Superpyth–22 equivalence continuum: Difference between revisions

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The '''Diaschismic-Porcupine continuum''' is a continuum of 5-limit temperaments which equate a number of [[Diaschisma|diaschismas]] with a [[250/243|porcupine comma]]. This continuum is theoretically interesting in that these are all 5-limit temperaments supported by [[22edo]].
The '''superpyth-22 equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of superpyth commas, 20480/19683 = {{monzo| 12 -9 1 }}, with the 22-comma, {{monzo| 35 -22 }}. This continuum is theoretically interesting in that these are all 5-limit temperaments supported by [[22edo]].


All temperaments in the continuum satisfy (2048/2025)<sup>''n''</sup> ~ 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.
All temperaments in the continuum satisfy (20480/19683)<sup>''n''</sup> ~ 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.


{| class="wikitable center-1 center-2"
{| class="wikitable center-1 center-2"
|+Temperaments in the continuum
|+Temperaments in the continuum
|-
|-
! rowspan="2" |''n''
! rowspan="2" | ''n''
! rowspan="2" |Temperament
! rowspan="2" | Temperament
! colspan="2" |Comma
! colspan="2" | Comma
|-
|-
!Ratio
! Ratio
!Monzo
! Monzo
|-
|-
| -3
| 0
|22 & 29c
| 22 & 22c
|
|  
|[34 -17 -3⟩
| {{Monzo| 35 -22 }}
|-
|-
| -2
| 1
|[[Quasisuper]]
| [[Quasisuper]]
|8388608/7971615
| 8388608/7971615
|[23 -13 -1⟩
| {{Monzo| 23 -13 -1 }}
|-
|-
| -1
| 2
|[[Superpyth]]
| [[Diaschismic]]
|20480/19683
| 2048/2025
|[12 -9 1⟩
| {{Monzo| 11 -4 -2 }}
|-
|-
|0
| 3
|[[Porcupine]]
| [[Porcupine]]
|250/243
| 250/243
|[1 -5 3⟩
| {{Monzo| 1 -5 3 }}
|-
|-
|1
| 4
|[[Magic]]
| [[Comic]]
|3125/3072
| 5120000/4782969
|[-10 -1 5⟩
| {{Monzo| 13 -14 4 }}
|-
|-
|2
|…
|[[Orson]]
|…
|2109375/2097152
|…
|[-21 3 7⟩
|…
|-
| ∞
| [[Superpyth]]
| 20480/19683
| {{monzo| 12 -9 1 }}
|}
 
We may also invert the continuum by setting ''m'' such that 1/''m'' + 1/''n'' = 1. The just value of ''m'' is 1.778495…
 
{| class="wikitable center-1 center-2"
|+Temperaments in the continuum
|-
|-
|3
! rowspan="2" | ''m''
|[[Escapade]]
! rowspan="2" | Temperament
|
! colspan="2" | Comma
|[32 -7 -9⟩
|-
|-
|4
! Ratio
|[[Hendecatonic]]
! Monzo
|
|[43 -11 -11⟩
|-
|-
|5
| 0
|22 & 111
| 22 & 22c
|
|  
|[54 -15 -13⟩
| {{Monzo| 35 -22 }}
|-
| 1
| [[Superpyth]]
| 20480/19683
| {{Monzo| 12 -9 1 }}
|-
| 2
| [[Diaschismic]]
| 2048/2025
| {{Monzo| 11 -4 -2 }}
|-
| 3
| 22 & 29c
|
| {{Monzo| 34 -17 -3 }}
|-
|-
|…
|…
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|…
|…
|-
|-
|∞
| ∞
|Diaschismic
| [[Quasisuper]]
|2048/2025
| 8388608/7971615
|[11 -4 -2⟩
| {{Monzo| 23 -13 -1 }}
|}
|}


To do: figure out fractional n  
To do: figure out fractional ''n''.
*
 
[[Category:22edo]]
[[Category:22edo]]
[[Category:Equivalence continua]]
[[Category:Equivalence continua]]

Revision as of 10:06, 9 May 2023

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.

Temperaments in the continuum
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⟩
… … … …
∞ Superpyth 20480/19683 [12 -9 1⟩

We may also invert the continuum by setting m such that 1/m + 1/n = 1. The just value of m is 1.778495…

Temperaments in the continuum
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⟩

To do: figure out fractional n.

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