# 296edo

 ← 295edo 296edo 297edo →
Prime factorization 23 × 37
Step size 4.05405¢
Fifth 173\296 (701.351¢)
Semitones (A1:m2) 27:23 (109.5¢ : 93.24¢)
Consistency limit 15
Distinct consistency limit 15

296 equal divisions of the octave (abbreviated 296edo or 296ed2), also called 296-tone equal temperament (296tet) or 296 equal temperament (296et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 296 equal parts of about 4.05 ¢ each. Each step represents a frequency ratio of 21/296, or the 296th root of 2.

## Theory

In the 5-limit, 296et not only tempers out the semicomma of 5-limit orwell (orson) temperament, 2109375/2097152, it also provides its optimal patent val, and tempers out the minortone comma, [-16 35 -17. It is also an interesting temperament in higher limits, being distinctly consistent through to the 15-odd-limit. In the 7-limit it tempers out 4375/4374 (ragisma), 16875/16807 (mirkwai), and 118098/117649 (stearnsma), supporting 7-limit octoid and sabric. In the 11-limit, 540/539, 1375/1372, 3025/3024, 4000/3993, 6250/6237 and 9801/9800; in the 13-limit, 625/624, 729/728, 1575/1573, 1716/1715, 2080/2079, and 6656/6655, so that it also supports the 11- and 13-limit versions of octoid. It allows swetismic chords and squbemic chords in the 13-odd-limit, in addition to nicolic chords in the 15-odd-limit.

### Prime harmonics

Approximation of prime harmonics in 296edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.60 -1.18 +0.09 +0.03 -1.34 +0.45 -1.57 +0.10 +0.15 -1.79
Relative (%) +0.0 -14.9 -29.1 +2.3 +0.8 -33.0 +11.1 -38.7 +2.6 +3.8 -44.2
Steps
(reduced)
296
(0)
469
(173)
687
(95)
831
(239)
1024
(136)
1095
(207)
1210
(26)
1257
(73)
1339
(155)
1438
(254)
1466
(282)

### Subsets and supersets

Since 296 factors into 23 × 37, 296edo has subset edos 2, 4, 8, 37, 74 and 148.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [-469 296 [296 469]] +0.1904 0.1905 4.70
2.3.5 2109375/2097152, [-16 35 -17 [296 469 687]] +0.2962 0.2158 5.32
2.3.5.7 4375/4374, 16875/16807, 2100875/2097152 [296 469 687 831]] +0.2138 0.2350 5.80
2.3.5.7.11 540/539, 1375/1372, 4000/3993, 2100875/2097152 [296 469 687 831 1024]] +0.1691 0.2284 5.63
2.3.5.7.11.13 540/539, 625/624, 729/728, 1375/1372, 15379/15360 [296 469 687 831 1024 1095]] +0.2012 0.2206 5.44

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 45\296 182.43 10/9 Mitonic
1 67\296 271.62 75/64 Sabric
1 105\296 425.68 2625/2048 Rainwell
2 57\296 231/08 8/7 Orga
8 144\296
(4\296)
583.78
(16.22)
7/5
(126/125)
Octoid
37 67\296
(3\296)
271.62
(12.16)
117/100
(?)
Dzelic

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