88cET
| ← 149ed2048 | 150ed2048 | 151ed2048 → |
88-cent equal temperament (also known as 1ed88¢ or APS88¢) 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 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.
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.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +32.0 | +34.0 | -24.0 | +29.7 | -22.0 | -24.8 | +8.0 | -19.9 | -26.3 | -15.3 |
| relative (%) | +36 | +39 | -27 | +34 | -25 | -28 | +9 | -23 | -30 | -17 | |
| Steps | 14 | 22 | 27 | 32 | 35 | 38 | 41 | 43 | 45 | 47 | |
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 68ed32 (taking every 5 steps of 68edo), the 109ed256 (taking every 8 steps of 109edo), the 150ed2048 (taking every 11 steps of 150edo), 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
| Degree | 11ed7/4 | 88cET | 41ed8 | 8edf | 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 |