Superpyth-22 equivalence continuum
The superpyth-22 equivalence continuum is a continuum of 5-limit temperaments which equate a number of superpyth commas, 20480/19683 = [12 -9 1⟩, with the 22-comma, [35 -22⟩. This continuum is theoretically interesting in that these are all 5-limit temperaments supported by 22edo.
All temperaments in the continuum satisfy (20480/19683)n ~ 250/243. Varying n results in different temperaments listed in the table below. It converges to 5-limit superpyth as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 5-limit temperaments supported by 22edo due to it being the unique equal temperament that tempers out both commas and thus tempers out all combinations of them. The just value of n is approximately 2.284531…, and temperaments having n near this value tend to be the most accurate ones.
The 22-comma is the characteristic 3-limit comma tempered out in 22edo, and has many advantages as a target. In each case, n equals the order of harmonic 5 in the corresponding comma, and equals the number of steps to obtain the interval class of harmonic 3 in the generator chain. For an n that is not coprime with 22, however, the corresponding temperament splits the octave into gcd (n, 22) parts, and splits the interval class of 3 into n/gcd (n, 22). For example:
- Quasisuper (n = 1) is generated by a fifth with an unsplit octave;
- Diaschismic (n = 2) splits the octave in two, as 2 divides 22;
- Porcupine (n = 3) splits the fourth in three, as 3 is coprime with 22;
- Etc.
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| 0 | 22 & 22c | (22 digits) | [35 -22⟩ |
| 1 | Quasisuper | 8388608/7971615 | [23 -13 -1⟩ |
| 2 | Diaschismic | 2048/2025 | [11 -4 -2⟩ |
| 3 | Porcupine | 250/243 | [1 -5 3⟩ |
| 4 | Comic | 5120000/4782969 | [13 -14 4⟩ |
| 5 | 22 & 3cc | (23 digits) | [25 -23 5⟩ |
| … | … | … | … |
| ∞ | Superpyth | 20480/19683 | [12 -9 1⟩ |
We may also invert the continuum by setting m such that 1/m + 1/n = 1. This may be called the quasisuper-22 equivalence continuum, which is essentially the same thing. The just value of m is 1.778495… The quasisuper comma is both larger and more complex than the superpyth comma. As such, this continuum does not contain as many useful temperaments, but still interesting nonetheless.
| m | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| 0 | 22 & 22c | (22 digits) | [35 -22⟩ |
| 1 | Superpyth | 20480/19683 | [12 -9 1⟩ |
| 2 | Diaschismic | 2048/2025 | [11 -4 -2⟩ |
| 3 | 22 & 29c | (22 digits) | [34 -17 -3⟩ |
| … | … | … | … |
| ∞ | Quasisuper | 8388608/7971615 | [23 -13 -1⟩ |
| n | m | Temperament | Comma |
|---|---|---|---|
| 11/5 = 2.2 | 11/6 = 1.83 | Hendecatonic | [43 -11 -11⟩ |
| 9/4 = 2.25 | 9/5 = 1.8 | Escapade | [32 -7 -9⟩ |
| 16/7 = 2.285714 | 16/9 = 1.8 | Kwazy | [-53 10 16⟩ |
| 7/3 = 2.3 | 7/4 = 1.75 | Orson | [-21 3 7⟩ |
| 5/2 = 2.5 | 5/3 = 1.6 | Magic | [-10 -1 5⟩ |