167edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|167}}
{{ED intro}}


== Theory ==
== 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]], 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 the 11-limit [[Breedsmic temperaments #Unthirds|unthirds temperament]].
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.
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== 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|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
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
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| {{monzo| 265 -167 }}
| {{monzo| 265 -167 }}
| {{mapping| 167 265 }}
| {{mapping| 167 265 }}
| -0.7056
| −0.7056
| 0.7052
| 0.7052
| 9.81
| 9.81
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| 393216/390625, {{monzo| 31 -21 1 }}
| 393216/390625, {{monzo| 31 -21 1 }}
| {{mapping| 167 265 388 }}
| {{mapping| 167 265 388 }}
| -0.7158
| −0.7158
| 0.5759
| 0.5759
| 8.01
| 8.01
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| 2401/2400, 3136/3125, 179200/177147
| 2401/2400, 3136/3125, 179200/177147
| {{mapping| 167 265 388 469 }}
| {{mapping| 167 265 388 469 }}
| -0.6467
| −0.6467
| 0.5129
| 0.5129
| 7.14
| 7.14
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| 896/891, 2200/2187, 2401/2400, 3136/3125
| 896/891, 2200/2187, 2401/2400, 3136/3125
| {{mapping| 167 265 388 469 578 }}
| {{mapping| 167 265 388 469 578 }}
| -0.6315
| −0.6315
| 0.4598
| 0.4598
| 6.40
| 6.40
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| 325/324, 352/351, 364/363, 1001/1000, 1716/1715
| 325/324, 352/351, 364/363, 1001/1000, 1716/1715
| {{mapping| 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
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| 256/255, 325/324, 352/351, 364/363, 442/441, 1001/1000
| 256/255, 325/324, 352/351, 364/363, 442/441, 1001/1000
| {{mapping| 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
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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 8ve
|-
! Generator<br>(reduced)
! Periods<br />per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|27\167
| 27\167
|194.01
| 194.01
|28/25
| 28/25
|[[Hemiwürschmidt]]
| [[Hemiwürschmidt]]
|-
|-
|1
| 1
|44\167
| 44\167
|316.17
| 316.17
|6/5
| 6/5
|[[Counterhanson]]
| [[Counterhanson]]
|-
|-
|1
| 1
|54\167
| 54\167
|388.02
| 388.02
|5/4
| 5/4
|[[Würschmidt]]
| [[Würschmidt]]
|-
|-
|1
| 1
|58\167
| 58\167
|416.77
| 416.77
|14/11
| 14/11
|[[Unthirds]]
| [[Unthirds]] (11-limit)
|-
|-
|1
| 1
|63\167
| 63\167
|452.69
| 452.69
|125/96
| 125/96
|[[Maja]]
| [[Majo]]
|-
|-
|1
| 1
|69\167
| 69\167
|495.81
| 495.81
|4/3
| 4/3
|[[Trisayo]] / [[polypyth]]
| [[Polypyth]]
|-
|-
|1
| 1
|70\167
| 70\167
|502.99
| 502.99
|147/110
| 147/110
|[[Quadrawürschmidt]]
| [[Quadrawürschmidt]]
|-
|-
|1
| 1
|78\167
| 78\167
|560.48
| 560.48
|864/625
| 242/175
|[[Whoosh]] / [[whoops]]
| [[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 ==
== Scales ==
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== Music ==
== Music ==
; [[User:Francium|Francium]]
; [[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] – helayo in 167edo tuning
* "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]
* "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]]

Latest revision as of 13:32, 13 March 2026

← 166edo 167edo 168edo →
Prime factorization 167 (prime)
Step size 7.18563 ¢ 
Fifth 98\167 (704.192 ¢)
Semitones (A1:m2) 18:11 (129.3 ¢ : 79.04 ¢)
Consistency limit 7
Distinct consistency limit 7

167 equal divisions of the octave (abbreviated 167edo or 167ed2), also called 167-tone equal temperament (167tet) or 167 equal temperament (167et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 167 equal parts of about 7.19 ¢ each. Each step represents a frequency ratio of 21/167, or the 167th root of 2.

Theory

167et tempers out the würschmidt comma, 393216/390625, and the leapday comma, [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 supports the 11-limit unthirds temperament.

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

Prime harmonics

Approximation of prime harmonics in 167edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37
Error Absolute (¢) +0.00 +2.24 +1.71 +1.23 +1.98 +0.19 +2.83 -2.90 -3.12 -2.03 -2.52 +0.15
Relative (%) +0.0 +31.1 +23.8 +17.2 +27.5 +2.7 +39.4 -40.4 -43.5 -28.3 -35.1 +2.1
Steps
(reduced) 		167
(0) 		265
(98) 		388
(54) 		469
(135) 		578
(77) 		618
(117) 		683
(15) 		709
(41) 		755
(87) 		811
(143) 		827
(159) 		870
(35)
Approximation of prime harmonics in 167edo (continued)
Harmonic 41 43 47 53 59 61 67 71 73 79 83 89
Error Absolute (¢) +2.08 -1.34 +2.76 +3.14 -2.88 -3.11 -0.27 -0.06 +2.15 +1.93 +2.65 -3.22
Relative (%) +28.9 -18.6 +38.4 +43.7 -40.1 -43.3 -3.7 -0.8 +29.9 +26.9 +36.8 -44.7
Steps
(reduced) 		895
(60) 		906
(71) 		928
(93) 		957
(122) 		982
(147) 		990
(155) 		1013
(11) 		1027
(25) 		1034
(32) 		1053
(51) 		1065
(63) 		1081
(79)

Subsets and supersets

167edo is the 39th prime edo.

Intervals

Main article: Table of 167edo intervals

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [265 -167⟩ [⟨167 265]] −0.7056 0.7052 9.81
2.3.5 393216/390625, [31 -21 1⟩ [⟨167 265 388]] −0.7158 0.5759 8.01
2.3.5.7 2401/2400, 3136/3125, 179200/177147 [⟨167 265 388 469]] −0.6467 0.5129 7.14
2.3.5.7.11 896/891, 2200/2187, 2401/2400, 3136/3125 [⟨167 265 388 469 578]] −0.6315 0.4598 6.40
2.3.5.7.11.13 325/324, 352/351, 364/363, 1001/1000, 1716/1715 [⟨167 265 388 469 578 618]] −0.5349 0.4721 6.57
2.3.5.7.11.13.17 256/255, 325/324, 352/351, 364/363, 442/441, 1001/1000 [⟨167 265 388 469 578 618 683]] −0.5573 0.4405 6.13

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
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

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

Scales

Music

Francium
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