# 320edo

 ← 319edo 320edo 321edo →
Prime factorization 26 × 5
Step size 3.75¢
Fifth 187\320 (701.25¢)
Semitones (A1:m2) 29:25 (108.8¢ : 93.75¢)
Consistency limit 19
Distinct consistency limit 19

320 equal divisions of the octave (abbreviated 320edo or 320ed2), also called 320-tone equal temperament (320tet) or 320 equal temperament (320et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 320 equal parts of exactly 3.75 ¢ each. Each step represents a frequency ratio of 21/320, or the 320th root of 2.

## Theory

320edo is consistent in the 19-odd-limit and a fairly good tuning for the 19-limit. It has a flat tendency for most prime harmonics from 3 to 19, with the sole exception of 17.

The equal temperament tempers out 65625/65536 (horwell comma) and 420175/419904 (wizma) in the 7-limit and 441/440, 8019/8000 and 9801/9800 in the 11-limit, and so supports the varuna temperament, the rank-3 temperament tempering out 441/440, 8019/8000 and 9801/9800, for which it provides the optimal patent val. It also provides the optimal patent val for the rank-4 werckismic temperament tempering out 441/440. It tempers out 729/728, 1001/1000, 1575/1573, 4225/4224 and 6656/6655 in the 13-limit, leading to further temperaments for which it provides the optimal patent val, such as tempering out 441/440 with 729/728, 1001/1000 or both, or with 8019/8000, leading to an extension of varuna.

### Prime harmonics

Approximation of prime harmonics in 320edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.71 -0.06 -1.33 -0.07 -0.53 +0.04 -1.26 +1.73 +1.67 -1.29
Relative (%) +0.0 -18.8 -1.7 -35.4 -1.8 -14.1 +1.2 -33.7 +46.0 +44.6 -34.3
Steps
(reduced)
320
(0)
507
(187)
743
(103)
898
(258)
1107
(147)
1184
(224)
1308
(28)
1359
(79)
1448
(168)
1555
(275)
1585
(305)

### Subsets and supersets

Since 320 factors into 26 × 5, 320edo has subset edos 2, 4, 5, 10, 16, 20, 32, 40, 64, 80, and 160.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [-507 320 [320 507]] +0.2224 0.2224 5.93
2.3.5 [23 6 -14, [-28 25 -5 [320 507 743]] +0.1574 0.2036 5.43
2.3.5.7 65625/65536, 235298/234375, 321489/320000 [320 507 743 898]] +0.2361 0.2229 5.94
2.3.5.7.11 441/440, 8019/8000, 41503/41472, 65625/65536 [320 507 743 898 1107]] +0.1928 0.2173 5.80
2.3.5.7.11.13 441/440, 729/728, 1001/1000, 4225/4224, 6656/6655 [320 507 743 898 1107 1184]] +0.1845 0.1993 5.31
2.3.5.7.11.13.17 441/440, 729/728, 833/832, 1001/1000, 1089/1088, 4225/4224 [320 507 743 898 1107 1184 1308]] +0.1565 0.1968 5.25
2.3.5.7.11.13.17.19 441/440, 513/512, 729/728, 833/832, 969/968, 1001/1000, 1521/1520 [320 507 743 898 1107 1184 1308 1359]] +0.1741 0.1899 5.06

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 7\320 26.25 [-2 13 -8 Sfourth (5-limit)
1 131\320 491.25 3645/2744 Fifthplus
1 157\320 588.75 45/32 Untriton (5-limit)
1 93\320 348.75 6144/3757 Hectosaros leap week
2 19\320 71.25 25/24 Narayana
5 133\320
(5\320)
498.75
(18.75)
4/3
(81/80)
Pental
8 133\320
(9\320)
566.25
(33.75)
104/75
(55/54)
Octowerck
10 19\320
(13\320)
71.25
(48.75)
25/24
(36/35)
Decavish
10 133\320
(5\320)
498.75
(18.75)
4/3
(81/80)
Decal
20 151\320
(7\320)
566.25
(26.25)
165/119
(?)
Soviet ferris wheel
32 133\320
(3\320)
498.75
(11.25)
4/3
(?)
Bezique
80 99\320
(3\320)
371.25
(11.25)
2275/1836
(?)
Mercury
80 133\320
(1\320)
498.75
(3.75)
4/3
(245/243)
Octogintic

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct