Archytas–limmic equivalence continuum: Difference between revisions
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The '''Archytas–diatonic | The '''Archytas–limmic''' (or '''septimal–limmic''', '''Archytas–diatonic''', and '''septimal–diatonic''') '''equivalence continuum''' is a [[equivalence continuum|continuum]] of [[2.3.7 subgroup]] temperaments which equate a number of [[64/63|Archytas' commas (64/63)]] with the [[256/243|Pythagorean limma (256/243)]]. This continuum is theoretically interesting in that these are all [[2.3.7 subgroup|2.3.7-subgroup]] temperaments [[support]]ed by [[5edo]]. | ||
All temperaments in the continuum satisfy {{nowrap|(64/63)<sup>''n''</sup> ~ 256/243}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[archy]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 2.3.7 subgroup temperaments supported by 5edo 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 3.3093…, and temperaments near this tend to be the most accurate ones. | All temperaments in the continuum satisfy {{nowrap|(64/63)<sup>''n''</sup> ~ 256/243}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[archy]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 2.3.7 subgroup temperaments supported by 5edo 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 3.3093…, and temperaments near this tend to be the most accurate ones. | ||
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{| 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 16: | Line 16: | ||
|- | |- | ||
| 0 | | 0 | ||
| [https://sintel.pythonanywhere.com/result?subgroup=2.3.7&reduce=off&weights=weil&target=&edos=5+%26+5d&submit_edo=submit&commas= | | [https://sintel.pythonanywhere.com/result?subgroup=2.3.7&reduce=off&weights=weil&target=&edos=5+%26+5d&submit_edo=submit&commas= Blackwood variant] | ||
| [[256/243]] | | [[256/243]] | ||
| {{Monzo| 8 -5 }} | | {{Monzo| 8 -5 }} | ||
| Line 29: | Line 29: | ||
| [[49/48]] | | [[49/48]] | ||
| {{Monzo| -4 -1 2 }} | | {{Monzo| -4 -1 2 }} | ||
|- | |- | ||
| 3 | | 3 | ||
| Line 39: | Line 34: | ||
| [[1029/1024]] | | [[1029/1024]] | ||
| {{Monzo| -10 1 3 }} | | {{Monzo| -10 1 3 }} | ||
|- | |- | ||
| 4 | | 4 | ||
| Line 61: | Line 41: | ||
|- | |- | ||
| 5 | | 5 | ||
| [[Obscenity]] | | [[5th-octave temperaments#Obscenity|Obscenity]] | ||
| [[4194304/4084101]] | | [[4194304/4084101]] | ||
| {{Monzo| 22 -5 -5 }} | | {{Monzo| 22 -5 -5 }} | ||
| Line 68: | Line 48: | ||
| … | | … | ||
| … | | … | ||
| | | … | ||
|- | |- | ||
| ∞ | | ∞ | ||
| Line 74: | Line 54: | ||
| [[64/63]] | | [[64/63]] | ||
| {{Monzo| 6 -2 -1 }} | | {{Monzo| 6 -2 -1 }} | ||
|} | |||
We may invert the continuum by setting ''m'' such that {{nowrap| 1/''m'' + 1/''n'' {{=}} 1 }}. This may be called the ''trienstonic–limmic equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.4330…. The [[trienstonic comma]] is larger than the archytas comma. As such, this continuum does not contain as many useful temperaments, but still interesting nonetheless. | |||
{| class="wikitable center-1" | |||
|+ style="font-size: 105%;" | Temperaments with integer ''m'' | |||
|- | |||
! rowspan="2" | ''m'' | |||
! rowspan="2" | Temperament | |||
! colspan="2" | Comma | |||
|- | |||
! Ratio | |||
! S. monzo | |||
|- | |||
| 0 | |||
| Blackwood variant | |||
| [[256/243]] | |||
| {{Monzo| 8 -5 }} | |||
|- | |||
| 1 | |||
| [[Archy]] | |||
| [[64/63]] | |||
| {{Monzo| 6 -2 -1 }} | |||
|- | |||
| 2 | |||
| [[Semaphore]] | |||
| [[49/48]] | |||
| {{Monzo| -4 -1 2 }} | |||
|- | |||
| … | |||
| … | |||
| … | |||
| … | |||
|- | |||
| ∞ | |||
| [[Trienstonian]] | |||
| [[28/27]] | |||
| {{Monzo| 2 -3 1 }} | |||
|} | |||
{| class="wikitable" | |||
|+ style="font-size: 105%;" | Temperaments with fractional ''n'' and ''m'' | |||
|- | |||
! ''n'' !! ''m'' !! Temperament !! Comma | |||
|- | |||
| 5/2 = 2.5 || 5/3 = 1.{{overline|6}} || [[Cloudy]] retraction || {{Monzo| -14 0 5 }} | |||
|- | |||
| 33/10 = 3.3 || 33/23 = 1.434783… || 5 & 436 || {{Monzo| 118 -16 -33 }} | |||
|- | |||
| 10/3 = 3.{{overline|3}} || 10/7 = 1.{{overline|428571}} || [[Slendroschismic]] || {{Monzo| 36 -5 -10 }} | |||
|- | |||
| 7/2 = 3.5 || 7/5 = 1.4 || [[Septiness]] retraction || {{Monzo| 26 -4 -7 }} | |||
|} | |} | ||
[[Category:5edo]] | [[Category:5edo]] | ||
[[Category:Equivalence continua]] | [[Category:Equivalence continua]] | ||