369edo

← 368edo

369edo

370edo →

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

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

Template:Comma basis begin |- | 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 Template:Comma basis end

Rank-2 temperaments

Template:Rank-2 begin |- | 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 Template:Rank-2 end Template:Orf