Archytas–diatonic equivalence continuum: Difference between revisions
Cleanup |
+ m-continuum and separate fractional values |
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| Line 6: | Line 6: | ||
{| 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 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–diatonic 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]] | ||