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.
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:
- University (n = -1.5)
- Uncle (n = -0.5)
- Ultrapyth (n = 0.5)
- 5 & 56 (n = 1.5)
- Counterpental (n = 2.5)
- Septiquarter (n = 3.5)
- 559 & 2513 (n = 4.2)
- 5 & 118 (n = 4.5)
- 5 & 137 (n = 5.5)
Hemiseven
- See also: Gamelismic clan #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
Badness: 0.720465
Ultrapyth
- See also: Archytas clan #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
Badness: 0.795243
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
Badness: 1.323443
The temperament finder - 5-limit 5 & 56
Counterpental
- See also: Orwellismic temperaments #Pentorwell
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
Badness: 1.500224
Septiquarter
- See also: Hemifamity temperaments #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
Badness: 0.971284
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
Badness: 0.134523
The temperament finder - 5-limit 2513 & 559
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
Badness: 0.617683
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
Badness: 3.620981