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.
{{Infobox ET}}
{{ED intro}}


164 = 4 * 41, with divisors 2, 4, 41, 82
== Theory ==
164 = 4 × 41, and 164edo shares its [[perfect fifth|fifth]] with [[41edo]]. In the 5-limit, 164et tempers out the [[würschmidt comma]], 393216/390625, and the [[vulture comma]], {{monzo| 24 -21 4 }}. It supplies the [[optimal patent val]] for the [[würschmidt]] temperament.
 
In the [[patent val]] {{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 {{nowrap|41 & 123}} temperament, and the 13-limit [[Gamelismic family #Portent|momentous]] temperament, the rank-3 temperament tempering out 196/195, 352/351, 385/384 and 441/440.
 
In the alternative val 164de {{val| 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 ===
{{Harmonics in equal|164}}
 
=== Subsets and supersets ===
Since {{nowrap|164 {{=}} {{factorization|164}}}}, 164edo has subset edos {{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 ==
{| 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 }}
| {{mapping| 164 260 381 }}
| −0.316
| 0.262
| 3.58
|-
| 2.3.5.13
| 676/675, 256000/255879, 393216/390625
| {{mapping| 164 260 381 607 }}
| −0.300
| 0.229
| 3.13
|}
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br />per 8ve
! Generator*
! Cents*
! 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)
|-
| 2
| 25\164
| 182.93
| 10/9
| [[Unidecmic]]
|-
| 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)
| [[Countercomp]]
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 
[[Category:Würschmidt]]

Latest revision as of 18:03, 19 February 2025

← 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. 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

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 distinct

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