166edo: Difference between revisions

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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 2 × 83
{{ED intro}}
| Step size = 7.22892¢
| Fifth = 97\166 (701.20¢)
| Major 2nd = 28\166 (202¢)
| Semitones = 15:13 (108¢ : 94¢)
| Consistency = 13
}}
The '''166 equal divisions of the octave''' ('''166edo'''), or the '''166(-tone) equal temperament''' ('''166tet''', '''166et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 166 equal steps of about 7.23 [[cent]]s each, a size close to [[243/242]], the rastma.


== Theory ==
== Theory ==
166edo is consistent through the [[13-odd-limit]], yet its principle interest lies in the usefulness of its approximations. In addition to the 5-limit [[amity comma]], it tempers out [[225/224]], [[325/324]], [[385/384]], [[540/539]], and [[729/728]], hence being an excellent tuning for the [[rank-3 temperament]] [[marvel]], in both the [[11-limit]] and in the 13-limit extension [[hecate]], the [[rank-2 temperament]] [[wizard]], which also tempers out [[4000/3993]], and [[houborizic]], which also tempers out [[2200/2197]], giving the [[optimal patent val]] for all of these. In the [[13-limit]] it tempers out 325/324, leading to hecate, and 1573/1568, leading to marvell, and tempering out both gives [[gizzard]], the 72&94 temperament, for which 166 is an excellent tuning through the [[19-limit]].
166edo is [[consistent]] through the [[13-odd-limit]]. It has a flat tendency, with [[harmonic]]s 3 to 13 all tuned flat. Its principal interest lies in the usefulness of its approximations. In addition to the 5-limit [[amity comma]], it [[tempering out|tempers out]] [[225/224]], [[325/324]], [[385/384]], [[540/539]], and [[729/728]], hence being an excellent tuning for the [[rank-3 temperament]] [[marvel]], in both the [[11-limit]] and in the 13-limit extension [[hecate]], the [[rank-2 temperament]] [[wizard]], which also tempers out [[4000/3993]], and [[houborizic]], which also tempers out [[2200/2197]], giving the [[optimal patent val]] for all of these. In the [[13-limit]] it tempers out 325/324, leading to hecate, and [[1573/1568]], leading to marvell, and tempering out both gives [[gizzard]], the {{nowrap|72 & 94}} temperament, for which 166 is an excellent tuning through the [[19-limit]].  
 
Its prime factorization is 166 = [[2edo|2]] × [[83edo|83]].


166edo (as 83edo) contains a very good approximation of the [[7/4|harmonic 7th]], of which it is only flat by 0.15121 cent.
166edo (as 83edo) contains a very good approximation of the [[7/4|harmonic 7th]], of which it is only flat by 0.15121 cent.


=== Prime harmonics ===
=== Prime harmonics ===
{{Primes in edo|166}}
{{Harmonics in equal|166|intervals=prime}}
 
=== Octave stretch ===
166edo's approximated harmonics 3, 5, 7, 11, and 13 can all be improved by slightly [[stretched and compressed tuning|stretching the octave]], using tunings such as [[263edt]] or [[429ed6]].
 
=== Subsets and supersets ===
Since 166 factors into primes as 2 × 83, 166edo contains [[2edo]] and [[83edo]] as subsets.


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
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| 2.3
| 2.3
| {{monzo| -263 166 }}
| {{monzo| -263 166 }}
| [{{val| 166 263 }}]
| {{mapping| 166 263 }}
| +0.237
| +0.237
| 0.237
| 0.237
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| 2.3.5
| 2.3.5
| 1600000/1594323, {{monzo| -31 2 12 }}
| 1600000/1594323, {{monzo| -31 2 12 }}
| [{{val| 166 263 385 }}]
| {{mapping| 166 263 385 }}
| +0.615
| +0.615
| 0.568
| 0.568
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| 2.3.5.7
| 2.3.5.7
| 225/224, 118098/117649, 1250000/1240029
| 225/224, 118098/117649, 1250000/1240029
| [{{val| 166 263 385 466 }}]
| {{mapping| 166 263 385 466 }}
| +0.474
| +0.474
| 0.549
| 0.549
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| 2.3.5.7.11
| 2.3.5.7.11
| 225/224, 385/384, 4000/3993, 322102/321489
| 225/224, 385/384, 4000/3993, 322102/321489
| [{{val| 166 263 385 466 574 }}]
| {{mapping| 166 263 385 466 574 }}
| +0.490
| +0.490
| 0.492
| 0.492
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 225/224, 325/324, 385/384, 1573/1568, 2200/2197
| 225/224, 325/324, 385/384, 1573/1568, 2200/2197
| [{{val| 166 263 385 466 574 614 }}]
| {{mapping| 166 263 385 466 574 614 }}
| +0.498
| +0.498
| 0.449
| 0.449
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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per octave
|-
! Generator<br>(reduced)
! Periods<br />per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
Line 85: Line 84:
| 339.76
| 339.76
| 243/200
| 243/200
| [[Amity]] / [[houborizic]]
| [[Houborizic]]
|-
|-
| 1
| 1
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| 585.54
| 585.54
| 7/5
| 7/5
| [[Merman]]
| [[Merman]] (7-limit)
|-
|-
| 2
| 2
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| [[Wizard]] / gizzard
| [[Wizard]] / gizzard
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


== Scales ==
== Scales ==
Line 111: Line 111:
* [[Fifteentofourteen]]
* [[Fifteentofourteen]]


[[Category:166edo| ]] <!-- main article -->
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Wizard]]
[[Category:Wizard]]
[[Category:Gizzard]]
[[Category:Houborizic]]
[[Category:Houborizic]]
[[Category:Marvel]]
[[Category:Marvel]]

Latest revision as of 14:16, 24 March 2025

← 165edo 166edo 167edo →
Prime factorization 2 × 83
Step size 7.22892 ¢ 
Fifth 97\166 (701.205 ¢)
Semitones (A1:m2) 15:13 (108.4 ¢ : 93.98 ¢)
Consistency limit 13
Distinct consistency limit 13

166 equal divisions of the octave (abbreviated 166edo or 166ed2), also called 166-tone equal temperament (166tet) or 166 equal temperament (166et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 166 equal parts of about 7.23 ¢ each. Each step represents a frequency ratio of 21/166, or the 166th root of 2.

Theory

166edo is consistent through the 13-odd-limit. It has a flat tendency, with harmonics 3 to 13 all tuned flat. Its principal interest lies in the usefulness of its approximations. In addition to the 5-limit amity comma, it tempers out 225/224, 325/324, 385/384, 540/539, and 729/728, hence being an excellent tuning for the rank-3 temperament marvel, in both the 11-limit and in the 13-limit extension hecate, the rank-2 temperament wizard, which also tempers out 4000/3993, and houborizic, which also tempers out 2200/2197, giving the optimal patent val for all of these. In the 13-limit it tempers out 325/324, leading to hecate, and 1573/1568, leading to marvell, and tempering out both gives gizzard, the 72 & 94 temperament, for which 166 is an excellent tuning through the 19-limit.

166edo (as 83edo) contains a very good approximation of the harmonic 7th, of which it is only flat by 0.15121 cent.

Prime harmonics

Approximation of prime harmonics in 166edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.75 -3.18 -0.15 -1.92 -1.97 +3.48 -1.13 +0.64 -3.07 -2.87
Relative (%) +0.0 -10.4 -44.0 -2.1 -26.6 -27.3 +48.1 -15.6 +8.9 -42.5 -39.7
Steps
(reduced) 		166
(0) 		263
(97) 		385
(53) 		466
(134) 		574
(76) 		614
(116) 		679
(15) 		705
(41) 		751
(87) 		806
(142) 		822
(158)

Octave stretch

166edo's approximated harmonics 3, 5, 7, 11, and 13 can all be improved by slightly stretching the octave, using tunings such as 263edt or 429ed6.

Subsets and supersets

Since 166 factors into primes as 2 × 83, 166edo contains 2edo and 83edo as subsets.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-263 166⟩ [⟨166 263]] +0.237 0.237 3.27
2.3.5 1600000/1594323, [-31 2 12⟩ [⟨166 263 385]] +0.615 0.568 7.86
2.3.5.7 225/224, 118098/117649, 1250000/1240029 [⟨166 263 385 466]] +0.474 0.549 7.59
2.3.5.7.11 225/224, 385/384, 4000/3993, 322102/321489 [⟨166 263 385 466 574]] +0.490 0.492 6.80
2.3.5.7.11.13 225/224, 325/324, 385/384, 1573/1568, 2200/2197 [⟨166 263 385 466 574 614]] +0.498 0.449 6.21

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 33\166 238.55 147/128 Tokko
1 47\166 339.76 243/200 Houborizic
1 81\166 585.54 7/5 Merman (7-limit)
2 30\166 216.87 17/15 Wizard / gizzard

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

Scales

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