# 292edo

 ← 291edo 292edo 293edo →
Prime factorization 22 × 73
Step size 4.10959¢
Fifth 171\292 (702.74¢)
Semitones (A1:m2) 29:21 (119.2¢ : 86.3¢)
Consistency limit 9
Distinct consistency limit 9

292 equal divisions of the octave (abbreviated 292edo or 292ed2), also called 292-tone equal temperament (292tet) or 292 equal temperament (292et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 292 equal parts of about 4.11 ¢ each. Each step represents a frequency ratio of 21/292, or the 292nd root of 2.

## Theory

292edo is closely related to 146edo, but the patent vals differ on the mapping for 3. The equal temperament tempers out [3 -18 11 (quartonic comma) and [38 -2 -15 (luna/hemithirds comma) in the 5-limit; 5120/5103 (hemifamity), 390625/388962 (dimcomp), 420175/419904 (wizma), and 4802000/4782969 (canousma) in the 7-limit; 1375/1372, 5632/5625, 6250/6237, 9801/9800 and 14641/14580 in the 11-limit; 352/351, 625/624, 847/845, 1716/1715, and 2080/2079 in the 13-limit.

It provides the optimal patent val for the undim temperament in the 7-, 11-, and 13-limit, and notably supports semiseptiquarter and semiluna.

### Prime harmonics

Approximation of prime harmonics in 292edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.78 -0.01 +1.04 -0.63 +1.94 +1.89 -1.62 +0.49 +1.93 +1.54
Relative (%) +0.0 +19.1 -0.3 +25.2 -15.4 +47.2 +46.1 -39.5 +12.0 +47.0 +37.5
Steps
(reduced)
292
(0)
463
(171)
678
(94)
820
(236)
1010
(134)
1081
(205)
1194
(26)
1240
(72)
1321
(153)
1419
(251)
1447
(279)

### Subsets and supersets

Since 292 factors into 22 × 73, 292edo has subset edos 2, 4, 73, and 146.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [463 -292 [292 463]] -0.2476 0.2475 6.02
2.3.5 [3 -18 11, [38 -2 -15 [292 463 678]] -0.1633 0.2346 5.71
2.3.5.7 5120/5103, 390625/388962, 420175/419904 [292 463 678 820]] -0.2148 0.2219 5.40
2.3.5.7.11 1375/1372, 5120/5103, 5632/5625, 14641/14580 [292 463 678 820 1010]] -0.1353 0.2544 6.19
2.3.5.7.11.13 352/351, 625/624, 847/845, 1716/1715, 14641/14580 [292 463 678 820 1010 1081]] -0.3480 0.2736 6.66
2.3.5.7.11.13.17 352/351, 625/624, 715/714, 847/845, 1225/1224, 2025/2023 [292 463 678 820 1010 1081 1194]] -0.2376 0.2696 6.56

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 11\292 45.21 250/243 Quartonic (5-limit)
1 47\292 193.15 262144/234375 Luna
1 59\292 242.47 147/128 Septiquarter
1 111\292 456.16 125/96 Qak
2 47\292 193.15 121/108 Semiluna
2 59\292 242.47 121/105 Semiseptiquarter
4 121\292 497.26 4/3 Undim

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