369edo

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← 368edo369edo370edo →
Prime factorization 32 × 41
Step size 3.25203¢ 
Fifth 216\369 (702.439¢) (→24\41)
Semitones (A1:m2) 36:27 (117.1¢ : 87.8¢)
Consistency limit 11
Distinct consistency limit 11

369 equal divisions of the octave (abbreviated 369edo or 369ed2), also called 369-tone equal temperament (369tet) or 369 equal temperament (369et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 369 equal parts of about 3.25 ¢ each. Each step represents a frequency ratio of 21/369, or the 369th root of 2.

Theory

369 = 9 × 41, and 369edo shares its fifth with 41edo. It has a sharp tendency, with harmonics 3 through 11 all tuned sharp. The equal temperament tempers out 2401/2400 and 4375/4374 in the 7-limit, so that it supports the ennealimmal temperament; in the 11-limit, 4000/3993, 5632/5625 and 16384/16335. It provides the optimal patent val for the 11-limit 130 & 239 temperament, the 65 & 152 temperament, and the rank-4 temperament tempering out 16384/16335, the semiporwellisma, as well as semiporwellic, the no-7 subgroup version thereof.

Extension to the 13-limit is viable by the 369f val, tempering out 1575/1573, 2080/2079, 2200/2197, and 3584/3575. The optimal tuning of this temperament is consistent in the 15-integer-limit.

Prime harmonics

Approximation of prime harmonics in 369edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.48 +0.68 +0.28 +1.53 -1.50 -0.89 -1.58 -0.63 +1.32 -0.32
Relative (%) +0.0 +14.9 +20.9 +8.6 +47.0 -46.2 -27.4 -48.5 -19.4 +40.5 -9.8
Steps
(reduced)
369
(0)
585
(216)
857
(119)
1036
(298)
1277
(170)
1365
(258)
1508
(32)
1567
(91)
1669
(193)
1793
(317)
1828
(352)

Divisors

Since 369 factors into 32 × 41, 369edo has subset edos 3, 9, 41, and 123.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3.5 [32 -7 -9, [1 -27 18 [369 585 857]] -0.1991 0.1409 4.33
2.3.5.7 2401/2400, 4375/4374, [32 -7 -9 [369 585 857 1036]] -0.1743 0.1294 3.98
2.3.5.7.11 2401/2400, 4000/3993, 4375/4374, 5632/5625 [369 585 857 1036 1277]] -0.2277 0.1576 4.85
2.3.5.7.11.13 1575/1573, 2080/2079, 2200/2197, 2401/2400, 3584/3575 [369 585 857 1036 1277 1366]] (369f) -0.2685 0.1703 5.24

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 17\369 55.28 33/32 Escapade
1 172\369 559.35 864/625 Tritriple (5-limit)
1 181\369 588.62 128/91 Countritonic
9 77\369
(5\369)
250.41
(16.26)
140/121
(100/99)
Semiennealimmal
9 97\369
(15\369)
315.45
(48.78)
6/5
(36/35)
Ennealimmal
9 68\369
(14\369)
221.14
(45.53)
25/22
(77/75)
Quadraennealimmal
41 55\369
(1\369)
178.86
(3.25)
567/512
(352/351)
Hemicountercomp

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