87edo

The 87 equal temperament, often abbreviated 87-tET, 87-EDO, or 87-ET, is the scale derived by dividing the octave into 87 equally-sized steps, where each step represents a frequency ratio of 13.79 cents. It is solid as both a 13-limit (or 15 odd limit) and as a 5-limit system, and of course does well enough in any limit in between. It represents the 13-limit tonality diamond both uniquely and consistently, and is the smallest equal temperament to do so.

87et tempers out 196/195, 325/324, 352/351, 364/363, 385/384, 441/440, 625/624, 676/675, and 1001/1000 as well as the 29-comma, <46 -29|, the misty comma, <26 -12 -3|, the kleisma, 15625/15552, 245/243, 1029/1024, 3136/3125, and 5120/5103.

87et is a particularly good tuning for rodan temperament. The 8/7 generator of 17\87 is a remarkable 0.00062 cents sharper than the 13-limit POTE generator and is close to the 11-limit POTE generator also. Also, the 32\87 generator for clyde temperament is 0.04455 cents sharp of the 7-limit POTE generator.

Rank two temperaments

Periods per octave	Generator	Cents	Associated ratio	Temperament
1	4\87	55.172	33/32	Sensa
1	10\87	137.931	13/12	Quartemka
1	14\87	193.103	28/25	Luna/hemithirds
1	17\87	234.483	8/7	Rodan
1	23\87	317.241	6/5	Hanson/countercata/metakleismic
1	32\87	441.379	9/7	Clyde
1	38\87	524.138	65/48	Widefourth
1	40\87	551.724	11/8	Emkay
3	23\87	317.241	6/5	Tritikleismic
29	28\87	386.207	5/4	Mystery

87 can serve as a MOS in these:

270&87 <<24 -9 -66 12 27 ... ||

494&87 <<51 -1 -133 11 32 ... ||

13-limit detempering of 87et

See detempering.

[91/90, 49/48, 40/39, 28/27, 25/24, 21/20, 35/33, 16/15, 14/13, 13/12, 12/11, 11/10, 10/9, 28/25, 9/8, 25/22, 8/7, 15/13, 7/6, 75/64, 13/11, 25/21, 6/5, 40/33, 11/9, 16/13, 26/21, 5/4, 44/35, 14/11, 32/25, 9/7, 13/10, 21/16, 33/25, 4/3, 35/26, 27/20, 15/11, 11/8, 18/13, 7/5, 45/32, 64/45, 10/7, 13/9, 16/11, 22/15, 40/27, 52/35, 3/2, 50/33, 32/21, 20/13, 14/9, 25/16, 11/7, 35/22, 8/5, 21/13, 13/8, 18/11, 33/20, 5/3, 42/25, 22/13, 75/44, 12/7, 26/15, 7/4, 44/25, 16/9, 25/14, 9/5, 20/11, 11/6, 24/13, 13/7, 15/8, 66/35, 21/11, 25/13, 27/14, 39/20, 55/28, 99/50, 2]

Steps of 87	Cents	Detempered Interval
1	13.79310	91/90
2	27.58621	49/48
3	41.37931	40/39
4	55.17241	28/27
5	68.96552	25/24
6	82.75862	21/20
7	96.55172	35/33
8	110.34483	16/15
9	124.13793	14/13
10	137.93103	13/12
11	151.72414	12/11
12	165.51724	11/10
13	179.31035	10/9
14	193.10345	28/25
15	206.89655	9/8
16	220.68966	25/22
17	234.48276	8/7
18	248.27586	15/13
19	262.06897	7/6
20	275.86207	75/64
21	289.65517	13/11
22	303.44828	25/21
23	317.24138	6/5
24	331.03448	40/33
25	344.82759	11/9
26	358.62069	16/13
27	372.41379	26/21
28	386.20690	5/4
29	400.00000	44/35
30	413.79310	14/11
31	427.58621	32/25
32	441.37931	9/7
33	455.17241	13/10
34	468.96552	21/16
35	482.75862	33/25
36	496.55172	4/3
37	510.34483	35/26
38	524.13793	27/20
39	537.93103	15/11
40	551.72414	11/8
41	565.51724	18/13
42	579.31035	7/5
43	593.10345	45/32
44	606.89655	64/45
45	620.68966	10/7
46	634.48276	13/9
47	648.27586	16/11
48	662.06897	22/15
49	675.86207	40/27
50	689.65517	52/35
51	703.44828	3/2
52	717.24138	50/33
53	731.03448	32/21
54	744.82759	20/13
55	758.62069	14/9
56	772.41379	25/16
57	786.20690	11/7
58	800.00000	35/22
59	813.79310	8/5
60	827.58621	21/13
61	841.37931	13/8
62	855.17241	18/11
63	868.96552	33/20
64	882.75862	5/3
65	896.55172	42/25
66	910.34483	22/13
67	924.13793	75/44
68	937.93103	12/7
69	951.72414	26/15
70	965.51724	7/4
71	979.31035	44/25
72	993.10345	16/9
73	1006.89655	25/14
74	1020.68966	9/5
75	1034.48276	20/11
76	1048.27586	11/6
77	1062.06897	24/13
78	1075.86207	13/7
79	1089.65517	15/8
80	1103.44828	66/35
81	1117.24138	21/11
82	1131.03448	25/13
83	1144.82759	27/14
84	1158.62069	39/20
85	1172.41379	55/28
86	1186.20690	99/50
87	1200.00000	2/1

Music

Pianodactyl play by Gene Ward Smith