editing disabled

88cET


Theory

88 cent equal temperament 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 cents equal temperament are very closely related, and the chords of 88 cents temperament are listed on the page chords of octacot. From this it may be seen that octacot, and hence 88 cents 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.

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 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
(41ed8)
8th root
of 3/2
Solfege
Some Nearby





syllable
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
878.444
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)