164edo
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. The 164dg val is a good tuning for 7- to 19-limit buzzard temperament, although if harmonic 11 is desired it is only easily accessible through the patent mapping.
Prime harmonics
| 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
Template:Comma basis begin |- | 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 Template:Comma basis end
Rank-2 temperaments
Template:Rank-2 begin
|-
| 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
Template:Rank-2 end
Template:Orf