327edo

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← 326edo327edo328edo →
Prime factorization 3 × 109
Step size 3.66972¢ 
Fifth 191\327 (700.917¢)
Semitones (A1:m2) 29:26 (106.4¢ : 95.41¢)
Consistency limit 7
Distinct consistency limit 7

327 equal divisions of the octave (abbreviated 327edo or 327ed2), also called 327-tone equal temperament (327tet) or 327 equal temperament (327et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 327 equal parts of about 3.67 ¢ each. Each step represents a frequency ratio of 21/327, or the 327th root of 2.

Theory

327edo is consistent to the 7-odd-limit, though it has a reasonable approximation to the full 13-limit in its patent val. The equal temperament tempers out the semicomma in the 5-limit; 16875/16807, 19683/19600, 250047/250000, and 2100875/2097152 in the 7-limit; 540/539, 1375/1372, 3025/3024, 8019/8000, 35937/35840, 46656/46585, 102487/102400, 137781/137500, and 160083/160000 in the 11-limit; and 625/624, 1575/1573, 1716/1715, 2200/2197, 4225/4224, and 10648/10647 in the 13-limit. It supports mirkat, pnict, and the subgroup temperament petrtri.

Odd harmonics

Approximation of odd harmonics in 327edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -1.04 -0.99 -0.02 +1.59 -0.86 -0.16 +1.64 +1.47 -0.27 -1.06 -0.75
Relative (%) -28.3 -27.0 -0.5 +43.5 -23.4 -4.4 +44.7 +40.0 -7.2 -28.8 -20.5
Steps
(reduced)
518
(191)
759
(105)
918
(264)
1037
(56)
1131
(150)
1210
(229)
1278
(297)
1337
(29)
1389
(81)
1436
(128)
1479
(171)

Subsets and supersets

Since 327 factors into 3 × 109, 327edo has 3edo and 109edo as its subsets.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [-518 327 [327 518]] 0.3273 0.3274 8.92
2.3.5 2109375/2097152, [-20 39 -18 [327 518 759]] 0.3608 0.2715 7.40
2.3.5.7 16875/16807, 19683/19600, 2100875/2097152 [327 518 759 918]] 0.2722 0.2807 7.65
2.3.5.7.11 540/539, 1375/1372, 8019/8000, 2100875/2097152 [327 518 759 918 1131]] 0.2674 0.2512 6.85
2.3.5.7.11.13 540/539, 625/624, 1575/1573, 2200/2197, 8019/8000 [327 518 759 918 1131 1210]] 0.2301 0.2441 6.65

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 74\327 271.56 75/64 Orson
3 44\327 161.47 192/175 Pnict
3 50\327 183.49 10/9 Mirkat

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