Syntonic-Archytas equivalence continuum
The syntonic-Archytas equivalence continuum is a continuum of 7-limit rank-3 temperament families which equate a number of syntonic commas (81/80) with an Archytas comma (64/63). This continuum is theoretically interesting in that these are all 7-limit rank-3 temperament families supported by dominant temperament.
All temperaments in the continuum satisfy (81/80)n ~ 64/63. Varying n results in different temperament families listed in the table below. It converges to didymus 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 1.267726433120519…, and temperaments having n near this value will be more accurate.
| n | Temperament family | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| 0 | Archytas | 64/63 | [6 -2 0 -1⟩ |
| 1/2 | 63 & 68 & 80 | 321489/327680 | [16 -8 1 -2⟩ |
| 1 | Hemifamity | 5120/5103 | [1 5 1 -4⟩ |
| 5/4 | 894 & 441 & 1106 | [44 -28 5 -4⟩ | |
| 19/15 | 5 & 12 & 836 | [166 -106 19 -15⟩ | |
| 4/3 | 159 & 166 & 171 | 10763703445887/10737418240000 | [-34 22 -4 3⟩ |
| 3/2 | 118 & 125 & 130 | 2109289329/2097152000 | [-24 16 -3 2⟩ |
| 2 | 72 & 77 & 79 | 413343/409600 | [-14 10 -2 1⟩ |
| ∞ | Didymus | 81/80 | [-4 4 -1 0⟩ |