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 both commas and thus tempers 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.
n | Temperament | Comma | |
---|---|---|---|
Ratio | Monzo | ||
0 | 22 & 22c | [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 | [25 -23 5⟩ | |
… | … | … | … |
∞ | Superpyth | 20480/19683 | [12 -9 1⟩ |
Examples of temperaments with fractional values of n:
- 22 & 39cc (n = 1/2 = 0.5)
- 22 & 29c (n = 3/2 = 1.5)
- Hendecatonic (n = 11/5 = 2.2)
- Escapade (n = 9/4 = 2.25)
- Kwazy (n = 16/7 = 2.285714...)
- Orson (n = 7/3 = 2.333...)
- Magic (n = 5/2 = 2.5)
We may also invert the continuum by setting m such that 1/m + 1/n = 1. The just value of m is 1.778495…
m | Temperament | Comma | |
---|---|---|---|
Ratio | Monzo | ||
0 | 22 & 22c | [35 -22⟩ | |
1 | Superpyth | 20480/19683 | [12 -9 1⟩ |
2 | Diaschismic | 2048/2025 | [11 -4 -2⟩ |
3 | 22 & 29c | [34 -17 -3⟩ | |
… | … | … | … |
∞ | Quasisuper | 8388608/7971615 | [23 -13 -1⟩ |