164edo: Difference between revisions

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The ''164 equal division'' divides the octave into 164 equal parts of 7.317 cents each. In the 5-limit it tempers out the würschmidt comma, 393216/390625, and supplies the [[Optimal_patent_val|optimal patent val]] for [[Würschmidt_family|würschmidt temperament]]. In higher limits, also supplies the optimal patent val for the 7-limit, 1/41 octave period 41&123 temperament, and the 13-limit [[Gamelismic_family#Portent|momentous temperament]], the rank-three temperament tempering out 196/195, 352/351, 385/384 and 441/440.
The ''164 equal division'' divides the octave into 164 equal parts of 7.317 cents each. In the 5-limit it tempers out the [[würschmidt comma]], 393216/390625, and supplies the [[optimal patent val]] for the [[würschmidt]] temperament. In higher limits, also supplies the optimal patent val for the 7-limit, 1/41 octave period 41&123 temperament, and the 13-limit the [[Gamelismic family #Portent|momentous]] temperament, the rank-3 temperament tempering out 196/195, 352/351, 385/384 and 441/440.


164 = 4 * 41, with divisors 2, 4, 41, 82
164 = 4 × 41, with divisors 2, 4, 41, 82
 
=== Prime harmonics ===
{{Harmonics in equal|164}}
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
 
|-
| 2.3.5
| 393216/390625, {{monzo| 24 -21 4 }}
| [{{val| 164 260 381 }}]
| -0.316
| 0.262
| 3.58
|-
| 2.3.5.13
| 676/675, 256000/255879, 393216/390625
| [{{val| 164 260 381 607 }}]
| -0.300
| 0.229
| 3.13
|}
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods<br>per Otave
! Generator<br>(Reduced)
! Cents<br>(Reduced)
! Associated<br>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)
|-
| 4
| 68\164<br>(14\164)
| 497.56<br>(102.44)
| 4/3<br>(35/33)
| [[Undim]] (164deff) / [[unlit]] (164f)
|-
| 41
| 53\164<br>(1\164)
| 387.80<br>(7.32)
| 5/4<br>(32805/32768)
| [[Counterpyth]]
|}


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Würschmidt]]

Revision as of 12:29, 26 August 2022

The 164 equal division divides the octave into 164 equal parts of 7.317 cents each. In the 5-limit it tempers out the würschmidt comma, 393216/390625, and supplies the optimal patent val for the würschmidt temperament. In higher limits, also supplies the optimal patent val for the 7-limit, 1/41 octave period 41&123 temperament, and the 13-limit the momentous temperament, the rank-3 temperament tempering out 196/195, 352/351, 385/384 and 441/440.

164 = 4 × 41, with divisors 2, 4, 41, 82

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)

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 Otave 		Generator
(Reduced) 		Cents
(Reduced) 		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)
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) 		Counterpyth
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