345edo: Difference between revisions
Created page with "{{Infobox ET}} {{EDO intro|345}} == Theory == 345et is only consistent to the 5-limit. Using the patent val, it tempers out 2460375/2458624, 134217728/133984375, [[5120/5103]..." |
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== Theory == | == Theory == | ||
345et is only consistent to the 5-limit | 345et is only [[consistent]] to the [[5-odd-limit]], though it has a reasonable [[13-limit]] interpretation using the [[patent val]]. It [[tempering out|tempers out]] {{monzo| 3 -18 11 }} (quartonic comma) and {{monzo| 47 -15 -10 }} (quintosec comma) in the 5-limit; [[5120/5103]], [[16875/16807]], 2460375/2458624, and 68359375/68024448 in the 7-limit; [[540/539]], 1375/1372, [[3025/3024]], [[16384/16335]], [[19712/19683]], 46656/46585, [[200704/200475]], and [[532400/531441]] in the 11-limit; and [[625/624]] and [[4225/4224]] in the 13-limit. It provides the [[optimal patent val]] for 7-limit [[kwai]]. | ||
=== Odd harmonics === | === Odd harmonics === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
345 factors into 3 × 5 × 23, | Since 345 factors into 3 × 5 × 23, 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|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|547 -345}} | | {{monzo| 547 -345 }} | ||
|{{mapping|345 547}} | | {{mapping| 345 547 }} | ||
| -0.2062 | | -0.2062 | ||
| 0.2062 | | 0.2062 | ||
| 5.93 | | 5.93 | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|{{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 | | -0.1050 | ||
| 0.2210 | | 0.2210 | ||
| 6.35 | | 6.35 | ||
|- | |- | ||
|2.3.5.7 | | 2.3.5.7 | ||
|5120/5103, 16875/16807, | | 5120/5103, 16875/16807, 68359375/68024448 | ||
|{{mapping|345 547 801 969}} | | {{mapping| 345 547 801 969 }} | ||
| -0.2220 | | -0.2220 | ||
| 0.2788 | | 0.2788 | ||
| 8.02 | | 8.02 | ||
|- | |- | ||
|2.3.5.7.11 | | 2.3.5.7.11 | ||
|540/539, 3025/3024, 5120/5103, 26796875/26763264 | | 540/539, 3025/3024, 5120/5103, 26796875/26763264 | ||
|{{mapping|345 547 801 969 1194}} | | {{mapping| 345 547 801 969 1194 }} | ||
| -0.2773 | | -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, 3025/3024, 5120/5103, 4225/4224 | | 540/539, 625/624, 3025/3024, 5120/5103, 4225/4224 | ||
|{{mapping|345 547 801 969 1194 1277}} | | {{mapping| 345 547 801 969 1194 1277 }} | ||
| -0.2857 | | -0.2857 | ||
| 0.2497 | | 0.2497 | ||
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|+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 | ||
|38\345 | | 38\345 | ||
|132.17 | | 132.17 | ||
|{{monzo|-38 5 13}} | | {{monzo| -38 5 13 }} | ||
|[[Astro]] | | [[Astro]] | ||
|- | |- | ||
|1 | | 1 | ||
|143\345 | | 143\345 | ||
|497.39 | | 497.39 | ||
|4/3 | | 4/3 | ||
|[[Kwai]] | | [[Kwai]] | ||
|- | |- | ||
|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) | ||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
[[Category:Kwai]] | |||