392edo

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← 391edo392edo393edo →
Prime factorization 23 × 72
Step size 3.06122¢
Fifth 229\392 (701.02¢)
Semitones (A1:m2) 35:31 (107.1¢ : 94.9¢)
Consistency limit 7
Distinct consistency limit 7

392 equal divisions of the octave (abbreviated 392edo), or 392-tone equal temperament (392tet), 392 equal temperament (392et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 392 equal parts of about 3.06 ¢ each. Each step of 392edo represents a frequency ratio of 21/392, or the 392nd root of 2.

Theory

392et is consistent to the 7-odd-limit with a flat tendency in the prime harmonics. The equal temperament tempers out the parakleisma in the 5-limit; 321489/320000 (varunisma), 420175/419904 (wizma), 703125/702464 (meter), and 823543/819200 (quince comma) in the 7-limit; and 441/440, 8019/8000, 9801/9800, and 41503/41472 in the 11-limit. It supports qak and octowerck.

Odd harmonics

Approximation of odd harmonics in 392edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error absolute (¢) -0.93 -0.60 -1.48 +1.19 -0.30 +1.31 +1.53 -0.87 -0.57 +0.65 -0.72
relative (%) -31 -20 -48 +39 -10 +43 +50 -29 -19 +21 -24
Steps
(reduced)
621
(229)
910
(126)
1100
(316)
1243
(67)
1356
(180)
1451
(275)
1532
(356)
1602
(34)
1665
(97)
1722
(154)
1773
(205)

Subsets and supersets

392 factors into 23 × 72, with subset edos 2, 4, 7, 8, 14, 28, 49, 56, 98, and 196.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [-621 392 [392 621]] +0.2948 0.2949 9.63
2.3.5 [8 14 -13, [-49 28 2 [392 621 910]] +0.2826 0.2414 7.89
2.3.5.7 321489/320000, 420175/419904, 703125/702464 [392 621 910 1100]] +0.3437 0.2343 7.65
2.3.5.7.11 441/440, 8019/8000, 41503/41472, 703125/702464 [392 621 910 1100 1356]] +0.2922 0.2335 7.63

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 37\392 113.27 16/15 Misneb (5-limit)
1 103\392 315.31 6/5 Parakleismic (5-limit)
1 149\392 456.12 125/96 Qak
8 185\392
(11\392)
566.33
(33.67)
104/75
(55/54)
Octowerck (392f)
28 163\392
(5\392)
498.98
(15.31)
4/3
(105/104)
Oquatonic (5-limit)

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