Superpyth–22 equivalence continuum: Difference between revisions
m FloraC moved page Diaschismic-Porcupine equivalence continuum to Diaschismic-porcupine equivalence continuum without leaving a redirect: WP:NCCAPS |
m FloraC moved page Diaschismic-porcupine equivalence continuum to Superpyth-22 equivalence continuum without leaving a redirect: See talk |
(No difference)
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Revision as of 09:32, 9 May 2023
The Diaschismic-Porcupine continuum is a continuum of 5-limit temperaments which equate a number of diaschismas with a porcupine comma. This continuum is theoretically interesting in that these are all 5-limit temperaments supported by 22edo.
All temperaments in the continuum satisfy (2048/2025)n ~ 250/243. Varying n results in different temperaments listed in the table below. It converges to diaschismic 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 2.5145615263.., and temperaments having n near this value tend to be the most accurate ones.
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| -3 | 22 & 29c | [34 -17 -3⟩ | |
| -2 | Quasisuper | 8388608/7971615 | [23 -13 -1⟩ |
| -1 | Superpyth | 20480/19683 | [12 -9 1⟩ |
| 0 | Porcupine | 250/243 | [1 -5 3⟩ |
| 1 | Magic | 3125/3072 | [-10 -1 5⟩ |
| 2 | Orson | 2109375/2097152 | [-21 3 7⟩ |
| 3 | Escapade | [32 -7 -9⟩ | |
| 4 | Hendecatonic | [43 -11 -11⟩ | |
| 5 | 22 & 111 | [54 -15 -13⟩ | |
| … | … | … | … |
| ∞ | Diaschismic | 2048/2025 | [11 -4 -2⟩ |
To do: figure out fractional n