263edo: Difference between revisions
+regular temperament properties |
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" Tags: Mobile edit Mobile web edit |
||
| (6 intermediate revisions by 3 users not shown) | |||
| Line 1: | Line 1: | ||
{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
Using the | == Theory == | ||
263et [[tempering out|tempers out]] 393216/390625 ([[würschmidt comma]]) and {{monzo| 50 -33 1 }} in the 5-limit. Using the [[patent val]], it tempers out [[4375/4374]], [[50421/50000]], and 458752/455625 in the 7-limit; [[441/440]], [[3388/3375]], [[16384/16335]], and 26411/26244 in the 11-limit; [[364/363]], [[2080/2079]], [[2197/2187]], and 3584/3575 in the 13-limit; [[595/594]], [[833/832]], [[936/935]], and [[1156/1155]] in the 17-limit. | |||
Using the 263df val, it tempers out 352/351, 640/637, 729/728, and 3584/3575 in the 13-limit. | Using the 263d val, it tempers out [[5120/5103]], [[16875/16807]], and 1959552/1953125 in the 7-limit; [[540/539]], 1375/1372, 16384/16335, and 43923/43750 in the 11-limit; [[351/350]], [[1001/1000]], [[1573/1568]], 2197/2187, and [[4225/4224]] in the 13-limit. | ||
===Prime harmonics=== | |||
Using the 263df val, it tempers out [[352/351]], [[640/637]], [[729/728]], and 3584/3575 in the 13-limit. | |||
Finally, it is accurate for the 17th harmonic, as the denominator of a convergent to log<sub>2</sub>17, after [[80edo|80]] and before [[343edo|343]]. | |||
=== Prime harmonics === | |||
{{Harmonics in equal|263}} | {{Harmonics in equal|263}} | ||
===Subsets and supersets=== | |||
263edo is the 56th [[prime | === Subsets and supersets === | ||
==Regular temperament properties== | 263edo is the 56th [[prime edo]]. | ||
Notable supersets include [[789edo]], which triples it to achieve extreme accuracy in the [[2.5.7 subgroup]], and [[1578edo]], which sextuples it to be extremely strong in the [[11-limit]] add-17 and in higher limits. | |||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
|- | |- | ||
|2.3 | ! rowspan="2" | [[Subgroup]] | ||
|{{monzo|417 -263}} | ! rowspan="2" | [[Comma list]] | ||
|{{val|263 417}} | ! rowspan="2" | [[Mapping]] | ||
| | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3 | |||
| {{monzo| 417 -263 }} | |||
| {{val| 263 417 }} | |||
| −0.2229 | |||
| 0.2229 | | 0.2229 | ||
| 4.89 | | 4.89 | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|393216/390625, {{monzo|50 -33 1}} | | 393216/390625, {{monzo| 50 -33 1 }} | ||
|{{val|263 417 611}} | | {{val| 263 417 611 }} | ||
| | | −0.3666 | ||
| 0.2728 | | 0.2728 | ||
| 5.98 | | 5.98 | ||
|} | |} | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | |- | ||
! Generator | ! Periods<br />per 8ve | ||
! Cents | ! Generator* | ||
! Associated<br>ratio | ! Cents* | ||
! Associated<br />ratio* | |||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|40\263 | | 40\263 | ||
|182.51 | | 182.51 | ||
|10/9 | | 10/9 | ||
|[[Minortone]] | | [[Minortone]] | ||
|- | |- | ||
|1 | | 1 | ||
|85\263 | | 85\263 | ||
|387.83 | | 387.83 | ||
|5/4 | | 5/4 | ||
|[[Würschmidt]] | | [[Würschmidt]] | ||
|} | |} | ||
[[ | <nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | ||
[[ | |||