167edo: Difference between revisions

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== 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 [[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.

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.

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== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list~~|Comma List~~]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve ~~Stretch~~ (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

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| {{monzo| 265 -167 }}

| {{mapping| 167 265 }}

| -0.7056

| −0.7056

| 0.7052

| 9.81

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| 393216/390625, {{monzo| 31 -21 1 }}

| {{mapping| 167 265 388 }}

| -0.7158

| −0.7158

| 0.5759

| 8.01

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| 2401/2400, 3136/3125, 179200/177147

| {{mapping| 167 265 388 469 }}

| -0.6467

| −0.6467

| 0.5129

| 7.14

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| 896/891, 2200/2187, 2401/2400, 3136/3125

| {{mapping| 167 265 388 469 578 }}

| -0.6315

| −0.6315

| 0.4598

| 6.40

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| 325/324, 352/351, 364/363, 1001/1000, 1716/1715

| {{mapping| 167 265 388 469 578 618 }}

| -0.5349

| −0.5349

| 0.4721

| 6.57

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| 256/255, 325/324, 352/351, 364/363, 442/441, 1001/1000

| {{mapping| 167 265 388 469 578 618 683 }}

| -0.5573

| −0.5573

| 0.4405

| 6.13

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=== Rank-2 temperaments ===

{| 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 per 8ve

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ~~Ratio~~*

! Associated ratio*

! Temperaments

|-

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| 78\167

| 560.48

| ~~864~~/~~625~~

| 242/175

| [[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

<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 ==

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== 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]

* "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]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|167}}
+{{ED intro}}
 == Theory ==
-et [[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.
+et [[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.
 edo also has a very close approximation to the [[golden magic]] scale.
@@ Line 19: / Line 19: @@
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! 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]] (¢)
@@ Line 31: / Line 32: @@
 | {{monzo| 265 -167 }}
 | {{mapping| 167 265 }}
-| -0.7056
+| −0.7056
 | 0.7052
 | 9.81
@@ Line 38: / Line 39: @@
 | 393216/390625, {{monzo| 31 -21 1 }}
 | {{mapping| 167 265 388 }}
-| -0.7158
+| −0.7158
 | 0.5759
 | 8.01
@@ Line 45: / Line 46: @@
 | 2401/2400, 3136/3125, 179200/177147
 | {{mapping| 167 265 388 469 }}
-| -0.6467
+| −0.6467
 | 0.5129
 | 7.14
@@ Line 52: / Line 53: @@
 | 896/891, 2200/2187, 2401/2400, 3136/3125
 | {{mapping| 167 265 388 469 578 }}
-| -0.6315
+| −0.6315
 | 0.4598
 | 6.40
@@ Line 59: / Line 60: @@
 | 325/324, 352/351, 364/363, 1001/1000, 1716/1715
 | {{mapping| 167 265 388 469 578 618 }}
-| -0.5349
+| −0.5349
 | 0.4721
 | 6.57
@@ Line 66: / Line 67: @@
 | 256/255, 325/324, 352/351, 364/363, 442/441, 1001/1000
 | {{mapping| 167 265 388 469 578 618 683 }}
-| -0.5573
+| −0.5573
 | 0.4405
 | 6.13
@@ Line 73: / Line 74: @@
 === Rank-2 temperaments ===
 {| 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
+|-
+! Periods<br />per 8ve
 ! Generator*
 ! Cents*
-! Associated<br>Ratio*
+! Associated<br />ratio*
 ! Temperaments
 |-
@@ Line 125: / Line 127: @@
 | 78\167
 | 560.48
-| 864/625
+| 242/175
 | [[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
+<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 ==
@@ Line 134: / Line 136: @@
 == 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]
 * "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]]