457edo: Difference between revisions
Created page with "{{Infobox ET}} {{EDO intro|457}} == Theory == 457et tempers out 283115520/282475249, 1220703125/1219784832, 26873856/26796875, 65625/65536 and 200120949/200000000 in the 7..." |
Cleanup; clarify the title row of the rank-2 temp table |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|457}} | {{EDO intro|457}} | ||
== Theory == | == Theory == | ||
457edo is [[consistent]] to the [[7-odd-limit]], but the error of [[harmonic]] [[3/1|3]] is quite large. The equal temperament [[tempering out|tempers out]] [[19683/19600]] and [[65625/65536]] in the 7-limit; [[540/539]], [[8019/8000]], and 43923/43904 in the 11-limit. | |||
===Odd harmonics=== | |||
=== Odd harmonics === | |||
{{Harmonics in equal|457}} | {{Harmonics in equal|457}} | ||
===Subsets and supersets=== | |||
=== Subsets and supersets === | |||
457edo is the 88th [[prime edo]]. | 457edo is the 88th [[prime edo]]. | ||
==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|-724 457}} | | {{monzo| -724 457 }} | ||
|{{ | | {{mapping| 457 724 }} | ||
| 0.2716 | | 0.2716 | ||
| 0.2716 | | 0.2716 | ||
| 10.34 | | 10.34 | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|{{monzo|-36 11 8}}, {{monzo|-5 31 -19}} | | {{monzo| -36 11 8 }}, {{monzo| -5 31 -19 }} | ||
|{{ | | {{mapping| 457 724 1061 }} | ||
| 0.2267 | | 0.2267 | ||
| 0.2307 | | 0.2307 | ||
| 8.79 | | 8.79 | ||
|- | |- | ||
|2.3.5.7 | | 2.3.5.7 | ||
|19683/19600, 65625/65536, 7381125/7340032 | | 19683/19600, 65625/65536, 7381125/7340032 | ||
|{{ | | {{mapping| 457 724 1061 1283 }} | ||
| 0.1609 | | 0.1609 | ||
| 0.2300 | | 0.2300 | ||
| 8.76 | | 8.76 | ||
|- | |- | ||
|2.3.5.7.11 | | 2.3.5.7.11 | ||
|540/539, 8019/8000, 19683/19600, 43923/43904 | | 540/539, 8019/8000, 19683/19600, 43923/43904 | ||
|{{ | | {{mapping| 457 724 1061 1283 1581 }} | ||
| 0.1227 | | 0.1227 | ||
| 0.2194 | | 0.2194 | ||
| 8.36 | | 8.36 | ||
|- | |- | ||
|2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
|540/539, 1716/1715, 4225/4224, 41067/40960, 43940/43923 | | 540/539, 1716/1715, 4225/4224, 41067/40960, 43940/43923 | ||
|{{ | | {{mapping| 457 724 1061 1283 1581 1691 }} | ||
| 0.1142 | | 0.1142 | ||
| 0.2012 | | 0.2012 | ||
| 7.66 | | 7.66 | ||
|- | |- | ||
|2.3.5.7.11.13.17 | | 2.3.5.7.11.13.17 | ||
|936/935, 1089/1088, 1275/1274, 1575/1573, 2601/2600, 4225/4224 | | 936/935, 1089/1088, 1275/1274, 1575/1573, 2601/2600, 4225/4224 | ||
|{{ | | {{mapping| 457 724 1061 1283 1581 1691 1868 }} | ||
| 0.0952 | | 0.0952 | ||
| 0.1920 | | 0.1920 | ||
| Line 65: | Line 69: | ||
|+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> | ! Associated<br>Ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|10\457 | | 10\457 | ||
|26.258 | | 26.258 | ||
|49/48 | | 49/48 | ||
|[[Sfourth]] | | [[Sfourth]] | ||
|- | |- | ||
|1 | | 1 | ||
|136\457 | | 136\457 | ||
|357.11 | | 357.11 | ||
|49/40 | | 49/40 | ||
|[[Dodifo]] | | [[Dodifo]] | ||
|- | |- | ||
|1 | | 1 | ||
|213\457 | | 213\457 | ||
|559.30 | | 559.30 | ||
|864/625 | | 864/625 | ||
|[[Tritriple]] | | [[Tritriple]] | ||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||