432edo: Difference between revisions
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Created page with "{{Infobox ET}} {{EDO intro|432}} == Theory == 432et tempers out 283115520/282475249, 703125/702464, 102760448/102515625 and 40353607/40310784 in the 7-limit. It provi..." |
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== Theory == | == Theory == | ||
432edo has a reasonable approximation to the 7-limit, where the equal temperament [[tempering out|tempers out]] [[5120/5103], [[703125/702464]], [[40353607/40310784]], 102760448/102515625, and 283115520/282475249. It [[support]]s and provides a good tuning for the 5-limit [[maja]] temperament. | |||
=== Odd harmonics === | === Odd harmonics === | ||
{{Harmonics in equal|432}} | {{Harmonics in equal|432}} | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
432 is a highly factorable number | 432 is a highly factorable number, factoring into {{factorization|432}}, so 432edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 36, 48, 54, 72, 108, 144, and 216 }}. | ||
== 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|685 -432}} | | {{monzo| 685 -432 }} | ||
|{{mapping|432 685}} | | {{mapping| 432 685 }} | ||
| -0.2596 | | -0.2596 | ||
| 0.2595 | | 0.2595 | ||
| 9.34 | | 9.34 | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|{{monzo|41 -20 -4}}, {{monzo|-3 -23 17}} | | {{monzo| 41 -20 -4 }}, {{monzo| -3 -23 17 }} | ||
|{{mapping|432 685 1003}} | | {{mapping| 432 685 1003 }} | ||
| -0.1440 | | -0.1440 | ||
| 0.2676 | | 0.2676 | ||
| 9.63 | | 9.63 | ||
|- | |- | ||
|2.3.5.7 | | 2.3.5.7 | ||
|5120/5103, 703125/702464, 6565234375/6530347008 | | 5120/5103, 703125/702464, 6565234375/6530347008 | ||
|{{mapping|432 685 1003 1213}} | | {{mapping| 432 685 1003 1213 }} | ||
| -0.1631 | | -0.1631 | ||
| 0.2341 | | 0.2341 | ||
| Line 51: | Line 53: | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|163\432 | | 163\432 | ||
|452.78 | | 452.78 | ||
|125/96 | | 125/96 | ||
|[[Maja]] | | [[Maja]] | ||
|- | |- | ||
|4 | | 4 | ||
|179\432<br>(37\432) | | 179\432<br>(37\432) | ||
|497.22<br>(102.78) | | 497.22<br>(102.78) | ||
|4/3 | | 4/3 | ||
|[[Undim]] | | [[Undim]] | ||
|} | |} | ||
<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 | ||
Revision as of 13:59, 9 November 2023
| ← 431edo | 432edo | 433edo → |
Theory
432edo has a reasonable approximation to the 7-limit, where the equal temperament tempers out [[5120/5103], 703125/702464, 40353607/40310784, 102760448/102515625, and 283115520/282475249. It supports and provides a good tuning for the 5-limit maja temperament.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.82 | -0.20 | +0.62 | -1.13 | -1.32 | +1.14 | +0.62 | +0.60 | -0.29 | -1.34 | -0.50 |
| Relative (%) | +29.6 | -7.3 | +22.3 | -40.8 | -47.4 | +41.0 | +22.3 | +21.6 | -10.5 | -48.1 | -17.9 | |
| Steps (reduced) |
685 (253) |
1003 (139) |
1213 (349) |
1369 (73) |
1494 (198) |
1599 (303) |
1688 (392) |
1766 (38) |
1835 (107) |
1897 (169) |
1954 (226) | |
Subsets and supersets
432 is a highly factorable number, factoring into 24 × 33, so 432edo has subset edos 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 36, 48, 54, 72, 108, 144, and 216.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [685 -432⟩ | [⟨432 685]] | -0.2596 | 0.2595 | 9.34 |
| 2.3.5 | [41 -20 -4⟩, [-3 -23 17⟩ | [⟨432 685 1003]] | -0.1440 | 0.2676 | 9.63 |
| 2.3.5.7 | 5120/5103, 703125/702464, 6565234375/6530347008 | [⟨432 685 1003 1213]] | -0.1631 | 0.2341 | 8.43 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 163\432 | 452.78 | 125/96 | Maja |
| 4 | 179\432 (37\432) |
497.22 (102.78) |
4/3 | Undim |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct