335edo
| ← 334edo | 335edo | 336edo → |
335 equal divisions of the octave (abbreviated 335edo or 335ed2), also called 335-tone equal temperament (335tet) or 335 equal temperament (335et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 335 equal parts of about 3.58 ¢ each. Each step represents a frequency ratio of 21/335, or the 335th root of 2.
Theory
335edo only is consistent to the 5-odd-limit. As an equal temperament, it tempers out [8 14 -13⟩ (parakleisma) and [39 -29 3⟩ (tricot comma), and is a quite efficient 5-limit system.
The 335d val (⟨335 531 778 941 1159 1240]), which scores the best, tempers out 6144/6125, 16875/16807 and 14348907/14336000 in the 7-limit; 540/539, 1375/1372, 3025/3024, 5632/5625 in the 11-limit; and 729/728, 2080/2079, 2200/2197, and 6656/6655 in the 13-limit. It supports grendel.
The patent val ⟨335 531 778 940] tempers out the 3136/3125 and 4375/4374 and in the 7-limit, supporting septimal parakleismic. This extension tempers out 441/440, 5632/5625, and 19712/19683 in the 11-limit. The 13-limit version of this, ⟨335 531 778 940 1159 1240], tempers out 847/845, 1001/1000, 1575/1573, 2200/2197, 4096/4095, 6656/6655, and 10648/10647. Another 13-limit extension is ⟨335 531 778 940 1159 1239] (335f), where it adds 364/363 and 2080/2079 to the comma list.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.13 | +0.55 | -1.66 | +0.32 | +1.26 | -1.07 | -0.20 | -1.41 | -1.52 | +1.23 |
| Relative (%) | +0.0 | +3.8 | +15.4 | -46.4 | +9.0 | +35.3 | -30.0 | -5.6 | -39.3 | -42.4 | +34.4 | |
| Steps (reduced) |
335 (0) |
531 (196) |
778 (108) |
940 (270) |
1159 (154) |
1240 (235) |
1369 (29) |
1423 (83) |
1515 (175) |
1627 (287) |
1660 (320) | |
Subsets and supersets
Since 335 factors into 5 × 67, 335edo has 5edo and 67edo as its subsets. 670edo, which doubles it, gives a good correction to the harmonic 7.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [531 -335⟩ | [⟨335 531]] | −0.0424 | 0.0424 | 1.18 |
| 2.3.5 | [8 14 -13⟩, [47 -15 -10⟩ | [⟨335 531 778]] | −0.1075 | 0.0984 | 2.75 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 88\335 | 315.22 | 6/5 | Parakleismic (335) |
| 1 | 108\335 | 386.87 | 5/4 | Counterwürschmidt |
| 1 | 158\335 | 565.97 | 81920/59049 | Trident (335d) Trillium / pseudotrillium (335) |
| 5 | 103\335 (31\335) |
368.96 (111.04) |
99/80 (16/15) |
Quintosec |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct