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 1.23 cents. Like 36edo, 101ed7 is consistent to the 8-integer-limit.

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 5/1's 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.

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)

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