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 [[~~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.

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|+Table of rank-2 temperaments by generator

! Periods per 8ve

! Generator~~ (reduced)~~

! Generator*

! Cents~~ (reduced)~~

! Cents*

! Associated ~~ratio~~

! Associated Ratio*

! 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 ==

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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

* "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]]

@@ Line 3: / Line 3: @@
 == 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 [[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]].
+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.
 edo also has a very close approximation to the [[golden magic]] scale.
@@ Line 75: / Line 75: @@
 |+Table of rank-2 temperaments by generator
 ! Periods<br>per 8ve
-! Generator<br>(reduced)
+! Generator*
-! Cents<br>(reduced)
+! Cents*
-! Associated<br>ratio
+! Associated<br>Ratio*
 ! 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 ==
@@ Line 136: / Line 137: @@
 * "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]
-* "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]]