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.

101ed7 is related to 36edo (sixth-tone tuning), but with the 7/1 rather than the 2/1 being just. The octave is stretched by about 1.2347 cents. It tempers out the syntonic comma and is consistent to the 8-integer-limit.

Lookalikes: 21edf, 36edo, 57edt, 93ed6, 129ed12

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

Harmonics

Compared to 36edo, 101ed7’s harmonics are almost exactly the same, but it has a slightly better 3/1, 7/1, and 13/1, and a slightly worse 2/1 and 5/1 versus 36edo.

Overall this means 36edo is still better in the 5-limit, but 101ed7 is better in the 13-limit. (The 7-limit and 11-limit could go either way.)

36edo’s 5-limit dominance flips on its head, though, if one approaches it as a dual-5 tuning. In that case, the fact that 101ed7’s 5/1 is closer to 50% relative error is actually a good thing, because it means the error on the worse of the two 5/1s is less.

So as a single-5 5-limit tuning, 36edo is better. But as a dual-5 5-limit tuning, 101ed7 is better.

And as a dual-5, dual-11 31-limit tuning, 101ed7 is exceptional for its size. It is very accurate.

Approximation of prime harmonics in 101ed7

Harmonic		2	3	5	7	11	13	17	19	23	29	31	37
Error	Absolute (¢)	+0.8	-0.7	+15.5	+0.0	-15.3	-4.4	-1.8	+5.8	+8.5	+7.5	-7.9	-14.0
	Relative (%)	+2.3	-2.2	+46.4	+0.0	-46.0	-13.0	-5.4	+17.3	+25.6	+22.5	-23.7	-42.0
Steps (reduced)		36 (36)	57 (57)	84 (84)	101 (0)	124 (23)	133 (32)	147 (46)	153 (52)	163 (62)	175 (74)	178 (77)	187 (86)

36edo for comparsion:

Approximation of prime harmonics in 36edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31	37
Error	Absolute (¢)	+0.0	-2.0	+13.7	-2.2	+15.3	-7.2	-5.0	+2.5	+5.1	+3.8	-11.7	+15.3
	Relative (%)	+0.0	-5.9	+41.1	-6.5	+46.0	-21.6	-14.9	+7.5	+15.2	+11.3	-35.1	+46.0
Steps (reduced)		36 (0)	57 (21)	84 (12)	101 (29)	125 (17)	133 (25)	147 (3)	153 (9)	163 (19)	175 (31)	178 (34)	188 (8)