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{{Infobox ET}}
{{Infobox ET}}
'''167edo''' is the [[EDO|equal division of the octave]] into 167 parts of 7.18562874251 [[cent]]s each. It [[tempering_out|tempers out]] the [[Würschmidt family|würschmidt comma]], 393216/390625 and 10737418240/10460353203 in the [[5-limit]]; [[2401/2400]], [[3136/3125]], and 179200/177147 in the [[7-limit]]; [[896/891]], 2200/2187, and 3388/3375 in the [[11-limit]]; [[325/324]], [[352/351]], [[364/363]], [[1001/1000]], and 1716/1715 in the [[13-limit]], providing the [[optimal patent val]] for 11- and 13-limit [[Porwell temperaments|polypyth temperament]]; [[256/255]], 442/441, [[595/594]], [[715/714]], and [[936/935]] in the [[17-limit]]. It also [[support]]s 11-limit [[Breedsmic temperaments|unthirds temperament]].
{{ED intro}}
 
== Theory ==
167et [[tempering out|tempers out]] the [[würschmidt comma]], 393216/390625, and the leapday comma, {{monzo| 31 -21 1 }}, in the [[5-limit]]; [[2401/2400]], [[3136/3125]], [[6144/6125]], and 179200/177147 in the [[7-limit]]; [[896/891]], [[2200/2187]], [[3025/3024]], [[3388/3375]], and [[4000/3993]] in the [[11-limit]]; [[325/324]], [[352/351]], [[364/363]], [[1001/1000]], and [[1716/1715]] in the [[13-limit]], providing the [[optimal patent val]] for 11- and 13-limit [[polypyth]] temperament; [[256/255]], [[442/441]], [[595/594]], [[715/714]], and [[936/935]] in the [[17-limit]]. It also [[support]]s the 11-limit [[unthirds]] temperament.


167edo also has a very close approximation to the [[golden magic]] scale.
167edo also has a very close approximation to the [[golden magic]] scale.


167edo is the 39th [[prime EDO]].
=== Prime harmonics ===
{{Harmonics in equal|167|intervals=prime|columns=12}}
{{Harmonics in equal|167|intervals=prime|columns=12|start=13|title=Approximation of prime harmonics in 167edo (continued)|collapsed=1}}
 
=== Subsets and supersets ===
167edo is the 39th [[prime edo]].


{{Harmonics in equal|167|intervals=prime|columns=13}}
== Intervals ==
{{Harmonics in equal|167|intervals=prime|start=14|columns=12}}
{{Main|Table of 167edo intervals}}


==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" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
|-
![[TE error|Absolute]] (¢)
! rowspan="2" | [[Subgroup]]
![[TE simple badness|Relative]] (%)
! 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
| 2.3
|{{monzo|265 -167}}
| {{monzo| 265 -167 }}
|{{val|167 265}}
| {{mapping| 167 265 }}
| -0.7056
| −0.7056
| 0.7052
| 0.7052
| 9.81
| 9.81
|-
|-
|2.3.5
| 2.3.5
|{{monzo|17 1 -8}}, {{14 -22 9}}
| 393216/390625, {{monzo| 31 -21 1 }}
|{{val|167 265 388}}
| {{mapping| 167 265 388 }}
| -0.7158
| −0.7158
| 0.5759
| 0.5759
| 8.01
| 8.01
|-
|-
|2.3.5.7
| 2.3.5.7
|6144/6125, 3136/3125, 179200/177147
| 2401/2400, 3136/3125, 179200/177147
|{{val|167 265 388 469}}
| {{mapping| 167 265 388 469 }}
| -0.6467
| −0.6467
| 0.5129
| 0.5129
| 7.14
| 7.14
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|896/891, 2200/2187, 6144/6125, 6250/6237
| 896/891, 2200/2187, 2401/2400, 3136/3125
|{{val|167 265 388 469 578}}
| {{mapping| 167 265 388 469 578 }}
| -0.6315
| −0.6315
| 0.4598
| 0.4598
| 6.40
| 6.40
|-
|-
|2.3.5.7.11.13
| 2.3.5.7.11.13
|325/324, 352/351, 896/891, 1001/1000, 6656/6615
| 325/324, 352/351, 364/363, 1001/1000, 1716/1715
|{{val|167 265 388 469 578 618}}
| {{mapping| 167 265 388 469 578 618 }}
| -0.5349
| −0.5349
| 0.4721
| 0.4721
| 6.57
| 6.57
|-
|-
|2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
|325/324, 352/351, 896/891, 256/255, 1001/1000, 1225/1224
| 256/255, 325/324, 352/351, 364/363, 442/441, 1001/1000
|{{val|167 265 388 469 578 618 683}}
| {{mapping| 167 265 388 469 578 618 683 }}
| -0.5573
| −0.5573
| 0.4405
| 0.4405
| 6.13
| 6.13
|}
|}


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
=== Rank-2 temperaments ===
[[Category:Prime EDO]]
{| 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
| 27\167
| 194.01
| 28/25
| [[Hemiwürschmidt]]
|-
| 1
| 44\167
| 316.17
| 6/5
| [[Counterhanson]]
|-
| 1
| 54\167
| 388.02
| 5/4
| [[Würschmidt]]
|-
| 1
| 58\167
| 416.77
| 14/11
| [[Unthirds]] (11-limit)
|-
| 1
| 63\167
| 452.69
| 125/96
| [[Majo]]
|-
| 1
| 69\167
| 495.81
| 4/3
| [[Polypyth]]
|-
| 1
| 70\167
| 502.99
| 147/110
| [[Quadrawürschmidt]]
|-
| 1
| 78\167
| 560.48
| 242/175
| [[Whoops]]
|}
<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
 
== Scales ==
* [[Helayo19]]
 
== Music ==
; [[Francium]]
* "way too random partying" from ''Helayo EP'' (2023) – [https://open.spotify.com/track/4yf5R4eVOxK2fgZEZRfCqU Spotify] | [https://francium223.bandcamp.com/track/way-too-random-partying Bandcamp] | [https://youtu.be/33T11NI7EQQ?si=mZ57p2EN4uvPCVo7 YouTube] – in Helayo, 167edo tuning
* "moving on" from ''hope in dark times'' (2024) – [https://open.spotify.com/track/5h0JcJ4YTQV20CB9N8S8Af Spotify] | [https://francium223.bandcamp.com/track/moving-on Bandcamp] | [https://www.youtube.com/watch?v=FSjU0-w6XVE YouTube]
* "ordering the universal theme on wish" from ''End of Sartorius Membranes'' (2024) – [https://open.spotify.com/track/00S85fGWQBI19kRwC9GrJ2 Spotify] | [https://francium223.bandcamp.com/track/ordering-the-universal-theme-on-wish Bandcamp] | [https://www.youtube.com/watch?v=g70V2NIPq1I YouTube]
* "Funky Man's Love" from ''Abbreviations Gone Wrong'' (2024) – [https://open.spotify.com/track/0ILOgCY4pzx7S3B51wA9ee Spotify] | [https://francium223.bandcamp.com/track/funky-mans-love Bandcamp] | [https://www.youtube.com/watch?v=4Evj3vX8ZDY YouTube]
* "Don't Bother" from ''Don't'' (2025) – [https://open.spotify.com/track/5B9LMtfG3wTNgQX0PKBFO3 Spotify] | [https://francium223.bandcamp.com/track/dont-bother Bandcamp] | [https://www.youtube.com/watch?v=kzlP4bWfQf8 YouTube]
 
[[Category:Listen]]
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