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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
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
|-
|-
|Zeta integral
! Record type
|
! 2
|
! 3
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! 5
|
! 7
|
! 12
|
! 19
|
! 31
|
! 41
|
! 46
|
! 53
|
! 72
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! 130
|
! 171
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! 224
|
! 270
|
! 311
|
! 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 integral
| ★
|  
| ★
| ★
| ★
| ★
| ★
| ★
|  
| ★
| ★
| ★
| ★
| ★
| ★
|  
| ★
| ★
|  
|  
|  
|  
|  
|  
|  
|  
|  
|  
|-
|-
|Zeta gap
| Zeta gap
|
| ★
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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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|}
|}


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


== 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
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
|
| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
|
| ★
| ★
| ★
|-
| Absolute zeta integer
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
| ★
|
|
| ★
| ★
| ★
| ★
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| ★
| ★
|
| ★
|
|
|
| ★
|
| ★
| ★
|
| ★
|
| ★
| ★
|-
| Real zeta peak
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
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| ★
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| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
|
| ★
| ★
| ★
|-
| Real zeta integer
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
| ★
|
|
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
|
|
|
| ★
| ★
| ★
| ★
|
| ★
|
|
| ★
|}
 
=== 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 integer
| ★
|
| ★
| ★
| ★
|
|
|
| ★
| ★
| ★
|
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
| ★
|
|
|
| ★
| ★
| ★
|
| ★
|-
| Real zeta peak
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
|
|
| ★
| ★
|
| ★
| ★
|
|
| ★
|
| ★
|-
| Real zeta integer
| ★
|
| ★
|
| ★
|
|
|
| ★
| ★
| ★
|
| ★
|
| ★
| ★
|
| ★
| ★
| ★
| ★
| ★
|
|
| ★
| ★
|
| ★
|
|}
 
=== 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
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
|
| ★
| ★
|
|
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
| ★
|
|-
| Absolute zeta integer
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
|
| ★
|
|
| ★
|
| ★
| ★
| ★
| ★
| ★
|
| ★
|
| ★
|
| ★
|-
| Real zeta peak
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
|
| ★
| ★
|
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
|-
| Real zeta integer
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
|
| ★
| ★
|
| ★
|
|
| ★
|
| ★
| ★
| ★
| ★
| ★
|
| ★
|
| ★
|
| ★
|}
 
== σ = 3/2 ==
{| class="wikitable center-all"
|-
! Record type
! [[1edo|1]]
! [[2edo|2]]
! [[3edo|3]]
! [[5edo|5]]
! [[7edo|7]]
! [[10edo|10]]
! [[12edo|12]]
! [[19edo|19]]
! [[31edo|31]]
! [[41edo|41]]
! [[53edo|53]]
! [[72edo|72]]
! [[130edo|130]]
! [[171edo|171]]
! [[224edo|224]]
! [[270edo|270]]
! [[494edo|494]]
! [[1106edo|1106]]
! [[1578edo|1578]]
! [[2460edo|2460]]
! [[2684edo|2684]]
! [[3395edo|3395]]
! [[5585edo|5585]]
! [[8269edo|8269]]
! [[8539edo|8539]]
|-
| Absolute zeta peak
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
|
|
| ★
| ★
| ★
| ★
|
| ★
| ★
|-
| Absolute zeta integer
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
|
| ★
| ★
|
| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
|-
| Real zeta peak
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
|
|
| ★
| ★
| ★
| ★
|
| ★
| ★
|-
| Real zeta integer
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
|
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
|
| ★
|
| ★
|}
 
=== 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 integer
| ★
|
| ★
| ★
|
| ★
| ★
| ★
|
| ★
|
| ★
| ★
|
| ★
|
| ★
| ★
|
| ★
|}
 
=== 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
| Absolute zeta peak
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
|  
|
| ★
|
|  
|
| ★
|
|  
|
|  
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
|
|
|
|
|
|
|-
|-
|Absolute zeta integer
| Absolute zeta integer
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
|  
|
| ★
|
| ★
|
| ★
|
|  
|
|  
|
| ★
|
|  
|
| ★
|
| ★
|
| ★
|
| ★
|
|  
|
| ★
|
|
|
|
|
|
|
|-
|-
|Real zeta peak
| Real zeta peak
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
|  
|
| ★
|
|  
|
| ★
|
| ★
|
| ★
|
| ★
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| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
|
|
|
|
|
|
|-
|-
|Real zeta integer
| Real zeta integer
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
|  
|
| ★
|
| ★
|
| ★
|
| ★
|
|  
|
| ★
|
|  
|
| ★
|
| ★
|
| ★
|
| ★
|
|  
|
| ★
|
|
|
|
|
|
|
|}
|}


== σ = 2 ==
== σ = 2 ==
{| class="wikitable"
{| class="wikitable center-all"
!Record type
|-
!1
! Record type
!2
! [[1edo|1]]
!3
! [[2edo|2]]
!5
! [[3edo|3]]
!7
! [[5edo|5]]
!12
! [[7edo|7]]
!19
! [[10edo|10]]
!31
! [[12edo|12]]
!41
! [[19edo|19]]
!53
! [[31edo|31]]
!130
! [[41edo|41]]
!171
! [[53edo|53]]
!270
! [[118edo|118]]
!1578
! [[130edo|130]]
!2460
! [[171edo|171]]
!2684
! [[270edo|270]]
!3395
! [[494edo|494]]
!6079
! [[1106edo|1106]]
!8269
! [[1578edo|1578]]
!
! [[2460edo|2460]]
!
! [[2684edo|2684]]
!
! [[3395edo|3395]]
!
! [[5585edo|5585]]
!
! [[6079edo|6079]]
!
! [[8269edo|8269]]
!
! [[8539edo|8539]]
!
|-
!
| Absolute zeta peak
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
|
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
|-
| Absolute zeta integer
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
|
| ★
| ★
| ★
| ★
| ★
|
| ★
|
|
| ★
|
|
| ★
|-
| Real zeta peak (cosine mu peak)
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
|
|
| ★
| ★
| ★
| ★
|
| ★
| ★
|
|-
| Real zeta integer (cosine mu integer)
| ★
| ★
|
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
|
| ★
| ★
| ★
| ★
| ★
|
| ★
|
|
| ★
|
|
| ★
|}
 
=== No-threes ===
{| class="wikitable center-all"
|-
! Record type
! [[1edo|1]]
! [[2edo|2]]
! [[3edo|3]]
! [[6edo|6]]
! [[31edo|31]]
! [[37edo|37]]
! [[93edo|93]]
! [[124edo|124]]
! [[233edo|233]]
! [[581edo|581]]
! [[1199edo|1199]]
! [[2917edo|2917]]
! [[3535edo|3535]]
! [[5352edo|5352]]
! [[6312edo|6312]]
|-
| Absolute zeta peak
|
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
| ★
| ★
|
| ★
|
|-
| Absolute zeta integer
| ★
|
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
|
| ★
|
| ★
|-
| Real zeta peak
|
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
| ★
| ★
|
| ★
|
|-
| Real zeta integer
| ★
|
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
|
| ★
|
| ★
|}
 
=== 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]]
! [[800edt|800]]
! [[914edt|914]]
! [[1342edt|1342]]
! [[1757edt|1757]]
! [[2426edt|2426]]
! [[3624edt|3624]]
! [[3755edt|3755]]
! [[6181edt|6181]]
|-
|-
|Absolute zeta peak
| Absolute zeta peak
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
|
|
|
|
|
|
|
|
|-
|-
|Absolute zeta integer
| Absolute zeta integer
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
|  
|
| ★
|
| ★
|
| ★
|
| ★
|
|  
|
| ★
|
| ★
|
| ★
|
| ★
|
|  
|
| ★
|
|
|
|
|
|
|
|
|
|-
|-
|Real zeta peak (cosine mu peak)
| Real zeta peak (cosine mu peak)
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
|
|
|
|
|
|
|
|
|-
|-
|Real zeta integer (cosine mu integer)
| Real zeta integer (cosine mu integer)
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
|  
|
| ★
|
| ★
|
| ★
|
| ★
|
|  
|
| ★
|
| ★
|
|  
|
| ★
|
|  
|
| ★
|
|
|
|
|
|
|
|
|
|}
|}
[[Category:Zeta| ]]
[[Category:Math]]
[[Category:Tuning]]
Retrieved from "https://en.xen.wiki/w/Riemann_zeta_function/Record_lists"