Archytas–diatonic equivalence continuum
The Archytas–diatonic equivalence continuum, or septimal–diatonic equivalence continuum, is a continuum of 2.3.7 subgroup temperaments which equate a number of Archytas' commas (64/63) with the Pythagorean limma (256/243). This continuum is theoretically interesting in that these are all 2.3.7-subgroup temperaments supported by 5edo.
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, and has many advantages as a target. In each case, n equals the order of harmonic 7 in the corresponding comma, and equals the number of generators to obtain a harmonic 3 in the generator chain.
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | S. monzo | ||
| 0 | Blackwood variant | 256/243 | [8 -5⟩ |
| 1 | Trienstonian | 28/27 | [2 -3 1⟩ |
| 2 | Semaphore | 49/48 | [-4 -1 2⟩ |
| 3 | Slendric | 1029/1024 | [-10 1 3⟩ |
| 4 | Buzzard | 65536/64827 | [16 -3 -4⟩ |
| 5 | Obscenity | 4194304/4084101 | [22 -5 -5⟩ |
| … | … | … | … |
| ∞ | Archy | 64/63 | [6 -2 -1⟩ |
We may invert the continuum by setting m such that 1/m + 1/n = 1. This may be called the trienstonic–diatonic equivalence continuum, which is essentially the same thing. The just value of m is 1.4330…. The trienstonic comma is larger than the archytas comma. As such, this continuum does not contain as many useful temperaments, but still interesting nonetheless.
| m | Temperament | Comma | |
|---|---|---|---|
| Ratio | S. monzo | ||
| 0 | Blackwood variant | 256/243 | [8 -5⟩ |
| 1 | Archy | 64/63 | [6 -2 -1⟩ |
| 2 | Semaphore | 49/48 | [-4 -1 2⟩ |
| … | … | … | … |
| ∞ | Trienstonian | 28/27 | [2 -3 1⟩ |
| n | m | Temperament | Comma |
|---|---|---|---|
| 5/2 = 2.5 | 5/3 = 1.6 | Cloudy retraction | [-14 0 5⟩ |
| 33/10 = 3.3 | 33/23 = 1.434783… | 5 & 436 | [118 -16 -33⟩ |
| 10/3 = 3.3 | 10/7 = 1.428571 | Slendroschismic | [36 -5 -10⟩ |
| 7/2 = 3.5 | 7/5 = 1.4 | Septiness retraction | [26 -4 -7⟩ |