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'''Division of the 5th harmonic into 67 equal parts''' (67ed5) is related to [[29edo|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 [[7-odd-limit|8-integer-limit]], with discrepancy for the 9th harmonic.
{{Infobox ET}}
'''[[Ed5|Division of the 5th harmonic]] into 67 equal parts''' (67ed5) is related to [[29edo]], but with the [[5/1]] rather than the [[2/1]] being [[just]]. The octave is about 6.0164 cents [[Stretched octave|stretched]] and the step size is about 41.5868 cents.  


{| class="wikitable"
== Theory ==
67ed5 has a generally sharp tendency for [[harmonic]]s up to 28. Unlike 29edo, it is only [[consistent]] up to the 8-[[integer-limit]], with discrepancy for the 9th harmonic. As an equal temperament, it [[tempering out|tempers out]] 49/48 in the [[7-limit]]; 55/54 in the 11-limit; 65/64 and 91/90 in the 13-limit; 85/84 in the 17-limit; 77/76 in the 19-limit; 70/69 in the 23-limit; 58/57 in the 29-limit; and 93/92 in the 31-limit.
 
=== Prime harmonics ===
Compared to 29edo, 67ed5 has a much better 5/1, 7/1, 11/1, 13/1, and 17/1, at the expense of a much worse 3/1.
 
The biggest argument in favor of this trade-off is that 29edo’s 7/1 is so inaccurate as to be unusable for many. So, the fact that 67ed5 makes the 3/1 not as good, but still definitely useable, and in return, replaces that unusable 7/1 with almost perfectly in-tune one, could be seen as a worthwhile trade-off.
{{Harmonics in equal|67|5|1|intervals=prime|columns=11}}
 
 
29edo for comparison:
{{Harmonics in equal|29|columns=11}}
 
=== 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]].
 
== Intervals ==
{| class="wikitable mw-collapsible"
|+ Intervals of 67ed5
|-
|-
! | degree
! | degree
Line 20: Line 39:
| | 2
| | 2
| | 83.1735
| | 83.1735
| | [[21/20]]
| | 22/21, [[21/20]]
| |  
| |  
|-
|-
| | 3
| | 3
| | 124.7603
| | 124.7603
| |  
| | 3375/3136
| |  
| |  
|-
|-
Line 35: Line 54:
| | 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 104:
| | 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
| |  
| |  
|-
|-
Line 130: Line 149:
| | 24
| | 24
| | 998.0825
| | 998.0825
| |  
| | [[16/9]], 57/32
| |  
| |  
|-
|-
Line 140: Line 159:
| | 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
| |  
| | 154/75
| |  
| |  
|-
|-
Line 170: Line 189:
| | 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
| |  
| |  
|-
|-
Line 225: Line 244:
| | 43
| | 43
| | 1788.2312
| | 1788.2312
| |  
| | 160/57, [[45/32|45/16]]
| |  
| |  
|-
|-
Line 235: Line 254:
| | 45
| | 45
| | 1871.4047
| | 1871.4047
| |  
| | 165/56, [[28/19|56/19]]
| |  
| |  
|-
|-
| | 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
| |  
| |  
|-
|-
Line 295: Line 314:
| | 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 367:
| | just major third plus two octaves
| | just major third plus two octaves
|}
|}
[[Category:29edo]]

Latest revision as of 19:23, 1 August 2025

← 66ed5 67ed5 68ed5 →
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 29edo, 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.

Theory

67ed5 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. As an equal temperament, it tempers out 49/48 in the 7-limit; 55/54 in the 11-limit; 65/64 and 91/90 in the 13-limit; 85/84 in the 17-limit; 77/76 in the 19-limit; 70/69 in the 23-limit; 58/57 in the 29-limit; and 93/92 in the 31-limit.

Prime harmonics

Compared to 29edo, 67ed5 has a much better 5/1, 7/1, 11/1, 13/1, and 17/1, at the expense of a much worse 3/1.

The biggest argument in favor of this trade-off is that 29edo’s 7/1 is so inaccurate as to be unusable for many. So, the fact that 67ed5 makes the 3/1 not as good, but still definitely useable, and in return, replaces that unusable 7/1 with almost perfectly in-tune one, could be seen as a worthwhile trade-off.

Approximation of prime harmonics in 67ed5
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +6.0 +11.0 +0.0 -0.3 +7.4 +9.3 +2.3 +17.7 +19.6 -7.4 +1.9
Relative (%) +14.5 +26.5 +0.0 -0.7 +17.7 +22.3 +5.5 +42.5 +47.1 -17.9 +4.5
Steps
(reduced) 		29
(29) 		46
(46) 		67
(0) 		81
(14) 		100
(33) 		107
(40) 		118
(51) 		123
(56) 		131
(64) 		140
(6) 		143
(9)


29edo for comparison:

Approximation of prime harmonics in 29edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 +1.5 -13.9 -17.1 -13.4 -12.9 +19.2 -7.9 -7.6 +4.9 +13.6
Relative (%) +0.0 +3.6 -33.6 -41.3 -32.4 -31.3 +46.4 -19.0 -18.3 +11.9 +32.8
Steps
(reduced) 		29
(0) 		46
(17) 		67
(9) 		81
(23) 		100
(13) 		107
(20) 		119
(3) 		123
(7) 		131
(15) 		141
(25) 		144
(28)

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.

Intervals

Intervals of 67ed5
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
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