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{{Infobox ET}}
{{Infobox ET}}
'''101 divisions of the seventh harmonic''' ('''101ed7''') is related to [[36edo]] (sixth-tone tuning), but with the 7/1 rather than the 2/1 being just. The octave is about 1.2347 [[cent]]s stretched and the step size is about 33.3547 cents. It is consistent to the 8-[[integer-limit]]. This tuning tempers out the [[syntonic comma]].
{{ED intro}}


Lookalikes: [[21edf]], [[36edo]], [[57edt]], [[93ed6]]
== Theory ==
101ed7 is closely related to [[36edo]] (sixth-tone tuning), but with the 7th harmonic rather than the [[2/1|octave]] being just. The octave is stretched by about 0.770 [[cent]]s (almost identical to [[93ed6]], where the octave is stretched by about 0.757 cents). Like 36edo, 101ed7 is [[consistent]] to the [[integer limit|8-integer-limit]].
 
Compared to 36edo, 101ed7 is pretty well optimized for the 2.3.7.13.17 [[subgroup]], with slightly better [[3/1|3]], [[7/1|7]], [[13/1|13]] and [[17/1|17]], and a slightly worse 2 versus 36edo. Using the [[patent val]], the [[5/1|5]] is also less accurate. Overall this means 36edo is still better in the [[5-limit]], but 101ed7 is better in the [[13-limit|13-]] and [[17-limit]], especially when treating it as a dual-5 dual-11 tuning.
 
=== Harmonics ===
{{Harmonics in equal|101|7|1|intervals=integer|columns=11}}
{{Harmonics in equal|101|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 101ed7 (continued)}}
 
=== Subsets and supersets ===
101ed7 is the 26th [[prime equal division|prime ed7]], so it does not contain any nontrivial subset ed7's.


== Intervals ==
== Intervals ==
{{Interval table}}
{{Interval table}}


== Harmonics ==
== See also ==
{{Harmonics in equal|101|7|1|intervals=prime}}
* [[21edf]] – relative edf
{{Harmonics in equal|101|7|1|intervals=prime|collapsed=1|start=12}}
* [[36edo]] – relative edo
 
* [[57edt]] – relative edt
* [[93ed6]] – relative ed6
* [[129ed12]] – relative ed12, close to the zeta-optimized tuning for 36edo


{{stub}}
[[Category:36edo]]
[[Category:Edonoi]]

Latest revision as of 19:13, 25 June 2025

← 100ed7 101ed7 102ed7 →
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 71/101, 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
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