67ed5

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← 66ed567ed568ed5 →
Prime factorization 67 (prime)
Step size 41.5868¢ 
Octave 29\67ed5 (1206.02¢)
Twelfth 46\67ed5 (1912.99¢)
Consistency limit 8
Distinct consistency limit 7

Division of the 5th harmonic into 67 equal parts (67ed5) is related to 29 edo, but with the 5/1 rather than the 2/1 being just. The octave is about 6.0164 cents stretched and the step size is about 41.5868 cents. The patent val has a generally sharp tendency for harmonics up to 28. Unlike 29edo, it is only consistent up to the 8-integer-limit, with discrepancy for the 9th harmonic.

Intervals

degree cents value corresponding
JI intervals
comments
0 0.0000 exact 1/1
1 41.5868
2 83.1735 22/21, 21/20
3 124.7603 3375/3136
4 166.3471 11/10
5 207.9339 150/133
6 249.5206 800/693, 231/200
7 291.1074 45/38
8 332.6942 40/33
9 374.2809 4455/3584
10 415.8677 80/63, 14/11
11 457.4545
12 499.0413 4/3
13 540.6280
14 582.2148 7/5
15 623.8016 1125/784
16 665.3883 22/15, 147/100
17 706.9751 200/133 pseudo-3/2
18 748.5619 77/50
19 790.1487 30/19
20 831.7354 160/99
21 873.3222 63/38
22 914.9090 95/56, 56/33
23 956.4958
24 998.0825 16/9, 57/32
25 1039.6693
26 1081.2561 28/15
27 1122.8428 375/196
28 1164.4296 49/25
29 1206.0164 800/399, 225/112 pseudo-octave
30 1247.6032 154/75
31 1289.1899 40/19
32 1330.7767 640/297
33 1372.3635 495/224, 42/19
34 1413.9502 95/42, 224/99
35 1455.5370 297/128
36 1497.1238 19/8
37 1538.7106 375/154
38 1580.2973 112/45, 399/160 pseudo-5/2
39 1621.8841 125/49
40 1663.4709 196/75
41 1705.0576 75/28
42 1746.6444
43 1788.2312 160/57, 45/16
44 1829.8180
45 1871.4047 165/56, 56/19
46 1912.9915 190/63
47 1954.5783 99/32
48 1996.1650 19/6
49 2037.7518 250/77
50 2079.3386 133/40 pseudo-10/3
51 2120.9254 500/147, 75/22
52 2162.5121 784/225
53 2204.0989 25/7
54 2245.6857
55 2287.2725 15/4
56 2328.8592
57 2370.4460 55/14, 63/16
58 2412.0328 3584/891
59 2453.6195 33/8
60 2495.2063 38/9
61 2536.7931 1000/231, 693/160
62 2578.3799 133/30
63 2619.9666 50/11
64 2661.5534 3136/675
65 2703.1402 100/21
66 2744.7269
67 2786.3137 exact 5/1 just major third plus two octaves

Harmonics

Approximation of harmonics in 67ed5
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +6.0 +11.0 +12.0 +0.0 +17.1 -0.3 +18.0 -19.5 +6.0 +7.4 -18.5
Relative (%) +14.5 +26.5 +28.9 +0.0 +41.0 -0.7 +43.4 -46.9 +14.5 +17.7 -44.5
Steps
(reduced)
29
(29)
46
(46)
58
(58)
67
(0)
75
(8)
81
(14)
87
(20)
91
(24)
96
(29)
100
(33)
103
(36)
Approximation of harmonics in 67ed5
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) +9.3 +5.7 +11.0 -17.5 +2.3 -13.5 +17.7 +12.0 +10.7 +13.4 +19.6
Relative (%) +22.3 +13.8 +26.5 -42.1 +5.5 -32.5 +42.5 +28.9 +25.8 +32.2 +47.1
Steps
(reduced)
107
(40)
110
(43)
113
(46)
115
(48)
118
(51)
120
(53)
123
(56)
125
(58)
127
(60)
129
(62)
131
(64)

67ed5 as a generator

67ed5 can also be thought of as a generator of the 2.3.5.7.11.19 subgroup temperament which tempers out 441/440, 513/512, 4000/3993, and 10125/10108, which is a cluster temperament with 29 clusters of notes in an octave. The small chroma interval between adjacent notes in each cluster is very versatile, representing 205821/204800 ~ 210/209 ~ 225/224 ~ 7448/7425 ~ 361/360 ~ 400/399 ~ 1375/1372 ~ 200704/200475 all tempered together. This temperament is supported by 29edo, 202edo, and 231edo.