1ed88c: Difference between revisions

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~~=88cET=~~

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

==Theory==

88 cent [[Equal|equal ~~temperament~~]] uses 88 cents, or 11\150 of an octave, to generate a [[~~nonoctave|~~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 ~~cents~~ equal ~~temperament~~ are very closely related, and the chords of 88 ~~cents temperament~~ are listed on the page [[Chords_of_octacot|chords of octacot]]. From this it may be seen that octacot, and hence 88 cents ~~temperament~~, share an abundance of [[Dyadic_chord|essentially tempered chords]].

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|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.

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==The 88cET family==

[[Gary_Morrison|Gary Morrison]] originally conceived of 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|~~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|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|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|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==

Line 16:

|-

! | Degree

! | 11th root

! | 11th root of 7/4

of 7/4

! | 88cET

! | 41st root of 8

! | 41st root of 8

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

~~(41ed8)~~

! | Solfege syllable

! | 8th root

! | Some Nearby JI Intervals

of 3/2

! | Solfege

! | Some Nearby

|-

~~! |~~

~~! | syllable~~

~~! |~~ JI Intervals

|-

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

Line 62:

Line 48:

| | 175.489

| | reh

| | [[~~11/10|~~11/10]]=165.004, 21/19=173.268, [[~~10/9|~~10/9]]=182.404

| | [[11/10]]=165.004, 21/19=173.268, [[10/9]]=182.404

|-

| | 3

Line 70:

Line 56:

| | 263.233

| | ma

| | [[~~7/6|~~7/6]]=266.871

| | [[7/6]]=266.871

|-

| | 4

Line 78:

Line 64:

| | 350.978

| | mu

| | [[~~11/9|~~11/9]]= 347.408, 27/22=354.547, 16/13=359.472

| | [[11/9]]=347.408, 27/22=354.547, 16/13=359.472

|-

| | 5

Line 86:

Line 72:

| | 438.722

| | mo

| | 32/25=427.373, [[~~9/7|~~9/7]]=435.084, 22/17 446.363

| | 32/25=427.373, [[9/7]]=435.084, [[22/17]]=446.363

|-

| | 6

Line 94:

Line 80:

| | 526.466

| | fih

| | 19/14=528.687, 49/36=533.742, [[~~15/11|~~15/11]]=536.95

| | [[19/14]]=528.687, 49/36=533.742, [[15/11]]=536.95

|-

| | 7

Line 102:

Line 88:

| | 614.211

| | se

| | [[~~10/7|~~10/7]]=617.488

| | [[10/7]]=617.488

|-

| | 8

Line 110:

Line 96:

| | 701.955

| | sol

| | [[~~3/2|~~3/2]]=701.955

| | [[3/2]]=701.955

|-

| | 9

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| | 789.699

| | leh

| | [[~~11/7|~~11/7]]=782.492, 30/19=790.756, 128/81=792.180, 19/12=795.558, 27/17=800.910, [[~~8/5|~~8/5]]=813.686

| | [[11/7]]=782.492, 30/19=790.756, 128/81=792.180, [[19/12]]=795.558, 27/17=800.910, [[8/5]]=813.686

|-

| | 10

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| | 880

| | 878.049

| | ~~878~~.444

| | 877.444

| | la

| | [[~~5/3|~~5/3]]=884.359

| | [[5/3]]=884.359

|-

| | 11

Line 134:

Line 120:

| | 965.188

| | ta

| | [[~~7/4|~~7/4]]=968.826

| | [[7/4]]=968.826

|-

| | 12

Line 142:

Line 128:

| | 1052.933

| | tu

| | [[~~11/6|~~11/6]]=1049.363, 35/19=1057.627, 24/13=1061.427

| | [[11/6]]=1049.363, 35/19=1057.627, 24/13=1061.427

|-

| | 13

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[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:~~88c~~]]

[[Category:Equal-step tuning]]

[[Category:~~edonoi~~]]

[[Category:Edonoi]]

@@ Line 1: / Line 1: @@
-=88cET=
+'''88-cent equal tuning''' uses equal steps of 88 cents each. It is equivalent to 13.6364edo, and is a subset of [[150edo]] (every eleventh step).
 ==Theory==
-cent [[Equal|equal temperament]] uses 88 cents, or 11\150 of an octave, to generate a [[nonoctave|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 cents equal temperament are very closely related, and the chords of 88 cents temperament are listed on the page [[Chords_of_octacot|chords of octacot]]. From this it may be seen that octacot, and hence 88 cents temperament, share an abundance of [[Dyadic_chord|essentially tempered chords]].
+-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|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.
@@ Line 9: / Line 9: @@
 ==The 88cET family==
-[[Gary_Morrison|Gary Morrison]] originally conceived of 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|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|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|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|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==
@@ Line 16: / Line 16: @@
 |-
 ! | Degree
-! | 11th root
+! | 11th root <br>of 7/4
-of 7/4
 ! | 88cET
-! | 41st root of 8
+! | 41st root <br>of 8
+! | [[8edf|8th root <br>of 3/2]]
-(41ed8)
+! | Solfege <br>syllable
-! | 8th root
+! | Some Nearby <br>JI Intervals
-of 3/2
-! | Solfege
-! | Some Nearby
-|-
-! |
-! |
-! |
-! |
-! |
-! | syllable
-! | JI Intervals
 |-
 ! colspan="6" | '''''first octave'''''
@@ Line 62: / Line 48: @@
 | | 175.489
 | | reh
-| | [[11/10|11/10]]=165.004, 21/19=173.268, [[10/9|10/9]]=182.404
+| | [[11/10]]=165.004, 21/19=173.268, [[10/9]]=182.404
 |-
 | | 3
@@ Line 70: / Line 56: @@
 | | 263.233
 | | ma
-| | [[7/6|7/6]]=266.871
+| | [[7/6]]=266.871
 |-
 | | 4
@@ Line 78: / Line 64: @@
 | | 350.978
 | | mu
-| | [[11/9|11/9]]= 347.408, 27/22=354.547, 16/13=359.472
+| | [[11/9]]=347.408, 27/22=354.547, 16/13=359.472
 |-
 | | 5
@@ Line 86: / Line 72: @@
 | | 438.722
 | | mo
-| | 32/25=427.373, [[9/7|9/7]]=435.084, 22/17 446.363
+| | 32/25=427.373, [[9/7]]=435.084, [[22/17]]=446.363
 |-
 | | 6
@@ Line 94: / Line 80: @@
 | | 526.466
 | | fih
-| | 19/14=528.687, 49/36=533.742, [[15/11|15/11]]=536.95
+| | [[19/14]]=528.687, 49/36=533.742, [[15/11]]=536.95
 |-
 | | 7
@@ Line 102: / Line 88: @@
 | | 614.211
 | | se
-| | [[10/7|10/7]]=617.488
+| | [[10/7]]=617.488
 |-
 | | 8
@@ Line 110: / Line 96: @@
 | | 701.955
 | | sol
-| | [[3/2|3/2]]=701.955
+| | [[3/2]]=701.955
 |-
 | | 9
@@ Line 118: / Line 104: @@
 | | 789.699
 | | leh
-| | [[11/7|11/7]]=782.492, 30/19=790.756, 128/81=792.180, 19/12=795.558, 27/17=800.910, [[8/5|8/5]]=813.686
+| | [[11/7]]=782.492, 30/19=790.756, 128/81=792.180, [[19/12]]=795.558, 27/17=800.910, [[8/5]]=813.686
 |-
 | | 10
@@ Line 124: / Line 110: @@
 | | 880
 | | 878.049
-| | 878.444
+| | 877.444
 | | la
-| | [[5/3|5/3]]=884.359
+| | [[5/3]]=884.359
 |-
 | | 11
@@ Line 134: / Line 120: @@
 | | 965.188
 | | ta
-| | [[7/4|7/4]]=968.826
+| | [[7/4]]=968.826
 |-
 | | 12
@@ Line 142: / Line 128: @@
 | | 1052.933
 | | tu
-| | [[11/6|11/6]]=1049.363, 35/19=1057.627, 24/13=1061.427
+| | [[11/6]]=1049.363, 35/19=1057.627, 24/13=1061.427
 |-
 | | 13
@@ Line 401: / Line 387: @@
 <span style=""><span style="">[http://micro.soonlabel.com/88cent_nonoctave/Prelude_in_88_Cent_Tuning.mp3 A Simple Prelude for 88 Cent Piano]</span></span> by [http://chrisvaisvil.com/?p=951 Chris Vaisvil] ([http://micro.soonlabel.com/88cent_nonoctave/A_Simple_Prelude_in_88_Cent_Tuning.pdf scordata])
-[[Category:88c]]
+[[Category:Equal-step tuning]]
-[[Category:edonoi]]
+[[Category:Edonoi]]