345edo

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← 344edo345edo346edo →
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 345-tone equal temperament (345tet), 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 of 345edo 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 (%) +19 -7 +46 +38 +50 +35 +12 -17 +47 -35 +37
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