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'''71 zeta peak index''' (abbreviated '''71zpi'''), is the [[Equal-step tuning|equal-step]] [[tuning system]] obtained from the 71st [[Zeta peak index|peak]] of the [[The Riemann zeta function and tuning|Riemann zeta function]].
'''71 zeta peak index''' (abbreviated '''71zpi'''), is the [[Equal-step tuning|equal-step]] [[tuning system]] obtained from the 71st [[Zeta peak index|peak]] of the [[The Riemann zeta function and tuning|Riemann zeta function]].


{|class="wikitable"
{| class="wikitable"
!colspan="3"|Tuning
! colspan="3" | Tuning
!colspan="3"|Strength
! colspan="3" | Strength
!colspan="2"|Closest EDO
! colspan="2" | Closest EDO
!colspan="2"|Integer limit
! colspan="2" | Integer limit
|-
|-
!ZPI
! ZPI
!Steps per octave
! Steps per octave
!Step size (cents)
! Step size (cents)
!Height
! Height
!Integral
! Integral
!Gap
! Gap
!EDO
! EDO
!Octave (cents)
! Octave (cents)
!Consistent
! Consistent
!Distinct
! Distinct
|-
|-
|[[71zpi]]
| [[71zpi]]
|20.2248393119540
| 20.2248393119540
|59.3329806724710
| 59.3329806724710
|3.531097
| 3.531097
|0.613581
| 0.613581
|12.986080
| 12.986080
|[[20edo]]
| [[20edo]]
|1186.65961344942
| 1186.65961344942
|6
| 6
|6
| 6
|}
|}


[[File:71zpi.png|thumb|The Riemann zeta function around 71zpi]]
[[File:71zpi.png|thumb|right|The Riemann zeta function around 71zpi]]


== Theory ==
== Theory ==
Line 54: Line 54:
{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-all right-2 left-3"
|-
|-
!Step
! Step
!Cents
! Cents
!Ratios
! Ratios
! colspan="3" |[[Ups and Downs Notation]] from [[20edo|20EDO]]
! colspan="3" | [[Ups and Downs Notation]] from [[20edo|20EDO]]
! colspan="3" |[[Ups and Downs Notation]] from [[182edo|182EDO]]
! colspan="3" | [[Ups and Downs Notation]] from [[182edo|182EDO]]
|-
|-
|0
| 0
|0.000
| 0.000
|[[1/1]]
| [[1/1]]
|unison
| unison
|P1
| P1
|D
| D
|unison
| unison
|P1
| P1
|D
| D
|-
|-
|1
| 1
|59.333
| 59.333
|[[30/29]], [[29/28]]
| [[30/29]], [[29/28]]
|up unison, upminor 2nd
| up unison, upminor 2nd
|^1, ^m2
| ^1, ^m2
|^D, ^Eb
| ^D, ^Eb
|
|  
|
|  
|
|  
|-
|-
|2
| 2
|118.666
| 118.666
|[[15/14]]
| [[15/14]]
|dup unison, mid 2nd
| dup unison, mid 2nd
|^^1, ~2
| ^^1, ~2
|^^D, vvE
| ^^D, vvE
|
|  
|
|  
|
|  
|-
|-
|3
| 3
|177.999
| 177.999
|[[10/9]]
| [[10/9]]
|downmajor 2nd
| downmajor 2nd
|vM2
| vM2
|vE
| vE
|
|  
|
|  
|
|  
|-
|-
|4
| 4
|237.332
| 237.332
|[[8/7]]
| [[8/7]]
|major 2nd, minor 3rd
| major 2nd, minor 3rd
|M2, m3
| M2, m3
|E, F
| E, F
|
|  
|
|  
|
|  
|-
|-
|5
| 5
|296.665
| 296.665
|[[13/11]], [[19/16]], [[6/5]]
| [[13/11]], [[19/16]], [[6/5]]
|upminor 3rd
| upminor 3rd
|^m3
| ^m3
|^F
| ^F
|
|  
|
|  
|
|  
|-
|-
|6
| 6
|355.998
| 355.998
|[[11/9]], [[27/22]], [[16/13]]
| [[11/9]], [[27/22]], [[16/13]]
|mid 3rd
| mid 3rd
|~3
| ~3
|^^F, vvF#
| ^^F, vvF#
|
|  
|
|  
|
|  
|-
|-
|7
| 7
|415.331
| 415.331
|[[5/4]], [[14/11]]
| [[5/4]], [[14/11]]
|downmajor 3rd
| downmajor 3rd
|vM3
| vM3
|vF#
| vF#
|
|  
|
|  
|
|  
|-
|-
|8
| 8
|474.664
| 474.664
|[[25/19]], [[4/3]]
| [[25/19]], [[4/3]]
|major 3rd, perfect fourth
| major 3rd, perfect fourth
|M3, P4
| M3, P4
|F#, G
| F#, G
|
|  
|
|  
|
|  
|-
|-
|9
| 9
|533.997
| 533.997
|[[15/11]]
| [[15/11]]
|up-fourth
| up-fourth
|^4
| ^4
|^G
| ^G
|
|  
|
|  
|
|  
|-
|-
|10
| 10
|593.330
| 593.330
|[[7/5]], [[31/22]]
| [[7/5]], [[31/22]]
|mid fourth, mid fifth
| mid fourth, mid fifth
|~4, ~5
| ~4, ~5
|^^G, vvA
| ^^G, vvA
|
|  
|
|  
|
|  
|-
|-
|11
| 11
|652.663
| 652.663
|[[16/11]], [[19/13]]
| [[16/11]], [[19/13]]
|down-fifth
| down-fifth
|v5
| v5
|vA
| vA
|
|  
|
|  
|
|  
|-
|-
|12
| 12
|711.996
| 711.996
|[[3/2]]
| [[3/2]]
|fifth
| fifth
|P5, m6
| P5, m6
|A
| A
|
|  
|
|  
|
|  
|-
|-
|13
| 13
|771.329
| 771.329
|[[14/9]], [[25/16]], [[11/7]]
| [[14/9]], [[25/16]], [[11/7]]
|upfifth, upminor 6th
| upfifth, upminor 6th
|^5, ^m6
| ^5, ^m6
|^A, ^Bb
| ^A, ^Bb
|
|  
|
|  
|
|  
|-
|-
|14
| 14
|830.662
| 830.662
|[[8/5]], [[21/13]], [[13/8]]
| [[8/5]], [[21/13]], [[13/8]]
|mid 6th
| mid 6th
|~6
| ~6
|^^A, vvB
| ^^A, vvB
|
|  
|
|  
|
|  
|-
|-
|15
| 15
|889.995
| 889.995
|[[5/3]]
| [[5/3]]
|downmajor 6th
| downmajor 6th
|vM6
| vM6
|vB
| vB
|
|  
|
|  
|
|  
|-
|-
|16
| 16
|949.328
| 949.328
|[[19/11]], [[26/15]], [[7/4]]
| [[19/11]], [[26/15]], [[7/4]]
|major 6th, minor 7th
| major 6th, minor 7th
|M6, m7
| M6, m7
|B, C
| B, C
|
|  
|
|  
|
|  
|-
|-
|17
| 17
|1008.661
| 1008.661
|[[9/5]]
| [[9/5]]
|upminor 7th
| upminor 7th
|^m7
| ^m7
|^C
| ^C
|
|  
|
|  
|
|  
|-
|-
|18
| 18
|1067.994
| 1067.994
|[[13/7]]
| [[13/7]]
|mid 7th
| mid 7th
|~7
| ~7
|^^C, vvD
| ^^C, vvD
|
|  
|
|  
|
|  
|-
|-
|19
| 19
|1127.327
| 1127.327
|[[23/12]]
| [[23/12]]
|downmajor 7th
| downmajor 7th
|vM7
| vM7
|vD
| vD
|
|  
|
|  
|
|  
|-
|-
|20
| 20
|1186.660
| 1186.660
|[[2/1]]
| [[2/1]]
|octave
| octave
|P8
| P8
|D
| D
|
|  
|
|  
|
|  
|-
|-
|22
| 22
|1305.326
| 1305.326
|[[17/8]]
| [[17/8]]
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|-
|-
|23
| 23
|1364.659
| 1364.659
|[[11/5]]
| [[11/5]]
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|-
|-
|25
| 25
|1483.325
| 1483.325
|[[7/3]]
| [[7/3]]
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|-
|-
|27
| 27
|1601.990
| 1601.990
|[[5/2]]
| [[5/2]]
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|-
|-
|28
| 28
|1661.323
| 1661.323
|[[13/5]]
| [[13/5]]
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|-
|-
|29
| 29
|1720.656
| 1720.656
|[[8/3]], [[27/10]]
| [[8/3]], [[27/10]]
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|-
|-
|30
| 30
|1779.989
| 1779.989
|[[14/5]]
| [[14/5]]
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|-
|-
|32
| 32
|1898.655
| 1898.655
|[[3/1]]
| [[3/1]]
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|-
|-
|33
| 33
|1957.988
| 1957.988
|[[31/10]]
| [[31/10]]
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|-
|-
|34
| 34
|2017.321
| 2017.321
|[[16/5]]
| [[16/5]]
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|-
|-
|35
| 35
|2076.654
| 2076.654
|[[10/3]]
| [[10/3]]
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|-
|-
|36
| 36
|2135.987
| 2135.987
|[[24/7]]
| [[24/7]]
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|-
|-
|37
| 37
|2195.320
| 2195.320
|[[7/2]], [[32/9]]
| [[7/2]], [[32/9]]
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|-
|-
|38
| 38
|2254.653
| 2254.653
|[[11/3]]
| [[11/3]]
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|-
|-
|39
| 39
|2313.986
| 2313.986
|[[19/5]]
| [[19/5]]
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|-
|-
|40
| 40
|2373.319
| 2373.319
|[[4/1]]
| [[4/1]]
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|-
|-
|44
| 44
|2610.651
| 2610.651
|[[9/2]]
| [[9/2]]
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|-
|-
|45
| 45
|2669.984
| 2669.984
|[[14/3]]
| [[14/3]]
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|-
|-
|46
| 46
|2729.317
| 2729.317
|[[29/6]]
| [[29/6]]
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|-
|-
|47
| 47
|2788.650
| 2788.650
|[[5/1]]
| [[5/1]]
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|-
|-
|51
| 51
|3025.982
| 3025.982
|[[23/4]]
| [[23/4]]
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|-
|-
|52
| 52
|3085.315
| 3085.315
|[[6/1]]
| [[6/1]]
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|-
|-
|57
| 57
|3381.980
| 3381.980
|[[7/1]]
| [[7/1]]
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|-
|-
|61
| 61
|3619.312
| 3619.312
|[[8/1]]
| [[8/1]]
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|-
|-
|63
| 63
|3737.978
| 3737.978
|[[26/3]]
| [[26/3]]
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|-
|-
|64
| 64
|3797.311
| 3797.311
|[[9/1]]
| [[9/1]]
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|-
|-
|67
| 67
|3975.310
| 3975.310
|[[10/1]]
| [[10/1]]
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|-
|-
|70
| 70
|4153.309
| 4153.309
|[[11/1]]
| [[11/1]]
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|-
|-
|75
| 75
|4449.974
| 4449.974
|[[13/1]]
| [[13/1]]
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|-
|-
|77
| 77
|4568.640
| 4568.640
|[[14/1]]
| [[14/1]]
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|-
|-
|78
| 78
|4627.972
| 4627.972
|[[29/2]]
| [[29/2]]
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|-
|-
|79
| 79
|4687.305
| 4687.305
|[[15/1]]
| [[15/1]]
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|-
|-
|80
| 80
|4746.638
| 4746.638
|[[31/2]]
| [[31/2]]
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|-
|-
|81
| 81
|4805.971
| 4805.971
|[[16/1]]
| [[16/1]]
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|}
|}


== Approximation to JI ==
== Approximation to JI ==
The following table illustrates the representation of the 32-integer limit intervals in 71zpi. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.  
The following table illustrates the representation of the 32-integer limit intervals in 71zpi. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.  


Line 618: Line 617:


{| class="wikitable center-all mw-collapsible mw-collapsed"
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space:nowrap" |Intervals by direct approximation (even if inconsistent)
|+ style="white-space: nowrap;" | Intervals by direct approximation (even if inconsistent)
|-
|-
! Ratio
! Ratio
! Error (abs, [[Cent|¢]])
! Error (abs, [[Cent| ¢]])
! Error (rel, [[Relative cent|%]])
! Error (rel, [[Relative cent| %]])
|-
|-
|[[14/1]]
| [[14/1]]
|0.186
| 0.186
|0.314
| 0.314
|-
|-
|[[11/5]]
| [[11/5]]
|0.346
| 0.346
|0.583
| 0.583
|-
|-
|''[[17/8]]''
| ''[[17/8]]''
|''0.370''
| ''0.370''
|''0.624''
| ''0.624''
|-
|-
|[[31/22]]
| [[31/22]]
|0.388
| 0.388
|0.654
| 0.654
|-
|-
|[[21/13]]
| [[21/13]]
|0.408
| 0.408
|0.688
| 0.688
|-
|-
|[[25/19]]
| [[25/19]]
|0.451
| 0.451
|0.759
| 0.759
|-
|-
|[[26/3]]
| [[26/3]]
|0.595
| 0.595
|1.003
| 1.003
|-
|-
|[[30/29]]
| [[30/29]]
|0.641
| 0.641
|1.081
| 1.081
|-
|-
|[[31/10]]
| [[31/10]]
|0.733
| 0.733
|1.236
| 1.236
|-
|-
|''[[32/9]]''
| ''[[32/9]]''
|''0.770''
| ''0.770''
|''1.297''
| ''1.297''
|-
|-
|[[15/14]]
| [[15/14]]
|0.777
| 0.777
|1.309
| 1.309
|-
|-
|''[[19/16]]''
| ''[[19/16]]''
|''0.848''
| ''0.848''
|''1.429''
| ''1.429''
|-
|-
|[[15/1]]
| [[15/1]]
|0.963
| 0.963
|1.623
| 1.623
|-
|-
|[[23/12]]
| [[23/12]]
|1.007
| 1.007
|1.698
| 1.698
|-
|-
|[[27/10]]
| [[27/10]]
|1.105
| 1.105
|1.863
| 1.863
|-
|-
|''[[25/16]]''
| ''[[25/16]]''
|''1.299''
| ''1.299''
|''2.189''
| ''2.189''
|-
|-
|[[29/28]]
| [[29/28]]
|1.418
| 1.418
|2.390
| 2.390
|-
|-
|[[27/22]]
| [[27/22]]
|1.451
| 1.451
|2.445
| 2.445
|-
|-
|[[31/2]]
| [[31/2]]
|1.603
| 1.603
|2.702
| 2.702
|-
|-
|[[29/2]]
| [[29/2]]
|1.605
| 1.605
|2.705
| 2.705
|-
|-
|[[29/6]]
| [[29/6]]
|1.695
| 1.695
|2.857
| 2.857
|-
|-
|'''[[11/1]]'''
| '''[[11/1]]'''
|'''1.991'''
| '''1.991'''
|'''3.355'''
| '''3.355'''
|-
|-
|[[14/11]]
| [[14/11]]
|2.177
| 2.177
|3.669
| 3.669
|-
|-
|[[23/4]]
| [[23/4]]
|2.292
| 2.292
|3.864
| 3.864
|-
|-
|'''[[5/1]]'''
| '''[[5/1]]'''
|'''2.336'''
| '''2.336'''
|'''3.938'''
| '''3.938'''
|-
|-
|[[14/5]]
| [[14/5]]
|2.523
| 2.523
|4.252
| 4.252
|-
|-
|[[19/5]]
| [[19/5]]
|2.787
| 2.787
|4.697
| 4.697
|-
|-
|''[[24/7]]''
| ''[[24/7]]''
|''2.858''
| ''2.858''
|''4.817''
| ''4.817''
|-
|-
|[[26/15]]
| [[26/15]]
|2.931
| 2.931
|4.940
| 4.940
|-
|-
|[[15/11]]
| [[15/11]]
|2.954
| 2.954
|4.979
| 4.979
|-
|-
|[[14/3]]
| [[14/3]]
|3.113
| 3.113
|5.247
| 5.247
|-
|-
|[[19/11]]
| [[19/11]]
|3.133
| 3.133
|5.280
| 5.280
|-
|-
|'''[[3/1]]'''
| '''[[3/1]]'''
|'''3.300'''
| '''3.300'''
|'''5.561'''
| '''5.561'''
|-
|-
|''[[16/13]]''
| ''[[16/13]]''
|''3.474''
| ''3.474''
|''5.856''
| ''5.856''
|-
|-
|''[[16/5]]''
| ''[[16/5]]''
|''3.635''
| ''3.635''
|''6.127''
| ''6.127''
|-
|-
|[[13/7]]
| [[13/7]]
|3.708
| 3.708
|6.250
| 6.250
|-
|-
|''[[16/11]]''
| ''[[16/11]]''
|''3.981''
| ''3.981''
|''6.709''
| ''6.709''
|-
|-
|[[19/13]]
| [[19/13]]
|4.323
| 4.323
|7.285
| 7.285
|-
|-
|[[10/9]]
| [[10/9]]
|4.405
| 4.405
|7.424
| 7.424
|-
|-
|[[11/3]]
| [[11/3]]
|5.290
| 5.290
|8.916
| 8.916
|-
|-
|[[5/3]]
| [[5/3]]
|5.636
| 5.636
|9.499
| 9.499
|-
|-
|''[[16/1]]''
| ''[[16/1]]''
|''5.971''
| ''5.971''
|''10.064''
| ''10.064''
|-
|-
|''[[8/7]]''
| ''[[8/7]]''
|''6.158''
| ''6.158''
|''10.378''
| ''10.378''
|-
|-
|[[14/9]]
| [[14/9]]
|6.413
| 6.413
|10.808
| 10.808
|-
|-
|[[9/1]]
| [[9/1]]
|6.599
| 6.599
|11.122
| 11.122
|-
|-
|[[9/2]]
| [[9/2]]
|6.741
| 6.741
|11.362
| 11.362
|-
|-
|[[13/5]]
| [[13/5]]
|7.110
| 7.110
|11.982
| 11.982
|-
|-
|[[13/11]]
| [[13/11]]
|7.455
| 7.455
|12.565
| 12.565
|-
|-
|[[10/3]]
| [[10/3]]
|7.704
| 7.704
|12.985
| 12.985
|-
|-
|[[11/9]]
| [[11/9]]
|8.590
| 8.590
|14.478
| 14.478
|-
|-
|[[9/5]]
| [[9/5]]
|8.936
| 8.936
|15.060
| 15.060
|-
|-
|'''[[13/1]]'''
| '''[[13/1]]'''
|'''9.446'''
| '''9.446'''
|'''15.920'''
| '''15.920'''
|-
|-
|''[[13/8]]''
| ''[[13/8]]''
|''9.866''
| ''9.866''
|''16.628''
| ''16.628''
|-
|-
|[[3/2]]
| [[3/2]]
|10.041
| 10.041
|16.923
| 16.923
|-
|-
|[[7/5]]
| [[7/5]]
|10.818
| 10.818
|18.232
| 18.232
|-
|-
|[[10/1]]
| [[10/1]]
|11.004
| 11.004
|18.546
| 18.546
|-
|-
|[[11/7]]
| [[11/7]]
|11.163
| 11.163
|18.815
| 18.815
|-
|-
|'''[[7/1]]'''
| '''[[7/1]]'''
|'''13.154'''
| '''13.154'''
|'''22.170'''
| '''22.170'''
|-
|-
|'''[[2/1]]'''
| '''[[2/1]]'''
|'''13.340'''
| '''13.340'''
|'''22.484'''
| '''22.484'''
|-
|-
|[[5/2]]
| [[5/2]]
|15.677
| 15.677
|26.422
| 26.422
|-
|-
|[[7/3]]
| [[7/3]]
|16.454
| 16.454
|27.731
| 27.731
|-
|-
|[[6/1]]
| [[6/1]]
|16.640
| 16.640
|28.045
| 28.045
|-
|-
|''[[8/5]]''
| ''[[8/5]]''
|''16.975''
| ''16.975''
|''28.610''
| ''28.610''
|-
|-
|[[6/5]]
| [[6/5]]
|18.976
| 18.976
|31.983
| 31.983
|-
|-
|''[[8/1]]''
| ''[[8/1]]''
|''19.312''
| ''19.312''
|''32.548''
| ''32.548''
|-
|-
|''[[7/4]]''
| ''[[7/4]]''
|''19.498''
| ''19.498''
|''32.862''
| ''32.862''
|-
|-
|''[[8/3]]''
| ''[[8/3]]''
|''22.611''
| ''22.611''
|''38.109''
| ''38.109''
|-
|-
|[[4/3]]
| [[4/3]]
|23.381
| 23.381
|39.407
| 39.407
|-
|-
|[[7/2]]
| [[7/2]]
|26.494
| 26.494
|44.654
| 44.654
|-
|-
|[[4/1]]
| [[4/1]]
|26.681
| 26.681
|44.968
| 44.968
|-
|-
|[[5/4]]
| [[5/4]]
|29.017
| 29.017
|48.906
| 48.906
|}
|}


[[Category:Zeta peak indexes]]
[[Category:Zeta peak indexes]]
Retrieved from "https://en.xen.wiki/w/71zpi"