72edo: Difference between revisions
Dave Keenan (talk | contribs) →Sagittal notation: updated the Sagittal notation section |
m →Regular temperament properties: cleanup |
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! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br | ! rowspan="2" | Optimal<br>8ve stretch (¢) | ||
! colspan="2" | Tuning error | ! colspan="2" | Tuning error | ||
|- | |- | ||
| Line 1,097: | Line 1,097: | ||
| 3.59 | | 3.59 | ||
|} | |} | ||
72et has lower relative errors than any previous equal temperaments in the 7-, 11-, 13-, 17-, and 19-limit. The next equal temperaments doing better in these subgroups are [[99edo|99]], [[270edo|270]], [[224edo|224]], [[494edo|494]], and [[217edo|217]], respectively. | * 72et has lower relative errors than any previous equal temperaments in the 7-, 11-, 13-, 17-, and 19-limit. The next equal temperaments doing better in these subgroups are [[99edo|99]], [[270edo|270]], [[224edo|224]], [[494edo|494]], and [[217edo|217]], respectively. | ||
=== Commas === | === Commas === | ||
| Line 1,103: | Line 1,103: | ||
{| class="commatable wikitable center-1 center-2 right-4" | {| class="commatable wikitable center-1 center-2 right-4" | ||
! [[Harmonic limit|Prime<br | ! [[Harmonic limit|Prime<br>limit]] | ||
! [[Ratio]]<ref group="note">{{rd}}</ref> | ! [[Ratio]]<ref group="note">{{rd}}</ref> | ||
! [[Monzo]] | ! [[Monzo]] | ||
| Line 1,111: | Line 1,111: | ||
| 3 | | 3 | ||
| [[531441/524288|(12 digits)]] | | [[531441/524288|(12 digits)]] | ||
| {{Monzo|-19 12 }} | | {{Monzo| -19 12 }} | ||
| 23.46 | | 23.46 | ||
| Pythagorean comma | | Pythagorean comma | ||
| Line 1,167: | Line 1,167: | ||
| {{Monzo| 0 3 4 -5 }} | | {{Monzo| 0 3 4 -5 }} | ||
| 6.99 | | 6.99 | ||
| Mirkwai | | Mirkwai comma | ||
|- | |- | ||
| 7 | | 7 | ||
| Line 1,173: | Line 1,173: | ||
| {{Monzo| -4 9 -2 -2 }} | | {{Monzo| -4 9 -2 -2 }} | ||
| 7.32 | | 7.32 | ||
| Cataharry | | Cataharry comma | ||
|- | |- | ||
| 7 | | 7 | ||
| Line 1,215: | Line 1,215: | ||
| {{Monzo| -2 0 3 -3 1 }} | | {{Monzo| -2 0 3 -3 1 }} | ||
| 3.78 | | 3.78 | ||
| Moctdel | | Moctdel comma | ||
|- | |- | ||
| 11 | | 11 | ||
| Line 1,227: | Line 1,227: | ||
| {{Monzo| 5 -1 3 0 -3 }} | | {{Monzo| 5 -1 3 0 -3 }} | ||
| 3.03 | | 3.03 | ||
| Wizardharry | | Wizardharry comma | ||
|- | |- | ||
| 11 | | 11 | ||
| Line 1,269: | Line 1,269: | ||
| {{Monzo| 2 -1 0 1 -2 1 }} | | {{Monzo| 2 -1 0 1 -2 1 }} | ||
| 4.76 | | 4.76 | ||
| | | Minor minthma | ||
|- | |- | ||
| 13 | | 13 | ||
| Line 1,328: | Line 1,328: | ||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
|- | |- | ||
! Periods<br | ! Periods<br>per 8ve | ||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br | ! Associated<br>ratio* | ||
! Temperament | ! Temperament | ||
|- | |- | ||
| Line 1,407: | Line 1,407: | ||
|- | |- | ||
| 2 | | 2 | ||
| 21\72<br | | 21\72<br>(19\72) | ||
| 316.7<br | | 316.7<br>(283.3) | ||
| 6/5<br | | 6/5<br>(13/11) | ||
| [[Bikleismic]] | | [[Bikleismic]] | ||
|- | |- | ||
| 2 | | 2 | ||
| 23\72<br | | 23\72<br>(13\72) | ||
| 383.3<br | | 383.3<br>(216.7) | ||
| 5/4<br | | 5/4<br>(17/15) | ||
| [[Wizard]] / lizard / gizzard | | [[Wizard]] / lizard / gizzard | ||
|- | |- | ||
| Line 1,425: | Line 1,425: | ||
|- | |- | ||
| 3 | | 3 | ||
| 19\72<br | | 19\72<br>(5\72) | ||
| 316.7<br | | 316.7<br>(83.3) | ||
| 6/5<br | | 6/5<br>(21/20) | ||
| [[Tritikleismic]] | | [[Tritikleismic]] | ||
|- | |- | ||
| 4 | | 4 | ||
| 19\72<br | | 19\72<br>(1\72) | ||
| 316.7<br | | 316.7<br>(16.7) | ||
| 6/5<br | | 6/5<br>(105/104) | ||
| [[Quadritikleismic]] | | [[Quadritikleismic]] | ||
|- | |- | ||
| 8 | | 8 | ||
| 34\72<br | | 34\72<br>(2\72) | ||
| 566.7<br | | 566.7<br>(33.3) | ||
| 168/121<br | | 168/121<br>(55/54) | ||
| [[Octowerck]] / octowerckis | | [[Octowerck]] / octowerckis | ||
|- | |- | ||
| 8 | | 8 | ||
| 35\72<br | | 35\72<br>(1\72) | ||
| 583.3<br | | 583.3<br>(16.7) | ||
| 7/5<br | | 7/5<br>(100/99) | ||
| [[Octoid]] / octopus | | [[Octoid]] / octopus | ||
|- | |- | ||
| 9 | | 9 | ||
| 19\72<br | | 19\72<br>(3\72) | ||
| 316.7<br | | 316.7<br>(50.0) | ||
| 6/5<br | | 6/5<br>(36/35) | ||
| [[Ennealimmal]] / ennealimnic | | [[Ennealimmal]] / ennealimnic | ||
|- | |- | ||
| 9 | | 9 | ||
| 23\72<br | | 23\72<br>(1\72) | ||
| 383.3<br | | 383.3<br>(16.7) | ||
| 5/4<br | | 5/4<br>(105/104) | ||
| [[Enneaportent]] | | [[Enneaportent]] | ||
|- | |- | ||
| 12 | | 12 | ||
| 23\72<br | | 23\72<br>(1\72) | ||
| 383.3<br | | 383.3<br>(16.7) | ||
| 5/4<br | | 5/4<br>(100/99) | ||
| [[Compton]] / comptone | | [[Compton]] / comptone | ||
|- | |- | ||
| 18 | | 18 | ||
| 19\72<br | | 19\72<br>(1\72) | ||
| 316.7<br | | 316.7<br>(16.7) | ||
| 6/5<br | | 6/5<br>(105/104) | ||
| [[Hemiennealimmal]] | | [[Hemiennealimmal]] | ||
|- | |- | ||
| 24 | | 24 | ||
| 23\72<br | | 23\72<br>(1\72) | ||
| 383.3<br | | 383.3<br>(16.7) | ||
| 5/4<br | | 5/4<br>(105/104) | ||
| [[Hours]] | | [[Hours]] | ||
|- | |- | ||
| 36 | | 36 | ||
| 23\72<br | | 23\72<br>(1\72) | ||
| 383.3<br | | 383.3<br>(16.7) | ||
| 5/4<br | | 5/4<br>(81/80) | ||
| [[Decades]] | | [[Decades]] | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if | <nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
== Scales == | == Scales == | ||