161edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
The '''161 equal divisions of the octave''' ('''161edo'''), or the '''161(-tone) equal temperament''' ('''161tet''', '''161et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 161 [[equal]] parts of about 7.45 [[cent]]s each.
{{EDO intro}}


== Theory ==
== Theory ==
161edo tempers out the [[Würschmidt comma]], 393216/390625, in the 5-limit; [[3136/3125]], [[6144/6125]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]], [[540/539]] and 5632/5625 in the 11-limit; and [[1188/1183]], [[351/350]], [[847/845]], [[1575/1573]], [[1001/1000]] and [[1716/1715]] in the 13-limit. It serves as the [[optimal patent val]] for the [[mintone]] temperament in the 5-, 7-, 11- and 13-limits.
161et [[tempering out|tempers out]] the [[würschmidt comma]], 393216/390625, in the 5-limit; [[3136/3125]], [[6144/6125]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]], [[540/539]] and [[5632/5625]] in the 11-limit; and [[351/350]], [[847/845]], [[1001/1000]], [[1188/1183]], [[1575/1573]] and [[1716/1715]] in the 13-limit. It serves as the [[optimal patent val]] for the [[mintone]] temperament in the 5-, 7-, 11- and 13-limit.


=== Prime harmonics ===
=== Prime harmonics ===
Line 11: Line 11:
== 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|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
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
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| 2.3
| 2.3
| {{monzo| -255 161 }}
| {{monzo| -255 161 }}
| [{{val| 161 255 }}]
| {{mapping| 161 255 }}
| +0.421
| +0.421
| 0.421
| 0.421
Line 29: Line 29:
| 2.3.5
| 2.3.5
| 393216/390625, {{monzo| -17 21 -7 }}
| 393216/390625, {{monzo| -17 21 -7 }}
| [{{val| 161 255 374 }}]
| {{mapping| 161 255 374 }}
| +0.099
| +0.099
| 0.570
| 0.570
Line 36: Line 36:
| 2.3.5.7
| 2.3.5.7
| 2401/2400, 3136/3125, 177147/175000
| 2401/2400, 3136/3125, 177147/175000
| [{{val| 161 255 374 452 }}]
| {{mapping| 161 255 374 452 }}
| +0.064
| +0.064
| 0.498
| 0.498
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| 2.3.5.7.11
| 2.3.5.7.11
| 243/242, 441/440, 3136/3125, 35937/35840
| 243/242, 441/440, 3136/3125, 35937/35840
| [{{val| 161 255 374 452 557 }}]
| {{mapping| 161 255 374 452 557 }}
| +0.037
| +0.037
| 0.448
| 0.448
Line 50: Line 50:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 243/242, 351/350, 441/440, 847/845, 3136/3125
| 243/242, 351/350, 441/440, 847/845, 3136/3125
| [{{val| 161 255 374 452 557 596 }}]
| {{mapping| 161 255 374 452 557 596 }}
| -0.046
| -0.046
| 0.449
| 0.449
Line 57: Line 57:
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 243/242, 351/350, 441/440, 561/560, 847/845, 1089/1088
| 243/242, 351/350, 441/440, 561/560, 847/845, 1089/1088
| [{{val| 161 255 374 452 557 596 658 }}]
| {{mapping| 161 255 374 452 557 596 658 }}
| -0.018
| -0.018
| 0.422
| 0.422
Line 64: Line 64:
| 2.3.5.7.11.13.17.19
| 2.3.5.7.11.13.17.19
| 243/242, 324/323, 351/350, 441/440, 456/455, 495/494, 513/512
| 243/242, 324/323, 351/350, 441/440, 456/455, 495/494, 513/512
| [{{val| 161 255 374 452 557 596 658 684 }}]
| {{mapping| 161 255 374 452 557 596 658 684 }}
| -0.034
| -0.034
| 0.397
| 0.397
| 5.32
| 5.32
|}
|}
 
* 161et has a lower [[TE error|absolute error]] than any previous equal temperaments in the 19-limit, even though it is inconsistent in the corresponding odd limit. The same subgroup is only better tuned by [[183edo|183et]].  
161et has lower [[TE error|absolute error]] than any previous equal temperaments in the 19-limit, even though it is inconsistent. The same subgroup is only better tuned by [[183edo|183et]].  


=== 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
|+Table of rank-2 temperaments by generator
! Periods<br>per octave
! Periods<br>per 8ve
! Generator<br>(reduced)
! Generator*
! Cents<br>(reduced)
! Cents*
! Associated<br>ratio
! Associated<br>Ratio*
! Temperaments
! Temperaments
|-
|-
Line 135: Line 134:
| [[Absurdity]]
| [[Absurdity]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct


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

Revision as of 11:16, 7 May 2024

← 160edo 161edo 162edo →
Prime factorization 7 × 23
Step size 7.45342 ¢ 
Fifth 94\161 (700.621 ¢)
Semitones (A1:m2) 14:13 (104.3 ¢ : 96.89 ¢)
Consistency limit 7
Distinct consistency limit 7

Template:EDO intro

Theory

161et tempers out the würschmidt comma, 393216/390625, in the 5-limit; 3136/3125, 6144/6125 and 2401/2400 in the 7-limit; 243/242, 441/440, 540/539 and 5632/5625 in the 11-limit; and 351/350, 847/845, 1001/1000, 1188/1183, 1575/1573 and 1716/1715 in the 13-limit. It serves as the optimal patent val for the mintone temperament in the 5-, 7-, 11- and 13-limit.

Prime harmonics

In the range of edos from 100 to 200, 161edo is notable as being low in 29-limit relative error.

Approximation of prime harmonics in 161edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -1.33 +1.26 +0.12 +0.23 +1.71 -0.61 +0.62 -2.19 -1.01 +2.79
Relative (%) +0.0 -17.9 +17.0 +1.6 +3.2 +22.9 -8.2 +8.4 -29.3 -13.5 +37.4
Steps
(reduced) 		161
(0) 		255
(94) 		374
(52) 		452
(130) 		557
(74) 		596
(113) 		658
(14) 		684
(40) 		728
(84) 		782
(138) 		798
(154)

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [-255 161⟩ [⟨161 255]] +0.421 0.421 5.65
2.3.5 393216/390625, [-17 21 -7⟩ [⟨161 255 374]] +0.099 0.570 7.65
2.3.5.7 2401/2400, 3136/3125, 177147/175000 [⟨161 255 374 452]] +0.064 0.498 6.67
2.3.5.7.11 243/242, 441/440, 3136/3125, 35937/35840 [⟨161 255 374 452 557]] +0.037 0.448 6.01
2.3.5.7.11.13 243/242, 351/350, 441/440, 847/845, 3136/3125 [⟨161 255 374 452 557 596]] -0.046 0.449 6.03
2.3.5.7.11.13.17 243/242, 351/350, 441/440, 561/560, 847/845, 1089/1088 [⟨161 255 374 452 557 596 658]] -0.018 0.422 5.66
2.3.5.7.11.13.17.19 243/242, 324/323, 351/350, 441/440, 456/455, 495/494, 513/512 [⟨161 255 374 452 557 596 658 684]] -0.034 0.397 5.32
  • 161et has a lower absolute error than any previous equal temperaments in the 19-limit, even though it is inconsistent in the corresponding odd limit. The same subgroup is only better tuned by 183et.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
Ratio* 		Temperaments
1 15\161 111.80 16/15 Vavoom
1 16\161 119.25 15/14 Septidiasemi
1 17\161 126.71 14/13 Mowglic
1 25\161 186.34 10/9 Mintone
1 26\161 193.79 28/25 Hemiwürschmidt
1 38\161 283.23 33/28 Neominor (161f)
1 52\161 387.58 5/4 Würschmidt (5-limit)
1 79\161 588.82 45/32 Aufo
7 67\161
(2\161) 		499.38
(14.91) 		4/3
(81/80) 		Absurdity

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

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