Superpyth–22 equivalence continuum: Difference between revisions
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Collect factional temps in a table |
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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. | 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 | {| class="wikitable center-1" | ||
|+Temperaments in the continuum | |+Temperaments in the continuum | ||
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| {{monzo| 12 -9 1 }} | | {{monzo| 12 -9 1 }} | ||
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We may also invert the continuum by setting ''m'' such that 1/''m'' + 1/''n'' = 1. The just value of ''m'' is 1.778495… | 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 | {| class="wikitable center-1" | ||
|+Temperaments in the continuum | |+Temperaments in the continuum | ||
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| {{Monzo| 23 -13 -1 }} | | {{Monzo| 23 -13 -1 }} | ||
|} | |} | ||
{| class="wikitable" | |||
|+ Temperaments with fractional ''n'' and ''m'' | |||
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! Temperament !! ''n'' !! ''m'' | |||
|- | |||
| [[Hendecatonic]] || 11/5 = 2.2 || 11/6 = 1.8{{overline|3}} | |||
|- | |||
| [[Escapade]] || 9/4 = 2.25 || 9/5 = 1.8 | |||
|- | |||
| [[Kwazy]] || 16/7 = 2.{{overline|285714}} || 16/9 = 1.{{overline|8}} | |||
|- | |||
| [[Orson]] || 7/3 = 2.{{overline|3}} || 7/4 = 1.75 | |||
|- | |||
| [[Magic]] || 5/2 = 2.5 || 5/3 = 1.{{overline|6}} | |||
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[[Category:22edo]] | [[Category:22edo]] | ||
[[Category:Equivalence continua]] | [[Category:Equivalence continua]] | ||