1. Each blue cell can move to a purple cell by a 7:1 interval. So an example of a traversal of 245:243 would be to start at a (blue) 0 cell and leap to a (purple) 92 cell, moving with a 7:1 interval. Then move by two perfect fourths, stepping left two cells, to a (blue) 72 cell. Leap by 7:1 again to a (purple) 50 cell. From there, move left three cells, three perfect fourths, to a 77 cell, and then up one cell, a major third, which brings one back to a 0 cell where the traversal started.

Here's what the scale sounds like: 114edo scale 24.

90edo Diaschismic

I have posted some music in the past that uses 34edo, a tuning system that has very good approximations for conventional intervals like perfect fifths and major thirds. But 34edo does not temper out the syntonic comma 81:80; instead it tempers out the diaschisma 2048:2025. The diagram above is a tonnetz diagram for diaschismic tuning of the conventional 12 notes of a piano.

Conventional music, in the Palestrina - Wagner tradition, is built on tempering the syntonic comma, which is the foundation of meantone tunings. The tonnetz diagram for meantone tuning looks quite different:

There is a spectrum of meantone tunings, where the flatness of the perfect fifth is traded against the sharpness of the major third. 31edo, or quarter comma meantone, are at one end of the spectrum, where the major thirds are quite precise while the perfect fifth is rather flat. It occurred to me that the same sort of spectrum should exist for diaschismic tunings.

Perfect fifths are a bit sharper than just, in diaschismic tunings. There are two wolf fifths, D-A and Ab-Eb in the tonnetz diagram. These are flat, to make for the sharpness of most of the fifths. As the fifths are sharpened, most of the major thirds get flatter. When the fifths get to around 707 cents (versus the just fifth of 702 cents), the major thirds become just, at 386 cents. There are wolf major thirds in this tuning though, such as A-C# in the tonnetz diagram, that sharpen as the fifths sharpen.

It turns out that 90edo has fifths that are close to the value needed to make the major thirds very exact... so of course I had to see what it sounded like: 90edo scale 12.

13edo

To divide octaves into thirteen equal steps: thirteen is very close to twelve, and twelve is the conventional and excellent way to divide octaves, so it might seem that thirteen should work very well too.

Somehow, though, that's not how the math works! This is a table of tuning errors for 13edo: how far off it is from just intonation for a variety of intervals. The number in the upper left, 0.395, is the most fundamental error, the error for perfect fifths. A just tuned perfect fifth lies 39.5% of the way between two of the notes of 13edo. It could hardly be worse!

So, maybe we can try to make some music using intervals that 13edo approximates well. Still focusing on the region of simpler intervals toward the upper left, 9:5 and 11:1 are approximated quite well. Let's try using those as building blocks!

So here is an unconventional tonnetz diagram based on these two intervals. 9:5 is equivalent to 10:9 which is one form of a conventional whole step. 2 steps of 13edo form a interval a bit flatter than 2 steps of 12edo; 10:9 is a bit flatter than 9:8. So far, so good! In the diagram, moving right or left from a cell is moving by 10:9 or 9:5. Moving down is moving by 11:8; moving up is moving by 16:11. These intervals are not conventional at all! They are close to the dissonant tritone: 11:8 is a bit flatter; 16:11 is a bit sharper. Then again, 11:8 is a bit sharper than a perfect fourth, 4:3, and 16:11 is a bit flatter than a perfect fifth, 3:2. This is how 13edo works: it chops the pitch spectrum up in a very different way!

Cells labeled 0 occur in many places. A tonnetz diagram like this is really an unrolled torus. A path from one 0 cell to another 0 cell represents a loop on the torus, which is a traversal of a comma. Starting from one 0 cell, each other 0 cell represents a different comma, a just interval that is quite close to unison. There is a path 0, 2, 4, 6, 0 that corresponding to a traversal of the comma 8019:8000. Extending this path looks like a nice way to build a scale with six notes per octave.

Here's what it sounds like: 13edo steer

A Kind of Pentatonic

Here is a new piece in 36edo, using a scale with five notes per octave.

This is built from tempered intervals approximating 7:4, or seven steps of 36edo. This is very close to 5edo, i.e. dividing octaves into five equal intervals. Here four of the intervals are seven steps of 36edo, and the fifth intervals is eight steps of 36edo. In the diagram above, this eight step interval is omitted since it acts as a sort of wolf tone.

This scale supports traversal of the comma 1029:1024, i.e. moving by 8:7 three times brings one to a perfect fifth 3:2, folding octaves along the way. The red arrows in the diagram represent intervals of 7:4, and the green arrows are the 3:2 perfect fifths.

Conventionally Unconventional

Here is a new piece in the 31edo tuning system, using a 19 note scale. This scale is built from a chain of perfect fifths, so conventional notation is applicable.

31edo is very close to quarter-comma meantone; that, and a 19 note scale, were used in Renaissance times. But here I am exploiting intervals that approximate just tuned intervals involving the prime number 7. For example, 8:7 is very close to 6 steps of 31edo. This 19 note scale includes many 6 step intervals.

This diagram shows the primary intervals available in this 19 note scale. Green arrows are perfect fifth, blue arrows are major thirds, and red arrows are 7:4 intervals that don't fit well into conventional terminology. Rather than arranging the notes into a circle of fifth, I used a spiral of fifths that prioritizes chains of major thirds. The conventional diatonic scale can be arranged this way, too:

12 Note Scale in 31edo

Here is a new piece in 31edo, using a conventional 12 note scale built from a chain of perfect fifths. 31edo has flatter fifths than conventional 12edo, which makes available a reasonable accurate 7:4 interval.

The green arrows here are perfect fifths, the blue arrows are major thirds, and the red arrows are 7:4 intervals which are not conventional diatonic intervals. These red arrows create many new comma traversals, e.g. Eb - G - D - A -C# - Eb.