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 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 |
second nonet | ||||||
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 |
third nonet | ||||||
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 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)