Archytas–chromatic 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 and highly contrived. |
The Archytas–chromatic equivalence continuum, or septimal–chromatic equivalence continuum, is a continuum of 7-limit temperaments which equate a number of Archytas commas (64/63) with the Pythagorean apotome (2187/2048).
All temperaments in the continuum satisfy (64/63)n ~ 2187/2048. Varying n results in different temperaments listed in the table below. It converges to archy as n approaches infinity. The just value of n is 4.169771, and temperaments near this tend to be the most accurate ones.
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| −2 | Dicot | 54/49 | [1 3 0 -2⟩ |
| −1 | Armodue | 243/224 | [-5 5 0 -1⟩ |
| 0 | Whitewood | 2187/2048 | [-11 7⟩ |
| 1 | Flattone | 137781/131072 | [-17 9 0 1⟩ |
| 2 | Mohajira | 8680203/8388608 | [-23 11 0 2⟩ |
| 3 | (29 & 36) | 546852789/536870912 | [-29 13 0 3⟩ |
| 4 | Sesquiquartififths | 34451725707/34359738368 | [-35 15 0 4⟩ |
| 41⁄6 | (1848 & 3431) | [very long] | [-216 92 0 25⟩ |
| … | … | … | |
| ∞ | Archy | 64/63 | [6 -2 0 -1⟩ |