95ed5: Difference between revisions

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== Intervals ==
== Intervals ==
{| class="wikitable mw-collapsible"
{| class="wikitable center-1 right-2"
|-
|-
! | degree
! #
! | cents value
! Cents
! | corresponding <br>JI intervals
! Approximate ratios
! | comments
|-
|-
| | 0
| 0
| | 0.0000
| 0.0
| | '''exact [[1/1]]'''
| [[1/1]]
| |
|-
|-
| | 1
| 1
| | 29.3296
| 29.3
| |
| [[49/48]], [[50/49]], [[64/63]], [[81/80]]
| |
|-
|-
| | 2
| 2
| | 58.6592
| 58.7
| | 931/900
| [[25/24]], [[28/27]], [[33/32]], [[36/35]]
| |
|-
|-
| | 3
| 3
| | 87.9889
| 88.0
| | 81/77, [[20/19]]
| [[19/18]], [[20/19]], [[21/20]], [[22/21]]
| |
|-
|-
| | 4
| 4
| | 117.3185
| 117.3
| | 1280/1197
| [[14/13]], [[15/14]], [[16/15]]
| |
|-
|-
| | 5
| 5
| | 146.6481
| 146.6
| | 209/192, [[49/45]]
| [[12/11], [[13/12]]
| |
|-
|-
| | 6
| 6
| | 175.9777
| 176.0
| | 448/405
| [[10/9]], [[11/10]], [[21/19]]
| |
|-
|-
| | 7
| 7
| | 205.3073
| 205.3
| |
| [[9/8]]
| |
|-
|-
| | 8
| 8
| | 234.6369
| 234.6
| | [[63/55]], [[55/48]]
| [[8/7]], [[15/13]]
| |
|-
|-
| | 9
| 9
| | 263.9666
| 264.0
| | 220/189
| [[7/6]], [[22/19]]
| |
|-
|-
| | 10
| 10
| | 293.2962
| 293.3
| |
| [[13/11]], [[19/16]], [[32/27]]
| |
|-
|-
| | 11
| 11
| | 322.6258
| 322.6
| | 135/112
| [[6/5]]
| |
|-
|-
| | 12
| 12
| | 351.9554
| 352.0
| | [[60/49]], 256/209
| [[11/9]], [[16/13]]
| |
|-
|-
| | 13
| 13
| | 381.2850
| 381.3
| | 399/320
| [[5/4]], [[26/21]]
| | pseudo-[[5/4]]
|-
|-
| | 14
| 14
| | 410.6147
| 410.6
| | [[19/15]]
| [[19/15]]
| |
|-
|-
| | 15
| 15
| | 439.9443
| 439.9
| | 1200/931
| [[9/7]], [[32/25]]
| |
|-
|-
| | 16
| 16
| | 469.2739
| 469.3
| | [[21/16]]
| [[21/16]], [[13/10]]
| |
|-
|-
| | 17
| 17
| | 498.6035
| 498.6
| | [[4/3]]
| [[4/3]]
| |
|-
|-
| | 18
| 18
| | 527.9331
| 527.9
| |
| [[15/11]], [[19/14]], [[27/20]]
| |
|-
|-
| | 19
| 19
| | 557.2627
| 557.3
| | 243/176
| [[11/8]], [[18/13]], [[26/19]]
| |
|-
|-
| | 20
| 20
| | 586.5924
| 586.6
| | 108/77, 275/196, 80/57
| [[7/5]], [[45/32]]
| |
|-
|-
| | 21
| 21
| | 615.9220
| 615.9
| |
| [[10/7]], [[64/45]]
| |
|-
|-
| | 22
| 22
| | 645.2516
| 645.3
| | 209/144, 196/135
| [[13/9]], [[16/11]], [[19/13]]
| |
|-
|-
| | 23
| 23
| | 674.5812
| 674.6
| | 2025/1372
| [[22/15]], [[28/19]], [[40/27]]
| |
|-
|-
| | 24
| 24
| | 703.9108
| 703.9
| | 1539/1024
| [[3/2]]
| | pseudo-[[3/2]]
|-
|-
| | 25
| 25
| | 733.2405
| 733.2
| | 171/112, 84/55, 55/36
| [[20/13]], [[32/21]]
| |
|-
|-
| | 26
| 26
| | 762.5701
| 762.6
| |
| [[14/9]], [[25/16]]
| |
|-
|-
| | 27
| 27
| | 791.8997
| 791.9
| |
| [[11/7]], [[19/12]], [[30/19]]
| |
|-
|-
| | 28
| 28
| | 821.2293
| 821.2
| | 45/28
| [[8/5]], [[21/13]]
| |
|-
|-
| | 29
| 29
| | 850.5589
| 850.6
| | 1024/627
| [[13/8]], [[18/11]]
| |
|-
|-
| | 30
| 30
| | 879.8885
| 879.9
| | 133/80
| [[5/3]]
| | pseudo-[[5/3]]
|-
|-
| | 31
| 31
| | 909.2182
| 909.2
| | 1232/729, 3645/2156, 225/133
| [[22/13]], [[27/16]], [[32/19]]
| |
|-
|-
| | 32
| 32
| | 938.5478
| 938.5
| |
| [[12/7]], [[19/11]]
| |
|-
|-
| | 33
| 33
| | 967.8774
| 967.9
| | [[7/4]]
| [[7/4]], [[26/15]]
| |
|-
|-
| | 34
| 34
| | 997.2070
| 997.2
| | [[16/9]]
| [[16/9]]
| |
|-
|-
| | 35
| 35
| | 1026.5366
| 1026.5
| |
| [[9/5]]
| |
|-
|-
| | 36
| 36
| | 1055.8662
| 1055.9
| | 81/44
| [[11/6]]
| |
|-
|-
| | 37
| 37
| | 1085.1959
| 1085.2
| | 144/77, 275/147, 320/171
| [[13/7]], [[15/8]]
| |
|-
|-
| | 38
| 38
| | 1114.5255
| 1114.5
| |
| [[19/10]], [[21/11]]
| |
|-
|-
| | 39
| 39
| | 1143.8551
| 1143.9
| | 209/108, 405/209
| [[27/14]], [[35/18]]
| |
|-
|-
| | 40
| 40
| | 1173.1847
| 1173.2
| | 675/343
| [[49/25]], [[55/28]], [[63/32]]
| |
|-
|-
| | 41
| 41
| | 1202.5143
| 1202.5
| | 513/256, 441/220
| [[2/1]]
| | pseudo-[[octave]]
|-
|-
| | 42
| 42
| | 1231.8440
| 1231.8
| | 57/28, [[56/55|112/55]], [[55/54|55/27]]
| [[45/22]], [[49/24]], [[55/27]], [[81/40]]
| |
|-
|-
| | 43
| 43
| | 1261.1736
| 1261.2
| |
| [[25/12]], [[33/16]]
| |
|-
|-
| | 44
| 44
| | 1290.5032
| 1290.5
| |
| [[19/9]], [[21/10]]
| |
|-
|-
| | 45
| 45
| | 1319.8328
| 1319.8
| | [[15/14|15/7]]
| [[15/7]]
| |
|-
|-
| | 46
| 46
| | 1349.1624
| 1349.2
| |
| [[13/6]]
| |
|-
|-
| | 47
| 47
| | 1378.4920
| 1378.5
| | 133/60, 539/243
| [[11/5]]
| |
|-
|-
| | 48
| 48
| | 1407.8217
| 1407.8
| | 1215/539, 300/133
| [[9/4]]
| |
|-
|-
| | 49
| 49
| | 1437.1513
| 1437.2
| |
| [[16/7]]
| |
|-
|-
| | 50
| 50
| | 1466.4809
| 1466.5
| | [[7/3]]
| [[7/3]]
| |
|-
|-
| | 51
| 51
| | 1495.8105
| 1495.8
| |
| [[19/8]]
| |
|-
|-
| | 52
| 52
| | 1525.1401
| 1525.1
| |
| [[12/5]]
| |
|-
|-
| | 53
| 53
| | 1554.4698
| 1554.5
| | [[27/22|27/11]], 275/112, 140/57
| [[22/9]], [[27/11]]
| |
|-
|-
| | 54
| 54
| | 1583.7994
| 1583.8
| | 1100/441, 1280/513
| [[5/2]]
| | pseudo-[[5/2]]
|-
|-
| | 55
| 55
| | 1613.1290
| 1613.1
| | 343/135
| [[28/11]], [[33/13]]
| |
|-
|-
| | 56
| 56
| | 1642.4586
| 1642.5
| | 209/81, 540/209
| [[18/7]]
| |
|-
|-
| | 57
| 57
| | 1671.7882
| 1671.8
| |
| [[21/8]]
| |
|-
|-
| | 58
| 58
| | 1701.1178
| 1701.1
| | 171/64, 147/55, 385/144
| [[8/3]]
| |
|-
|-
| | 59
| 59
| | 1730.4475
| 1730.4
| | 220/81
| [[19/7]]
| |
|-
|-
| | 60
| 60
| | 1759.7771
| 1759.8
| |
| [[11/4]]
| |
|-
|-
| | 61
| 61
| | 1789.1067
| 1789.1
| | [[45/32|45/16]]
| [[14/5]]
| |
|-
|-
| | 62
| 62
| | 1818.4363
| 1818.4
| | [[10/7|20/7]]
| [[20/7]]
| |
|-
|-
| | 63
| 63
| | 1847.7659
| 1847.8
| |
| [[26/9]]
| |
|-
|-
| | 64
| 64
| | 1877.0956
| 1877.1
| | 133/45, 2156/729, 3645/1232
| [[44/15]]
| |
|-
|-
| | 65
| 65
| | 1906.4252
| 1906.4
| | 400/133
| [[3/1]]
| | pseudo-[[3/1]]
|-
|-
| | 66
| 66
| | 1935.7548
| 1935.8
| | 3135/1024
| [[40/13]]
| |
|-
|-
| | 67
| 67
| | 1965.0844
| 1965.1
| | [[14/9|28/9]]
| [[25/8]], [[28/9]]
| |
|-
|-
| | 68
| 68
| | 1994.4140
| 1994.4
| |
| [[19/6]], [[22/7]]
| |
|-
|-
| | 69
| 69
| | 2023.7436
| 2023.7
| |
| [[16/5]]
| |
|-
|-
| | 70
| 70
| | 2053.0733
| 2053.1
| | [[18/11|36/11]], 275/84, 560/171
| [[13/4]]
| |
|-
|-
| | 71
| 71
| | 2082.4029
| 2082.4
| | 5120/1539
| [[10/3]]
| | pseudo-[[10/3]]
|-
|-
| | 72
| 72
| | 2111.7325
| 2111.7
| | 1372/405
| [[27/8]]
| |
|-
|-
| | 73
| 73
| | 2141.0621
| 2141.1
| | 675/196, 720/209
| [[24/7]]
| |
|-
|-
| | 74
| 74
| | 2170.3917
| 2170.4
| |
| [[7/2]]
| |
|-
|-
| | 75
| 75
| | 2199.7214
| 2199.7
| | 57/16, 196/55, 385/108
| [[25/7]]
| |
|-
|-
| | 76
| 76
| | 2229.0510
| 2229.1
| | 880/243
| [[18/5]]
| |
|-
|-
| | 77
| 77
| | 2258.3806
| 2258.4
| |
| [[11/3]]
| |
|-
|-
| | 78
| 78
| | 2287.7102
| 2287.7
| | [[15/4]]
| [[15/4]]
| |
|-
|-
| | 79
| 79
| | 2317.0398
| 2317.0
| | [[40/21|80/21]]
| [[19/5]]
| |
|-
|-
| | 80
| 80
| | 2346.3694
| 2346.4
| | 931/240
| [[27/7]], [[35/9]]
| |
|-
|-
| | 81
| 81
| | 2375.6991
| 2375.7
| | 75/19
| [[55/14]], [[63/16]]
| |
|-
|-
| | 82
| 82
| | 2405.0287
| 2405.0
| | 1600/399
| [[4/1]]
| | pseudo-[[4/1]]
|-
|-
| | 83
| 83
| | 2434.3583
| 2434.4
| | 1045/256, [[49/48|49/12]]
| [[49/12]], [[81/20]]
| |
|-
|-
| | 84
| 84
| | 2463.6879
| 2463.7
| | [[28/27|112/27]]
| [[25/6]], [[33/8]]
| |
|-
|-
| | 85
| 85
| | 2493.0175
| 2493.0
| |
| [[21/5]]
| |
|-
|-
| | 86
| 86
| | 2522.3472
| 2522.3
| | 189/44
| [[30/7]]
| |
|-
|-
| | 87
| 87
| | 2551.6768
| 2551.7
| | [[12/11|48/11]], 275/63
| [[13/3]]
| |
|-
|-
| | 88
| 88
| | 2581.0064
| 2581.0
| |
| [[22/5]]
| |
|-
|-
| | 89
| 89
| | 2610.3360
| 2610.3
| | 2025/448
| [[9/2]]
| |
|-
|-
| | 90
| 90
| | 2639.6656
| 2639.7
| | 225/49, 960/209
| [[16/7]]
| |
|-
|-
| | 91
| 91
| | 2668.9952
| 2669.0
| | 1197/256
| [[14/3]]
| |
|-
|-
| | 92
| 92
| | 2698.3249
| 2698.3
| | 19/4, 385/81
| [[19/4]]
| |
|-
|-
| | 93
| 93
| | 2727.6545
| 2727.7
| | 4500/931
| [[24/5]]
| |
|-
|-
| | 94
| 94
| | 2756.9841
| 2757.0
| |
| [[39/8]]
| |
|-
|-
| | 95
| 95
| | 2786.3137
| 2786.3
| | '''exact [[5/1]]'''
| [[5/1]]
| | just major third plus two octaves
|}
|}


== As a generator ==
== As a generator ==
95ed5 can also be thought of as a [[generator]] of the 2.3.5.7.11.19-subgroup temperament which tempers out 1540/1539, 3025/3024, 6875/6859, and 184877/184320, which is a [[cluster temperament]] with 41 clusters of notes in an octave. While the small chroma interval between adjacent notes in each cluster represents 385/384 ~ 441/440 ~ 1479016/1476225 ~ 194579/194400 ~ 204800/204687 ~ 176000/175959 tempered together, the step interval is very versatile, representing 16807/16500 ~ 19551/19200 ~ 18000/17689 ~ 72900/71687 ~ 273375/268912 ~ 295245/290521 ~ 12100/11907 ~ 64/63 all tempered together. This temperament is supported by [[41edo]], [[491edo]] (491e val), and [[532edo]] (532d val) among others.
95ed5 can also be thought of as a [[generator]] of the 2.3.5.7.11.19-subgroup temperament which tempers out 1540/1539, 3025/3024, 6875/6859, and 184877/184320, which is a [[cluster temperament]] with 41 clusters of notes in an octave. While the small chroma interval between adjacent notes in each cluster represents 385/384 ~ 441/440 ~ 1479016/1476225 ~ 194579/194400 ~ 204800/204687 ~ 176000/175959 tempered together, the step interval is very versatile, representing 16807/16500 ~ 19551/19200 ~ 18000/17689 ~ 72900/71687 ~ 273375/268912 ~ 295245/290521 ~ 12100/11907 ~ 64/63 all tempered together. This temperament is supported by [[41edo]], [[491edo]] (491e val), and [[532edo]] (532d val) among others.

Revision as of 17:47, 20 March 2025

← 94ed5 95ed5 96ed5 →
Prime factorization 5 × 19
Step size 29.3296 ¢ 
Octave 41\95ed5 (1202.51 ¢)
Twelfth 65\95ed5 (1906.43 ¢) (→ 13\19ed5)
Consistency limit 12
Distinct consistency limit 10

95 equal divisions of the 5th harmonic (abbreviated 95ed5) is a nonoctave tuning system that divides the interval of 5/1 into 95 equal parts of about 29.3 ¢ each. Each step represents a frequency ratio of 51/95, or the 95th root of 5.

Theory

95ed5 is related to 41edo, but with the 5/1 rather than the 2/1 being just. The octave is about 2.5143 cents stretched. This tuning has a generally sharp tendency for harmonics up to 12. Unlike 41edo, it is only consistent up to the 12-integer-limit, with discrepancy for the 13th harmonic.

Harmonics

Approximation of harmonics in 95ed5
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +2.5 +4.5 +5.0 +0.0 +7.0 +4.1 +7.5 +8.9 +2.5 +13.5 +9.5
Relative (%) +8.6 +15.2 +17.1 +0.0 +23.8 +13.9 +25.7 +30.5 +8.6 +46.0 +32.4
Steps
(reduced) 		41
(41) 		65
(65) 		82
(82) 		95
(0) 		106
(11) 		115
(20) 		123
(28) 		130
(35) 		136
(41) 		142
(47) 		147
(52)
Approximation of harmonics in 95ed5 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -11.8 +6.6 +4.5 +10.1 -6.9 +11.5 +5.8 +5.0 +8.6 -13.3 -2.3 +12.0
Relative (%) -40.1 +22.5 +15.2 +34.3 -23.6 +39.1 +19.9 +17.1 +29.2 -45.4 -7.8 +41.0
Steps
(reduced) 		151
(56) 		156
(61) 		160
(65) 		164
(69) 		167
(72) 		171
(76) 		174
(79) 		177
(82) 		180
(85) 		182
(87) 		185
(90) 		188
(93)

Intervals

# Cents Approximate ratios
0 0.0 1/1
1 29.3 49/48, 50/49, 64/63, 81/80
2 58.7 25/24, 28/27, 33/32, 36/35
3 88.0 19/18, 20/19, 21/20, 22/21
4 117.3 14/13, 15/14, 16/15
5 146.6 [[12/11], 13/12
6 176.0 10/9, 11/10, 21/19
7 205.3 9/8
8 234.6 8/7, 15/13
9 264.0 7/6, 22/19
10 293.3 13/11, 19/16, 32/27
11 322.6 6/5
12 352.0 11/9, 16/13
13 381.3 5/4, 26/21
14 410.6 19/15
15 439.9 9/7, 32/25
16 469.3 21/16, 13/10
17 498.6 4/3
18 527.9 15/11, 19/14, 27/20
19 557.3 11/8, 18/13, 26/19
20 586.6 7/5, 45/32
21 615.9 10/7, 64/45
22 645.3 13/9, 16/11, 19/13
23 674.6 22/15, 28/19, 40/27
24 703.9 3/2
25 733.2 20/13, 32/21
26 762.6 14/9, 25/16
27 791.9 11/7, 19/12, 30/19
28 821.2 8/5, 21/13
29 850.6 13/8, 18/11
30 879.9 5/3
31 909.2 22/13, 27/16, 32/19
32 938.5 12/7, 19/11
33 967.9 7/4, 26/15
34 997.2 16/9
35 1026.5 9/5
36 1055.9 11/6
37 1085.2 13/7, 15/8
38 1114.5 19/10, 21/11
39 1143.9 27/14, 35/18
40 1173.2 49/25, 55/28, 63/32
41 1202.5 2/1
42 1231.8 45/22, 49/24, 55/27, 81/40
43 1261.2 25/12, 33/16
44 1290.5 19/9, 21/10
45 1319.8 15/7
46 1349.2 13/6
47 1378.5 11/5
48 1407.8 9/4
49 1437.2 16/7
50 1466.5 7/3
51 1495.8 19/8
52 1525.1 12/5
53 1554.5 22/9, 27/11
54 1583.8 5/2
55 1613.1 28/11, 33/13
56 1642.5 18/7
57 1671.8 21/8
58 1701.1 8/3
59 1730.4 19/7
60 1759.8 11/4
61 1789.1 14/5
62 1818.4 20/7
63 1847.8 26/9
64 1877.1 44/15
65 1906.4 3/1
66 1935.8 40/13
67 1965.1 25/8, 28/9
68 1994.4 19/6, 22/7
69 2023.7 16/5
70 2053.1 13/4
71 2082.4 10/3
72 2111.7 27/8
73 2141.1 24/7
74 2170.4 7/2
75 2199.7 25/7
76 2229.1 18/5
77 2258.4 11/3
78 2287.7 15/4
79 2317.0 19/5
80 2346.4 27/7, 35/9
81 2375.7 55/14, 63/16
82 2405.0 4/1
83 2434.4 49/12, 81/20
84 2463.7 25/6, 33/8
85 2493.0 21/5
86 2522.3 30/7
87 2551.7 13/3
88 2581.0 22/5
89 2610.3 9/2
90 2639.7 16/7
91 2669.0 14/3
92 2698.3 19/4
93 2727.7 24/5
94 2757.0 39/8
95 2786.3 5/1

As a generator

95ed5 can also be thought of as a generator of the 2.3.5.7.11.19-subgroup temperament which tempers out 1540/1539, 3025/3024, 6875/6859, and 184877/184320, which is a cluster temperament with 41 clusters of notes in an octave. While the small chroma interval between adjacent notes in each cluster represents 385/384 ~ 441/440 ~ 1479016/1476225 ~ 194579/194400 ~ 204800/204687 ~ 176000/175959 tempered together, the step interval is very versatile, representing 16807/16500 ~ 19551/19200 ~ 18000/17689 ~ 72900/71687 ~ 273375/268912 ~ 295245/290521 ~ 12100/11907 ~ 64/63 all tempered together. This temperament is supported by 41edo, 491edo (491e val), and 532edo (532d val) among others.

Retrieved from "https://en.xen.wiki/index.php?title=95ed5&oldid=187419"