1323edo

 ← 1322edo 1323edo 1324edo →
Prime factorization 33 × 72
Step size 0.907029¢
Fifth 774\1323 (702.041¢) (→86\147)
Semitones (A1:m2) 126:99 (114.3¢ : 89.8¢)
Consistency limit 29
Distinct consistency limit 29

1323 equal divisions of the octave (abbreviated 1323edo or 1323ed2), also called 1323-tone equal temperament (1323tet) or 1323 equal temperament (1323et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1323 equal parts of about 0.907 ¢ each. Each step represents a frequency ratio of 21/1323, or the 1323rd root of 2.

Theory

1323edo is the smallest edo distinctly consistent in the 29-odd-limit. It is enfactored in the 7-limit, sharing the same excellent 7-limit approximation with 441edo, but it makes for a great higher-limit system by splitting each step of 441edo into three.

It provides the optimal patent val for the 11-limit trinealimmal temperament, which has a period of 1\27 octave.

Prime harmonics

Approximation of prime harmonics in 1323edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 +0.086 +0.081 -0.118 +0.156 +0.289 +0.260 -0.007 +0.297 -0.099 -0.364
Relative (%) +0.0 +9.5 +8.9 -13.1 +17.2 +31.8 +28.7 -0.8 +32.8 -10.9 -40.2
Steps
(reduced)
1323
(0)
2097
(774)
3072
(426)
3714
(1068)
4577
(608)
4896
(927)
5408
(116)
5620
(328)
5985
(693)
6427
(1135)
6554
(1262)

Subsets and supersets

Since 1323 factors into 33 × 72, 1323edo has subset edos 3, 7, 9, 21, 27, 49, 63, 147, 189, 441, of which 441edo is a member of the zeta edos.

Regular temperament properties

Rank-2 temperaments

Note: 7-limit temperaments supported by 441et are not included.

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
27 299\1323
(5\1323)
271.201
(4.535)
1375/1176
(?)
Trinealimmal

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