Breedsmic-syntonic equivalence continuum

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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.

Temperament families in the continuum
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 GPV sequence1677, 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 GPV sequence3, 14c, 17c, 17d, 20c, 31, 34, 45, 62, 65

The temperament finder - 7-limit 34p & 31 & 14c