432edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|432}}
{{ED intro}}


== Theory ==
== Theory ==
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== 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]]
! 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]] (%)
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| {{monzo| 685 -432 }}
| {{monzo| 685 -432 }}
| {{mapping| 432 685 }}
| {{mapping| 432 685 }}
| -0.2596
| −0.2596
| 0.2595
| 0.2595
| 9.34
| 9.34
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| {{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
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| 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
| 8.43
| 8.43
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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>Ratio*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
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|-
|-
| 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 />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

Latest revision as of 06:20, 21 February 2025

← 431edo 432edo 433edo →
Prime factorization 24 × 33
Step size 2.77778 ¢ 
Fifth 253\432 (702.778 ¢)
Semitones (A1:m2) 43:31 (119.4 ¢ : 86.11 ¢)
Consistency limit 7
Distinct consistency limit 7

432 equal divisions of the octave (abbreviated 432edo or 432ed2), also called 432-tone equal temperament (432tet) or 432 equal temperament (432et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 432 equal parts of about 2.78 ¢ each. Each step represents a frequency ratio of 21/432, or the 432nd root of 2.

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

Approximation of odd harmonics in 432edo
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

Table of rank-2 temperaments by generator
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 distinct

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