# 164edo

 ← 163edo 164edo 165edo →
Prime factorization 22 × 41
Step size 7.31707¢
Fifth 96\164 (702.439¢) (→24\41)
Semitones (A1:m2) 16:12 (117.1¢ : 87.8¢)
Consistency limit 5
Distinct consistency limit 5

164 equal divisions of the octave (abbreviated 164edo or 164ed2), also called 164-tone equal temperament (164tet) or 164 equal temperament (164et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 164 equal parts of about 7.32 ¢ each. Each step represents a frequency ratio of 21/164, or the 164th root of 2.

## Theory

164 = 4 × 41, and 164edo shares its fifth with 41edo. In the 5-limit, 164et tempers out the würschmidt comma, 393216/390625, and the vulture comma, [24 -21 4. It supplies the optimal patent val for the würschmidt temperament.

In the patent val 164 260 381 460 567 607], it tempers out 196/195, 352/351, 385/384, 441/440, 676/675, and supplies the optimal patent val for the 7-limit, 1/41 octave period 41&123 temperament, and the 13-limit momentous temperament, the rank-3 temperament tempering out 196/195, 352/351, 385/384 and 441/440.

In the alternative val 164de 164 260 381 461 568 607], it tempers out 243/242, 351/350, 364/363, 640/637, 676/675, 729/728, and 1575/1573.

### Prime harmonics

Approximation of prime harmonics in 164edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.48 +1.49 -2.97 -2.54 +0.94 -2.52 +2.49 +0.99 +2.13 -3.57
Relative (%) +0.0 +6.6 +20.4 -40.6 -34.7 +12.8 -34.4 +34.0 +13.6 +29.1 -48.8
Steps
(reduced)
164
(0)
260
(96)
381
(53)
460
(132)
567
(75)
607
(115)
670
(14)
697
(41)
742
(86)
797
(141)
812
(156)

### Subsets and supersets

Since 164 = 22 × 41, 164edo has subset edos 2, 4, 41, 82. 328edo, which doubles it, provides good correction for the approximation to harmonics 7 and 11, and is consistent in the 13-odd-limit.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3.5 393216/390625, [24 -21 4 [164 260 381]] -0.316 0.262 3.58
2.3.5.13 676/675, 256000/255879, 393216/390625 [164 260 381 607]] -0.300 0.229 3.13

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 47\164 343.90 8000/6561 Geb
1 49\164 358.54 16/13 Restles (164)
1 53\164 387.80 5/4 Würschmidt
1 53\164 475.61 320/243 Vulture
1 69\164 504.88 104976/78125 Countermeantone
2 17\164 124.39 275/256 Semivulture (164)
2 25\164 182.93 10/9 Unidecmic
4 68\164
(14\164)
497.56
(102.44)
4/3
(35/33)
Undim (164deff) / unlit (164f)
41 53\164
(1\164)
387.80
(7.32)
5/4
(32805/32768)
Countercomp

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