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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..."
 
Francium (talk | contribs)
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|432}}
{{ED intro}}


== Theory ==
== Theory ==
432et tempers out 283115520/282475249, [[703125/702464]], 102760448/102515625 and [[40353607/40310784]] in the 7-limit. It provides the optimal patent val for the [[maja]] temperament.
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. It factors into 2<sup>4</sup> × 3<sup>3</sup>, with subset edos {{EDOs|2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 36, 48, 54, 72, 108, 144, and 216}}.
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" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|-
|2.3
! rowspan="2" | [[Subgroup]]
|{{monzo|685 -432}}
! rowspan="2" | [[Comma list]]
|{{mapping|432 685}}
! rowspan="2" | [[Mapping]]
| -0.2596
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 685 -432 }}
| {{mapping| 432 685 }}
| −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
| 8.43
| 8.43
Line 44: Line 47:
=== 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
|-
|-
|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 />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if 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
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