Xenharmonic Wiki

Riemann zeta function/Record lists: Difference between revisions

← Older edit
VisualWikitext
Lériendil (talk | contribs)
8,046 edits
fixed stuff
m Fredg999 moved page The Riemann zeta function and tuning/Record lists to Riemann zeta function/Record lists: (Fake) parent page was renamed
 
(47 intermediate revisions by 3 users not shown)
Line 1: Line 1:
{{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
|
! 5
|
! 7
|
! 12
|
! 19
|
! 31
|
! 41
|
! 46
|
! 53
|
! 72
|
! 130
|
! 171
|
! 224
|
! 270
|
! 311
|
! 764
|
! 954
|
!
|
!
|
!
|
!
|
!
|
!
|
!
|
!
|
!
|
!
|-
| Zeta integral
| ★
|  
| ★
| ★
| ★
| ★
| ★
| ★
|  
| ★
| ★
| ★
| ★
| ★
| ★
|  
| ★
| ★
|  
|  
|  
|  
|  
|  
|  
|  
|  
|  
|-
|-
|Zeta gap
| Zeta gap
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
|  
|
| ★
|
| ★
|
| ★
|
|  
|
|  
|
|  
|
| ★
|
| ★
|
|  
|
| ★
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|}
|}


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


== 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 peak
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
|  
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
|  
|
| ★
|
|  
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
|  
| ★
| ★
| ★
|
| ★
| ★
| ★
|-
|-
|Absolute zeta integer
| Absolute zeta integer
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
|  
|
| ★
|
| ★
|
| ★
|
| ★
|
|  
|
|  
|
| ★
|
| ★
|
| ★
|
| ★
|
|  
|
| ★
|
| ★
|
|  
|
| ★
|
|  
|
|  
|
|  
|
| ★
|
|  
| ★
| ★
|
| ★
|
| ★
| ★
|-
|-
|Real zeta peak
| Real zeta peak
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
|  
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
|  
|
| ★
|
|  
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
|  
| ★
| ★
| ★
|
| ★
| ★
| ★
|-
|-
|Real zeta integer
| Real zeta integer
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
|  
|
| ★
|
| ★
|
| ★
|
| ★
|
|  
|
|  
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
| ★
|
|  
|
| ★
|
|  
|
|  
|
|  
|
| ★
|
| ★
| ★
| ★
|
| ★
|
|
| ★
|}
|}


== σ = 3/2 ==
=== 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"
{| 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
! 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
| ★
| ★
| ★
| ★
| ★
| ★
| ★
Line 439: Line 652:
| ★
| ★
| ★
| ★
|
| ★
| ★
|
|
| ★
| ★
| ★
| ★
Line 445: Line 663:
| ★
| ★
|  
|  
| ★
| ★
| ★
| ★
| ★
|  
|  
|-
| Absolute zeta integer
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|  
|  
| ★
| ★
| ★
| ★
| ★
|  
|  
| ★
|  
|  
|  
|  
| ★
|  
|  
| ★
| ★
| ★
| ★
| ★
|
| ★
|
| ★
|  
|  
| ★
|-
|-
|Absolute zeta integer
| Real zeta peak
| ★
| ★
| ★
| ★
| ★
Line 472: Line 716:
| ★
| ★
|  
|  
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
|-
| Real zeta integer
| ★
| ★
| ★
| ★
| ★
Line 478: Line 737:
| ★
| ★
| ★
| ★
|
|  
|  
| ★
| ★
| ★
| ★
|
| ★
| ★
|
|
| ★
|
| ★
| ★
| ★
| ★
| ★
|
| ★
|
| ★
|
| ★
|}
== 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]]
|-
|-
|Real zeta peak
| Absolute zeta peak
| ★
| ★
| ★
| ★
Line 510: Line 817:
| ★
| ★
|-
|-
|Real zeta integer
| Absolute zeta integer
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
|
| ★
| ★
|
| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
|-
| Real zeta peak
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
|
|
| ★
| ★
| ★
| ★
|
| ★
| ★
|-
| Real zeta integer
| ★
| ★
| ★
| ★
Line 538: Line 899:
|}
|}


=== No-twos tritave divisions ===
=== No-threes ===
{| class="wikitable center-all"
{| 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
! 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
|
| ★
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
| ★
| ★
Line 571: Line 941:
| ★
| ★
| ★
| ★
|
|
| ★
| ★
|
|-
| 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 integer
| Absolute zeta integer
| ★
| ★
| ★
| ★
Line 609: Line 1,085:
| ★
| ★
|-
|-
|Real zeta peak
| Real zeta peak
| ★
| ★
| ★
| ★
Line 632: Line 1,108:
| ★
| ★
|-
|-
|Real zeta integer
| Real zeta integer
| ★
| ★
| ★
| ★
Line 658: Line 1,134:
== σ = 2 ==
== σ = 2 ==
{| class="wikitable center-all"
{| 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]]
![[118edo|118]]
![[130edo|130]]
![[171edo|171]]
![[270edo|270]]
![[494edo|494]]
![[1106edo|1106]]
![[1578edo|1578]]
![[2460edo|2460]]
![[2684edo|2684]]
![[3395edo|3395]]
![[5585edo|5585]]
![[6079edo|6079]]
![[8269edo|8269]]
![[8539edo|8539]]
|-
|-
|Absolute zeta peak
! Record type
! [[1edo|1]]
! [[2edo|2]]
! [[3edo|3]]
! [[5edo|5]]
! [[7edo|7]]
! [[10edo|10]]
! [[12edo|12]]
! [[19edo|19]]
! [[31edo|31]]
! [[41edo|41]]
! [[53edo|53]]
! [[118edo|118]]
! [[130edo|130]]
! [[171edo|171]]
! [[270edo|270]]
! [[494edo|494]]
! [[1106edo|1106]]
! [[1578edo|1578]]
! [[2460edo|2460]]
! [[2684edo|2684]]
! [[3395edo|3395]]
! [[5585edo|5585]]
! [[6079edo|6079]]
! [[8269edo|8269]]
! [[8539edo|8539]]
|-
| Absolute zeta peak
| ★
| ★
| ★
| ★
Line 712: Line 1,189:
| ★
| ★
|-
|-
|Absolute zeta integer
| Absolute zeta integer
| ★
| ★
| ★
| ★
Line 739: Line 1,216:
| ★
| ★
|-
|-
|Real zeta peak (cosine mu peak)
| Real zeta peak (cosine mu peak)
| ★
| ★
| ★
| ★
Line 766: Line 1,243:
|  
|  
|-
|-
|Real zeta integer (cosine mu integer)
| Real zeta integer (cosine mu integer)
| ★
| ★
| ★
| ★
Line 794: Line 1,271:
|}
|}


=== No-twos tritave divisions ===
=== No-threes ===
{| class="wikitable center-all"
{| 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
! 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
| ★
| ★
|
| ★
| ★
| ★
| ★
Line 835: Line 1,323:
| ★
| ★
|  
|  
| ★
|-
| Real zeta peak
|  
|  
| ★
| ★
| ★
| ★
| ★
|
| ★
| ★
| ★
| ★
| ★
|
| ★
|  
|  
|-
|-
|Absolute zeta integer
| 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 integer
| ★
| ★
| ★
| ★
Line 859: Line 1,426:
| ★
| ★
|-
|-
|Real zeta peak (cosine mu peak)
| Real zeta peak (cosine mu peak)
| ★
| ★
| ★
| ★
| ★
Line 878: Line 1,446:
| ★
| ★
| ★
| ★
|
|-
|-
|Real zeta integer (cosine mu integer)
| Real zeta integer (cosine mu integer)
| ★
| ★
| ★
| ★
Line 901: Line 1,468:
| ★
| ★
|}
|}
[[Category:Zeta| ]]
[[Category:Math]]
[[Category:Tuning]]
Retrieved from "https://en.xen.wiki/w/Riemann_zeta_function/Record_lists"