Superpyth–22 equivalence continuum: Difference between revisions
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{{Mathematical interest}} | |||
The '''superpyth-22 equivalence continuum''' is a [[equivalence continuum|continuum]] of [[5-limit]] [[regular temperament|temperaments]] which equate a number of [[superpyth comma]]s, 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 [[support]]ed by [[22edo]]. | |||
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 [[5/1|harmonic 5]] in the corresponding comma, and equals the number of steps to obtain the interval class of [[3/1|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'', | All temperaments in the continuum satisfy {{nowrap| (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 [[tempering out|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. | ||
* [[Quasisuper]] ({{nowrap|''n'' {{=}} 1}}) is generated by a fifth with an unsplit octave; | |||
* [[Diaschismic]] ({{nowrap|''n'' {{=}} 2}}) splits the octave in two, as 2 divides 22; | 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 [[5/1|harmonic 5]] in the corresponding comma, and equals the number of steps to obtain the interval class of [[3/1|harmonic 3]] in the generator chain. For an ''n'' that is not coprime with 22, however, the corresponding temperament splits the [[octave]] into {{nowrap| gcd(''n'', 22) }} parts, and splits the interval class of 3 into {{nowrap| ''n''/gcd(''n'', 22) }}. For example: | ||
* [[Porcupine]] ({{nowrap|''n'' {{=}} 3}}) splits the fourth in three, as 3 is coprime with 22; | * [[Quasisuper]] ({{nowrap| ''n'' {{=}} 1 }}) is generated by a fifth with an unsplit octave; | ||
* [[Diaschismic]] ({{nowrap| ''n'' {{=}} 2 }}) splits the octave in two, as 2 divides 22; | |||
* [[Porcupine]] ({{nowrap| ''n'' {{=}} 3 }}) splits the fourth in three, as 3 is coprime with 22; | |||
* Etc. | * Etc. | ||
{| class="wikitable center-1" | {| class="wikitable center-1" | ||
|+ style="font-size: 105%;" | Temperaments | |+ style="font-size: 105%;" | Temperaments with integer ''n'' | ||
|- | |- | ||
! rowspan="2" | ''n'' | ! rowspan="2" | ''n'' | ||
| Line 60: | Line 62: | ||
|} | |} | ||
We may also invert the continuum by setting ''m'' such that {{nowrap|1/''m'' + 1/''n'' {{=}} 1}}. This may be called the '' | We may also invert the continuum by setting ''m'' such that {{nowrap| 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. | ||
{| class="wikitable center-1" | {| class="wikitable center-1" | ||
|+ style="font-size: 105%;" | Temperaments | |+ style="font-size: 105%;" | Temperaments with integer ''m'' | ||
|- | |- | ||
! rowspan="2" | ''m'' | ! rowspan="2" | ''m'' | ||
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| 5/2 = 2.5 || 5/3 = 1.{{overline|6}} || [[Magic]] || {{monzo| -10 -1 5 }} | | 5/2 = 2.5 || 5/3 = 1.{{overline|6}} || [[Magic]] || {{monzo| -10 -1 5 }} | ||
|} | |} | ||
[[Category:22edo]] | [[Category:22edo]] | ||
[[Category:Equivalence continua]] | [[Category:Equivalence continua]] | ||
Latest revision as of 11:20, 23 August 2025
|
This page presents a topic of primarily mathematical interest.
While it is derived from sound mathematical principles, its applications in terms of utility for actual music may be limited, highly contrived, or as yet unknown. |
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⟩ |
| 12/5 = 2.4 | 12/7 = 1.714285 | Wizard | [-31 2 12⟩ |
| 5/2 = 2.5 | 5/3 = 1.6 | Magic | [-10 -1 5⟩ |