96ed5: Difference between revisions

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Line 24: Line 24:
! Steps
! Steps
! Cents
! Cents
! Approximate ratios
! 8.9.5.7.11.13.17.23 ratios
|-
|-
| 0
| 0
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| 2
| 2
| 58
| 58
| [[30/29]], [[31/30]]
|  
|-
|-
| 3
| 3
| 87.1
| 87.1
| [[41/39]]
|  
|-
|-
| 4
| 4
| 116.1
| 116.1
| [[31/29]]
|  
|-
|-
| 5
| 5
| 145.1
| 145.1
| [[25/23]], [[37/34]]
| [[25/23]]
|-
|-
| 6
| 6
| 174.1
| 174.1
| [[21/19]]
|  
|-
|-
| 7
| 7
| 203.2
| 203.2
|  
| [[9/8]]
|-
|-
| 8
| 8
| 232.2
| 232.2
|  
| [[8/7]]
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| 9
| 9
| 261.2
| 261.2
| [[36/31]], [[43/37]]
|  
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| 10
| 10
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| 12
| 12
| 348.3
| 348.3
|  
| [[11/9]]
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| 13
| 13
| 377.3
| 377.3
| [[41/33]]
|  
|-
|-
| 14
| 14
| 406.3
| 406.3
| [[43/34]]
|  
|-
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| 15
| 15
| 435.4
| 435.4
|  
| [[9/7]]
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| 16
| 16
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| 19
| 19
| 551.5
| 551.5
|  
| [[11/8]]
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| 20
| 20
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| 21
| 21
| 609.5
| 609.5
| [[37/26]]
|  
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| 22
| 22
| 638.5
| 638.5
|  
| [[13/9]]
|-
|-
| 23
| 23
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| 25
| 25
| 725.6
| 725.6
| [[35/23]], [[38/25]]
| [[35/23]]
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| 26
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| 28
| 28
| 812.7
| 812.7
|  
| [[8/5]]
|-
|-
| 29
| 29
| 841.7
| 841.7
|  
| [[13/8]]
|-
|-
| 30
| 30
| 870.7
| 870.7
| [[38/23]], [[43/26]]
|  
|-
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| 31
| 31
| 899.7
| 899.7
| [[37/22]], [[42/25]]
|  
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| 32
| 32
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| 33
| 33
| 957.8
| 957.8
| [[33/19]]
| [[40/23]]
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| 34
| 34
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| 35
| 35
| 1015.8
| 1015.8
|  
| [[9/5]]
|-
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| 36
| 36
| 1044.9
| 1044.9
| [[42/23]]
|  
|-
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| 37
| 37
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| 38
| 38
| 1102.9
| 1102.9
|  
| [[17/9]]
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| 39
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| 40
| 40
| 1161
| 1161
| [[43/22]]
| [[45/23]]
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| 41
| 41
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| 45
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| 1306.1
| 1306.1
|  
| [[17/8]], [[49/23]]
|-
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| 48
| 48
| 1393.2
| 1393.2
| [[38/17]]
|  
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| 49
| 49
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| 51
| 51
| 1480.2
| 1480.2
|  
| [[40/17]]
|-
|-
| 52
| 52
| 1509.3
| 1509.3
|  
| [[55/23]]
|-
|-
| 53
| 53
| 1538.3
| 1538.3
| [[17/7]]
| [[17/7]], [[56/23]]
|-
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| 54
| 54
| 1567.3
| 1567.3
| [[42/17]]
|  
|-
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| 55
| 55
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| 56
| 56
| 1625.3
| 1625.3
|  
| [[23/9]]
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| 57
| 57
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| 58
| 58
| 1683.4
| 1683.4
| [[37/14]]
| [[45/17]]
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| 59
| 59
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| 60
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| 1741.4
| 1741.4
| [[41/15]]
| [[63/23]]
|-
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| 61
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| 1770.5
| 1770.5
|  
| [[25/9]], [[64/23]]
|-
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| 62
| 62
| 1799.5
| 1799.5
|  
| [[65/23]]
|-
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| 63
| 63
| 1828.5
| 1828.5
|  
| [[23/8]], [[49/17]]
|-
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| 64
| 64
| 1857.5
| 1857.5
| [[38/13]]
|  
|-
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| 65
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| 67
| 67
| 1944.6
| 1944.6
| [[43/14]]
| [[40/13]]
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| 68
| 1973.6
| 1973.6
|  
| [[25/8]], [[72/23]]
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| 2031.7
| 2031.7
| [[42/13]]
| [[55/17]]
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| 2060.7
| 2060.7
| [[23/7]]
| [[23/7]], [[56/17]]
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| 2089.7
| 2089.7
|  
| [[77/23]]
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| 2147.8
| 2147.8
| [[38/11]]
| [[45/13]]
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| 2176.8
| 2176.8
|  
| [[81/23]]
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| 77
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| 2234.9
| 2234.9
|  
| [[40/11]]
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| 2263.9
| 2263.9
| [[37/10]]
| [[63/17]], [[85/23]]
|-
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| 2292.9
| 2292.9
|  
| [[64/17]], [[49/13]]
|-
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| 80
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| 2321.9
| 2321.9
| [[42/11]]
| [[65/17]]
|-
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| 2351
| 2351
|  
| [[35/9]]
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| 2380
| 2380
|  
| [[91/23]]
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| 2438
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|  
| [[45/11]]
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| 86
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| 2496.1
| 2496.1
|  
| [[55/13]], [[72/17]]
|-
|-
| 87
| 87
| 2525.1
| 2525.1
| [[43/10]]
| [[56/13]], [[99/23]]
|-
|-
| 88
| 88
| 2554.1
| 2554.1
|  
| [[35/8]]
|-
|-
| 89
| 89
| 2583.1
| 2583.1
|  
| [[40/9]], [[49/11]]
|-
|-
| 90
| 90
| 2612.2
| 2612.2
|  
| [[77/17]], [[104/23]]
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| 91
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| 93
| 93
| 2699.2
| 2699.2
|  
| [[81/17]]
|-
|-
| 94
| 94
| 2728.3
| 2728.3
| [[29/6]]
| [[63/13]]
|-
|-
| 95
| 95
| 2757.3
| 2757.3
|  
| [[64/13]]
|-
|-
| 96
| 96
| 2786.3
| 2786.3
| [[5/1]]
| [[5/1]]
|}{{Todo|replace auto-generated table of intervals with manually curated table}}
|}


== Optimization ==
== Optimization ==

Revision as of 23:23, 23 June 2026

← 95ed5 96ed5 97ed5 →
Prime factorization 25 × 3
Step size 29.0241 ¢ 
Octave 41\96ed5 (1189.99 ¢)
Twelfth 66\96ed5 (1915.59 ¢) (→ 11\16ed5)
Consistency limit 2
Distinct consistency limit 2

96 equal divisions of the 5th harmonic (abbreviated 96ed5) is a nonoctave tuning system that divides the interval of 5/1 into 96 equal parts of about 29 ¢ each. Each step represents a frequency ratio of 51/96, or the 96th root of 5.

Theory

This non-octave, non-tritave scale features a well-balanced harmonic series segment from 5 to 9, and performs exceptionally well across all prime harmonics from 5 to 23, with the exception of 19.

This system can be approximated as 41.34495 EDO, meaning each step of 96ed5 corresponds roughly to three steps of 124edo, or 124ed8.

96ed5 sets a height record on the Riemann zeta function with primes 2 and 3 removed, approximating 41.3478 EDO. This record remains unbeaten until approximately 98.62575 EDO (~229ed5).

Additionally, 96ed5 is related to 186zpi.

Harmonic series

Approximation of harmonics in 96ed5
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Error Absolute (¢) -10.0 +13.6 +9.0 +0.0 +3.6 -2.0 -1.0 -1.8 -10.0 -0.9 -6.4 +0.2 -12.0 +13.6 -11.0
Relative (%) -34.5 +47.0 +31.0 +0.0 +12.5 -7.0 -3.5 -6.0 -34.5 -3.0 -22.0 +0.6 -41.5 +47.0 -38.0
Steps
(reduced) 		41
(41) 		66
(66) 		83
(83) 		96
(0) 		107
(11) 		116
(20) 		124
(28) 		131
(35) 		137
(41) 		143
(47) 		148
(52) 		153
(57) 		157
(61) 		162
(66) 		165
(69)
Approximation of harmonics in 96ed5
Harmonic 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
Error Absolute (¢) +0.1 -11.8 +10.7 +9.0 +11.6 -10.9 -0.8 +12.6 +0.0 -9.9 +11.9 +7.0 +4.3 +3.6 +4.9 +8.0
Relative (%) +0.4 -40.5 +37.0 +31.0 +40.0 -37.5 -2.6 +43.5 +0.0 -33.9 +40.9 +24.0 +14.7 +12.5 +16.9 +27.5
Steps
(reduced) 		169
(73) 		172
(76) 		176
(80) 		179
(83) 		182
(86) 		184
(88) 		187
(91) 		190
(94) 		192
(0) 		194
(2) 		197
(5) 		199
(7) 		201
(9) 		203
(11) 		205
(13) 		207
(15)

Intervals

Steps Cents 8.9.5.7.11.13.17.23 ratios
0 0 1/1
1 29
2 58
3 87.1
4 116.1
5 145.1 25/23
6 174.1
7 203.2 9/8
8 232.2 8/7
9 261.2
10 290.2 13/11
11 319.3
12 348.3 11/9
13 377.3
14 406.3
15 435.4 9/7
16 464.4 17/13
17 493.4
18 522.4 23/17
19 551.5 11/8
20 580.5 7/5
21 609.5
22 638.5 13/9
23 667.6 25/17
24 696.6
25 725.6 35/23
26 754.6 17/11
27 783.7 11/7
28 812.7 8/5
29 841.7 13/8
30 870.7
31 899.7
32 928.8
33 957.8 40/23
34 986.8 23/13
35 1015.8 9/5
36 1044.9
37 1073.9 13/7
38 1102.9 17/9
39 1131.9 25/13
40 1161 45/23
41 1190
42 1219
43 1248 35/17
44 1277.1 23/11
45 1306.1 17/8, 49/23
46 1335.1
47 1364.1 11/5
48 1393.2
49 1422.2 25/11
50 1451.2
51 1480.2 40/17
52 1509.3 55/23
53 1538.3 17/7, 56/23
54 1567.3
55 1596.3
56 1625.3 23/9
57 1654.4 13/5
58 1683.4 45/17
59 1712.4 35/13
60 1741.4 63/23
61 1770.5 25/9, 64/23
62 1799.5 65/23
63 1828.5 23/8, 49/17
64 1857.5
65 1886.6
66 1915.6
67 1944.6 40/13
68 1973.6 25/8, 72/23
69 2002.7 35/11
70 2031.7 55/17
71 2060.7 23/7, 56/17
72 2089.7 77/23
73 2118.8 17/5
74 2147.8 45/13
75 2176.8 81/23
76 2205.8 25/7
77 2234.9 40/11
78 2263.9 63/17, 85/23
79 2292.9 64/17, 49/13
80 2321.9 65/17
81 2351 35/9
82 2380 91/23
83 2409
84 2438 45/11
85 2467
86 2496.1 55/13, 72/17
87 2525.1 56/13, 99/23
88 2554.1 35/8
89 2583.1 40/9, 49/11
90 2612.2 77/17, 104/23
91 2641.2 23/5
92 2670.2
93 2699.2 81/17
94 2728.3 63/13
95 2757.3 64/13
96 2786.3 5/1

Optimization

The local maxima for the finite Euler product over the primes 5.7.11.13.17.23 is 29.0283 cents.


Approximation of harmonics in optimized 96ed5
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Error Absolute (¢) -9.8 +13.9 +9.3 +0.4 +4.1 -1.5 -0.5 -1.2 -9.4 -0.3 -5.8 +0.8 -11.4 +14.3 -10.3
Relative (%) -33.9 +47.9 +32.2 +1.4 +14.0 -5.3 -1.7 -4.1 -32.5 -0.9 -19.9 +2.8 -39.2 +49.3 -35.6
Step 41 66 83 96 107 116 124 131 137 143 148 153 157 162 165
Approximation of harmonics in optimized 96ed5
Harmonic 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
Error Absolute (¢) +0.8 -11.0 +11.5 +9.8 +12.4 -10.1 +0.0 +13.4 +0.8 -9.0 +12.7 +7.8 +5.1 +4.5 +5.8 +8.9
Relative (%) +2.8 -38.0 +39.5 +33.6 +42.6 -34.8 +0.1 +46.2 +2.8 -31.1 +43.8 +26.9 +17.6 +15.4 +19.9 +30.5
Step 169 172 176 179 182 184 187 190 192 194 197 199 201 203 205 207

Intervals

Steps Cents Approximate ratios
0 0 1/1
1 29
2 58 30/29, 31/30
3 87.1 41/39
4 116.1 31/29
5 145.1 25/23, 37/34
6 174.1 21/19
7 203.2
8 232.2
9 261.2 36/31, 43/37
10 290.2 13/11
11 319.3
12 348.3
13 377.3 41/33
14 406.3 43/34
15 435.4
16 464.4 17/13
17 493.4
18 522.4 23/17
19 551.5
20 580.5 7/5
21 609.5 37/26
22 638.5
23 667.6 25/17
24 696.6
25 725.6 35/23, 38/25
26 754.6 17/11
27 783.7 11/7
28 812.7
29 841.7
30 870.7 38/23, 43/26
31 899.7 37/22, 42/25
32 928.8
33 957.8 33/19
34 986.8 23/13
35 1015.8
36 1044.9 42/23
37 1073.9 13/7
38 1102.9
39 1131.9 25/13
40 1161 43/22
41 1190
42 1219
43 1248 35/17
44 1277.1 23/11
45 1306.1
46 1335.1
47 1364.1 11/5
48 1393.2 38/17
49 1422.2 25/11
50 1451.2
51 1480.2
52 1509.3
53 1538.3 17/7
54 1567.3 42/17
55 1596.3
56 1625.3
57 1654.4 13/5
58 1683.4 37/14
59 1712.4 35/13
60 1741.4 41/15
61 1770.5
62 1799.5
63 1828.5
64 1857.5 38/13
65 1886.6
66 1915.6
67 1944.6 43/14
68 1973.6
69 2002.7 35/11
70 2031.7 42/13
71 2060.7 23/7
72 2089.7
73 2118.8 17/5
74 2147.8 38/11
75 2176.8
76 2205.8 25/7
77 2234.9
78 2263.9 37/10
79 2292.9
80 2321.9 42/11
81 2351
82 2380
83 2409
84 2438
85 2467
86 2496.1
87 2525.1 43/10
88 2554.1
89 2583.1
90 2612.2
91 2641.2 23/5
92 2670.2
93 2699.2
94 2728.3 29/6
95 2757.3
96 2786.3 5/1
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