Riemann zeta function/Record lists: Difference between revisions

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{{breadcrumb}}
== Definitions ==
=== Absolute ===
Based on the absolute value of zeta(s), which is more mathematically convenient to calculate at σ = 0.5.
=== Real ===
Based on the real part of zeta(s), which directly follows from the [[The Riemann zeta function and tuning/Vector's derivation|derivation of zeta as an equal tuning badness metric]].
== Zero-based metrics ==
== Zero-based metrics ==
Zeta metrics corresponding to regions between zeros on the critical line (σ = 0.5). The Riemann hypothesis is the equivalent to the statement that these correspond directly to peaks.  
Zeta metrics corresponding to regions between zeros on the critical line (σ = 0.5). The Riemann hypothesis is the equivalent to the statement that regions between zeros correspond directly to peaks.  
{| class="wikitable"
{| class="wikitable"
|+
!Record type
!
!
!
!
!
!
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!
!
!
!
!
!
!
!
!
!
!
!
!
!
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!
!
|-
|-
|Zeta integral
! Record type
|
! 2
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! 3
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! 5
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! 7
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! 12
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! 19
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! 31
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! 41
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! 46
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! 53
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! 72
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! 130
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! 171
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! 224
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! 270
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! 311
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! 764
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! 954
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!
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!
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!
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!
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!
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!
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!
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!
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!
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!
|-
|-
|Zeta gap
| Zeta integral
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| ★
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|-
| Zeta gap
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|  
| ★
| ★
| ★
|  
|  
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| ★
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|  
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|  
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|}
|}


== σ = 0.5 ==
== σ = 1/2 ==
{| class="wikitable"
{| class="wikitable"
!Record type
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
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!
!
|-
|-
|Absolute zeta peak
! Record type
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!
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!
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!
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!
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!
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!
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!
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!
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!
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!
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!
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!
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!
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!
|-
|-
|Absolute zeta integer
| Absolute zeta peak
|
|  
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|  
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|  
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|  
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|  
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|  
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|  
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|  
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|  
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|  
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|  
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|  
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|  
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|  
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|  
|-
|-
|Real zeta peak
| Absolute zeta integer
|
|  
|
|  
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|  
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|  
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|  
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|  
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|  
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|  
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|  
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|  
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|  
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|-
|-
|Real zeta integer
| Real zeta peak
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|-
| Real zeta integer
|  
|  
|  
|  
|  
|  
|  
|  
|  
|  
|  
|  
|  
|  
|  
|  
|  
|  
|  
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|  
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|  
|  
|  
|  
|  
|  
|}
|}


== σ = 1 ==
== σ = 1 ==
{| class="wikitable"
{| class="wikitable center-all"
!Record type
|-
!
! Record type
!
! [[1edo|1]]
!
! [[2edo|2]]
!
! [[3edo|3]]
!
! [[5edo|5]]
!
! [[7edo|7]]
!
! [[10edo|10]]
!
! [[12edo|12]]
!
! [[19edo|19]]
!
! [[22edo|22]]
!
! [[31edo|31]]
!
! [[41edo|41]]
!
! [[53edo|53]]
!
! [[72edo|72]]
!
! [[118edo|118]]
!
! [[130edo|130]]
!
! [[171edo|171]]
!
! [[224edo|224]]
!
! [[270edo|270]]
!
! [[311edo|311]]
!
! [[494edo|494]]
!
! [[742edo|742]]
!
! [[764edo|764]]
!
! [[1106edo|1106]]
!
! [[1236edo|1236]]
!
! [[1395edo|1395]]
!
! [[1448edo|1448]]
!
! [[1578edo|1578]]
! [[2072edo|2072]]
! [[2460edo|2460]]
! [[2684edo|2684]]
! [[3395edo|3395]]
! [[5585edo|5585]]
! [[7033edo|7033]]
! [[8269edo|8269]]
! [[8539edo|8539]]
|-
| Absolute zeta peak
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
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| ★
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| ★
| ★
| ★
| ★
| ★
| ★
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| ★
| ★
| ★
|
| ★
| ★
| ★
|-
| Absolute zeta integer
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
| ★
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| ★
| ★
| ★
| ★
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| ★
| ★
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| ★
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| ★
|
| ★
| ★
|
| ★
|
| ★
| ★
|-
| Real zeta peak
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
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| ★
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| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
|
| ★
| ★
| ★
|-
| Real zeta integer
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
| ★
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| ★
| ★
| ★
| ★
| ★
| ★
| ★
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| ★
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| ★
| ★
| ★
| ★
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| ★
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|
| ★
|}
 
=== No-threes ===
{| class="wikitable center-all"
|-
! Record type
! [[1edo|1]]
! [[2edo|2]]
! [[3edo|3]]
! [[4edo|4]]
! [[6edo|6]]
! [[16edo|16]]
! [[21edo|21]]
! [[25edo|25]]
! [[31edo|31]]
! [[37edo|37]]
! [[93edo|93]]
! [[109edo|109]]
! [[124edo|124]]
! [[233edo|233]]
! [[457edo|457]]
! [[581edo|581]]
! [[618edo|618]]
! [[851edo|851]]
! [[1028edo|1028]]
! [[1121edo|1121]]
! [[1199edo|1199]]
! [[1702edo|1702]]
! [[1963edo|1963]]
! [[2917edo|2917]]
! [[3535edo|3535]]
! [[4880edo|4880]]
! [[5352edo|5352]]
! [[6312edo|6312]]
! [[8232edo|8232]]
|-
|-
|Absolute zeta peak
| Absolute zeta peak
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
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| ★
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| ★
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|  
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| ★
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|  
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|  
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|  
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| ★
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| ★
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|  
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| ★
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| ★
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|  
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|  
|
| ★
|
|  
| ★
|-
|-
|Absolute zeta integer
| Absolute zeta integer
|
| ★
|
|  
|
| ★
|
| ★
|
| ★
|
|  
|
|  
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|  
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| ★
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| ★
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| ★
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|  
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| ★
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|  
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|  
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|  
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| ★
|
| ★
|
| ★
|
|  
| ★
|-
|-
|Real zeta peak
| Real zeta peak
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
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| ★
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|  
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| ★
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| ★
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| ★
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| ★
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|  
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| ★
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|  
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| ★
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|  
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| ★
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| ★
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|  
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|  
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| ★
|
|  
| ★
|-
|-
|Real zeta integer
| Real zeta integer
|
| ★
|
|  
|
| ★
|
|  
|
| ★
|
|  
|
|  
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|  
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| ★
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| ★
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| ★
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|  
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| ★
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|  
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| ★
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| ★
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|  
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| ★
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| ★
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| ★
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| ★
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| ★
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|  
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|  
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| ★
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| ★
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|  
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| ★
|
|}
 
=== No-twos tritave divisions ===
{| class="wikitable center-all"
|-
! Record type
! [[1edt|1]]
! [[2edt|2]]
! [[4edt|4]]
! [[9edt|9]]
! [[13edt|13]]
! [[17edt|17]]
! [[26edt|26]]
! [[39edt|39]]
! [[71edt|71]]
! [[88edt|88]]
! [[131edt|131]]
! [[245edt|245]]
! [[415edt|415]]
! [[630edt|630]]
! [[712edt|712]]
! [[800edt|800]]
! [[1123edt|1123]]
! [[1342edt|1342]]
! [[1473edt|1473]]
! [[1757edt|1757]]
! [[2426edt|2426]]
! [[3624edt|3624]]
! [[3755edt|3755]]
! [[4424edt|4424]]
! [[5136edt|5136]]
! [[6181edt|6181]]
! [[8682edt|8682]]
! [[9805edt|9805]]
|-
| Absolute zeta peak
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
|
| ★
| ★
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|
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
| ★
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|-
| Absolute zeta integer
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
|
| ★
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| ★
|
| ★
| ★
| ★
| ★
| ★
|
| ★
|
| ★
|
| ★
|-
| Real zeta peak
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
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| ★
| ★
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| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
|-
| Real zeta integer
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
|
| ★
| ★
|
| ★
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| ★
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| ★
| ★
| ★
| ★
| ★
|
| ★
|
| ★
|
| ★
|}
|}


== σ = 3/2 ==
== σ = 3/2 ==
{| class="wikitable"
{| class="wikitable center-all"
!Record type
|-
![[1edo|1]]
! Record type
![[2edo|2]]
! [[1edo|1]]
![[3edo|3]]
! [[2edo|2]]
![[5edo|5]]
! [[3edo|3]]
![[7edo|7]]
! [[5edo|5]]
![[10edo|10]]
! [[7edo|7]]
![[12edo|12]]
! [[10edo|10]]
![[19edo|19]]
! [[12edo|12]]
![[31edo|31]]
! [[19edo|19]]
![[41edo|41]]
! [[31edo|31]]
![[53edo|53]]
! [[41edo|41]]
![[72edo|72]]
! [[53edo|53]]
![[130edo|130]]
! [[72edo|72]]
![[171edo|171]]
! [[130edo|130]]
![[224edo|224]]
! [[171edo|171]]
![[270edo|270]]
! [[224edo|224]]
![[494edo|494]]
! [[270edo|270]]
![[1106edo|1106]]
! [[494edo|494]]
![[1578edo|1578]]
! [[1106edo|1106]]
![[2460edo|2460]]
! [[1578edo|1578]]
![[2684edo|2684]]
! [[2460edo|2460]]
![[3395edo|3395]]
! [[2684edo|2684]]
![[5585edo|5585]]
! [[3395edo|3395]]
![[8269edo|8269]]
! [[5585edo|5585]]
![[8539edo|8539]]
! [[8269edo|8269]]
! [[8539edo|8539]]
|-
| Absolute zeta peak
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
|
|
| ★
| ★
| ★
| ★
|
| ★
| ★
|-
| Absolute zeta integer
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
|
| ★
| ★
|
| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
|-
|-
|Absolute zeta peak
| Real zeta peak
| ★
| ★
| ★
| ★
Line 441: Line 856:
| ★
| ★
| ★
| ★
| s
| ★
| ★
| ★
| ★
| t
| ★
|
| ★
| ★
|  
|  
Line 456: Line 871:
| ★
| ★
|-
|-
|Absolute zeta integer
| Real zeta integer
| ★
| ★
| ★
| ★
Line 468: Line 883:
| ★
| ★
| ★
| ★
| s
|  
| ★
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| ★
| ★
| t
| ★
| ★
| ★
| ★
Line 482: Line 897:
|  
|  
| ★
| ★
|}
=== No-threes ===
{| class="wikitable center-all"
|-
! Record type
! [[1edo|1]]
! [[2edo|2]]
! [[3edo|3]]
! [[6edo|6]]
! [[25edo|25]]
! [[31edo|31]]
! [[37edo|37]]
! [[93edo|93]]
! [[109edo|109]]
! [[124edo|124]]
! [[233edo|233]]
! [[457edo|457]]
! [[581edo|581]]
! [[618edo|618]]
! [[1199edo|1199]]
! [[2917edo|2917]]
! [[3535edo|3535]]
! [[4880edo|4880]]
! [[5352edo|5352]]
! [[6312edo|6312]]
|-
| Absolute zeta peak
|
| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
|
| ★
|
|-
| Absolute zeta integer
| ★
|
| ★
| ★
|
| ★
| ★
| ★
|
| ★
|
| ★
| ★
|
| ★
|
| ★
| ★
| ★
| ★
|-
| Real zeta peak
|
| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
|
| ★
|
|-
|-
|Real zeta peak
| Real zeta integer
| ★
|
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
| ★
Line 491: Line 1,001:
|  
|  
| ★
| ★
|
| ★
| ★
| ★
| ★
|
| ★
| ★
|
| ★
| ★
| s
| ★
| ★
|
| ★
| ★
| t
|}
 
=== No-twos tritave divisions ===
{| class="wikitable center-all"
|-
! Record type
! [[1edt|1]]
! [[2edt|2]]
! [[4edt|4]]
! [[9edt|9]]
! [[13edt|13]]
! [[17edt|17]]
! [[26edt|26]]
! [[39edt|39]]
! [[88edt|88]]
! [[114edt|114]]
! [[131edt|131]]
! [[415edt|415]]
! [[712edt|712]]
! [[800edt|800]]
! [[914edt|914]]
! [[1342edt|1342]]
! [[1757edt|1757]]
! [[2426edt|2426]]
! [[3624edt|3624]]
! [[3755edt|3755]]
! [[6181edt|6181]]
|-
| Absolute zeta peak
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
|
| ★
| ★
|  
|  
Line 505: Line 1,058:
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|-
| Absolute zeta integer
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
|
|
| ★
| ★
|  
|  
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
|-
|-
|Real zeta integer
| Real zeta peak
| ★
| ★
| ★
| ★
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== σ = 2 ==
== σ = 2 ==
{| class="wikitable"
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[[Category:Zeta| ]]
[[Category:Math]]
[[Category:Tuning]]

Latest revision as of 19:25, 16 July 2026

< Riemann zeta function

Definitions

Absolute

Based on the absolute value of zeta(s), which is more mathematically convenient to calculate at σ = 0.5.

Real

Based on the real part of zeta(s), which directly follows from the derivation of zeta as an equal tuning badness metric.

Zero-based metrics

Zeta metrics corresponding to regions between zeros on the critical line (σ = 0.5). The Riemann hypothesis is the equivalent to the statement that regions between zeros correspond directly to peaks.

Record type 2 3 5 7 12 19 31 41 46 53 72 130 171 224 270 311 764 954
Zeta integral ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★
Zeta gap ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★

σ = 1/2

Record type
Absolute zeta peak
Absolute zeta integer
Real zeta peak
Real zeta integer

σ = 1

Record type 1 2 3 5 7 10 12 19 22 31 41 53 72 118 130 171 224 270 311 494 742 764 1106 1236 1395 1448 1578 2072 2460 2684 3395 5585 7033 8269 8539
Absolute zeta peak ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★
Absolute zeta integer ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★
Real zeta peak ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★
Real zeta integer ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★

No-threes

Record type 1 2 3 4 6 16 21 25 31 37 93 109 124 233 457 581 618 851 1028 1121 1199 1702 1963 2917 3535 4880 5352 6312 8232
Absolute zeta peak ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★
Absolute zeta integer ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★
Real zeta peak ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★
Real zeta integer ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★

No-twos tritave divisions

Record type 1 2 4 9 13 17 26 39 71 88 131 245 415 630 712 800 1123 1342 1473 1757 2426 3624 3755 4424 5136 6181 8682 9805
Absolute zeta peak ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★
Absolute zeta integer ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★
Real zeta peak ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★
Real zeta integer ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★

σ = 3/2

Record type 1 2 3 5 7 10 12 19 31 41 53 72 130 171 224 270 494 1106 1578 2460 2684 3395 5585 8269 8539
Absolute zeta peak ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★
Absolute zeta integer ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★
Real zeta peak ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★
Real zeta integer ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★

No-threes

Record type 1 2 3 6 25 31 37 93 109 124 233 457 581 618 1199 2917 3535 4880 5352 6312
Absolute zeta peak ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★
Absolute zeta integer ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★
Real zeta peak ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★
Real zeta integer ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★

No-twos tritave divisions

Record type 1 2 4 9 13 17 26 39 88 114 131 415 712 800 914 1342 1757 2426 3624 3755 6181
Absolute zeta peak ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★
Absolute zeta integer ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★
Real zeta peak ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★
Real zeta integer ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★

σ = 2

Record type 1 2 3 5 7 10 12 19 31 41 53 118 130 171 270 494 1106 1578 2460 2684 3395 5585 6079 8269 8539
Absolute zeta peak ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★
Absolute zeta integer ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★
Real zeta peak (cosine mu peak) ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★
Real zeta integer (cosine mu integer) ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★

No-threes

Record type 1 2 3 6 31 37 93 124 233 581 1199 2917 3535 5352 6312
Absolute zeta peak ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★
Absolute zeta integer ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★
Real zeta peak ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★
Real zeta integer ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★

No-twos tritave divisions

Record type 1 2 4 9 13 17 26 39 88 114 131 800 914 1342 1757 2426 3624 3755 6181
Absolute zeta peak ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★
Absolute zeta integer ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★
Real zeta peak (cosine mu peak) ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★
Real zeta integer (cosine mu integer) ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★
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