Superpyth–22 equivalence continuum: Difference between revisions
+alt name |
+commas for fractional-numbered temps |
||
| Line 15: | Line 15: | ||
| 0 | | 0 | ||
| 22 & 22c | | 22 & 22c | ||
| | | (22 digits) | ||
| {{Monzo| 35 -22 }} | | {{Monzo| 35 -22 }} | ||
|- | |- | ||
| 1 | | 1 | ||
| [[Quasisuper]] | | [[Quasisuper]] | ||
| 8388608/7971615 | | [[8388608/7971615]] | ||
| {{Monzo| 23 -13 -1 }} | | {{Monzo| 23 -13 -1 }} | ||
|- | |- | ||
| 2 | | 2 | ||
| [[Diaschismic]] | | [[Diaschismic]] | ||
| 2048/2025 | | [[2048/2025]] | ||
| {{Monzo| 11 -4 -2 }} | | {{Monzo| 11 -4 -2 }} | ||
|- | |- | ||
| 3 | | 3 | ||
| [[Porcupine]] | | [[Porcupine]] | ||
| 250/243 | | [[250/243]] | ||
| {{Monzo| 1 -5 3 }} | | {{Monzo| 1 -5 3 }} | ||
|- | |- | ||
| 4 | | 4 | ||
| [[Comic]] | | [[Comic]] | ||
| 5120000/4782969 | | [[5120000/4782969]] | ||
| {{Monzo| 13 -14 4 }} | | {{Monzo| 13 -14 4 }} | ||
|- | |- | ||
|5 | | 5 | ||
|22 & 3cc | | 22 & 3cc | ||
| | | (23 digits) | ||
| {{Monzo| 25 -23 5 }} | | {{Monzo| 25 -23 5 }} | ||
|- | |- | ||
| Line 50: | Line 50: | ||
| ∞ | | ∞ | ||
| [[Superpyth]] | | [[Superpyth]] | ||
| 20480/19683 | | [[20480/19683]] | ||
| {{monzo| 12 -9 1 }} | | {{monzo| 12 -9 1 }} | ||
|} | |} | ||
| Line 68: | Line 68: | ||
| 0 | | 0 | ||
| 22 & 22c | | 22 & 22c | ||
| | | (22 digits) | ||
| {{Monzo| 35 -22 }} | | {{Monzo| 35 -22 }} | ||
|- | |- | ||
| 1 | | 1 | ||
| [[Superpyth]] | | [[Superpyth]] | ||
| 20480/19683 | | [[20480/19683]] | ||
| {{Monzo| 12 -9 1 }} | | {{Monzo| 12 -9 1 }} | ||
|- | |- | ||
| 2 | | 2 | ||
| [[Diaschismic]] | | [[Diaschismic]] | ||
| 2048/2025 | | [[2048/2025]] | ||
| {{Monzo| 11 -4 -2 }} | | {{Monzo| 11 -4 -2 }} | ||
|- | |- | ||
| 3 | | 3 | ||
| 22 & 29c | | 22 & 29c | ||
| | | (22 digits) | ||
| {{Monzo| 34 -17 -3 }} | | {{Monzo| 34 -17 -3 }} | ||
|- | |- | ||
| Line 93: | Line 93: | ||
| ∞ | | ∞ | ||
| [[Quasisuper]] | | [[Quasisuper]] | ||
| 8388608/7971615 | | [[8388608/7971615]] | ||
| {{Monzo| 23 -13 -1 }} | | {{Monzo| 23 -13 -1 }} | ||
|} | |} | ||
| Line 100: | Line 100: | ||
|+ Temperaments with fractional ''n'' and ''m'' | |+ Temperaments with fractional ''n'' and ''m'' | ||
|- | |- | ||
! ''n'' !! ''m'' !! Temperament !! Comma | |||
|- | |- | ||
| 11/5 = 2.2 || 11/6 = 1.8{{overline|3}} || [[Hendecatonic]] || {{monzo| 43 -11 -11 }} | |||
|- | |- | ||
| 9/4 = 2.25 || 9/5 = 1.8 || [[Escapade]] || {{monzo| 32 -7 -9 }} | |||
|- | |- | ||
| 16/7 = 2.{{overline|285714}} || 16/9 = 1.{{overline|8}} || [[Kwazy]] || {{monzo| -53 10 16 }} | |||
|- | |- | ||
| 7/3 = 2.{{overline|3}} || 7/4 = 1.75 || [[Orson]] || {{monzo| -21 3 7 }} | |||
|- | |- | ||
| 5/2 = 2.5 || 5/3 = 1.{{overline|6}} || [[Magic]] || {{monzo| -10 -1 5 }} | |||
|} | |} | ||
[[Category:22edo]] | [[Category:22edo]] | ||
[[Category:Equivalence continua]] | [[Category:Equivalence continua]] | ||
Revision as of 09:07, 23 July 2024
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 | (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…
| 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⟩ |