# 392edo

 ← 391edo 392edo 393edo →
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 392ed2), also called 392-tone equal temperament (392tet) or 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 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 (%) -30.5 -19.6 -48.3 +38.9 -9.7 +42.8 +49.9 -28.5 -18.8 +21.2 -23.6
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