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'''Division of the 5th harmonic into 95 equal parts''' (95ed5) is related to [[41edo|41 edo]], but with the 5/1 rather than the 2/1 being just. The octave is about 2.5143 cents stretched and the step size about 29.3296 cents. This tuning has a generally sharp tendency for harmonics up to 12. Unlike 41edo, it is only consistent up to the [[11-limit|12-integer-limit]], with discrepancy for the 13th harmonic.
{{Infobox ET}}
{{ED intro}}


{| class="wikitable"
== Theory ==
95ed5 is related to [[41edo]], but with the 5th harmonic rather than the [[2/1|octave]] being just. The octave is about 2.51 cents stretched. This tuning has a generally sharp tendency for [[harmonic]]s up to 12. Unlike 41edo, it is only [[consistent]] up to the [[integer limit|12-integer-limit]], with discrepancy for the [[13/1|13th harmonic]].
 
=== Harmonics ===
{{Harmonics in equal|95|5|1|intervals=integer}}
{{Harmonics in equal|95|5|1|intervals=integer|columns=12|start=12|collapsed=1|title=Approximation of harmonics in 95ed5 (continued)}}
 
=== Subsets and supersets ===
Since 95 factors into primes as {{nowrap| 5 × 19 }}, 95ed5 contains [[5ed5]] and [[19ed5]] as subset ed5's.
 
== Intervals ==
{| 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
| |
| [[25/24]], [[28/27]], [[33/32]], [[36/35]]
| |
|-
|-
| | 3
| 3
| | 87.9889
| 88.0
| | 81/77
| [[19/18]], [[20/19]], [[21/20]], [[22/21]]
| |
|-
|-
| | 4
| 4
| | 117.3185
| 117.3
| |
| [[14/13]], [[15/14]], [[16/15]]
| |
|-
|-
| | 5
| 5
| | 146.6481
| 146.6
| | [[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
| | [[55/48]], [[63/55]]
| [[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]]
| [[11/9]], [[16/13]]
| |
|-
|-
| | 13
| 13
| | 381.2850
| 381.3
| |
| [[5/4]], [[26/21]]
| | pseudo-[[5/4]]
|-
|-
| | 14
| 14
| | 410.6147
| 410.6
| | 308/243
| [[19/15]]
| |
|-
|-
| | 15
| 15
| | 439.9443
| 439.9
| |
| [[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
| [[7/5]], [[45/32]]
| |
|-
|-
| | 21
| 21
| | 615.9220
| 615.9
| |
| [[10/7]], [[64/45]]
| |
|-
|-
| | 22
| 22
| | 645.2516
| 645.3
| | 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
| |
| [[3/2]]
| | pseudo-[[3/2]]
|-
|-
| | 25
| 25
| | 733.2405
| 733.2
| | 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
| |
| [[13/8]], [[18/11]]
| |
|-
|-
| | 30
| 30
| | 879.8885
| 879.9
| | 539/324
| [[5/3]]
| | pseudo-[[5/3]]
|-
|-
| | 31
| 31
| | 909.2182
| 909.2
| | 1232/729, 3645/2156
| [[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
| [[13/7]], [[15/8]]
| |
|-
|-
| | 38
| 38
| | 1114.5255
| 1114.5
| |
| [[19/10]], [[21/11]]
| |
|-
|-
| | 39
| 39
| | 1143.8551
| 1143.9
| | 784/405
| [[27/14]], [[35/18]]
| |
|-
|-
| | 40
| 40
| | 1173.1847
| 1173.2
| | 675/343
| [[49/25]], [[55/28]], [[63/32]]
| |
|-
|-
| | 41
| 41
| | 1202.5143
| 1202.5
| | 441/220
| [[2/1]]
| | pseudo-[[octave]]
|-
|-
| | 42
| 42
| | 1231.8440
| 1231.8
| | [[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
| | 539/243
| [[11/5]]
| |
|-
|-
| | 48
| 48
| | 1407.8217
| 1407.8
| | 1215/539
| [[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
| [[22/9]], [[27/11]]
| |
|-
|-
| | 54
| 54
| | 1583.7994
| 1583.8
| | 1100/441
| [[5/2]]
| | pseudo-[[5/2]]
|-
|-
| | 55
| 55
| | 1613.1290
| 1613.1
| | 343/135
| [[28/11]], [[33/13]]
| |
|-
|-
| | 56
| 56
| | 1642.4586
| 1642.5
| | 2025/784
| [[18/7]]
| |
|-
|-
| | 57
| 57
| | 1671.7882
| 1671.8
| |
| [[21/8]]
| |
|-
|-
| | 58
| 58
| | 1701.1178
| 1701.1
| | 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
| | 2156/729, 3645/1232
| [[44/15]]
| |
|-
|-
| | 65
| 65
| | 1906.4252
| 1906.4
| | 1620/539
| [[3/1]]
| | pseudo-[[3/1]]
|-
|-
| | 66
| 66
| | 1935.7548
| 1935.8
| |
| [[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
| [[13/4]]
| |
|-
|-
| | 71
| 71
| | 2082.4029
| 2082.4
| |
| [[10/3]]
| | pseudo-[[10/3]]
|-
|-
| | 72
| 72
| | 2111.7325
| 2111.7
| | 1372/405
| [[27/8]]
| |
|-
|-
| | 73
| 73
| | 2141.0621
| 2141.1
| | 675/196
| [[24/7]]
| |
|-
|-
| | 74
| 74
| | 2170.3917
| 2170.4
| |
| [[7/2]]
| |
|-
|-
| | 75
| 75
| | 2199.7214
| 2199.7
| | 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
| |
| [[27/7]], [[35/9]]
| |
|-
|-
| | 81
| 81
| | 2375.6991
| 2375.7
| | 1215/308
| [[55/14]], [[63/16]]
| |
|-
|-
| | 82
| 82
| | 2405.0287
| 2405.0
| |
| [[4/1]]
| | pseudo-[[4/1]]
|-
|-
| | 83
| 83
| | 2434.3583
| 2434.4
| | [[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
| [[16/7]]
| |
|-
|-
| | 91
| 91
| | 2668.9952
| 2669.0
| |
| [[14/3]]
| |
|-
|-
| | 92
| 92
| | 2698.3249
| 2698.3
| | 385/81
| [[19/4]]
| |
|-
|-
| | 93
| 93
| | 2727.6545
| 2727.7
| |
| [[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
|}
|}


95ed5 can also be thought of as a generator of the 11-limit temperament which tempers out 3025/3024, 184877/184320, and 2460375/2458624, which is a [[cluster temperament]] with 41 clusters of notes in an octave. The small chroma interval between adjacent notes in each cluster is very versatile, representing 64/63 ~ 12100/11907 ~ 295245/290521 ~ 273375/268912 ~ 16807/16500 all tempered together. This temperament is supported by [[41edo]], [[491edo]] (491e val), and [[532edo]] (532d val) among others.
== 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.
 
== See also ==
* [[24edf]] – relative edf
* [[41edo]] – relative edo
* [[65edt]] – relative edt
* [[106ed6]] – relative ed6
* [[147ed12]] – relative ed12
* [[361ed448]] – close to the zeta-optimized tuning for 41edo
 
[[Category:41edo]]

Latest revision as of 14:00, 20 June 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 5th harmonic rather than the octave being just. The octave is about 2.51 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)

Subsets and supersets

Since 95 factors into primes as 5 × 19, 95ed5 contains 5ed5 and 19ed5 as subset ed5's.

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.

See also

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