345edo: Difference between revisions
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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" |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
Since 345 factors into | Since 345 factors into {{factorisation|345}}, 345edo has subset edos {{EDOs| 3, 5, 15, 23, 69, and 115 }}. | ||
== 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 | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
! [[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
| Line 25: | Line 26: | ||
| {{monzo| 547 -345 }} | | {{monzo| 547 -345 }} | ||
| {{mapping| 345 547 }} | | {{mapping| 345 547 }} | ||
| | | −0.2062 | ||
| 0.2062 | | 0.2062 | ||
| 5.93 | | 5.93 | ||
| Line 32: | Line 33: | ||
| {{monzo| 3 -18 11 }}, {{monzo| 47 -15 -10 }} | | {{monzo| 3 -18 11 }}, {{monzo| 47 -15 -10 }} | ||
| {{mapping| 345 547 801 }} | | {{mapping| 345 547 801 }} | ||
| | | −0.1050 | ||
| 0.2210 | | 0.2210 | ||
| 6.35 | | 6.35 | ||
| Line 39: | Line 40: | ||
| 5120/5103, 16875/16807, 68359375/68024448 | | 5120/5103, 16875/16807, 68359375/68024448 | ||
| {{mapping| 345 547 801 969 }} | | {{mapping| 345 547 801 969 }} | ||
| | | −0.2220 | ||
| 0.2788 | | 0.2788 | ||
| 8.02 | | 8.02 | ||
|- | |- | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 540/539, | | 540/539, 1375/1372, 5120/5103, 1953125/1940598 | ||
| {{mapping| 345 547 801 969 1194 }} | | {{mapping| 345 547 801 969 1194 }} | ||
| | | −0.2773 | ||
| 0.2728 | | 0.2728 | ||
| 7.84 | | 7.84 | ||
|- | |- | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 540/539, 625/624, | | 540/539, 625/624, 1375/1372, 4225/4224, 5120/5103 | ||
| {{mapping| 345 547 801 969 1194 1277 }} | | {{mapping| 345 547 801 969 1194 1277 }} | ||
| | | −0.2857 | ||
| 0.2497 | | 0.2497 | ||
| 7.18 | | 7.18 | ||
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=== 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 | |- | ||
! Periods<br />per 8ve | |||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br> | ! Associated<br />ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |||
| 1 | |||
| 13\345 | |||
| 45.22 | |||
| 250/243 | |||
| [[Quartonic]] (5-limit) | |||
|- | |- | ||
| 1 | | 1 | ||
| Line 80: | Line 88: | ||
|- | |- | ||
| 5 | | 5 | ||
| 106\345<br>(32\345) | | 106\345<br />(32\345) | ||
| 368.70<br>(111.30) | | 368.70<br />(111.30) | ||
| 1024/891<br>(16/15) | | 1024/891<br />(16/15) | ||
| [[Quintosec]] (5-limit) | | [[Quintosec]] (5-limit) | ||
|} | |} | ||
<nowiki>* | <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 | ||
[[Category:Kwai]] | [[Category:Kwai]] | ||