335edo: Difference between revisions
Created page with "{{Infobox ET}} {{EDO intro|335}} == Theory == 335et only is consistent to the 5-odd-limit. Using the patent val, it tempers out the parakleisma in the 5-limit; 4375/43..." |
Review |
||
| Line 3: | Line 3: | ||
== Theory == | == Theory == | ||
335edo only is [[consistent]] to the [[5-odd-limit]]. The equal temperament [[tempering out|tempers out]] {{monzo| 8 14 -13 }} ([[parakleisma]]) and {{monzo| 39 -29 3 }} ([[tricot comma]]), and is a quite efficient [[5-limit]] system. | |||
The 335d [[val]] ({{val| 335 531 778 '''941''' 1159 1240 }}), which scores the best, tempers out [[6144/6125]], [[16875/16807]] and [[14348907/14336000]] in the 7-limit; [[540/539]], 1375/1372, [[3025/3024]], [[5632/5625]] in the 11-limit; and [[729/728]], [[2080/2079]], [[2200/2197]], and [[6656/6655]] in the 13-limit. It [[support]]s [[grendel]]. | |||
The [[patent val]] {{val| 335 531 778 940 }} tempers out the [[3136/3125]] and [[4375/4374]] and in the 7-limit, supporting septimal [[parakleismic]]. This extension tempers out [[441/440]], 5632/5625, and [[19712/19683]] in the 11-limit. The 13-limit version of this, {{val| 335 531 778 940 1159 1240 }}, tempers out [[847/845]], [[1001/1000]], [[1575/1573]], 2200/2197, [[4096/4095]], [[6656/6655]], and [[10648/10647]]. Another 13-limit extension is {{val| 335 531 778 940 1159 '''1239''' }} (335f), where it adds [[364/363]] and 2080/2079 to the comma list. | |||
=== Prime harmonics === | === Prime harmonics === | ||
| Line 12: | Line 13: | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
335 factors into 5 × 67 | Since 335 factors into 5 × 67, 335edo has [[5edo]] and [[67edo]] as its subsets. [[670edo]], which doubles it, gives a good correction to the harmonic 7. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
! rowspan="2" |[[Subgroup]] | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" |[[Comma list|Comma List]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" |[[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" |Optimal<br>8ve Stretch (¢) | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
! colspan="2" |Tuning Error | ! colspan="2" | Tuning Error | ||
|- | |- | ||
![[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
![[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
|- | |- | ||
|2.3 | | 2.3 | ||
|{{monzo|531 -335}} | | {{monzo| 531 -335 }} | ||
|{{mapping|335 531}} | | {{mapping| 335 531 }} | ||
| -0.0424 | | -0.0424 | ||
| 0.0424 | | 0.0424 | ||
| 1.18 | | 1.18 | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|{{monzo|8 14 -13}}, {{monzo|47 -15 -10}} | | {{monzo| 8 14 -13 }}, {{monzo| 47 -15 -10 }} | ||
|{{mapping|335 531 778}} | | {{mapping| 335 531 778 }} | ||
| -0.1075 | | -0.1075 | ||
| 0.0984 | | 0.0984 | ||
| Line 44: | Line 45: | ||
|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br>Ratio* | ! Associated<br>Ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|88\335 | | 88\335 | ||
|315.22 | | 315.22 | ||
|6/5 | | 6/5 | ||
|[[Parakleismic]] | | [[Parakleismic]] (335) | ||
|- | |- | ||
|1 | | 1 | ||
| | | 108\335 | ||
| | | 386.87 | ||
| | | 5/4 | ||
|[[ | | [[Counterwürschmidt]] | ||
|- | |- | ||
|5 | | 1 | ||
|232\335<br>(31\335) | | 158\335 | ||
|831.04<br>(111.04) | | 565.97 | ||
|80/49<br>(16/15) | | 81920/59049 | ||
|[[ | | [[Trident]] (335d)<br>[[Trillium]] / pseudotrillium (335) | ||
|- | |||
| 5 | |||
| 232\335<br>(31\335) | |||
| 831.04<br>(111.04) | |||
| 80/49<br>(16/15) | |||
| [[Quintosec]] | |||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | <nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | ||