405edo: Difference between revisions
Created page with "{{Infobox ET}} {{EDO intro|405}} == Theory == 405et is consistent to the 7-odd-limit. Using the patent val, it tempers out 4096000/4084101, 2460375/2458624, 3355443..." |
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== Theory == | == Theory == | ||
405edo is [[enfactoring|enfactored]] in the 3-limit, with the same tuning as [[135edo]]. Like 135edo, it is [[consistent]] to the [[7-odd-limit]] with a poor approximation to the [[harmonic]] [[5/1|5]]. Using the [[patent val]], the equal temperament [[tempering out|tempers out]] [[15625/15552]] in the 5-limit; [[2100875/2097152]], and 2460375/2458624 in the 7-limit; 1375/1372, [[4000/3993]], [[19712/19683]], and [[41503/41472]] in the 11-limit. It [[support]]s [[marthirds]], [[novemkleismic]] and [[kleirtismic]]. It provides the [[optimal patent val]] for 7- and 11-limit novemkleismic. | |||
=== Prime harmonics === | === Prime harmonics === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
405 factors into 3<sup>4</sup> × 5, | Since 405 factors into 3<sup>4</sup> × 5, 405edo has subset edos {{EDOs| 3, 5, 9, 15, 27, 45, 81, and 135 }}. | ||
== 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]] (%) | ||
|- | |- | ||
| 2.3.5 | |||
| 15625/15552, {{monzo| 110 -65 -3 }} | |||
| {{mapping| 405 642 940 }} | |||
|2.3.5 | |||
|15625/15552, {{monzo|110 -65 -3}} | |||
|{{mapping|405 642 940}} | |||
| +0.1058 | | +0.1058 | ||
| 0.2776 | | 0.2776 | ||
| 9.37 | | 9.37 | ||
|- | |- | ||
|2.3.5.7 | | 2.3.5.7 | ||
|15625/15552, 2460375/2458624 | | 15625/15552, 2100875/2097152, 2460375/2458624 | ||
|{{mapping|405 642 940 1137}} | | {{mapping| 405 642 940 1137 }} | ||
| +0.0737 | | +0.0737 | ||
| 0.2467 | | 0.2467 | ||
| 8.33 | | 8.33 | ||
|- | |- | ||
|2.3.5.7.11 | | 2.3.5.7.11 | ||
|1375/1372, | | 1375/1372, 4000/3993, 19712/19683, 41503/41472 | ||
|{{mapping|405 642 940 1137 1401}} | | {{mapping| 405 642 940 1137 1401 }} | ||
| +0.0709 | | +0.0709 | ||
| 0.2207 | | 0.2207 | ||
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! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|56\405 | | 56\405 | ||
|165.93 | | 165.93 | ||
| | | 11/10 | ||
|[[Satin]] | | [[Satin]] | ||
|- | |- | ||
|1 | | 1 | ||
|107\405 | | 107\405 | ||
|317.04 | | 317.04 | ||
|6/5 | | 6/5 | ||
|[[Hanson]] | | [[Hanson]] | ||
|- | |- | ||
|9 | | 9 | ||
|107\405<br>(17\405) | | 107\405<br>(17\405) | ||
|317.04<br>(50.37) | | 317.04<br>(50.37) | ||
|6/5<br>(36/35) | | 6/5<br>(36/35) | ||
|[[Novemkleismic]] | | [[Novemkleismic]] | ||
|} | |} | ||
<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 | ||