393edo

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← 392edo393edo394edo →
Prime factorization 3 × 131
Step size 3.05344¢
Fifth 230\393 (702.29¢)
Semitones (A1:m2) 38:29 (116¢ : 88.55¢)
Consistency limit 5
Distinct consistency limit 5

393 equal divisions of the octave (abbreviated 393edo), or 393-tone equal temperament (393tet), 393 equal temperament (393et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 393 equal parts of about 3.05 ¢ each. Each step of 393edo represents a frequency ratio of 21/393, or the 393rd root of 2.

Theory

393et is only consistent to the 5-limit, with three mappings possible for the 7-limit:

  • 393 623 913 1103] (patent val),
  • 393 623 912 1103] (393c),
  • 393 623 913 1104] (393d).

Using the patent val, it tempers out 393216/390625 and [-46 51 -15 in the 5-limit; 10976/10935, 393216/390625 and 5250987/5242880 in the 7-limit.

Using the 393c val, it tempers out 2109375/2097152 and [32 -48 19 in the 5-limit; 2401/2400, 1071875/1062882 and 2109375/2097152 in the 7-limit.

Using the 393d val, it tempers out 393216/390625 and [-46 51 -15 in the 5-limit; 250047/250000, 2460375/2458624 and 2097152/2083725 in the 7-limit.

Odd harmonics

Approximation of odd harmonics in 393edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error absolute (¢) +0.34 +1.47 -0.89 +0.67 +1.35 -0.83 -1.25 -1.14 -1.33 -0.55 +0.73
relative (%) +11 +48 -29 +22 +44 -27 -41 -37 -44 -18 +24
Steps
(reduced)
623
(230)
913
(127)
1103
(317)
1246
(67)
1360
(181)
1454
(275)
1535
(356)
1606
(34)
1669
(97)
1726
(154)
1778
(206)

Subsets and supersets

393 factors into 3 × 131, with 3edo and 131edo as its subset edos. 786edo, which doubles it, gives a good correction to the harmonic 7.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [623 -393 [393 623]] -0.1057 0.1057 3.43
2.3.5 393216/390625, [-46 51 -15 -0.2819 0.2636 8.63

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 127\393 387.79 5/4 Würschmidt