432edo: Difference between revisions
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== Regular temperament properties == | == Regular temperament properties == | ||
{ | {| class="wikitable center-4 center-5 center-6" | ||
|- | |||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br />8ve stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |- | ||
| 2.3 | | 2.3 | ||
| Line 34: | Line 43: | ||
| 0.2341 | | 0.2341 | ||
| 8.43 | | 8.43 | ||
|} | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{ | {| class="wikitable center-all left-5" | ||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br />per 8ve | |||
! Generator* | |||
! Cents* | |||
! Associated<br />ratio* | |||
! Temperaments | |||
|- | |- | ||
| 1 | | 1 | ||
| Line 50: | Line 66: | ||
| 4/3 | | 4/3 | ||
| [[Undim]] | | [[Undim]] | ||
|} | |||
<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 13:16, 16 November 2024
| ← 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