# 345edo

 ← 344edo 345edo 346edo →
Prime factorization 3 × 5 × 23
Step size 3.47826¢
Fifth 202\345 (702.609¢)
Semitones (A1:m2) 34:25 (118.3¢ : 86.96¢)
Consistency limit 5
Distinct consistency limit 5

345 equal divisions of the octave (abbreviated 345edo or 345ed2), also called 345-tone equal temperament (345tet) or 345 equal temperament (345et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 345 equal parts of about 3.48 ¢ each. Each step represents a frequency ratio of 21/345, or the 345th root of 2.

## Theory

345et is only consistent to the 5-odd-limit, though it has a reasonable 13-limit interpretation using the patent val. It tempers out [3 -18 11 (quartonic comma) and [47 -15 -10 (quintosec comma) in the 5-limit; 5120/5103, 16875/16807, 2460375/2458624, and 68359375/68024448 in the 7-limit; 540/539, 1375/1372, 3025/3024, 16384/16335, 19712/19683, 46656/46585, 200704/200475, and 532400/531441 in the 11-limit; and 625/624 and 4225/4224 in the 13-limit. It provides the optimal patent val for 7-limit kwai.

### Odd harmonics

Approximation of odd harmonics in 345edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +0.65 -0.23 +1.61 +1.31 +1.73 +1.21 +0.43 -0.61 +1.62 -1.22 +1.29
Relative (%) +18.8 -6.5 +46.3 +37.6 +49.6 +34.8 +12.3 -17.5 +46.5 -35.0 +37.1
Steps
(reduced)
547
(202)
801
(111)
969
(279)
1094
(59)
1194
(159)
1277
(242)
1348
(313)
1410
(30)
1466
(86)
1515
(135)
1561
(181)

### Subsets and supersets

Since 345 factors into 3 × 5 × 23, 345edo has subset edos 3, 5, 15, 23, 69, and 115.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [547 -345 [345 547]] -0.2062 0.2062 5.93
2.3.5 [3 -18 11, [47 -15 -10 [345 547 801]] -0.1050 0.2210 6.35
2.3.5.7 5120/5103, 16875/16807, 68359375/68024448 [345 547 801 969]] -0.2220 0.2788 8.02
2.3.5.7.11 540/539, 1375/1372, 5120/5103, 1953125/1940598 [345 547 801 969 1194]] -0.2773 0.2728 7.84
2.3.5.7.11.13 540/539, 625/624, 1375/1372, 4225/4224, 5120/5103 [345 547 801 969 1194 1277]] -0.2857 0.2497 7.18

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 13\345 45.22 250/243 Quartonic (5-limit)
1 38\345 132.17 [-38 5 13 Astro
1 143\345 497.39 4/3 Kwai
5 106\345
(32\345)
368.70
(111.30)
1024/891
(16/15)
Quintosec (5-limit)

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