# 335edo

 ← 334edo 335edo 336edo →
Prime factorization 5 × 67
Step size 3.58209¢
Fifth 196\335 (702.09¢)
Semitones (A1:m2) 32:25 (114.6¢ : 89.55¢)
Consistency limit 5
Distinct consistency limit 5

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. The equal temperament 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

Approximation of prime harmonics in 335edo
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

Table of rank-2 temperaments by generator
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 it is distinct