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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==
==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.
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==
==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==
==Intervals==
Line 16: Line 16:
|-
|-
! | Degree
! | Degree
! | 11th root
! | 11th root <br>of 7/4
 
of 7/4
! | 88cET
! | 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'''''
! colspan="6" | '''''first octave'''''
Line 62: Line 48:
| | 175.489
| | 175.489
| | reh
| | 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
| | 3
Line 70: Line 56:
| | 263.233
| | 263.233
| | ma
| | ma
| | [[7/6|7/6]]=266.871
| | [[7/6]]=266.871
|-
|-
| | 4
| | 4
Line 78: Line 64:
| | 350.978
| | 350.978
| | mu
| | 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
| | 5
Line 86: Line 72:
| | 438.722
| | 438.722
| | mo
| | 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
| | 6
Line 94: Line 80:
| | 526.466
| | 526.466
| | fih
| | 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
| | 7
Line 102: Line 88:
| | 614.211
| | 614.211
| | se
| | se
| | [[10/7|10/7]]=617.488
| | [[10/7]]=617.488
|-
|-
| | 8
| | 8
Line 110: Line 96:
| | 701.955
| | 701.955
| | sol
| | sol
| | [[3/2|3/2]]=701.955
| | [[3/2]]=701.955
|-
|-
| | 9
| | 9
Line 118: Line 104:
| | 789.699
| | 789.699
| | leh
| | 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
| | 10
Line 124: Line 110:
| | 880
| | 880
| | 878.049
| | 878.049
| | 878.444
| | 877.444
| | la
| | la
| | [[5/3|5/3]]=884.359
| | [[5/3]]=884.359
|-
|-
| | 11
| | 11
Line 134: Line 120:
| | 965.188
| | 965.188
| | ta
| | ta
| | [[7/4|7/4]]=968.826
| | [[7/4]]=968.826
|-
|-
| | 12
| | 12
Line 142: Line 128:
| | 1052.933
| | 1052.933
| | tu
| | 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
| | 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])
<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]]

Revision as of 06:41, 13 February 2019

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 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 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)^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.

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 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

Degree 11th root
of 7/4 		88cET 41st root
of 8 		8th root
of 3/2 		Solfege
syllable 		Some Nearby
JI Intervals
first octave
0 0 0 0 0 do 1/1=0
1 88.075 88 87.805 87.744 rih 22/21=80.537, 21/20=84.467, 20/19=88.801, 19/18=93.603
2 176.15 176 175.610 175.489 reh 11/10=165.004, 21/19=173.268, 10/9=182.404
3 264.225 264 263.415 263.233 ma 7/6=266.871
4 352.3 352 351.220 350.978 mu 11/9=347.408, 27/22=354.547, 16/13=359.472
5 440.375 440 439.024 438.722 mo 32/25=427.373, 9/7=435.084, 22/17=446.363
6 528.45 528 526.829 526.466 fih 19/14=528.687, 49/36=533.742, 15/11=536.95
7 616.526 616 614.634 614.211 se 10/7=617.488
8 704.601 704 702.439 701.955 sol 3/2=701.955
9 792.676 792 790.244 789.699 leh 11/7=782.492, 30/19=790.756, 128/81=792.180, 19/12=795.558, 27/17=800.910, 8/5=813.686
10 880.751 880 878.049 877.444 la 5/3=884.359
11 968.826 968 965.854 965.188 ta 7/4=968.826
12 1056.901 1056 1053.659 1052.933 tu 11/6=1049.363, 35/19=1057.627, 24/13=1061.427
13 1144.976 1144 1141.463 1140.677 to 27/14=1137.039, 31/16=1145.036
second octave
14 33.051 32 29.268 28.421 di 65/64=26.841, 64/63=27.264, 63/62=27.700, 58/57=30.109
15 121.126 120 117.073 116.166 ra 16/15=111.731, 15/14=119.443, 14/13=128.298
16 209.201 208 204.878 203.910 re 9/8=203.910
17 297.276 296 292.683 291.654 meh 13/11=289.210, 32/27=294.135, 19/16=297.513
18 385.351 384 380.488 379.399 mi 5/4=386.314
19 473.427 472 468.293 467.143 fe 17/13=464.428, 21/16=470.781
20 561.502 560 556.098 554.888 fu 11/8=551.318, 18/13=563.382
21 649.577 648 643.902 642.632 su 16/11=648.682
22 737.652 736 731.707 730.376 si 32/21=729.219, 26/17=735.572, 49/32=737.652
23 825.727 824 819.512 818.121 le 8/5=813.686, 45/28=821.398, 21/13=830.253
24 913.802 912 907.317 905.865 laa 42/25=898.153, 32/19=902.487, 27/16=905.865, 22/13=910.790, 17/10=918.642
25 1001.877 1000 995.122 993.609 teh 39/22=991.165, 16/9=996.090, 25/14=1003.802, 34/19=1007.442
26 1089.952 1088 1082.927 1081.354 ti 28/15=1080.557, 15/8=1088.269
27 1178.027 1176 1170.732 1169.098 da 63/32=1172.736, 160/81=1178.494
third octave
28 66.102 64 58.537 56.843 ro 33/3253.273, 28/27=62.961, 80/77=66.170, 27/26=65.337
29 154.177 152 146.341 144.587 ru 49/45=147.428, 12/11=150.637, 35/32=155.140
30 242.252 240 234.146 232.331 ri 8/7=231.174, 23/20=241.961, 15/13=247.741
31 330.328 328 321.951 320.076 me 6/5=315.641, 23/19=330.761
32 418.403 416 409.756 407.820 maa 81/64=407.820, 33/26=412.745, 14/11=417.508
33 506.478 504 497.561 495.564 fa 85/64=491.269, 4/3=498.045, 75/56=505.757
34 594.553 592 585.366 583.309 fi 7/5=582.512, 45/32=590.224, 38/27=591.648
35 682.628 680 673.171 671.053 sih 28/19=671.313, 40/27=680.449
36 770.703 768 760.976 758.798 lo 17/11=753.637, 99/64=755.228, 14/9=764.916, 39/25=769.855, 25/16=772.627
37 858.778 856 848.780 846.542 lu 13/8=840.528, 18/11=852.592
38 946.853 944 936.585 934.286 li 12/7=933.129, 19/11=946.195
39 1034.928 1032 1024.390 1022.031 te 9/5=1017.596, 49/27=1031.787, 20/11=1034.996
40 1123.003 1120 1112.195 1109.775 taa 36/19=1106.397, 243/128=1109.775, 19/10=1111.199, 21/11=1119.463
fourth octave (near match)
41 11.078 8 0 1197.59 do 1/1=0, 2/1=1200

Scales

Compositions

88 East by Carlo Serafini

88 VocoEast by Carlo Serafini

88 Bulgarians by Carlo Serafini (blog entry)

88 Jingle Bells by Carlo Serafini (blog entry)

88 cent guitar improvisation by Chris Vaisvil

A Simple Prelude for 88 Cent Piano by Chris Vaisvil (scordata)

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