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.

171edo tertiaseptal 31

Here's a new piece in 171edo. The scale here has 31 notes per octave, spaced rather evenly: 5 or 6 steps of 171edo between the notes of the scale. I learned about this scale from a discussion on reddit.

This scale supports traversal of the commas 2401:2400 and 65536:65625. Poking around on the web, I found an email where these commas are discussed and the name tertiaseptal is given to a class of scales that support traversing these commas.

Here is a chart showing tuning errors for 171edo. The numbers on the left and upper margins are the numerators and denominators of ratios for intervals in just intonation. The cells in the center show the tuning error of 171edo, in terms of steps of 171edo. So the maximum error possible is 0.5, where the just interval falls exactly between two steps of 171edo. This chart shows that ratios involving 11 are not approximated very well in 171edo! So for this piece I was prioritizing intervals that approximate ratios involving the primes 2, 3, 5, and 7.

72edo

Here's a new piece in 72edo. This uses the scale I mentioned a couple days ago, 21 notes per octave generated by the 7 microstep that approximates the semitone 16:15. Conventionally a perfect fifth corresponds to 7 semitones. But if you compare the just tuned perfect fifth, 3:2, to the just tuned semitone, 16:15, the perfect fifth corresponds to 6.28 semitones. In 72edo, the perfect fifth corresponds to 6 semitones. The post here from a couple days ago gets into comma traversals supported by this scale.

This piece is generated by my usual thermodynamic process, where order spontaneously emerges at a phase transition that is marked by a sudden drop in energy or cost:

This new piece is a snapshot of the system at a temperature around 10000000, right at the phase transition.

Loop Search

Recently I was exploring the 270edo tuning system, and a 90 note scale. I wanted a symmetric scale, which this subset is. I also wanted a scale that would traverse the comma 2080:2079, which includes the primes 2, 3, 5, 7, 11, and 13. I was wondering whether that scale might support traversal of other commas, too. This is a lot of notes in a space of rather high dimensionality! So I wrote a program! I can switch the set of primes, the tuning system, and the scale, all very easily, to explore the possibilities for comma traversals.

The program looks at loops for each note in the scale: sometimes the loops that pass through one note are different than those that pass through a different note. Experimenting with this program, I see that happening sometimes. For this particular 90 note subset of 270edo, each note has the same set of loops:

• | 12 -2 -1 -1 0 -1 > = 4096:4095
• | -7 -1 2 0 -1 2 > = 4225:4224
• | -10 -1 -1 1 0 3 > = 15379:15360
• | 2 1 -1 -3 1 1 > = 1716:1715

These strings of numbers are the powers of the primes 2, 3, 5, 7, 11, and 13, so they give the prime factorization of the ratios. These commas can be combined arbitrarily to form a whole linear space of commas that can be traversed with this scale. My new program looks for a minimal basis for this space.

This 90 note subset is symmetric: the 270 notes of the tuning system are divided into 10 blocks of 27 notes each. The scale picks out 9 notes from each block, at the same positions in each block. I ran the program with only 8 notes per block: then the sets of commas traversable had a basis of three instead of four commas. When I increased the number of notes per block, some notes had a basis set of five commas. So 9 notes per block does seem like a nice threshold.

Another scale I have explored at some point - I don't remember when! - is a 21 note scale in 72edo, generated by the semitone 16:15. 72edo works well with primes 2, 3, 5, 7, and 11, but not with 13. So I ommitted 13 in this analysis. Again, all the notes in the scale had the same space of commas traversable:

• | -10 1 0 3 0 > = 1029:1024
• | -5 2 2 -1 0 > = 225:224
• | -7 -1 1 1 1 > = 385:384