Septimal-diatonic equivalence continuum
The septimal-diatonic equivalence continuum is a continuum of temperaments which equate a number of septimal 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. However, if we let k = n - 1 (meaning n = k + 1) so that k = 0 means n = 1, k = -1 means n = 0, etc. then the continuum corresponds to (64/63)k = 28/27, which might be a preferred way of conceptualising it because:
- 28/27 is the classic 2.3.7 diatonic semitone, notable in 2.3.7 as the difference between 9/8 and 7/6, so this shifted continuum could also logically be termed the "septimal-diatonic equivalence continuum". This means that at k = 0, 9/8, and 7/6 are mapped to the same interval while 64/63 becomes independent of 28/27 (meaning 64/63 may or may not be tempered) because the relation becomes (64/63)0 ~ 1/1 ~ 28/27.
- k = 1 and upwards (up to a point) represent temperaments with (the potential for) reasonably good accuracy as equating at least one 64/63 with 28/27 seems like a good lower bound for a temperament intended to model JI. A good upper bound might be k = 2, with the only exception being Archy (n = k = (unsigned) infinity). (Temperaments corresponding to k = 0, -1, -2 are comparatively low-accuracy to the point of developing various intriguing structures and consequences.)
- 28/27 is the simplest ratio to be tempered in the continuum, the second simplest being 49/48 at k = -2, which together are the two smallest 2.3.7 superparticular intervals and, along with 64/63, the only superparticular intervals in the continuum.
| k = n − 1 | n = k + 2 | Temperament | Comma | |
|---|---|---|---|---|
| Ratio | Monzo | |||
| -2 | 0 | 256/243 | [8, -5⟩ | |
| -1 | 1 | No-fives trienstonic | 28/27 | [2, -3, 1⟩ |
| 0 | 2 | Semaphore | 49/48 | [-4, -1, 2⟩ |
| 1 | 3 | Slendric | 1029/1024 | [-10, 1, 3⟩ |
| 2 | 4 | 2.3.7 Vulture | 65536/64827 | [16, -3, -4⟩ |
| 3 | 5 | 2.3.7 Saquinru | 4194304/4084101 | [22, -5, -5⟩ |
| … | … | … | … | |
| ∞ | ∞ | Archy | 64/63 | [6, -2, -1⟩ |