332edo: Difference between revisions

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Created page with "{{Infobox ET}} {{EDO intro}} == Theory == 332edo tempers out 118098/117649, 134217728/133984375, 29360128/29296875 and 2401/2400 in the 7-limit. It provides the optimal pa..."
 
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"
 
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro}}
{{ED intro}}
 
== Theory ==
== Theory ==
332edo tempers out 118098/117649, 134217728/133984375, 29360128/29296875 and [[2401/2400]] in the 7-limit. It provides the optimal patent val for [[majvamoid]] and [[sedia]].
332edo is [[consistent]] to the [[7-odd-limit]]. The equal temperament [[tempering out|tempers out]] [[2401/2400]], [[19683/19600]], 118098/117649, and 29360128/29296875 in the 7-limit. It provides the [[optimal patent val]] for 11-, 13-, and 17-limit [[sedia]].
 
=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal}}
{{Harmonics in equal|332}}
 
=== Subsets and supersets ===
=== Subsets and supersets ===
332 factors into 2<sup>2</sup> × 83, with subset edos {{EDOs|2, 4, 83, and 166}}.
Since 332 factors into {{factorisation|332}}, 332edo has subset edos {{EDOs| 2, 4, 83, and 166 }}.
 
== 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|-263 166}}
! rowspan="2" | [[Comma list]]
|{{mapping|332 526}}
! rowspan="2" | [[Mapping]]
| 0.2367
! rowspan="2" | Optimal<br />8ve stretch (¢)
| 0.2367
! colspan="2" | Tuning error
| 6.55
|-
|-
|2.3.5
! [[TE error|Absolute]] (¢)
|{{monzo|-13 17 -6}}, {{monzo|-53 10 16}}
! [[TE simple badness|Relative]] (%)
|{{mapping|332 526 771}}
|-
| 0.0955
| 2.3.5
| {{monzo| -13 17 -6 }}, {{monzo| -53 10 16 }}
| {{mapping| 332 526 771 }}
| +0.0955
| 0.2778
| 0.2778
| 7.69
| 7.69
|-
|-
|2.3.5.7
| 2.3.5.7
|2401/2400, 19683/19600, 29360128/29296875
| 2401/2400, 19683/19600, 29360128/29296875
|{{mapping|332 526 771 932}}
| {{mapping| 332 526 771 932 }}
| 0.0851
| +0.0851
| 0.2412
| 0.2412
| 6.67
| 6.67
Line 42: Line 40:
=== 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
|33\332
| 33\332
|119.28
| 119.28
|15/14
| 15/14
|[[Septidiasemi]]
| [[Septidiasemi]]
|-
|-
|1
| 1
|75\332
| 75\332
|271.08
| 271.08
|1024/875
| 1024/875
|[[Quasiorwell]]
| [[Quasiorwell]]
|-
|-
|1
| 1
|127\332
| 127\332
|459.04
| 459.04
|125/96
| 125/96
|[[Majvam]]
| [[Majvam]]
|-
|-
|1
| 1
|143\332
| 143\332
|516.87
| 516.87
|27/20
| 27/20
|[[Gravity]]
| [[Gravity]]
|-
|-
|2
| 2
|143\332<br>(23\332)
| 143\332<br />(23\332)
|516.87<br>(83.13)
| 516.87<br />(83.13)
|27/20<br>(21/20)
| 27/20<br />(21/20)
|[[Harry]]
| [[Harry]]
|-
|-
|2
| 2
|45\332
| 45\332
|162.65
| 162.65
|1125/1024
| 1125/1024
|[[Kwazy]]
| [[Kwazy]]
|}
|}
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|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
[[Category:Sedia]]

Latest revision as of 13:31, 13 March 2026

← 331edo 332edo 333edo →
Prime factorization 22 × 83
Step size 3.61446 ¢ 
Fifth 194\332 (701.205 ¢) (→ 97\166)
Semitones (A1:m2) 30:26 (108.4 ¢ : 93.98 ¢)
Consistency limit 7
Distinct consistency limit 7

332 equal divisions of the octave (abbreviated 332edo or 332ed2), also called 332-tone equal temperament (332tet) or 332 equal temperament (332et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 332 equal parts of about 3.61 ¢ each. Each step represents a frequency ratio of 21/332, or the 332nd root of 2.

Theory

332edo is consistent to the 7-odd-limit. The equal temperament tempers out 2401/2400, 19683/19600, 118098/117649, and 29360128/29296875 in the 7-limit. It provides the optimal patent val for 11-, 13-, and 17-limit sedia.

Prime harmonics

Approximation of prime harmonics in 332edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.75 +0.43 -0.15 +1.69 +1.64 -0.14 -1.13 +0.64 +0.54 +0.75
Relative (%) +0.0 -20.8 +12.0 -4.2 +46.9 +45.4 -3.8 -31.2 +17.7 +15.0 +20.7
Steps
(reduced) 		332
(0) 		526
(194) 		771
(107) 		932
(268) 		1149
(153) 		1229
(233) 		1357
(29) 		1410
(82) 		1502
(174) 		1613
(285) 		1645
(317)

Subsets and supersets

Since 332 factors into 22 × 83, 332edo has subset edos 2, 4, 83, and 166.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3.5 [-13 17 -6⟩, [-53 10 16⟩ [⟨332 526 771]] +0.0955 0.2778 7.69
2.3.5.7 2401/2400, 19683/19600, 29360128/29296875 [⟨332 526 771 932]] +0.0851 0.2412 6.67

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 33\332 119.28 15/14 Septidiasemi
1 75\332 271.08 1024/875 Quasiorwell
1 127\332 459.04 125/96 Majvam
1 143\332 516.87 27/20 Gravity
2 143\332
(23\332) 		516.87
(83.13) 		27/20
(21/20) 		Harry
2 45\332 162.65 1125/1024 Kwazy

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

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