45edo: Difference between revisions
+ subsets and supersets |
→Regular temperament properties: + data and table of rank-2 temps |
||
| Line 497: | Line 497: | ||
== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | |||
|- | |||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br>8ve stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3 | |||
| {{Monzo| -71 45 }} | |||
| {{Mapping| 45 71 }} | |||
| +2.72 | |||
| 2.73 | |||
| 10.2 | |||
|- | |||
| 2.3.5 | |||
| 81/80, {{monzo| -27 1 11 }} | |||
| {{Mapping| 45 71 104 }} | |||
| +3.68 | |||
| 2.61 | |||
| 9.75 | |||
|- | |||
| 2.3.5.7 | |||
| 81/80, 525/512, 2401/2400 | |||
| {{Mapping| 45 71 104 126 }} | |||
| +3.55 | |||
| 2.27 | |||
| 8.49 | |||
|- | |||
| 2.3.5.7.13 | |||
| 65/64, 81/80, 105/104, 2401/2400 | |||
| {{Mapping| 45 71 104 126 166 }} (45f) | |||
| +3.59 | |||
| 2.03 | |||
| 7.60 | |||
|} | |||
=== Commas === | === Commas === | ||
This is a partial list of the [[commas]] that 45et [[tempering out|tempers out]] with its | This is a partial list of the [[commas]] that 45et [[tempering out|tempers out]] with its [[patent val]], {{val| 45 71 104 126 143 156 167 }}. | ||
{| class="commatable wikitable center-1 center-2 right-4 center-5" | {| class="commatable wikitable center-1 center-2 right-4 center-5" | ||
| Line 511: | Line 551: | ||
| 5 | | 5 | ||
| [[81/80]] | | [[81/80]] | ||
| {{ | | {{Monzo| -4 4 -1 }} | ||
| 21.51 | | 21.51 | ||
| Gu | | Gu | ||
| Syntonic comma, Didymus comma, meantone comma | | Syntonic comma, Didymus' comma, meantone comma | ||
|- | |- | ||
| 5 | | 5 | ||
| <abbr title="7629394531250/7625597484987">(26 digits)</abbr> | | <abbr title="7629394531250/7625597484987">(26 digits)</abbr> | ||
| {{ | | {{Monzo| 1 -27 18 }} | ||
| 0.86 | | 0.86 | ||
| Satritribiyo | | Satritribiyo | ||
| Line 525: | Line 565: | ||
| 7 | | 7 | ||
| [[16807/16384]] | | [[16807/16384]] | ||
| {{ | | {{Monzo| -14 0 0 5 }} | ||
| 44.13 | | 44.13 | ||
| Laquinzo | | Laquinzo | ||
| Line 532: | Line 572: | ||
| 7 | | 7 | ||
| [[525/512]] | | [[525/512]] | ||
| {{ | | {{Monzo| -9 1 2 1 }} | ||
| 43.41 | | 43.41 | ||
| Lazoyoyo | | Lazoyoyo | ||
| Line 539: | Line 579: | ||
| 7 | | 7 | ||
| [[875/864]] | | [[875/864]] | ||
| {{ | | {{Monzo| -5 -3 3 1 }} | ||
| 21.90 | | 21.90 | ||
| Zotrigu | | Zotrigu | ||
| Line 546: | Line 586: | ||
| 7 | | 7 | ||
| [[3125/3087]] | | [[3125/3087]] | ||
| {{ | | {{Monzo| 0 -2 5 -3 }} | ||
| 21.18 | | 21.18 | ||
| Triru-aquinyo | | Triru-aquinyo | ||
| Line 553: | Line 593: | ||
| 7 | | 7 | ||
| <abbr title="40353607/40310784">(16 digits)</abbr> | | <abbr title="40353607/40310784">(16 digits)</abbr> | ||
| {{ | | {{Monzo| -11 -9 0 9 }} | ||
| 1.84 | | 1.84 | ||
| Tritrizo | | Tritrizo | ||
| Line 560: | Line 600: | ||
| 7 | | 7 | ||
| [[4375/4374]] | | [[4375/4374]] | ||
| {{ | | {{Monzo| -1 -7 4 1 }} | ||
| 0.40 | | 0.40 | ||
| Zoquadyo | | Zoquadyo | ||
| Line 567: | Line 607: | ||
| 11 | | 11 | ||
| [[45/44]] | | [[45/44]] | ||
| {{ | | {{Monzo| -2 2 1 0 -1 }} | ||
| 38.91 | | 38.91 | ||
| Luyo | | Luyo | ||
| Line 574: | Line 614: | ||
| 11 | | 11 | ||
| [[385/384]] | | [[385/384]] | ||
| {{ | | {{Monzo| -7 -1 1 1 1 }} | ||
| 4.50 | | 4.50 | ||
| Lozoyo | | Lozoyo | ||
| Line 581: | Line 621: | ||
| 11 | | 11 | ||
| <abbr title="117440512/117406179">(18 digits)</abbr> | | <abbr title="117440512/117406179">(18 digits)</abbr> | ||
| {{ | | {{Monzo| 24 -6 0 1 -5 }} | ||
| 0.51 | | 0.51 | ||
| Saquinlu-azo | | Saquinlu-azo | ||
| Line 587: | Line 627: | ||
|} | |} | ||
<references group="note" /> | <references group="note" /> | ||
=== Rank-2 temperaments === | |||
{| class="wikitable center-all left-5" | |||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br>per 8ve | |||
! Generator* | |||
! Cents* | |||
! Associated<br>ratio* | |||
! Temperament | |||
|- | |||
| 1 | |||
| 1\45 | |||
| 26.7 | |||
| 49/48 | |||
| [[Sfourth]] | |||
|- | |||
| 1 | |||
| 2\45 | |||
| 53.3 | |||
| 36/35 | |||
| [[Chromo]] | |||
|- | |||
| 1 | |||
| 7\45 | |||
| 186.7 | |||
| 10/9 | |||
| [[Mintone]] | |||
|- | |||
| 1 | |||
| 11\45 | |||
| 293.3 | |||
| 25/21 | |||
| [[Quasitemp]] | |||
|- | |||
| 1 | |||
| 14\45 | |||
| 373.3 | |||
| 5/4 | |||
| [[Submerged]] | |||
|- | |||
| 1 | |||
| 16\45 | |||
| 426.7 | |||
| 9/7 | |||
| [[Squares]] | |||
|- | |||
| 1 | |||
| 23\45 | |||
| 453.3 | |||
| 13/10 | |||
| [[Maja]] | |||
|- | |||
| 1 | |||
| 19\45 | |||
| 506.7 | |||
| 4/3 | |||
| [[Flattone]] | |||
|- | |||
| 3 | |||
| 19\45<br>(4\45) | |||
| 506.7<br>(106.7) | |||
| 4/3<br>(15/14) | |||
| [[Lithium]] | |||
|- | |||
| 5 | |||
| 19\45<br>(1\45) | |||
| 506.7<br>(26.7) | |||
| 4/3<br>(49/48) | |||
| [[Cloudtone]] | |||
|- | |||
| 9 | |||
| 12\45<br>(2\45) | |||
| 320.0<br>(53.3) | |||
| 6/5<br>(36/35) | |||
| [[Ennealimmal]] | |||
|- | |||
| 15 | |||
| 19\45<br>(1\45) | |||
| 506.7<br>(26.7) | |||
| 4/3<br>(126/125) | |||
| [[Pentadecal]] | |||
|} | |||
<nowiki/>* [[Normal forms|Octave-reduced form]], reduced to the first half-octave, and [[normal forms|minimal form]] in parentheses if distinct | |||
== Octave stretch and compression == | == Octave stretch and compression == | ||