339edo: Difference between revisions
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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|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 | ||
|- | |- | ||
| Line 34: | 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>Ratio* | ! Associated<br />Ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
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| [[Tritriple]] (339d) | | [[Tritriple]] (339d) | ||
|} | |} | ||
{{orf}} | |||
Revision as of 01:08, 16 November 2024
| ← 338edo | 339edo | 340edo → |
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) |