Breedsmic–syntonic equivalence continuum
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This page presents a topic of primarily mathematical interest.
While it is derived from sound mathematical principles, its applications in terms of utility for actual music may be limited, highly contrived, or as yet unknown. |
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 29.820259, 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 | Didymus | 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:
- 34 & 31 & 14c (n = −1⁄2 = −0.5)
- Skwares (n = 1⁄2 = 0.5)
1677 & 6691 & 41854
Comma list: [146 34 59 -120⟩
POTE generators: ~3 = 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
Badness (Sintel): 475.7
34 & 31 & 14c
Comma list: [-13 7 -4 4⟩ = 5250987/5120000
POTE generators: ~80/63 = 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
Badness (Sintel): 19.984