[P1, m2, M2, m3, M3, P4, P5, m6, M6, m7, M7, P8]
I've also argued at some length that rank-3 intervals and higher should be justly tuned so that the natural impure intervals
[m2, M2, m3, M3, m6, M6, m7, M7]
have 5-limit frequency ratios, rather than 3-limit (Pythagorean) frequency ratios.
The obvious next step is to give conditions for when an EDO is diatonic over rank-3 intervals, in the sense of putting the rank-3 natural intervals in the same usual order that we know and love from 12-TET.
Surprisingly, there's only one EDO that badly violates rank-3 diatonicity. When we tune
[P1, m2, M2, m3, M3, P4, P5, m6, M6, m7, M7, P8]
in 11-EDO we get these steps:
[0, 1, 3, 2, 4, 5, 6, 7, 9, 8, 10, 11]
in which (3 comes before 2) and (9 comes before 8). All of the other EDOs between 5-EDO and 500-EDO are non-decreasing from left to right in their tunings of the natural rank-3 intervals.
Also a little surprising to me is that most EDOs tune the natural rank-3 intervals to distinct steps. Obviously everything below 11 has too few steps to put all 12 natural intervals on differing steps, so those EDOs are indistinct over rank-3 natural intervals:
5-EDO: [0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5]
6-EDO: [0, 0, 0, 2, 2, 2, 4, 4, 4, 6, 6, 6]
7-EDO: [0, 1, 1, 2, 2, 3, 4, 5, 5, 6, 6, 7]
8-EDO: [0, 0, 1, 2, 3, 3, 5, 5, 6, 7, 8, 8]
9-EDO: [0, 1, 2, 2, 3, 4, 5, 6, 7, 7, 8, 9]
10-EDO: [0, 1, 1, 3, 3, 4, 6, 7, 7, 9, 9, 10]
The only other indistinct EDOs are:
13-EDO: [0, 1, 1, 4, 4, 5, 8, 9, 9, 12, 12, 13]
14-EDO: [0, 1, 3, 3, 5, 6, 8, 9, 11, 11, 13, 14]
17-EDO: [0, 2, 2, 5, 5, 7, 10, 12, 12, 15, 15, 17]
20-EDO: [0, 2, 2, 6, 6, 8, 12, 14, 14, 18, 18, 20]
Seeing that 17-EDO tunes the rank-3 minor second and major second together, as well as the minor third with the major third, was a bit of a surprise. With rank-2 intervals, 17-EDO is very well behaved, giving us these steps for the natural intervals:
17: [0, 1, 3, 4, 6, 7, 10, 11, 13, 14, 16, 17]
So here's a condition for rank-3 lax diatonicity: "Don't be 11-EDO".
And here's a condition for rank-3 distinction, which we might require for "strict diatonicity": "Be larger than 20-EDO or find yourself in the set (12, 15, 16, 18, 19)".
I can't help but feel that the rank-2 conditions were a lot harder to figure out, a lot mathier. Still, I'm happy to have made some progress.
Nothing I've said ensures that these well behaved EDOs don't collapse to a smaller EDO: for example, 24-EDO only tunes rank-3 intervals to even steps, equivalent to 12-EDO. I'm still figuring out the conditions for ... full occupancy of the EDO at a given rank.
Maybe I could figure out some rule that lets us see why [13, 14, 17, 20]-EDO are indistinct next.
It the EDO tunes the rank-3 A1 to 0 steps, then the scale is indistinct, but that's not necessary for it to be indistinct. If the EDO tunes the rank-3 d2 to -1 steps, then the scale is indistinct, but again, not necessary. If AcA1 is tuned to zero steps and d2 is tuned to zero steps, then it's indistinct. That literally only described 9-EDO but it's the last edge case. Man, I don't know.
...
I propose that these guys:
[9, 11, 14, 16, 21, 23, 28, 33, 35, 40, 47, 52, 64]-EDO
are secretly not so well behaved over rank-3 intervals. Those are the EDOs which tune the Acute Unison to a negative number of steps. Like a bunch of a noobs.
...
Let's talk about tuned orders of the once-modified rank-3 intervals that are induced by different EDOs.
There are very few EDOs that tune the augmented unison to zero steps: the full set between 5 and 100 is [6, 7, 10, 13, 17, 20]-EDO.
6-EDO, over the rank-3 intervals, reduced to 3 edo, and has this order:
[A1=A2=M2=P1=d2=m2, A3=A4=M3=P4=d3=d4=m3, A5=A6=M6=P5=d5=d6=m6, A7=M7=P8=d7=d8=m7]
The other members of the set all have this order:
[A1=P1, A2=M2=d2=m2, A3=M3=d3=m3, A4=P4=d4, A5=P5=d5, A6=M6=d6=m6, A7=M7=d7=m7, P8=d8]
Within an equivalence set connected by "="s, I have things sorted alphabetically, but it might be easier to see that e.g.
dd2 = d2 = m2 = M2 = A2 = AA2
The number of augmentations and diminutions doesn't affect the tuning of that interval since A1 is tempered out.
From 5-EDO up to 100-EDO, there are 22 different orders of the intervals [d2, P1, m2, A1, d3, M2, m3, A2, d4, M3, P4, A3, d5, A4, d6, P5, m6, A5, d7, M6, m7, A6, d8, M7, P8, A7], which I'll call the once modified intervals, although we could also modify intervals with acuteness and gravity, instead of just augmentation and diminution. Most of these are slightly degenerate orders in which two intervals are tuned to the same step. The only intervals that are not at all degenerate over the once-modified:
1. [P1, A1, d2, m2, M2, A2, d3, m3, M3, A3, d4, P4, A4, d5, P5, A5, d6, m6, M6, A6, d7, m7, M7, A7, d8, P8]
2. [P1, d2, A1, m2, M2, A2, d3, m3, M3, d4, A3, P4, A4, d5, P5, d6, A5, m6, M6, A6, d7, m7, M7, d8, A7, P8]
3. [P1, d2, A1, m2, M2, d3, A2, m3, M3, d4, A3, P4, A4, d5, P5, d6, A5, m6, M6, d7, A6, m7, M7, d8, A7, P8]
...
Order 1 is shared by [26, 29, 32, 45, 48, 51, 54]-EDO.
Order 2 is shared by [31, 43, 46, 47, 50, 55, 58, 61, 62, 65, 67, 69, 70, 73, 74, 77, 80, 81, 84, 86, 88, 89, 92, 93, 95, 96, 98, 99, 100]-EDO.
Order 3 is shared by [37, 56, 59, 71, 75, 78, 90, 94, 97]-EDO.
Now, Order 1 is the same as what I called tetracot ordering in the post on rank-2 orderings of once modified intervals. Order 2 is the meantone ordering. And order 3 is new, but it's really really close to the meantone ordering. It only swaps (d3 with A2) and (d7 with A6).
All three of these orders are strict in the sense of being ordered by ">". If we reinterpret them as being laxly ordered by ">=", what other EDOs can we describe as falling into these orders, and are there any other orders we'll need to describe the EDOs that are degenerate in the tuned orders they induce over rank-3 once modified intervals?
There's a great degenerate order I've notice
[P1, A1=d2, m2, M2, A2, d3, m3, M3, A3=d4, P4, A4, d5, P5, A5=d6, m6, M6, A6, d7, m7, M7, A7=d8, P8]
that's shared by [22, 25, 41, 44, 60, 63, 66, 79, 82, 85]-EDO. I think is is consistent with both the lax version of order1 and the lax version of order 2, in that this degenerate order equates all of the pairwise swaps from order 1 to order 2.
So now I'm curious if there's an EDO order that equates (d3 with A2) and (d7 with A6), which would of course be described by both lax order 2 and lax order 3.
Past 12 divisions, no EDO tuned multiple once-modified intervals to the same step.
I did this in kind of lazy way with text editing instead of programming, but these guys with degenerate orders over once modified rank-3 intervals are consistent with lax order 1:
[19, 22, 25, 38, 41, 44, 57, 60, 63, 66, 76, 79, 82, 85]-EDO
and maybe others as well. It's possible for a list to be consistent with multiple lax orders. These guys are consistent with lax order 2:
[15, 18, 30]-EDO
at least.
And these guys are consistent with lax order 3 at least:
[12, 24, 27, 28, 36, 39, 40, 42, 52, 64]
Doing that same cheap textual analysis in a slightly different way also tells me that these guys are consistent with lax order 2:
[22, 25, 41, 44, 60, 63, 66, 79, 82, 85]-EDO
and these guys are consistent with lax order 3:
[19, 35, 38, 57, 76]-EDO
.
So these EDOs are both ordered by lax order 1 and lax order 3:
[19, 38, 57, 76]-EDO
And maybe others should be here and maybe some of these are also lax order 3.
These guys are ordered by lax order 1 and lax order 2, at least:
[22, 25, 41, 44, 60, 63, 66, 79, 82, 85]
...
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