332edo: Difference between revisions
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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}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
| Line 9: | Line 9: | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
Since 332 factors into | 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" | ||
|- | |||
! 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]] (%) | |||
|- | |- | ||
| 2.3.5 | | 2.3.5 | ||
| {{monzo| -13 17 -6 }}, {{monzo| -53 10 16 }} | | {{monzo| -13 17 -6 }}, {{monzo| -53 10 16 }} | ||
| {{mapping| 332 526 771 }} | | {{mapping| 332 526 771 }} | ||
| 0.0955 | | +0.0955 | ||
| 0.2778 | | 0.2778 | ||
| 7.69 | | 7.69 | ||
| Line 24: | Line 33: | ||
| 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 | ||
|} | |||
=== 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 67: | Line 83: | ||
| 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 | |||
[[Category:Sedia]] | [[Category:Sedia]] | ||
Latest revision as of 13:31, 13 March 2026
| ← 331edo | 332edo | 333edo → |
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
| 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
| 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