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{{Infobox ET}}
{{ED intro}}
== Theory ==
== Theory ==
96ed5 is an [[Equal-step tuning|equal-step]] [[tuning system]] created by dividing the interval of [[5/1]] into 96 equal parts.


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 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]].
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 [[The Riemann zeta function and tuning|Riemann zeta function]] with [[The Riemann zeta function and tuning#Removing primes|primes 2 and 3 removed]], approximating 41.3478 EDO. This record remains unbeaten until approximately 98.62575 EDO.
96ed5 sets a height record on the [[Riemann zeta function]] with [[Riemann zeta function#Removing primes|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]].
Additionally, 96ed5 is related to [[186zpi]].
== Harmonic series ==
== Harmonic series ==


{{Harmonics in equal|96|5|1|prec=1|columns=15}}
{{Harmonics in equal|96|5|1|prec=1|columns=15}}
{{Harmonics in equal|96|5|1|prec=1|columns=16|start=16}}
{{Harmonics in equal|96|5|1|prec=1|columns=16|start=16}}
== Intervals ==
{| class="wikitable center-1 right-2 mw-collapsible mw-collapsed"
|-
! Steps
! Cents
! 8.9.5.7.11.13.17.23 ratios
|-
| 0
| 0
| [[1/1]]
|-
| 1
| 29
| [[56/55]], [[64/63]], [[65/64]], [[117/115]], [[119/117]], [[121/119]], [[637/625]]
|-
| 2
| 58
| [[65/63]], [[119/115]], [[121/117]], [[125/121]], [[175/169]]
|-
| 3
| 87.1
| [[81/77]], [[104/99]], [[121/115]], [[143/136]], [[343/325]], [[637/605]]
|-
| 4
| 116.1
| [[77/72]], [[91/85]], [[153/143]]
|-
| 5
| 145.1
| [[25/23]]
|-
| 6
| 174.1
| [[72/65]]
|-
| 7
| 203.2
| [[9/8]], [[55/49]]
|-
| 8
| 232.2
| [[8/7]], [[143/125]]
|-
| 9
| 261.2
| [[99/85]]
|-
| 10
| 290.2
| [[13/11]], [[77/65]]
|-
| 11
| 319.3
|
|-
| 12
| 348.3
| [[11/9]], [[104/85]], [[175/143]]
|-
| 13
| 377.3
|
|-
| 14
| 406.3
|
|-
| 15
| 435.4
| [[9/7]]
|-
| 16
| 464.4
| [[17/13]]
|-
| 17
| 493.4
| [[65/49]], [[121/91]]
|-
| 18
| 522.4
| [[23/17]], [[169/125]]
|-
| 19
| 551.5
| [[11/8]], [[125/91]]
|-
| 20
| 580.5
| [[7/5]], [[169/121]]
|-
| 21
| 609.5
|
|-
| 22
| 638.5
| [[13/9]], [[175/121]]
|-
| 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]], [[125/77]]
|-
| 30
| 870.7
| [[91/55]]
|-
| 31
| 899.7
|
|-
| 32
| 928.8
| [[245/143]]
|-
| 33
| 957.8
| [[40/23]]
|-
| 34
| 986.8
| [[23/13]]
|-
| 35
| 1015.8
| [[9/5]]
|-
| 36
| 1044.9
|
|-
| 37
| 1073.9
| [[13/7]], [[121/65]]
|-
| 38
| 1102.9
| [[17/9]]
|-
| 39
| 1131.9
| [[25/13]]
|-
| 40
| 1161
| [[45/23]], [[49/25]]
|-
| 41
| 1190
|
|-
| 42
| 1219
| [[245/121]], [[343/169]]
|-
| 43
| 1248
| [[35/17]]
|-
| 44
| 1277.1
| [[23/11]]
|-
| 45
| 1306.1
| [[17/8]], [[49/23]]
|-
| 46
| 1335.1
|
|-
| 47
| 1364.1
| [[11/5]], [[169/77]]
|-
| 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
| [[121/49]]
|-
| 55
| 1596.3
|
|-
| 56
| 1625.3
| [[23/9]], [[125/49]]
|-
| 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
| [[143/39]]
|-
| 65
| 1886.6
|
|-
| 66
| 1915.6
| [[275/91]]
|-
| 67
| 1944.6
| [[40/13]], [[77/25]], [[169/55]]
|-
| 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]], [[121/35]], [[169/49]]
|-
| 75
| 2176.8
| [[81/23]]
|-
| 76
| 2205.8
| [[25/7]]
|-
| 77
| 2234.9
| [[40/11]], [[91/25]]
|-
| 78
| 2263.9
| [[63/17]], [[85/23]]
|-
| 79
| 2292.9
| [[49/13]], [[64/17]]
|-
| 80
| 2321.9
| [[65/17]]
|-
| 81
| 2351
| [[35/9]]
|-
| 82
| 2380
| [[91/23]]
|-
| 83
| 2409
|
|-
| 84
| 2438
| [[45/11]], [[143/35]]
|-
| 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]], [[121/25]], [[169/35]]
|-
| 95
| 2757.3
| [[64/13]]
|-
| 96
| 2786.3
| [[5/1]]
|}


== Optimization ==
== Optimization ==
In the 32-integer-limit and 5.7.11.13.17.23 subgroup, the lowest relative error is 41.346437627379-edo, or 41<1189.94532112775>, or 29.023056612872 cents.
{{Harmonics in cet|29.023056612872|columns=15|title=Approximation of harmonics in optimized 96ed5}}
{{Harmonics in cet|29.023056612872|columns=16|start=16|title=Approximation of harmonics in optimized 96ed5}}


[[Category:Ed5]]
The local maxima for the finite Euler product over the primes 5.7.11.13.17.23 is 29.0283 cents.
 
{{Harmonics in cet|29.0283|columns=15|title=Approximation of harmonics in optimized 96ed5}}
{{Harmonics in cet|29.0283|columns=16|start=16|title=Approximation of harmonics in optimized 96ed5}}
 
== Intervals ==
{{Interval table}}
 
{{todo|expand}}
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