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The ''161 equal division'' divides the octave into 161 equal parts of 7.453 cents each. It 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 [[Breedsmic_temperaments#Mintone|mintone]] temperament in the 5-, 7-, 11- and 13-limits.
{{Infobox ET}}
{{ED intro}}


== Just approximation ==
== Theory ==
161edo is notable as being low in [[29-limit]] relative error in the 100 to 200 range.
161edo has a [[perfect fifth]] slightly sharp of that of [[12edo]], such that it maps the [[Pythagorean comma]] to one step. It approximates many of the low primes fairly well; however, it is only consistent to the [[7-odd-limit]], due to [[10/9]] being mapped too sharply from prime [[5/1|5]] being sharp, while [[3/1|3]] is flat. Nonetheless it does well for its size in higher limits, with the inconsistent intervals in the [[23-odd-limit]] being 9/5, [[13/9]], [[23/13]], and their [[octave complement]]s, and additional inconsistencies in the [[25-odd-limit]] include [[25/18]], [[25/23]], and their octave complements. Prime [[29/1|29]] is also accurate, though harmonic [[27/1|27]] is mapped inconsistently flat, causing many of its intervals to be inconsistent. Additionally, the flatness of 27 causes [[28/27]] to be mapped wider than [[27/26]], meaning 161edo is at most [[diamond monotone]] in the 25-odd-limit.
{{Primes in edo|161|columns=10}}
 
As an equal temperament, 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 ===
 
{{Harmonics in equal|161}}
 
=== Subsets and supersets ===
Since 161 factors into 7 × 23, 161edo contains [[7edo]] and [[23edo]] as its subsets.
 
== Intervals ==
{{Interval table}}
 
== 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
| {{monzo| -255 161 }}
| {{mapping| 161 255 }}
| +0.421
| 0.421
| 5.65
|-
| 2.3.5
| 393216/390625, {{monzo| -17 21 -7 }}
| {{mapping| 161 255 374 }}
| +0.099
| 0.570
| 7.65
|-
| 2.3.5.7
| 2401/2400, 3136/3125, 177147/175000
| {{mapping| 161 255 374 452 }}
| +0.064
| 0.498
| 6.67
|-
| 2.3.5.7.11
| 243/242, 441/440, 3136/3125, 35937/35840
| {{mapping| 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
| {{mapping| 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
| {{mapping| 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
| {{mapping| 161 255 374 452 557 596 658 684 }}
| −0.034
| 0.397
| 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]].
 
=== 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
| 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<br />(2\161)
| 499.38<br />(14.91)
| 4/3<br />(81/80)
| [[Absurdity]]
|}
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 
[[Category:Mintone]]
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