167edo: Difference between revisions

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== 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]], 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 [[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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|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(reduced)
! Generator*
! Cents<br>(reduced)
! Cents*
! Associated<br>ratio
! 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
| 864/625
|[[Whoosh]] / [[whoops]]
| [[Whoops]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct


== Scales ==
== Scales ==
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* "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] – helayo in 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]
* "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]


[[Category:Listen]]
[[Category:Listen]]

Revision as of 08:44, 5 May 2024

← 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

Template:EDO intro

Theory

167et tempers out the würschmidt comma, 393216/390625, and the leapday comma, [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 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\167 560.48 864/625 Whoops

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

Scales

Music

Francium
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