# 176edo

 ← 175edo 176edo 177edo →
Prime factorization 24 × 11
Step size 6.81818¢
Fifth 103\176 (702.273¢)
Semitones (A1:m2) 17:13 (115.9¢ : 88.64¢)
Consistency limit 11
Distinct consistency limit 11

The 176 equal divisions of the octave (176edo), or the 176(-tone) equal temperament (176tet, 176et) when viewed from a regular temperament perspective, is the equal division of the octave into 176 parts of about 6.82 cents each, a size close to 243/242, the rastma.

## Theory

176edo is consistent to the 11-odd-limit. The equal temperament tempers out 78732/78125 (sensipent comma) and [41 -20 -4 (undim comma) in the 5-limit; 6144/6125, 10976/10935, and 50421/50000 in the 7-limit; 441/440, 3388/3375, 6912/6875, 8019/8000, 9801/9800 and 16384/16335 in the 11-limit. Using the patent val, 351/350, 364/363, 2080/2079, 2197/2187, and 4096/4095 in the 13-limit.

It supports the bison temperament and the commatic temperament, and provides the optimal patent val for countermiracle in the 7- and 11-limit, and countermanna, one of the extensions, in the 13-limit.

### Prime harmonics

Approximation of prime harmonics in 176edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.32 +2.32 -0.64 +0.95 -1.89 -2.68 +2.49 -1.00 -0.03 +0.42
Relative (%) +0.0 +4.7 +34.1 -9.4 +14.0 -27.7 -39.3 +36.5 -14.7 -0.5 +6.1
Steps
(reduced)
176
(0)
279
(103)
409
(57)
494
(142)
609
(81)
651
(123)
719
(15)
748
(44)
796
(92)
855
(151)
872
(168)

### Subsets and supersets

Since 176 factors into 24 × 11, 176edo has subset edos 2, 4, 8, 11, 22, 44, and 88.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [279 -176 [176 279]] -0.100 0.100 1.47
2.3.5 78732/78125, [41 -20 -4 [176 279 409]] -0.400 0.432 6.34
2.3.5.7 6144/6125, 10976/10935, 50421/50000 [176 279 409 494]] -0.243 0.463 6.79
2.3.5.7.11 441/440, 3388/3375, 6144/6125, 8019/8000 [176 279 409 494 609]] -0.250 0.414 6.08
2.3.5.7.11.13 351/350, 364/363, 441/440, 2197/2187, 3146/3125 [176 279 409 494 609 651]] -0.123 0.473 6.93

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 17\176 115.91 77/72 Mercy / countermiracle / counterbenediction / countermanna
1 35\176 238.64 147/128 Tokko
1 65\176 443.18 162/125 Sensipent
1 73\176 497.73 4/3 Gary / cotoneum
1 83\176 565.91 13/9 Tricot / trident
2 23\176 20.45 81/80 Commatic
2 23\176 156.82 35/32 Bison
4 73\176
(15\176)
497.73
(102.27)
4/3
(35/33)
Undim
8 73\176
(7\176)
497.73
(47.73)
4/3
(36/35)
Twilight
8 83\176
(5\176)
565.91
(34.09)
168/121
(55/54)
Octowerck (176f) / octowerckis (176)
11 73\176
(7\176)
497.73
(47.73)
4/3
(36/35)
Hendecatonic
22 73\176
(1\176)
497.73
(6.82)
4/3
(385/384)
Icosidillic / major arcana

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