88cET[edit]

Theory[edit]

88 cent Equal-step Tuning, 150ed2048, 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[edit]

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[edit]

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