1ed88c: Difference between revisions

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~~{{Infobox ET|150ed2048}}~~

'''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).

'''88-cent equal temperament''' (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 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 ~~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.

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== 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 [[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 ==

{| class="wikitable"

|-

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! Some Nearby JI Intervals

|-

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

! colspan="6" | first octave

!

|-

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

|-

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

! colspan="6" | second octave

!

|-

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| re

| 9/8=203.910

|-

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

!

|-

| 17

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

|-

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

! colspan="6" | third octave

!

|-

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| maa

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

|-

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

!

|-

| 33

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| 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 ==

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

* [[Symmetrical scales of 88cET]]

== Music ==

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; [[Mundoworld]]

* [https://www.youtube.com/watch?v=RBv9c_qlFEk ~~To Become Water~~]

* "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]

== Further reading ==

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

[[Category:Equal-step tuning]]

~~[[Category:Edonoi]]~~

@@ Line 1: / Line 1: @@
-{{Infobox ET|150ed2048}}
+'''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).
-'''88-cent equal temperament''' (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 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 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.
@@ Line 15: / Line 12: @@
 == 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 [[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 ==
+{{todo|cleanup|inline=true}}
 {| class="wikitable"
 |-
@@ Line 28: / Line 26: @@
 ! Some Nearby <br>JI Intervals
 |-
-! colspan="6" | '''''first octave'''''
+! colspan="6" | first octave
 !
 |-
@@ Line 143: / Line 141: @@
 | 27/14=1137.039, 31/16=1145.036
 |-
-! colspan="6" | '''''second octave'''''
+! colspan="6" | second octave
 !
 |-
@@ Line 169: / Line 167: @@
 | re
 | 9/8=203.910
-|-
-! colspan="6" |''second nonet''
-!
 |-
 | 17
@@ Line 261: / Line 256: @@
 | 63/32=1172.736, 160/81=1178.494
 |-
-! colspan="6" | '''''third octave'''''
+! colspan="6" | third octave
 !
 |-
@@ Line 303: / Line 298: @@
 | maa
 | 81/64=407.820, 33/26=412.745, 14/11=417.508
-|-
-! colspan="6" |''third nonet''
-!
 |-
 | 33
@@ Line 371: / 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 384: / Line 376: @@
 == Scales ==
-* [[symmetrical scales of 88cET]]
+* [[Symmetrical scales of 88cET]]
 == Music ==
@@ Line 399: / Line 391: @@
 ; [[Mundoworld]]
-* [https://www.youtube.com/watch?v=RBv9c_qlFEk To Become Water]
+* "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]
+== Further reading ==
+* [[Gary Morrison]]’s 2001 [https://soundcloud.com/mr88cet/sets/88cet-lecture-demo-gary-morrison-june-2001 lecture about 88cET]
 [[Category:Equal-step tuning]]
-[[Category:Edonoi]]