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'''88-cent equal ~~tuning~~''' uses equal steps of 88 [[cent]]s each. It is equivalent to 13.6364edo, and is a subset of [[150edo]] (every eleventh step).

'''88-cent equal temperament''' ('''88cET''', also known as '''1ed88¢''' or '''APS88¢''') uses equal steps of 88 [[cent]]s each. It is equivalent to 13.6364edo, and is a subset of [[150edo]] (every eleventh step).

== Theory ==

88-cent [[Equal-step tuning|equal temperament]] uses 88 cents, or 11\150 of an octave, to generate a [[nonoctave]] rank-1 scale. Since the 88-cent step is an excellent generator for the [[octacot]] temperament, it can be viewed as the generator chain of octacot, stripped of octaves. However viewed, octacot and 88-cent equal temperament are very closely related, and the chords of 88-cent equal temperament are listed on the page [[Chords of octacot]]. From this it may be seen that octacot, and hence 88 cent equal temperament , share an abundance of [[essentially tempered chord]]s.

88~~-cent [[Equal-step tuning|equal tuning]] uses 88~~ cents, ~~or 11\150~~ of ~~an octave~~, ~~to generate a [[nonoctave]] rank one scale~~. ~~Since 88 cents is an excellent generator for~~ [[~~Tetracot family|octacot temperament~~]], ~~it can~~ be ~~viewed as the generator chain~~ of ~~octacot~~, ~~stripped~~ of ~~octaves. However viewed~~, ~~octacot~~ and ~~88-cent equal tuning are very closely related~~, and the ~~chords of 88-cent tuning are listed on the page~~ [[~~Chords of octacot~~]]. ~~From this it may be seen that~~ octacot~~, and hence 88 cents tuning, share an abundance of [[Dyadic chord|essentially tempered chords]]~~.

Eight steps of 88 cents gives 704 cents, two cents sharp of 3/2, and eighteen gives 1584 cents, two cents flat of 5/2. Taken together this tells us that (5/2)4/(3/2)9 = [[20000/19683]], the minimal diesis or tetracot comma, must be being tempered out. Eleven steps of 88 cents gives 968 cents, less than a cent flat of 7/4, and this tells us that (7/4)8/(3/2)11 = 5764801/5668704 must be tempered out also. Taking this, multiplying it by the tetracot comma and taking the fourth root yields [[245/243]], which therefore must be tempered out also. The tetracot comma and 245/243 taken together define 7-limit octacot.

~~Eight~~ steps of 88 cents gives ~~704~~ cents, ~~two cents sharp of~~ 3~~/2, and eighteen gives 1584 cents, two~~ cents ~~flat~~ of ~~5/2. Taken together this tells us that (5~~/~~2)^~~4/~~(3/2)^9 = 20000/19683, the minimal diesis or tetracot comma, must be~~ being tempered out~~. Eleven~~ steps ~~of 88 cents~~ gives ~~968~~ cents, ~~less than a cent flat~~ of 7/4, and ~~this tells us that (7~~/~~4)^8/(3/2)^11 = 5764801/5668704 must be~~ tempered out ~~also~~. ~~Taking this, multiplying it by the tetracot comma and taking the fourth root yields 245/243~~, which ~~therefore must~~ be ~~tempered out also. The tetracot comma and 245/243 taken together define 7-limit octacot~~.

Continuing on, twenty steps of 88 cents gives 1760 cents, which we may compare to the 1751.3 cents of 11/4 and suggests [[100/99]] being tempered out, and four steps gives 352 cents, which may be compared to the 359.5 cents of 16/13, and suggests [[325/324]] being tempered out. These would give an extended octacot, for which 88 cents would be an excellent generator tuning.

~~Continuing on, twenty steps of~~ 88 cents gives 1760 cents, which we may compare to the 1751.3 cents of 11/4 and suggests 100/99 being tempered out, and four steps gives 352 cents, which may be compared to the 359.5 cents of 16/13, and suggests 325/324 being tempered out. These would give an extended octacot, for which 88 cents would be an excellent generator tuning.

=== Harmonics ===

== The 88cET family ==

[[Gary Morrison]] originally conceived of 88-cent equal temperament as composed of steps of exactly 88¢. Nonetheless, composers have recognized a kinship between strict 88cET and some other scales – in particular, the 41ed8 (equivalent to taking three steps of [[41edo]] as a generator with no octaves), the 68ed32 (taking every 5 steps of [[68edo]]), the 109ed256 (taking every 8 steps of [[109edo]]), the 150ed2048 (taking every 11 steps of [[150edo]] i.e. the strict 88cET), the [[8edf]], and the 11ed7/4, the latter being a preferred variant of composer and software designer [[X. J. Scott]]. These cousins of strict 88cET have single steps of approximately 87.805¢, 88.235¢, 88.073¢, 88¢, 87.744¢, and 88.075¢, respectively. These small differences add up, as can be seen by examining the interval list below.

[[Gary Morrison]] originally conceived of 88-cent equal ~~tuning (88cET)~~ as composed of steps of exactly 88¢. Nonetheless, composers have recognized a kinship between strict 88cET and some other scales -- in particular, the ~~41st root of 8~~ (equivalent to taking three steps of [[41edo]] as a generator with no octaves), the ~~8th root~~ of ~~3/2~~, and the ~~11th root of 7~~/4, the latter being a preferred variant of composer and software designer [[X. J. Scott]]. These ~~three~~ cousins of strict 88cET have single steps of approximately 87.805¢, 87.744¢, and 88.075¢, respectively. These small differences add up, as can be seen by examining the interval list below.

== Intervals ==

{| class="wikitable"

|-

! Degree

! ~~11th root of 7~~/4

! 11ed7/4

! 88cET

! ~~41st root of 8~~

! 41ed8

! [[8edf~~|8th root of 3/2]]~~

! 8edf

! Solfege syllable

! Some Nearby JI Intervals

|-

! colspan="6" | ~~'''''~~first octave~~'''''~~

! colspan="6" | first octave

!

|-

Line 140:

Line 141:

| 27/14=1137.039, 31/16=1145.036

|-

! colspan="6" | ~~'''''~~second octave~~'''''~~

! colspan="6" | second octave

!

|-

Line 166:

Line 167:

| re

| 9/8=203.910

|-

~~! colspan="6" |''second nonet''~~

!

|-

| 17

Line 258:

Line 256:

| 63/32=1172.736, 160/81=1178.494

|-

! colspan="6" | ~~'''''~~third octave~~'''''~~

! colspan="6" | third octave

!

|-

Line 300:

Line 298:

| maa

| 81/64=407.820, 33/26=412.745, 14/11=417.508

|-

~~! colspan="6" |''third nonet''~~

!

|-

| 33

Line 368:

Line 363:

| 36/19=1106.397, 243/128=1109.775, 19/10=1111.199, 21/11=1119.463

|-

! colspan="6" | ~~'''''~~fourth octave~~'''''~~ (near match)

! colspan="6" | fourth octave (near match)

!

|-

Line 381:

Line 376:

== Scales ==

* [[Symmetrical scales of 88cET]]

== Music ==

; [[Carlo Serafini]]

* [http://www.seraph.it/dep/det/88east.mp3 88 East]

* [http://www.seraph.it/dep/det/88vocoeast.mp3 88 VocoEast]

* [http://www.seraph.it/dep/det/88Bulgarians.mp3 88 Bulgarians] ([http://www.seraph.it/blog_files/9660ca3450a996ea8b55713cbf36151f-15.html blog entry])

* [http://www.seraph.it/dep/int/88jinglebells.mp3 88 Jingle Bells] ([http://www.seraph.it/blog_files/495ec175ce56cf38cb399d1cd24db164-17.html blog entry])

* [http://www.seraph.it/dep/det/The88thDoor.mp3 The 88th Door] ([http://www.seraph.it/blog_files/927f59ac10125056bcf7871636f246a6-302.html blog entry])

* [[~~symmetrical scales of 88cET~~]]

; [[Chris Vaisvil]]

* [http://micro.soonlabel.com/88cent_nonoctave/STE-004_88_cent_guitar.mp3 88 cent guitar improvisation]

* [http://micro.soonlabel.com/88cent_nonoctave/Prelude_in_88_Cent_Tuning.mp3 A Simple Prelude for 88 Cent Piano] ([http://micro.soonlabel.com/88cent_nonoctave/A_Simple_Prelude_in_88_Cent_Tuning.pdf scordata])

== ~~Compositions ==~~

; [[Mundoworld]]

* "To Become Water" from ''Mundoworld III'' (2021) – [https://open.spotify.com/track/39gEeGXprXGbAnbq0iyjMF Spotify] | [https://www.youtube.com/watch?v=RBv9c_qlFEk YouTube]

* "Mirage Passage" from ''Mirage Passage'' (2024) – [https://open.spotify.com/track/2hAyfHr9XPG96SZPvBNHPP Spotify] | [https://www.youtube.com/watch?v=dWgmmK80I9U YouTube]

* [http://www.seraph.it/dep/det/88east.mp3 88 East] by [[Carlo Serafini]]

== Further reading ==

* ~~[http://www.seraph.it/dep/det/88vocoeast.mp3 88 VocoEast] by~~ [[~~Carlo Serafini~~]]

* [[Gary Morrison]]’s 2001 [https://soundcloud.com/mr88cet/sets/88cet-lecture-demo-gary-morrison-june-2001 lecture about 88cET]

* [~~http~~://~~www~~.~~seraph.it~~/~~dep~~/~~det~~/~~88Bulgarians.mp3 88 Bulgarians] by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/9660ca3450a996ea8b55713cbf36151f~~-~~15.html blog entry])~~

* [http://www.seraph.it/dep/int/88jinglebells.mp3 88 Jingle Bells] by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/495ec175ce56cf38cb399d1cd24db164-~~17.html blog entry])~~

* [http://www.seraph.it/dep/det/The88thDoor.mp3 The 88th Door] by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/927f59ac10125056bcf7871636f246a6-~~302.html blog entry])~~

* [http://micro.soonlabel.com/88cent_nonoctave/STE-~~004_88_cent_guitar.mp3 88 cent guitar improvisation] by [http://www.chrisvaisvil.com Chris Vaisvil]~~

* [http://micro.soonlabel.com/88cent_nonoctave/Prelude_in_88_Cent_Tuning.mp3 A Simple Prelude for 88 Cent Piano] by [http://chrisvaisvil.com/?p=951 Chris Vaisvil] ([http://micro.soonlabel.com/88cent_nonoctave/A_Simple_Prelude_in_88_Cent_Tuning.pdf scordata])

[[Category:Equal-step tuning]]

~~[[Category:Edonoi]]~~

@@ Line 1: / Line 1: @@
-'''88-cent equal tuning''' uses equal steps of 88 [[cent]]s each. It is equivalent to 13.6364edo, and is a subset of [[150edo]] (every eleventh step).
+'''88-cent equal temperament''' ('''88cET''', also known as '''1ed88¢''' or '''APS88¢''') uses equal steps of 88 [[cent]]s each. It is equivalent to 13.6364edo, and is a subset of [[150edo]] (every eleventh step).
 == Theory ==
+-cent [[Equal-step tuning|equal temperament]] uses 88 cents, or 11\150 of an octave, to generate a [[nonoctave]] rank-1 scale. Since the 88-cent step is an excellent generator for the [[octacot]] temperament, it can be viewed as the generator chain of octacot, stripped of octaves. However viewed, octacot and 88-cent equal temperament are very closely related, and the chords of 88-cent equal temperament are listed on the page [[Chords of octacot]]. From this it may be seen that octacot, and hence 88 cent equal temperament , share an abundance of [[essentially tempered chord]]s.
--cent [[Equal-step tuning|equal tuning]] uses 88 cents, or 11\150 of an octave, to generate a [[nonoctave]] rank one scale. Since 88 cents is an excellent generator for [[Tetracot family|octacot temperament]], it can be viewed as the generator chain of octacot, stripped of octaves. However viewed, octacot and 88-cent equal tuning are very closely related, and the chords of 88-cent tuning are listed on the page [[Chords of octacot]]. From this it may be seen that octacot, and hence 88 cents tuning, share an abundance of [[Dyadic chord|essentially tempered chords]].
+Eight steps of 88 cents gives 704 cents, two cents sharp of 3/2, and eighteen gives 1584 cents, two cents flat of 5/2. Taken together this tells us that (5/2)<sup>4</sup>/(3/2)<sup>9</sup> = [[20000/19683]], the minimal diesis or tetracot comma, must be being tempered out. Eleven steps of 88 cents gives 968 cents, less than a cent flat of 7/4, and this tells us that (7/4)<sup>8</sup>/(3/2)<sup>11</sup> = 5764801/5668704 must be tempered out also. Taking this, multiplying it by the tetracot comma and taking the fourth root yields [[245/243]], which therefore must be tempered out also. The tetracot comma and 245/243 taken together define 7-limit octacot.
-Eight steps of 88 cents gives 704 cents, two cents sharp of 3/2, and eighteen gives 1584 cents, two cents flat of 5/2. Taken together this tells us that (5/2)^4/(3/2)^9 = 20000/19683, the minimal diesis or tetracot comma, must be being tempered out. Eleven steps of 88 cents gives 968 cents, less than a cent flat of 7/4, and this tells us that (7/4)^8/(3/2)^11 = 5764801/5668704 must be tempered out also. Taking this, multiplying it by the tetracot comma and taking the fourth root yields 245/243, which therefore must be tempered out also. The tetracot comma and 245/243 taken together define 7-limit octacot.
+Continuing on, twenty steps of 88 cents gives 1760 cents, which we may compare to the 1751.3 cents of 11/4 and suggests [[100/99]] being tempered out, and four steps gives 352 cents, which may be compared to the 359.5 cents of 16/13, and suggests [[325/324]] being tempered out. These would give an extended octacot, for which 88 cents would be an excellent generator tuning.
-Continuing on, twenty steps of 88 cents gives 1760 cents, which we may compare to the 1751.3 cents of 11/4 and suggests 100/99 being tempered out, and four steps gives 352 cents, which may be compared to the 359.5 cents of 16/13, and suggests 325/324 being tempered out. These would give an extended octacot, for which 88 cents would be an excellent generator tuning.
+=== Harmonics ===
+{{Harmonics in cet|88}}
 == The 88cET family ==
+[[Gary Morrison]] originally conceived of 88-cent equal temperament as composed of steps of exactly 88¢. Nonetheless, composers have recognized a kinship between strict 88cET and some other scales – in particular, the 41ed8 (equivalent to taking three steps of [[41edo]] as a generator with no octaves), the 68ed32 (taking every 5 steps of [[68edo]]), the 109ed256 (taking every 8 steps of [[109edo]]), the 150ed2048 (taking every 11 steps of [[150edo]] i.e. the strict 88cET), the [[8edf]], and the 11ed7/4, the latter being a preferred variant of composer and software designer [[X. J. Scott]]. These cousins of strict 88cET have single steps of approximately 87.805¢, 88.235¢, 88.073¢, 88¢, 87.744¢, and 88.075¢, respectively. These small differences add up, as can be seen by examining the interval list below.
-[[Gary Morrison]] originally conceived of 88-cent equal tuning (88cET) as composed of steps of exactly 88¢. Nonetheless, composers have recognized a kinship between strict 88cET and some other scales -- in particular, the 41st root of 8 (equivalent to taking three steps of [[41edo]] as a generator with no octaves), the 8th root of 3/2, and the 11th root of 7/4, the latter being a preferred variant of composer and software designer [[X. J. Scott]]. These three cousins of strict 88cET have single steps of approximately 87.805¢, 87.744¢, and 88.075¢, respectively. These small differences add up, as can be seen by examining the interval list below.
 == Intervals ==
+{{todo|cleanup|inline=true}}
 {| class="wikitable"
 |-
 ! Degree
-! 11th root <br>of 7/4
+! 11ed7/4
 ! 88cET
-! 41st root <br>of 8
+! 41ed8
-! [[8edf|8th root <br>of 3/2]]
+! 8edf
 ! Solfege <br>syllable
 ! Some Nearby <br>JI Intervals
 |-
-! colspan="6" | '''''first octave'''''
+! colspan="6" | first octave
 !
 |-
@@ Line 140: / Line 141: @@
 | 27/14=1137.039, 31/16=1145.036
 |-
-! colspan="6" | '''''second octave'''''
+! colspan="6" | second octave
 !
 |-
@@ Line 166: / Line 167: @@
 | re
 | 9/8=203.910
-|-
-! colspan="6" |''second nonet''
-!
 |-
 | 17
@@ Line 258: / Line 256: @@
 | 63/32=1172.736, 160/81=1178.494
 |-
-! colspan="6" | '''''third octave'''''
+! colspan="6" | third octave
 !
 |-
@@ Line 300: / Line 298: @@
 | maa
 | 81/64=407.820, 33/26=412.745, 14/11=417.508
-|-
-! colspan="6" |''third nonet''
-!
 |-
 | 33
@@ Line 368: / Line 363: @@
 | 36/19=1106.397, 243/128=1109.775, 19/10=1111.199, 21/11=1119.463
 |-
-! colspan="6" | '''''fourth octave''''' (near match)
+! colspan="6" | fourth octave (near match)
 !
 |-
@@ Line 381: / Line 376: @@
 == Scales ==
+* [[Symmetrical scales of 88cET]]
+== Music ==
+; [[Carlo Serafini]]
+* [http://www.seraph.it/dep/det/88east.mp3 88 East]
+* [http://www.seraph.it/dep/det/88vocoeast.mp3 88 VocoEast]
+* [http://www.seraph.it/dep/det/88Bulgarians.mp3 88 Bulgarians] ([http://www.seraph.it/blog_files/9660ca3450a996ea8b55713cbf36151f-15.html blog entry])
+* [http://www.seraph.it/dep/int/88jinglebells.mp3 88 Jingle Bells] ([http://www.seraph.it/blog_files/495ec175ce56cf38cb399d1cd24db164-17.html blog entry])
+* [http://www.seraph.it/dep/det/The88thDoor.mp3 The 88th Door] ([http://www.seraph.it/blog_files/927f59ac10125056bcf7871636f246a6-302.html blog entry])
-* [[symmetrical scales of 88cET]]
+; [[Chris Vaisvil]]
+* [http://micro.soonlabel.com/88cent_nonoctave/STE-004_88_cent_guitar.mp3 88 cent guitar improvisation]
+* [http://micro.soonlabel.com/88cent_nonoctave/Prelude_in_88_Cent_Tuning.mp3 A Simple Prelude for 88 Cent Piano] ([http://micro.soonlabel.com/88cent_nonoctave/A_Simple_Prelude_in_88_Cent_Tuning.pdf scordata])
-== Compositions ==
+; [[Mundoworld]]
+* "To Become Water" from ''Mundoworld III'' (2021) – [https://open.spotify.com/track/39gEeGXprXGbAnbq0iyjMF Spotify] | [https://www.youtube.com/watch?v=RBv9c_qlFEk YouTube]
+* "Mirage Passage" from ''Mirage Passage'' (2024) – [https://open.spotify.com/track/2hAyfHr9XPG96SZPvBNHPP Spotify] | [https://www.youtube.com/watch?v=dWgmmK80I9U YouTube]
-* [http://www.seraph.it/dep/det/88east.mp3 88 East] by [[Carlo Serafini]]
+== Further reading ==
-* [http://www.seraph.it/dep/det/88vocoeast.mp3 88 VocoEast] by [[Carlo Serafini]]
+* [[Gary Morrison]]’s 2001 [https://soundcloud.com/mr88cet/sets/88cet-lecture-demo-gary-morrison-june-2001 lecture about 88cET]
-* [http://www.seraph.it/dep/det/88Bulgarians.mp3 88 Bulgarians] by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/9660ca3450a996ea8b55713cbf36151f-15.html blog entry])
-* [http://www.seraph.it/dep/int/88jinglebells.mp3 88 Jingle Bells] by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/495ec175ce56cf38cb399d1cd24db164-17.html blog entry])
-* [http://www.seraph.it/dep/det/The88thDoor.mp3 The 88th Door] by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/927f59ac10125056bcf7871636f246a6-302.html blog entry])
-* [http://micro.soonlabel.com/88cent_nonoctave/STE-004_88_cent_guitar.mp3 88 cent guitar improvisation] by [http://www.chrisvaisvil.com Chris Vaisvil]
-* [http://micro.soonlabel.com/88cent_nonoctave/Prelude_in_88_Cent_Tuning.mp3 A Simple Prelude for 88 Cent Piano] by [http://chrisvaisvil.com/?p=951 Chris Vaisvil] ([http://micro.soonlabel.com/88cent_nonoctave/A_Simple_Prelude_in_88_Cent_Tuning.pdf scordata])
 [[Category:Equal-step tuning]]
-[[Category:Edonoi]]