324edo

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← 323edo324edo325edo →
Prime factorization 22 × 34
Step size 3.7037¢
Fifth 190\324 (703.704¢) (→95\162)
Semitones (A1:m2) 34:22 (125.9¢ : 81.48¢)
Dual sharp fifth 190\324 (703.704¢) (→95\162)
Dual flat fifth 189\324 (700¢) (→7\12)
Dual major 2nd 55\324 (203.704¢)
Consistency limit 3
Distinct consistency limit 3

324 equal divisions of the octave (abbreviated 324edo or 324ed2), also called 324-tone equal temperament (324tet) or 324 equal temperament (324et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 324 equal parts of about 3.704 ¢ each. Each step represents a frequency ratio of 21/324, or the 324th root of 2.

324edo is inconsistent to the 5-odd-limit and harmonic 3 is about halfway between its steps, making it a dual-fifth system, with the flat fifth being the 700-cent fifth coming from 12edo, and the sharp fifth coming from 162edo.

It is nonetheless an excellent 2.9.15.21.11.13 subgroup tuning.

Odd harmonics

Approximation of odd harmonics in 324edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error absolute (¢) +1.75 -1.13 +1.54 -0.21 +0.53 +0.21 +0.62 -1.25 -1.22 -0.41 +1.36
relative (%) +47 -30 +42 -6 +14 +6 +17 -34 -33 -11 +37
Steps
(reduced)
514
(190)
752
(104)
910
(262)
1027
(55)
1121
(149)
1199
(227)
1266
(294)
1324
(28)
1376
(80)
1423
(127)
1466
(170)

Subsets and supersets

Since 324 factors into 22 × 34, 324edo has subset edos 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 81, 108, and 162. 648edo, which doubles it, gives a possible correction to its harmonic 3.