# 328edo

 ← 327edo 328edo 329edo →
Prime factorization 23 × 41
Step size 3.65854¢
Fifth 192\328 (702.439¢) (→24\41)
Semitones (A1:m2) 32:24 (117.1¢ : 87.8¢)
Consistency limit 13
Distinct consistency limit 13

328 equal divisions of the octave (abbreviated 328edo or 328ed2), also called 328-tone equal temperament (328tet) or 328 equal temperament (328et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 328 equal parts of about 3.66 ¢ each. Each step represents a frequency ratio of 21/328, or the 328th root of 2.

## Theory

328edo is enfactored in the 5-limit, with the same tuning as 164edo, but the approximation of higher harmonics are much improved. It has a sharp tendency, with harmonics 3 through 17 all tuned sharp. The equal temperament tempers out 2401/2400, 3136/3125, and 6144/6125 in the 7-limit, 9801/9800, 16384/16335 and 19712/19683 in the 11-limit, 676/675, 1001/1000, 1716/1715 and 2080/2079 in the 13-limit, 936/935, 1156/1155 and 2601/2600 in the 17-limit, so that it supports würschmidt and hemiwürschmidt, and provides the optimal patent val for 7-limit hemiwürschmidt, 11- and 13-limit semihemiwür, and 13-limit semiporwell.

### Prime harmonics

Approximation of prime harmonics in 328edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.48 +1.49 +0.69 +1.12 +0.94 +1.14 -1.17 +0.99 -1.53 +0.09
Relative (%) +0.0 +13.2 +40.8 +18.8 +30.6 +25.6 +31.2 -32.0 +27.2 -41.8 +2.4
Steps
(reduced)
328
(0)
520
(192)
762
(106)
921
(265)
1135
(151)
1214
(230)
1341
(29)
1393
(81)
1484
(172)
1593
(281)
1625
(313)

### Subsets and supersets

Since 328 factors into 23 × 41, 328edo has subset edos 2, 4, 8, 41, 82, and 164.

## Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3.5.7 2401/2400, 3136/3125, 589824/588245 [328 520 762 921]] -0.298 0.229 6.27
2.3.5.7.11 2401/2400, 3136/3125, 9801/9800, 19712/19683 [328 520 762 921 1135]] -0.303 0.205 5.61
2.3.5.7.11.13 676/675, 1001/1000, 1716/1715, 3136/3125, 10648/10647 [328 520 762 921 1135 1214]] -0.295 0.188 5.15
2.3.5.7.11.13.17 676/675, 936/935, 1001/1000, 1156/1155, 1716/1715, 3136/3125 [328 520 762 921 1135 1214 1341]] -0.293 0.174 4.77

### Rank-2 temperaments

Note: 5-limit temperaments supported by 164et are not listed.

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 53\328 193.90 28/25 Hemiwürschmidt
1 117\328 428.05 2800/2187 Osiris
2 17\328 62.20 28/27 Eagle
2 111\328
(53\328)
406.10
(193.90)
495/392
(28/25)
Semihemiwürschmidt
8 136\328
(13\328)
497.56
(47.56)
4/3
(36/35)
Twilight
41 49\328
(1\328)
179.27
(3.66)
567/512
(352/351)
Hemicountercomp

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