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 has the advantage of being the characteristic 3-limit comma tempered out in 5edo. For 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:
- 25/24 is the 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, the second simplest being 81/80 at (unsigned) infinity, which together are the two smallest 5-limit superparticular intervals and the only superparticular intervals in the continuum if we don't count non-integer k.
| k = n − 2 | n = k + 2 | 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 | 5 & 72 | [32 -29 6⟩ | |
| … | … | … | … | |
| ∞ | ∞ | Meantone | 81/80 | [-4 4 -1⟩ |
Examples of temperaments with fractional values of n:
- University (n = -0.5)
- 5 & 32p (n = 0.5)
- 5 & 56 (n = 1.5)
- Counterpental (n = 2.5)
- 99 & 94 (n = 3.5)
- 2513 & 559 (n = 4.2)
- 5 & 137 (n = 4.5)
5 & 72
Comma: [32 -29 6⟩
POTE generator: 483.2474 cents
Map: [<1 4 14|, <0 -6 -29|]
EDOs: 5, 10c, 67c, 72, 77, 82c, 139c, 144, 149, 154
The temperament finder - 5-limit 5 & 72
5 & 32p
Comma: [20 -14 1⟩ (5242880/4782969)
POTE generator: ~4/3 = 486.1713 cents
Map: [<1 2 8|, <0 -1 -14|]
The temperament finder - 5-limit 5 & 32p
5 & 56
Comma: [28 -22 3⟩ (33554432000/31381059609)
POTE generator: 235.8673 cents
Map: [<1 1 -2|, <0 3 22|]
The temperament finder - 5-limit 5 & 56
Counterpental
Comma: [36 -30 5⟩
POTE generator: 15.4278 cents
Map: [<5 8 12|, <0 -1 -6|]
The temperament finder - 5-limit 5 & 75
99 & 94
Comma: [44 -38 7⟩
POTE generator: 242.4567 cents
Map: [<1 3 10|, <0 -7 -38|]
EDOs: 5, 94, 99, 193, 198, 292, 297
The temperament finder - 5-limit 99 & 94
2513 & 559
Comma: [-124 109 -21⟩
POTE generator: 480.8595 cents
Map: [<1 10 46|, <0 -21 -109|]
EDOs: 559, 1118, 1395, 1954, 2513, 3072, 3631, 4467, 5026, 5585
The temperament finder - 5-limit 2153 & 559
5 & 137
Comma: [60 -54 11⟩
POTE generator: 481.7421 cents
Map: [<1 6 24|, <0 -11 -54|]