88cET

Prime factorization

2 × 3 × 5²

Step size

88¢

Octave

14\150ed2048 (1232¢) (→7\75ed2048)

Twelfth

22\150ed2048 (1936¢) (→11\75ed2048)

Consistency limit

3

Distinct consistency limit

3

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

Approximation of harmonics in 1ed88c
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
Error	Relative (%)	+36.4	+38.7	-27.3	+33.7	-24.9	-28.2	+9.1	-22.6	-29.9	-17.4
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 i.e. the strict 88cET), 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