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Created page with "{{Infobox ET}} {{EDO intro|339}} == Theory == 339et is only consistent to the 5-odd-limit. It can be used for the 2.3.5.13.19.23.31.41 subgroup, tempering out 780/779..." |
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 == | ||
339edo is [[enfactoring|enfactored]] in the [[3-limit]] with the same tuning as [[113edo]], and is only [[consistent]] to the [[5-odd-limit]]. Using the 339d val ({{val| 339 537 787 '''951''' }}) in the 7-limit and the 339de val ({{val| 339 537 787 '''951''' '''1172''' }}) in the 11-limit, it [[support]]s [[tritriple]]. | |||
It can be used for the 2.3.5.13.19.23.31.41 [[subgroup]], where it [[tempering out|tempers out]] [[621/620]], 780/779, 1026/1025, 1426/1425, 2945/2944, 14391/14375 and 73853/73800. | |||
=== Odd harmonics === | === Odd harmonics === | ||
| Line 9: | Line 11: | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
339 factors into 3 × 113, | Since 339 factors into 3 × 113, 339edo has [[3edo]] and 113edo as its subsets. [[1017edo]], which triples it, gives a good correction to the harmonics 7 and 11. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
|- | |- | ||
|2 | ! 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|-44 -3 21}} | | {{monzo| -13 17 -6 }}, {{monzo| -44 -3 21 }} | ||
|{{mapping|339 537 787}} | | {{mapping| 339 537 787 }} | ||
| 0.2930 | | +0.2930 | ||
| 0.2828 | | 0.2828 | ||
| 7.99 | | 7.99 | ||
| Line 39: | Line 35: | ||
=== 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> | ! Associated<br />ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|32\339 | | 32\339 | ||
|113.27 | | 113.27 | ||
|16/15 | | 16/15 | ||
|[[Misneb]] | | [[Misneb]] | ||
|- | |- | ||
|1 | | 1 | ||
|146\339 | | 146\339 | ||
|516.81 | | 516.81 | ||
|27/20 | | 27/20 | ||
|[[Gravity]] | | [[Gravity]] | ||
|- | |- | ||
|1 | | 1 | ||
|158\339 | | 158\339 | ||
|559.29 | | 559.29 | ||
|864/625 | | 864/625 | ||
|[[Tritriple]] (339d) | | [[Tritriple]] (339d) | ||
|} | |} | ||
<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>* | |||
Latest revision as of 13:31, 13 March 2026
| ← 338edo | 339edo | 340edo → |
339 equal divisions of the octave (abbreviated 339edo or 339ed2), also called 339-tone equal temperament (339tet) or 339 equal temperament (339et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 339 equal parts of about 3.54 ¢ each. Each step represents a frequency ratio of 21/339, or the 339th root of 2.
Theory
339edo is enfactored in the 3-limit with the same tuning as 113edo, and is only consistent to the 5-odd-limit. Using the 339d val (⟨339 537 787 951]) in the 7-limit and the 339de val (⟨339 537 787 951 1172]) in the 11-limit, it supports tritriple.
It can be used for the 2.3.5.13.19.23.31.41 subgroup, where it tempers out 621/620, 780/779, 1026/1025, 1426/1425, 2945/2944, 14391/14375 and 73853/73800.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.07 | -0.47 | +1.09 | +1.40 | +0.89 | -1.59 | -1.54 | +1.24 | -0.17 | +0.02 | -1.73 |
| Relative (%) | -30.2 | -13.4 | +30.7 | +39.5 | +25.3 | -44.9 | -43.6 | +35.0 | -4.7 | +0.4 | -48.8 | |
| Steps (reduced) |
537 (198) |
787 (109) |
952 (274) |
1075 (58) |
1173 (156) |
1254 (237) |
1324 (307) |
1386 (30) |
1440 (84) |
1489 (133) |
1533 (177) | |
Subsets and supersets
Since 339 factors into 3 × 113, 339edo has 3edo and 113edo as its subsets. 1017edo, which triples it, gives a good correction to the harmonics 7 and 11.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | [-13 17 -6⟩, [-44 -3 21⟩ | [⟨339 537 787]] | +0.2930 | 0.2828 | 7.99 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 32\339 | 113.27 | 16/15 | Misneb |
| 1 | 146\339 | 516.81 | 27/20 | Gravity |
| 1 | 158\339 | 559.29 | 864/625 | Tritriple (339d) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct