101ed7

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101ed7

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Prime factorization	101 (prime)
Step size	33.3547 ¢
Octave	36\101ed7 (1200.77 ¢)
Twelfth	57\101ed7 (1901.22 ¢)
Consistency limit	8
Distinct consistency limit	8

101 equal divisions of the 7th harmonic (abbreviated 101ed7) is a nonoctave tuning system that divides the interval of 7/1 into 101 equal parts of about 33.4 ¢ each. Each step represents a frequency ratio of 7^1/101, or the 101st root of 7.

Theory

101ed7 is closely related to 36edo (sixth-tone tuning), but with the 7th harmonic rather than the octave being just. The octave is stretched by about 0.770 cents (almost identical to 93ed6, where the octave is stretched by about 0.757 cents). Like 36edo, 101ed7 is consistent to the 8-integer-limit.

Compared to 36edo, 101ed7 is pretty well optimized for the 2.3.7.13.17 subgroup, with slightly better 3, 7, 13 and 17, and a slightly worse 2 versus 36edo. Using the patent val, the 5 is also less accurate. Overall this means 36edo is still better in the 5-limit, but 101ed7 is better in the 13- and 17-limit, especially when treating it as a dual-5 dual-11 tuning.

Harmonics

Approximation of harmonics in 101ed7

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+0.8	-0.7	+1.5	+15.5	+0.0	+0.0	+2.3	-1.5	+16.3	-15.3	+0.8
	Relative (%)	+2.3	-2.2	+4.6	+46.4	+0.1	+0.0	+6.9	-4.4	+48.7	-46.0	+2.4
Steps (reduced)		36 (36)	57 (57)	72 (72)	84 (84)	93 (93)	101 (0)	108 (7)	114 (13)	120 (19)	124 (23)	129 (28)

Approximation of harmonics in 101ed7 (continued)

Harmonic		13	14	15	16	17	18	19	20	21	22	23	24
Error	Absolute (¢)	-4.4	+0.8	+14.7	+3.1	-1.8	-0.7	+5.8	-16.3	-0.7	-14.6	+8.5	+1.6
	Relative (%)	-13.0	+2.3	+44.2	+9.2	-5.4	-2.1	+17.3	-49.0	-2.2	-43.7	+25.6	+4.7
Steps (reduced)		133 (32)	137 (36)	141 (40)	144 (43)	147 (46)	150 (49)	153 (52)	155 (54)	158 (57)	160 (59)	163 (62)	165 (64)

Subsets and supersets

101ed7 is the 26th prime ed7, so it does not contain any nontrivial subset ed7's.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	33.4
2	66.7	27/26
3	100.1	18/17
4	133.4	41/38
5	166.8	43/39
6	200.1	37/33
7	233.5	8/7
8	266.8	7/6
9	300.2	44/37
10	333.5
11	366.9	21/17
12	400.3	29/23, 34/27
13	433.6	9/7
14	467	38/29
15	500.3	4/3
16	533.7
17	567	43/31
18	600.4	41/29
19	633.7
20	667.1
21	700.4	3/2
22	733.8	26/17, 29/19
23	767.2	14/9
24	800.5	27/17
25	833.9	34/21
26	867.2	38/23
27	900.6	32/19, 37/22
28	933.9	12/7
29	967.3	7/4
30	1000.6	41/23
31	1034
32	1067.4
33	1100.7	17/9
34	1134.1
35	1167.4
36	1200.8	2/1
37	1234.1
38	1267.5	27/13
39	1300.8	36/17
40	1334.2
41	1367.5
42	1400.9
43	1434.3
44	1467.6	7/3
45	1501
46	1534.3	17/7
47	1567.7	42/17
48	1601
49	1634.4	18/7
50	1667.7
51	1701.1
52	1734.4
53	1767.8
54	1801.2	17/6
55	1834.5	26/9
56	1867.9
57	1901.2	3/1
58	1934.6
59	1967.9
60	2001.3
61	2034.6
62	2068
63	2101.3	37/11
64	2134.7	24/7
65	2168.1	7/2
66	2201.4
67	2234.8
68	2268.1
69	2301.5	34/9
70	2334.8	27/7
71	2368.2
72	2401.5	4/1
73	2434.9
74	2468.2
75	2501.6
76	2535
77	2568.3
78	2601.7	9/2
79	2635
80	2668.4	14/3
81	2701.7
82	2735.1	34/7
83	2768.4
84	2801.8
85	2835.2	36/7
86	2868.5	21/4
87	2901.9
88	2935.2
89	2968.6
90	3001.9	17/3
91	3035.3
92	3068.6
93	3102	6/1
94	3135.3
95	3168.7
96	3202.1
97	3235.4
98	3268.8
99	3302.1
100	3335.5
101	3368.8	7/1

See also

• 21edf – relative edf
• 36edo – relative edo
• 57edt – relative edt
• 93ed6 – relative ed6
• 129ed12 – relative ed12, close to the zeta-optimized tuning for 36edo