43-EDO can be defined over rank-3 interval space by a pure octave and tempering out the acute unison, justly tuned to 81/80, and the Grdddddd4, justly tuned to 50331648/48828125.
If we write these interval in rank-3 prime harmonic coordinates they are [[1, 0, 0], [-4, 4, -1], [24, 1, -11]]. We might call this an EDO basis.
This matrix isn't unimodular: it has determinant -43, from which we can infer that it defines 43 EDO.
What comma would we have to use in place of the octave to get a unimodular basis? There are tons of options. If we restrict ourself to intervals with just tunings between 1 and 2, we still have tons of options. Here are some options sorted by increasing numerator size:
32805/32768 _ [-15, 8, 1]
393216/390625 _ [17, 1, -8]
531441/524288 _ [-19, 12, 0]
1990656/1953125 _ [13, 5, -9]
10077696/9765625 _ [9, 9, -10]
43046721/41943040 _ [-23, 16, -1]
51018336/48828125 _ [5, 13, -11]
258280326/244140625 _ [1, 17, -12]
3486784401/3355443200 _ [-27, 20, -2]
10460353203/9765625000 _ [-3, 21, -13]
847288609443/781250000000 _ [-7, 25, -14]
68630377364883/62500000000000 _ [-11, 29, -15]
2567836929097728/2384185791015625 _ [29, 14, -22]
12999674453557248/11920928955078125 _ [25, 18, -23]
65810851921133568/59604644775390625 _ [21, 22, -24]
333167437850738688/298023223876953125 _ [17, 26, -25]
1686660154119364608/1490116119384765625 _ [13, 30, -26]
It seems obvious to me to use the simplest one in terms of frequency ratio. That interval, [-15, 8, 1], justly tuned to 32805/32768, is affectionately called the "schisma" in music theory. If we combine it with our other two commas we get a unimodular matrix:
[[-15, 8, 1], [-4, 4, -1], [24, 1, -11]]
which we could use to define any rank-3 interval in integer coordinates. The inverse of this unimodular matrix is:
[-43, 89, -12],
[-68, 141, -19],
[-100, 207, -28]
The columns of this matrix, up to a factor of -1, are the tunings of [P8, P12, M17] in [43, 89, and 12]-EDO. And those are justly tuned to the first three prime harmonics.
There's more structure here though.
12-EDO tempers out the schisma and Ac1, but tuns [24, 1, -11] to -1 step.
43-EDO tempers out Ac1 and [24, 1, -11] but tunes the schisma to -1 step.
89-EDO tempers out the schisma and [24, 1, -11], but tunes Ac1 to 1 step.
So ... are these three special friends? I don't know. I kind of doubt it, but let's find some other relations like these and see.
Here are some EDO bases with the octave replaced with whatever simple interval will produce a unimodular matrix:
"12-EDO": [[2, 9, -7], [-4, 4, -1], [7, 0, -3]],
"13-EDO": [[1, -2, 1], [-3, -1, 2], [9, -7, 1]],
"14-EDO": [[-4, 4, -1], [0, 3, -2], [11, -4, -2]],
"15-EDO": [[11, -4, -2], [7, 0, -3], [1, -5, 3]],
"16-EDO": [[-4, 4, -1], [-7, 3, 1], [3, 4, -4]],
"17-EDO": [[-4, 4, -1], [-3, -1, 2], [12, -9, 1]],
"18-EDO": [[1, -5, 3], [7, 0, -3], [5, -6, 2]],
"19-EDO": [[-3, -1, 2], [-4, 4, -1], [-10, -1, 5]],
"21-EDO": [[0, 3, -2], [7, 0, -3], [-11, 7, 0]],
"22-EDO": [[-3, -1, 2], [1, -5, 3], [11, -4, -2]],
"23-EDO": [[-4, 4, -1], [-7, 3, 1], [-1, 8, -5]],
"25-EDO": [[1, -5, 3], [8, -5, 0], [-10, -1, 5]],
"26-EDO": [[-3, -1, 2], [-4, 4, -1], [-13, -2, 7]],
"27-EDO": [[11, -4, -2], [7, 0, -3], [5, -9, 4]],
"28-EDO": [[7, 0, -3], [3, 4, -4], [-11, 7, 0]],
"29-EDO": [[-10, -1, 5], [1, -5, 3], [-14, 3, 4]],
"31-EDO": [[-15, 8, 1], [-4, 4, -1], [17, 1, -8]],
.
Let's take the unimodular matrices, invert, transpose, remove negative signs, and interpret as tunings of prime harmonics.
Here's the transpose of the inverse:
12-EDO [[-12, -19, -28], [27, 43, 63], [19, 30, 44]]
13-EDO [[13, 21, 30], [-5, -8, -11], [-3, -5, -7]]
14-EDO [[-14, -22, -33], [12, 19, 28], [-5, -8, -12]]
15-EDO [[-15, -24, -35], [22, 35, 51], [12, 19, 28]]
16-EDO [[-16, -25, -37], [12, 19, 28], [7, 11, 16]]
17-EDO [[17, 27, 39], [5, 8, 12], [7, 11, 16]]
18-EDO [[-18, -29, -42], [-8, -13, -19], [15, 24, 35]]
19-EDO [[19, 30, 44], [3, 5, 7], [-7, -11, -16]]
21-EDO [[21, 33, 49], [-14, -22, -33], [-9, -14, -21]]
22-EDO [[22, 35, 51], [-10, -16, -23], [7, 11, 16]]
23-EDO [[-23, -36, -53], [12, 19, 28], [7, 11, 16]]
25-EDO [[-25, -40, -58], [22, 35, 51], [15, 24, 35]]
26-EDO [[26, 41, 60], [3, 5, 7], [-7, -11, -16]]
27-EDO [[-27, -43, -63], [34, 54, 79], [12, 19, 28]]
28-EDO [[28, 44, 65], [-21, -33, -49], [12, 19, 28]]
29-EDO [[-29, -46, -67], [19, 30, 44], [22, 35, 51]]
31-EDO [[-31, -49, -72], [65, 103, 151], [-12, -19, -28]]
From which we learn that
12-EDO is friends with: 27 and 19 EDO
13-EDO is friends with: 5 and 3 EDO
14-EDO is friends with: 12 and 5 EDO
15-EDO is friends with: 22 and 12 EDO
16-EDO is friends with: 12 and 7 EDO
17-EDO is friends with: 5 and 7 EDO
18-EDO is friends with: 8 and 15 EDO
19-EDO is friends with: 3 and 7 EDO
21-EDO is friends with: 14 and 9 EDO
22-EDO is friends with: 10 and 7 EDO
23-EDO is friends with: 12 and 7 EDO
25-EDO is friends with: 22 and 15 EDO
26-EDO is friends with: 3 and 7 EDO
27-EDO is friends with: 34 and 12 EDO
28-EDO is friends with: 21 and 12 EDO
29-EDO is friends with: 19 and 22 EDO
31-EDO is friends with: 65 and 12 EDO
.
From this it seems that 12-EDO is friends with at least [5, 7, 12, 14, 15, 16, 19, 21, 22, 23, 27, 28, 31, 34, and 65]-EDO.
I think it's pretty clear that these little EDO triples are just alternative descriptions of the comma triplets, and not really specific to the EDO. Each EDO has lots of intervals that it tempers out, and lots of intervals that it tunes to -1 or 1 step, and within that large space there are ...overlaps? Like every pair of EDOs defined on rank-3 interval space has a shared comma. That doesn't mean that they have a special relationship; everyone has that relationship with everyone else. Now we're just saying, "These two EDOs have a shared tempered comma and also each one tunes one of the others' tempered commas to 1 or -1 steps." I don't know, man. It feels like there should be something deeper here, but I'm not seeing it.
How about this: It seems that in rank-n interval space, there's a unimodular matrix of {n} commas associated with every set of {n} distinct EDOs, such that each EDO tempers out {n - 1} of the commas and also tunes the remaining comma to -1 or 1 step.
And um....maybe we can use this to find EDO bases, the regular non-unimodular ones that include the octave? If you want a rank-7 basis for 87-EDO, you take 7-EDOs, including 87-EDO, and look at how they tune the intervals justly associated with the first 7 primes, and then you invert, and uh, all but one of those will be tempered commas of 87 EDO, and you can replace the last one with the octave for an EDO basis, and if the commas aren't to your liking, then roll the dice with a different set of EDOs? It's not a great procedure. I'll probably just stick with my own usual methods for finding EDO bases. But maybe it's something to keep considering.
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