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'''34 zeta peak index''' (abbreviated '''34zpi'''), is the [[Equal-step tuning|equal-step]] [[tuning system]] obtained from the 34th [[Zeta peak index|peak]] of the [[The Riemann zeta function and tuning|Riemann zeta function]].
'''34 zeta peak index''' (abbreviated '''34zpi'''), is the [[Equal-step tuning|equal-step]] [[tuning system]] obtained from the 34th [[Zeta peak index|peak]] of the [[The Riemann zeta function and tuning|Riemann zeta function]].


{|class="wikitable"
{{ZPI
!colspan="3"|Tuning
| zpi = 34
!colspan="3"|Strength
| steps = 12.0231830072926
!colspan="2"|Closest EDO
| step size = 99.8071807833375
!colspan="2"|Integer limit
| height = 5.193290
| integral = 1.269599
| gap = 15.899282
| edo = 12edo
| octave = 1197.68616940005
| consistent = 10
| distinct = 6
}}
 
== Intervals ==
{| class="wikitable center-1 right-2 left-3 center-4"
|+ style="font-size: 105%; white-space: nowrap;" | Intervals in 34zpi
|-
| colspan="3" style="text-align:left;" | JI ratios are comprised of 16-integer-limit ratios,<br>and are stylized as follows to indicate their accuracy:
* '''<u>Bold Underlined:</u>''' relative error < 8.333 %
* '''Bold:''' relative error < 16.667 %
* Normal: relative error < 25 %
* <small>Small:</small> relative error < 33.333 %
* <small><small>Small Small:</small></small> relative error < 41.667 %
* <small><small><small>Small Small Small:</small></small></small> relative error < 50 %
| style="text-align:right;" | <center>'''⟨12 19]'''</center><br>[[9/8|Whole tone]] = 2 steps<br>[[256/243|Limma]] = 1 step<br>[[2187/2048|Apotome]] = 1 step
|-
! Degree
! Cents
! Ratios
! Ups and downs notation
|-
| 0
| 0.000
|
| P1
|-
| 1
| 99.807
| '''[[16/15]]''', [[15/14]], <small>[[14/13]]</small>, <small><small>[[13/12]]</small></small>
| m2
|-
| 2
| 199.614
| <small><small><small>[[12/11]]</small></small></small>, <small><small>[[11/10]]</small></small>, [[10/9]], '''<u>[[9/8]]'''</u>, <small>[[8/7]]</small>, <small><small><small>[[15/13]]</small></small></small>
| M2
|-
| 3
| 299.422
| <small>[[7/6]]</small>, '''[[13/11]]''', '''[[6/5]]''', <small><small><small>[[11/9]]</small></small></small>
| m3
|-
| 4
| 399.229
| <small><small>[[16/13]]</small></small>, '''[[5/4]]''', [[14/11]], <small><small>[[9/7]]</small></small>
| M3
|-
| 5
| 499.036
| <small><small><small>[[13/10]]</small></small></small>, '''<u>[[4/3]]'''</u>, <small><small>[[15/11]]</small></small>
| P4
|-
| 6
| 598.843
| <small><small><small>[[11/8]]</small></small></small>, '''[[7/5]]''', [[10/7]], <small><small>[[13/9]]</small></small>, <small><small><small>[[16/11]]</small></small></small>
| A4, d5
|-
| 7
| 698.650
| '''<u>[[3/2]]'''</u>
| P5
|-
| 8
| 798.457
| <small><small>[[14/9]]</small></small>, '''[[11/7]]''', '''[[8/5]]''', <small><small><small>[[13/8]]</small></small></small>
| m6
|-
| 9
| 898.265
| '''[[5/3]]''', <small><small>[[12/7]]</small></small>
| M6
|-
| 10
| 998.072
| <small>[[7/4]]</small>, '''<u>[[16/9]]'''</u>, [[9/5]]
| m7
|-
| 11
| 1097.879
| <small><small><small>[[11/6]]</small></small></small>, <small>[[13/7]]</small>, '''[[15/8]]'''
| M7
|-
| 12
| 1197.686
| '''<u>[[2/1]]'''</u>
| P1 +1 oct
|-
| 13
| 1297.493
| [[15/7]], <small><small>[[13/6]]</small></small>
| m2 +1 oct
|-
| 14
| 1397.301
| <small>[[11/5]]</small>, '''<u>[[9/4]]'''</u>, <small><small>[[16/7]]</small></small>
| M2 +1 oct
|-
| 15
| 1497.108
| <small>[[7/3]]</small>, [[12/5]]
| m3 +1 oct
|-
| 16
| 1596.915
| '''[[5/2]]'''
| M3 +1 oct
|-
| 17
| 1696.722
| <small><small><small>[[13/5]]</small></small></small>, '''<u>[[8/3]]'''</u>
| P4 +1 oct
|-
| 18
| 1796.529
| <small><small><small>[[11/4]]</small></small></small>, '''[[14/5]]'''
| A4 +1 oct, d5 +1 oct
|-
| 19
| 1896.336
| '''<u>[[3/1]]'''</u>
| P5 +1 oct
|-
| 20
| 1996.144
| [[16/5]], <small><small><small>[[13/4]]</small></small></small>
| m6 +1 oct
|-
| 21
| 2095.951
| '''[[10/3]]'''
| M6 +1 oct
|-
| 22
| 2195.758
| <small>[[7/2]]</small>
| m7 +1 oct
|-
| 23
| 2295.565
| <small><small><small>[[11/3]]</small></small></small>, '''<u>[[15/4]]'''</u>
| M7 +1 oct
|-
| 24
| 2395.372
| '''<u>[[4/1]]'''</u>
| P1 +2 oct
|-
| 25
| 2495.180
| <small><small><small>[[13/3]]</small></small></small>
| m2 +2 oct
|-
| 26
| 2594.987
| '''[[9/2]]'''
| M2 +2 oct
|-
| 27
| 2694.794
| <small>[[14/3]]</small>
| m3 +2 oct
|-
| 28
| 2794.601
| '''<u>[[5/1]]'''</u>
| M3 +2 oct
|-
| 29
| 2894.408
| '''<u>[[16/3]]'''</u>
| P4 +2 oct
|-
| 30
| 2994.215
| <small><small><small>[[11/2]]</small></small></small>
| A4 +2 oct, d5 +2 oct
|-
| 31
| 3094.023
| '''<u>[[6/1]]'''</u>
| P5 +2 oct
|-
| 32
| 3193.830
| <small><small><small>[[13/2]]</small></small></small>
| m6 +2 oct
|-
| 33
| 3293.637
|
| M6 +2 oct
|-
| 34
| 3393.444
| [[7/1]]
| m7 +2 oct
|-
| 35
| 3493.251
| '''<u>[[15/2]]'''</u>
| M7 +2 oct
|-
| 36
| 3593.059
| '''<u>[[8/1]]'''</u>
| P1 +3 oct
|-
| 37
| 3692.866
|
| m2 +3 oct
|-
| 38
| 3792.673
| '''[[9/1]]'''
| M2 +3 oct
|-
| 39
| 3892.480
|
| m3 +3 oct
|-
| 40
| 3992.287
| '''<u>[[10/1]]'''</u>
| M3 +3 oct
|-
| 41
| 4092.094
|
| P4 +3 oct
|-
| 42
| 4191.902
| <small><small>[[11/1]]</small></small>
| A4 +3 oct, d5 +3 oct
|-
| 43
| 4291.709
| '''[[12/1]]'''
| P5 +3 oct
|-
| 44
| 4391.516
| <small><small><small>[[13/1]]</small></small></small>
| m6 +3 oct
|-
| 45
| 4491.323
|
| M6 +3 oct
|-
| 46
| 4591.130
| [[14/1]]
| m7 +3 oct
|-
|-
!ZPI
| 47
!Steps per octave
| 4690.937
!Step size (cents)
| '''<u>[[15/1]]'''</u>
!Height
| M7 +3 oct
!Integral
!Gap
!EDO
!Octave (cents)
!Consistent
!Distinct
|-
|-
|34zpi
| 48
|12.0231830072926
| 4790.745
|99.8071807833375
| '''[[16/1]]'''
|5.193290
| P1 +4 oct
|1.269599
|15.899282
|[[12edo]]
|1197.68616940005
|10
|6
|}
|}


== Intervals ==
== Approximation to JI ==
 
=== Interval mappings ===
 
The following tables show how 16-integer-limit intervals are represented in 34zpi. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italics''.
 
{| class="wikitable center-1 right-2 right-3 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | 16-integer-limit intervals in 34zpi (by direct approximation)
|-
! Ratio
! Error (abs, [[Cent|¢]])
! Error (rel, [[Relative cent|%]])
|-
| [[4/3]]
| +0.991
| +0.993
|-
| [[8/3]]
| -1.323
| -1.325
|-
| [[16/9]]
| +1.982
| +1.986
|-
| '''[[2/1]]'''
| '''-2.314'''
| '''-2.318'''
|-
| [[15/1]]
| +2.669
| +2.674
|-
| [[3/2]]
| -3.305
| -3.311
|-
| [[16/3]]
| -3.637
| -3.644
|-
| [[9/8]]
| -4.296
| -4.304
|-
| [[4/1]]
| -4.628
| -4.637
|-
| [[15/2]]
| +4.983
| +4.992
|-
| '''[[3/1]]'''
| '''-5.619'''
| '''-5.629'''
|-
| [[10/1]]
| +5.974
| +5.985
|-
| [[9/4]]
| -6.609
| -6.622
|-
| [[8/1]]
| -6.941
| -6.955
|-
| [[15/4]]
| +7.296
| +7.311
|-
| [[6/1]]
| -7.932
| -7.948
|-
| '''[[5/1]]'''
| '''+8.287'''
| '''+8.303'''
|-
| [[9/2]]
| -8.923
| -8.941
|-
| [[16/1]]
| -9.255
| -9.273
|-
| [[15/8]]
| +9.610
| +9.629
|- style="background-color: #cccccc;"
| ''[[13/11]]''
| ''+10.212''
| ''+10.232''
|-
| [[12/1]]
| -10.246
| -10.266
|-
| [[5/2]]
| +10.601
| +10.622
|-
| [[9/1]]
| -11.237
| -11.259
|-
| [[10/3]]
| +11.592
| +11.614
|-
| [[16/15]]
| -11.924
| -11.947
|-
| [[5/4]]
| +12.915
| +12.940
|-
| [[5/3]]
| +13.906
| +13.933
|-
| [[14/5]]
| +14.017
| +14.044
|-
| [[8/5]]
| -15.229
| -15.258
|-
| [[11/7]]
| +15.965
| +15.996
|-
| [[6/5]]
| -16.220
| -16.251
|-
| [[7/5]]
| +16.331
| +16.362
|-
| [[10/9]]
| +17.211
| +17.244
|-
| [[16/5]]
| -17.543
| -17.577
|-
| [[14/11]]
| -18.279
| -18.315
|-
| [[12/5]]
| -18.534
| -18.569
|-
| [[10/7]]
| -18.645
| -18.681
|-
| [[9/5]]
| -19.524
| -19.562
|-
| [[15/14]]
| -19.636
| -19.674
|-
| [[15/7]]
| -21.949
| -21.992
|-
| [[14/1]]
| +22.304
| +22.347
|-
| '''[[7/1]]'''
| '''+24.618'''
| '''+24.666'''
|- style="background-color: #cccccc;"
| ''[[13/7]]''
| ''+26.177''
| ''+26.228''
|-
| [[7/2]]
| +26.932
| +26.984
|-
| [[14/3]]
| +27.923
| +27.977
|- style="background-color: #cccccc;"
| ''[[14/13]]''
| ''-28.491''
| ''-28.546''
|-
| [[7/4]]
| +29.246
| +29.302
|-
| [[7/3]]
| +30.237
| +30.295
|-
| [[8/7]]
| -31.560
| -31.621
|-
| [[11/5]]
| +32.296
| +32.359
|-
| [[7/6]]
| +32.551
| +32.614
|-
| [[14/9]]
| +33.542
| +33.606
|-
| [[16/7]]
| -33.874
| -33.939
|-
| [[11/10]]
| +34.610
| +34.677
|-
| [[12/7]]
| -34.864
| -34.932
|-
| [[9/7]]
| -35.855
| -35.925
|-
| [[13/9]]
| -37.775
| -37.848
|-
| [[15/11]]
| -37.915
| -37.988
|-
| [[13/12]]
| -38.765
| -38.840
|-
| [[16/13]]
| +39.756
| +39.833
|-
| '''[[11/1]]'''
| '''+40.584'''
| '''+40.662'''
|-
| [[13/6]]
| -41.079
| -41.159
|-
| [[13/8]]
| -42.070
| -42.151
|- style="background-color: #cccccc;"
| ''[[13/5]]''
| ''+42.508''
| ''+42.590''
|-
| [[11/2]]
| +42.897
| +42.980
|-
| [[13/3]]
| -43.393
| -43.477
|-
| [[13/4]]
| -44.384
| -44.470
|- style="background-color: #cccccc;"
| ''[[13/10]]''
| ''+44.822''
| ''+44.909''
|-
| [[11/4]]
| +45.211
| +45.299
|-
| [[11/3]]
| +46.202
| +46.291
|-
| [[13/2]]
| -46.698
| -46.788
|-
| [[11/8]]
| +47.525
| +47.617
|- style="background-color: #cccccc;"
| ''[[11/9]]''
| ''-47.986''
| ''-48.079''
|- style="background-color: #cccccc;"
| ''[[15/13]]''
| ''-48.127''
| ''-48.220''
|-
| [[11/6]]
| +48.516
| +48.610
|- style="background-color: #cccccc;"
| ''[[12/11]]''
| ''+48.977''
| ''+49.072''
|-
| '''[[13/1]]'''
| '''-49.012'''
| '''-49.106'''
|-
| [[16/11]]
| -49.839
| -49.935
|}


== Approximation to JI ==
{| class="wikitable center-1 right-2 right-3 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | 16-integer-limit intervals in 34zpi (by patent val mapping)
|-
! Ratio
! Error (abs, [[Cent|¢]])
! Error (rel, [[Relative cent|%]])
|-
| [[4/3]]
| +0.991
| +0.993
|-
| [[8/3]]
| -1.323
| -1.325
|-
| [[16/9]]
| +1.982
| +1.986
|-
| '''[[2/1]]'''
| '''-2.314'''
| '''-2.318'''
|-
| [[15/1]]
| +2.669
| +2.674
|-
| [[3/2]]
| -3.305
| -3.311
|-
| [[16/3]]
| -3.637
| -3.644
|-
| [[9/8]]
| -4.296
| -4.304
|-
| [[4/1]]
| -4.628
| -4.637
|-
| [[15/2]]
| +4.983
| +4.992
|-
| '''[[3/1]]'''
| '''-5.619'''
| '''-5.629'''
|-
| [[10/1]]
| +5.974
| +5.985
|-
| [[9/4]]
| -6.609
| -6.622
|-
| [[8/1]]
| -6.941
| -6.955
|-
| [[15/4]]
| +7.296
| +7.311
|-
| [[6/1]]
| -7.932
| -7.948
|-
| '''[[5/1]]'''
| '''+8.287'''
| '''+8.303'''
|-
| [[9/2]]
| -8.923
| -8.941
|-
| [[16/1]]
| -9.255
| -9.273
|-
| [[15/8]]
| +9.610
| +9.629
|-
| [[12/1]]
| -10.246
| -10.266
|-
| [[5/2]]
| +10.601
| +10.622
|-
| [[9/1]]
| -11.237
| -11.259
|-
| [[10/3]]
| +11.592
| +11.614
|-
| [[16/15]]
| -11.924
| -11.947
|-
| [[5/4]]
| +12.915
| +12.940
|-
| [[5/3]]
| +13.906
| +13.933
|-
| [[14/5]]
| +14.017
| +14.044
|-
| [[8/5]]
| -15.229
| -15.258
|-
| [[11/7]]
| +15.965
| +15.996
|-
| [[6/5]]
| -16.220
| -16.251
|-
| [[7/5]]
| +16.331
| +16.362
|-
| [[10/9]]
| +17.211
| +17.244
|-
| [[16/5]]
| -17.543
| -17.577
|-
| [[14/11]]
| -18.279
| -18.315
|-
| [[12/5]]
| -18.534
| -18.569
|-
| [[10/7]]
| -18.645
| -18.681
|-
| [[9/5]]
| -19.524
| -19.562
|-
| [[15/14]]
| -19.636
| -19.674
|-
| [[15/7]]
| -21.949
| -21.992
|-
| [[14/1]]
| +22.304
| +22.347
|-
| '''[[7/1]]'''
| '''+24.618'''
| '''+24.666'''
|-
| [[7/2]]
| +26.932
| +26.984
|-
| [[14/3]]
| +27.923
| +27.977
|-
| [[7/4]]
| +29.246
| +29.302
|-
| [[7/3]]
| +30.237
| +30.295
|-
| [[8/7]]
| -31.560
| -31.621
|-
| [[11/5]]
| +32.296
| +32.359
|-
| [[7/6]]
| +32.551
| +32.614
|-
| [[14/9]]
| +33.542
| +33.606
|-
| [[16/7]]
| -33.874
| -33.939
|-
| [[11/10]]
| +34.610
| +34.677
|-
| [[12/7]]
| -34.864
| -34.932
|-
| [[9/7]]
| -35.855
| -35.925
|-
| [[13/9]]
| -37.775
| -37.848
|-
| [[15/11]]
| -37.915
| -37.988
|-
| [[13/12]]
| -38.765
| -38.840
|-
| [[16/13]]
| +39.756
| +39.833
|-
| '''[[11/1]]'''
| '''+40.584'''
| '''+40.662'''
|-
| [[13/6]]
| -41.079
| -41.159
|-
| [[13/8]]
| -42.070
| -42.151
|-
| [[11/2]]
| +42.897
| +42.980
|-
| [[13/3]]
| -43.393
| -43.477
|-
| [[13/4]]
| -44.384
| -44.470
|-
| [[11/4]]
| +45.211
| +45.299
|-
| [[11/3]]
| +46.202
| +46.291
|-
| [[13/2]]
| -46.698
| -46.788
|-
| [[11/8]]
| +47.525
| +47.617
|-
| [[11/6]]
| +48.516
| +48.610
|-
| '''[[13/1]]'''
| '''-49.012'''
| '''-49.106'''
|-
| [[16/11]]
| -49.839
| -49.935
|- style="background-color: #cccccc;"
| ''[[12/11]]''
| ''-50.830''
| ''-50.928''
|- style="background-color: #cccccc;"
| ''[[15/13]]''
| ''+51.680''
| ''+51.780''
|- style="background-color: #cccccc;"
| ''[[11/9]]''
| ''+51.821''
| ''+51.921''
|- style="background-color: #cccccc;"
| ''[[13/10]]''
| ''-54.985''
| ''-55.091''
|- style="background-color: #cccccc;"
| ''[[13/5]]''
| ''-57.299''
| ''-57.410''
|- style="background-color: #cccccc;"
| ''[[14/13]]''
| ''+71.316''
| ''+71.454''
|- style="background-color: #cccccc;"
| ''[[13/7]]''
| ''-73.630''
| ''-73.772''
|- style="background-color: #cccccc;"
| ''[[13/11]]''
| ''-89.595''
| ''-89.768''
|}


== See also ==
== See also ==
Retrieved from "https://en.xen.wiki/w/34zpi"