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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 ==
== 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-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 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 [[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)<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.
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== The 88cET family ==
== The 88cET family ==
[[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 41ed8 (equivalent to taking three steps of [[41edo]] as a generator with no octaves), the [[8edf]], and the 11ed7/4, 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.
[[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.


== Intervals ==
== Intervals ==
{{todo|cleanup|inline=true}}
{| class="wikitable"
{| class="wikitable"
|-
|-
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! Some Nearby <br>JI Intervals
! Some Nearby <br>JI Intervals
|-
|-
! colspan="6" | '''''first octave'''''
! colspan="6" | first octave
!  
!  
|-
|-
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| 27/14=1137.039, 31/16=1145.036
| 27/14=1137.039, 31/16=1145.036
|-
|-
! colspan="6" | '''''second octave'''''
! colspan="6" | second octave
!  
!  
|-
|-
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| re
| re
| 9/8=203.910
| 9/8=203.910
|-
! colspan="6" |''second nonet''
!
|-
|-
| 17
| 17
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| 63/32=1172.736, 160/81=1178.494
| 63/32=1172.736, 160/81=1178.494
|-
|-
! colspan="6" | '''''third octave'''''
! colspan="6" | third octave
!  
!  
|-
|-
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| maa
| maa
| 81/64=407.820, 33/26=412.745, 14/11=417.508
| 81/64=407.820, 33/26=412.745, 14/11=417.508
|-
! colspan="6" |''third nonet''
!
|-
|-
| 33
| 33
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| 36/19=1106.397, 243/128=1109.775, 19/10=1111.199, 21/11=1119.463
| 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)
!  
!  
|-
|-
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== Scales ==
== Scales ==
* [[symmetrical scales of 88cET]]
* [[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])
 
; [[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])
 
; [[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]


== Compositions ==
== Further reading ==
* [http://www.seraph.it/dep/det/88east.mp3 88 East] 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/88vocoeast.mp3 88 VocoEast] by [[Carlo Serafini]]
* [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:Equal-step tuning]]
[[Category:Edonoi]]
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