Archytas-diatonic equivalence continuum
The Archytas-diatonic equivalence continuum or the septimal-diatonic equivalence continuum is a continuum of 2.3.7 subgroup temperaments which equate a number of Archytas commas (64/63) with the limma (256/243).
All temperaments in the continuum satisfy (64/63)n ~ 256/243. Varying n results in different temperaments listed in the table below. It converges to archy as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 2.3.7 subgroup temperaments supported by 5edo (due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them). The just value of n is 3.3093…, and temperaments near this tend to be the most accurate ones.
256/243 is the characteristic 3-limit comma tempered out in 5edo. In each case, we notice that n equals the order of harmonic 7 in the corresponding comma, and equals the number of generators to obtain a harmonic 3 in the MOS scale.
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| 0 | 2.3.7 blackwood | 256/243 | [8 -5⟩ |
| 1 | No-fives trienstonic | 28/27 | [2 -3 1⟩ |
| 2 | Semaphore | 49/48 | [-4 -1 2⟩ |
| 2.5 | Cloudy | 16807/16384 | [-14 0 5⟩ |
| 3 | Slendric | 1029/1024 | [-10 1 3⟩ |
| 3.3 | Slendrismic* | [very long] | [118 -16 -33⟩ |
| 3.5 | Septiness | 67108864/66706983 | [26 -4 -7⟩ |
| 4 | Buzzard | 65536/64827 | [16 -3 -4⟩ |
| 5 | 5 & 75 | 4194304/4084101 | [22 -5 -5⟩ |
| … | … | … | |
| ∞ | Archy | 64/63 | [6 -2 -1⟩ |
* The name "slendrismic" may be changed to minimise confusion with confusingly named temperaments, or those temperaments may be changed instead, either way consensus would first need to be formed.