67ed5: Difference between revisions

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Created page with "'''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 cent..."
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Line 20: Line 20:
| | 2
| | 2
| | 83.1735
| | 83.1735
| | [[21/20]]
| | 22/21 ~ [[21/20]]
| |  
| |  
|-
|-
| | 3
| | 3
| | 124.7603
| | 124.7603
| |  
| | 3375/3136
| |  
| |  
|-
|-
Line 35: Line 35:
| | 5
| | 5
| | 207.9339
| | 207.9339
| |  
| | 150/133
| |  
| |  
|-
|-
| | 6
| | 6
| | 249.5206
| | 249.5206
| |  
| | 800/693, 231/200
| |  
| |  
|-
|-
| | 7
| | 7
| | 291.1074
| | 291.1074
| |  
| | 45/38
| |  
| |  
|-
|-
| | 8
| | 8
| | 332.6942
| | 332.6942
| |  
| | 40/33
| |  
| |  
|-
|-
| | 9
| | 9
| | 374.2809
| | 374.2809
| |  
| | 4455/3584
| |  
| |  
|-
|-
| | 10
| | 10
| | 415.8677
| | 415.8677
| |  
| | 80/63, [[14/11]]
| |  
| |  
|-
|-
Line 85: Line 85:
| | 15
| | 15
| | 623.8016
| | 623.8016
| |  
| | 1125/784
| |  
| |  
|-
|-
| | 16
| | 16
| | 665.3883
| | 665.3883
| |  
| | 22/15, 147/100
| |  
| |  
|-
|-
| | 17
| | 17
| | 706.9751
| | 706.9751
| |  
| | 200/133
| | pseudo-[[3/2]]
| | pseudo-[[3/2]]
|-
|-
| | 18
| | 18
| | 748.5619
| | 748.5619
| |  
| | 77/50
| |  
| |  
|-
|-
| | 19
| | 19
| | 790.1487
| | 790.1487
| |  
| | [[30/19]]
| |  
| |  
|-
|-
| | 20
| | 20
| | 831.7354
| | 831.7354
| |  
| | 160/99
| |  
| |  
|-
|-
| | 21
| | 21
| | 873.3222
| | 873.3222
| |  
| | 63/38
| |  
| |  
|-
|-
| | 22
| | 22
| | 914.9090
| | 914.9090
| |  
| | 95/56, 56/33
| |  
| |  
|-
|-
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| | 24
| | 24
| | 998.0825
| | 998.0825
| |  
| | [[16/9]], 57/32
| |  
| |  
|-
|-
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| | 26
| | 26
| | 1081.2561
| | 1081.2561
| |  
| | [[28/15]]
| |  
| |  
|-
|-
| | 27
| | 27
| | 1122.8428
| | 1122.8428
| |  
| | 375/196
| |  
| |  
|-
|-
| | 28
| | 28
| | 1164.4296
| | 1164.4296
| |  
| | [[49/25]]
| |  
| |  
|-
|-
| | 29
| | 29
| | 1206.0164
| | 1206.0164
| |  
| | 800/399, 225/112
| | pseudo-[[octave]]
| | pseudo-[[octave]]
|-
|-
| | 30
| | 30
| | 1247.6032
| | 1247.6032
| |  
| | [[77/75|154/75]]
| |  
| |  
|-
|-
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| | 32
| | 32
| | 1330.7767
| | 1330.7767
| |  
| | 640/297
| |  
| |  
|-
|-
| | 33
| | 33
| | 1372.3635
| | 1372.3635
| |  
| | 495/224, [[21/19|42/19]]
| |  
| |  
|-
|-
| | 34
| | 34
| | 1413.9502
| | 1413.9502
| |  
| | 95/42, 224/99
| |  
| |  
|-
|-
| | 35
| | 35
| | 1455.5370
| | 1455.5370
| |  
| | 297/128
| |  
| |  
|-
|-
| | 36
| | 36
| | 1497.1238
| | 1497.1238
| |  
| | [[19/16|19/8]]
| |  
| |  
|-
|-
| | 37
| | 37
| | 1538.7106
| | 1538.7106
| |  
| | 375/154
| |  
| |  
|-
|-
| | 38
| | 38
| | 1580.2973
| | 1580.2973
| |  
| | [[56/45|112/45]], 399/160
| | pseudo-[[5/2]]
| | pseudo-[[5/2]]
|-
|-
| | 39
| | 39
| | 1621.8841
| | 1621.8841
| |  
| | 125/49
| |  
| |  
|-
|-
| | 40
| | 40
| | 1663.4709
| | 1663.4709
| |  
| | 196/75
| |  
| |  
|-
|-
| | 41
| | 41
| | 1705.0576
| | 1705.0576
| |  
| | 75/28
| |  
| |  
|-
|-
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| | 43
| | 43
| | 1788.2312
| | 1788.2312
| |  
| | 160/57, [[45/32|45/16]]
| |  
| |  
|-
|-
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| | 45
| | 45
| | 1871.4047
| | 1871.4047
| |  
| | 165/56, [[28/19|56/19]]
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| |  
|-
|-
| | 46
| | 46
| | 1912.9915
| | 1912.9915
| |  
| | 190/63
| |  
| |  
|-
|-
| | 47
| | 47
| | 1954.5783
| | 1954.5783
| |  
| | 99/32
| |  
| |  
|-
|-
| | 48
| | 48
| | 1996.1650
| | 1996.1650
| |  
| | [[19/12|19/6]]
| |  
| |  
|-
|-
| | 49
| | 49
| | 2037.7518
| | 2037.7518
| |  
| | 250/77
| |  
| |  
|-
|-
| | 50
| | 50
| | 2079.3386
| | 2079.3386
| |  
| | 133/40
| | pseudo-[[10/3]]
| | pseudo-[[10/3]]
|-
|-
| | 51
| | 51
| | 2120.9254
| | 2120.9254
| |  
| | 500/147, 75/22
| |  
| |  
|-
|-
| | 52
| | 52
| | 2162.5121
| | 2162.5121
| |  
| | 784/225
| |  
| |  
|-
|-
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| | 57
| | 57
| | 2370.4460
| | 2370.4460
| |  
| | 55/14, 63/16
| |  
| |  
|-
|-
| | 58
| | 58
| | 2412.0328
| | 2412.0328
| |  
| | 3584/891
| |  
| |  
|-
|-
| | 59
| | 59
| | 2453.6195
| | 2453.6195
| |  
| | [[33/32|33/8]]
| |  
| |  
|-
|-
| | 60
| | 60
| | 2495.2063
| | 2495.2063
| |  
| | [[19/18|38/9]]
| |  
| |  
|-
|-
| | 61
| | 61
| | 2536.7931
| | 2536.7931
| |  
| | 1000/231, 693/160
| |  
| |  
|-
|-
| | 62
| | 62
| | 2578.3799
| | 2578.3799
| |  
| | 133/30
| |  
| |  
|-
|-
| | 63
| | 63
| | 2619.9666
| | 2619.9666
| |  
| | [[25/22|50/11]]
| |  
| |  
|-
|-
| | 64
| | 64
| | 2661.5534
| | 2661.5534
| |  
| | 3136/675
| |  
| |  
|-
|-
| | 65
| | 65
| | 2703.1402
| | 2703.1402
| |  
| | [[25/21|100/21]]
| |  
| |  
|-
|-
Line 348: Line 348:
| | just major third plus two octaves
| | just major third plus two octaves
|}
|}
==67ed5 as a generator==
67ed5 can also be thought of as a [[generator]] of the 2.3.5.7.11.19 [[Subgroup temperaments|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]].

Revision as of 02:51, 31 December 2018

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 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.

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

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.

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