Breedsmic-syntonic equivalence continuum
The breedsmic-syntonic equivalence continuum is a continuum of 7-limit temperament families which equate a number of breedsmas (2401/2400) with a syntonic comma (81/80). This continuum is theoretically interesting in that these are all 7-limit temperament families supported by squares temperament. In addition, 81/80 and 2401/2400 are the smallest 5-limit and 7-limit superparticular intervals to be tempered out by 31edo.
All temperaments in the continuum satisfy (2401/2400)n ~ 81/80. Varying n results in different temperament families listed in the table below. It converges to breedsmic as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 7-limit temperament families supported by squares (due to it being the unique rank-2 temperament that tempers both commas and thus tempers all combinations of them). The just value of n is approximately 29.82025..., and temperaments having n near this value tend to be the most accurate ones.
| n | Temperament family | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| -4 | 217 & 31 & 14c | [-24 0 -9 16⟩ | |
| -3 | 159 & 31 & 14c | [-19 1 -7 12⟩ | |
| -2 | 87 & 31 & 14c | 51883209/51200000 | [-14 2 -5 8⟩ |
| -1 | Squalentine | 64827/64000 | [-9 3 -3 4⟩ |
| 0 | Meantone | 81/80 | [-4 4 -1⟩ |
| 1 | Nuwell | 2430/2401 | [1 5 1 -4⟩ |
| 2 | 14c & 31 & 80 | 5832000/5764801 | [6 6 3 -8⟩ |
| 3 | 14c & 31 & 152 | 13996800000/13841287201 | [11 7 5 -12⟩ |
| 4 | 14c & 31 & 224 | [16 8 7 -16⟩ | |
| 5 | 265 & 31 & 282 | [21 9 9 -20⟩ | |
| 6 | 14c & 31 & 323 | [26 10 11 -24⟩ | |
| 7 | 17c & 395 & 364 | [31 11 13 -28⟩ | |
| 8 | 14c & 31 & 422 | [36 12 15 -32⟩ | |
| … | … | … | … |
| 30 | 1677 & 6691 & 41854 | [146 34 59 -120⟩ | |
| … | … | … | … |
| ∞ | Breedsmic | 2401/2400 | [-5 -1 -2 4⟩ |
Examples of temperaments with fractional values of n:
- 34p & 31 & 14c (n = -1/2 = -0.5)
- Skwares (n = 1/2 = 0.5)
1677 & 6691 & 41854
Comma list: [146 34 59 -120⟩
POTE generators: 1901.9549, -775.6679
Mapping: [⟨1 0 26 14], ⟨0 1 34 17], ⟨0 0 120 59]]
Optimal ET sequence: 1677, 5014, 6691, 41854, 43531, 46868, 48545, 50222, 55236, 97090
The temperament finder - 7-limit 1677 & 6691 & 41854
34p & 31 & 14c
Comma list: [-13 7 -4 4⟩ = 5250987/5120000
POTE generators: -425.3382, ~5 = 2785.5671
Mapping: [⟨1 3 0 -2], ⟨0 4 0 -7], ⟨0 0 1 1]]
Optimal ET sequence: 3, 14c, 17c, 17d, 20c, 31, 34, 45, 62, 65