Syntonic–Archytas 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 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 1.267726, and temperaments having n near this value will be the most accurate.
| n | Temperament family | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| −1 | Mint | 36/35 | [2 2 -1 -1⟩ |
| 0 | Archytas | 64/63 | [6 -2 0 -1⟩ |
| 1/2 | 63 & 68 & 80 | 327680/321489 | [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⟩ |