Syntonic-diatonic equivalence continuum

The syntonic-diatonic equivalence continuum is a continuum of temperaments which equate a number of syntonic commas (81/80) with the limma (256/243).

All temperaments in the continuum satisfy (81/80)n ~ 256/243. Varying n results in different temperaments listed in the table below. It converges to meantone as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 5-limit temperaments supported by 5edo due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them. The just value of n is 4.1952…, and temperaments near this tend to be the most accurate ones.

256/243 is the characteristic 3-limit comma tempered out in 5edo. In each case, we notice that n equals the order of harmonic 5 in the corresponding comma, and equals the number of generators to obtain a harmonic 3 in the MOS scale. However, if we let k = n + 1 (meaning n = k - 1) so that k = 0 means n = -1, k = 1 means n = 0, etc. then the continuum corresponds to (81/80)k = 16/15, which might be a preferred way of conceptualising it because:

• 16/15 is the classic diatonic semitone, notable in the 5-limit as the difference between 4/3 and 5/4, so this shifted continuum could also logically be termed the "syntonic-diatonic equivalence continuum". This means that at k = 0, 4/3 and 5/4 are mapped to the same interval while 81/80 becomes independent of 16/15 (meaning 81/80 may or may not be tempered) because the relation becomes (81/80)0 ~ 1/1 ~ 16/15.
• k = 1 and upwards (up to a point) represent temperaments with (the potential for) reasonably good accuracy as equating at least one 81/80 with 16/15 seems like a good lower bound for a temperament intended to model JI. A good upper bound might be rodan (k = 4), with the only exception being meantone (n = k = (unsigned) infinity). (Temperaments corresponding to k = 0, -1, -2 are comparatively low-accuracy to the point of developing various intriguing structures and consequences.)
• 16/15 is the simplest ratio to be tempered in the continuum.
Temperaments in the continuum
k = n + 1 n = k − 1 Temperament Comma
Ratio Monzo
-3 -4 Laquadgu 177147/160000 [-8 11 -4
-2 -3 Laconic 2187/2000 [-4 7 -3
-1 -2 Bug 27/25 [0 3 -2
0 -1 Father 16/15 [4 -1 -1
1 0 Blackwood 256/243 [8 -5
2 1 Superpyth 20480/19683 [12 -9 1
3 2 Immunity 1638400/1594323 [16 -13 2
4 3 Rodan 131072000/129140163 [20 -17 3
5 4 Vulture 10485760000/10460353203 [24 -21 4
6 5 Pental [-28 25 -5
7 6 Hemiseven [-32 29 -6
Meantone 81/80 [-4 4 -1

Examples of temperaments with fractional values of n:

Hemiseven

Subgroup: 2.3.5

Comma list: [32 -29 6

Mapping: [1 4 14], 0 -6 -29]]

Mapping generators: ~2, ~320/243

POTE generator: ~320/243 = 483.2474

Vals: 5, 62c, 67c, 72, 149, 221, 370, 591b, 961bb

Ultrapyth

Subgroup: 2.3.5

Comma list: 5242880/4782969

Mapping: [1 0 -20], 0 -1 -14]]

Mapping generators: ~2, ~3

POTE generator: ~3/2 = 713.8287

Vals: 5, 27c, 32, 37, 79bc, 116bbc

Trisatriyo (5 & 56)

Subgroup: 2.3.5

Comma list: [28 -22 3 = 33554432000/31381059609

Mapping: [1 1 -2], 0 3 22]]

Mapping generators: ~2, ~2560/2187

POTE generator: ~2560/2187 = 235.8673

Vals: 5, 56, 61

Counterpental

Subgroup: 2.3.5

Comma list: [36 -30 5

Mapping: [5 0 -36], 0 1 6]]

Mapping generators: ~729/640, ~3

POTE generator: ~3/2 = 704.5722

Vals: 5, 75, 80

Septiquarter

Subgroup: 2.3.5

Comma list: [44 -38 7

Mapping: [1 3 10], 0 -7 -38]]

Mapping generators: ~2, ~204800/177147

POTE generator: ~204800/177147 = 242.4567

Vals: 5, 89c, 94, 99, 193, 292, 391

559 & 2513

Subgroup: 2.3.5

Comma list: [-124 109 -21

Mapping: [1 10 46], 0 -21 -109]]

Mapping generators: ~2, ~3355443200000/2541865828329

POTE generator: ~3355443200000/2541865828329 = 480.8595

Vals: 5, 267c, 272c, 277, 559, 1395, 1954, 2513, 40767, 43280, 45793, 48306, 50819, 53332, 55845, 58358, 60871, 63384, 65897, 68410, 70923, 73436, 75949, 78462

Quinla-tritrigu (5 & 118)

Subgroup: 2.3.5

Comma list: [-52 46 -9

Mapping: [1 -2 -16], 0 9 46]]

Mapping generators: ~2, ~320/243

POTE generator: ~320/243 = 477.9609

Vals: 5, 108c, 113, 118, 1057, 1175, 1293, 1411, 1529, 1647, 1765, 1883, 2001b, 3884b

Tribilalegu (5 & 137)

Subgroup: 2.3.5

Comma list: [-60 54 -11

Mapping: [1 6 24], 0 -11 -54]]

Mapping generators: ~2, ~320/243

POTE generator: ~320/243 = 481.7421

Vals: 5, 127c, 132, 137, 553, 690b, 827b, 964b