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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
!ZPI
| gap = 15.899282
!Steps per octave
| edo = 12edo
!Step size (cents)
| octave = 1197.68616940005
!Height
| consistent = 10
!Integral
| distinct = 6
!Gap
}}
!EDO
!Octave (cents)
!Consistent
!Distinct
|-
|34zpi
|12.0231830072926
|99.8071807833375
|5.193290
|1.269599
|15.899282
|[[12edo]]
|1197.68616940005
|10
|6
|}


== Intervals ==
== Intervals ==
{| class="wikitable center-1 right-2 left-3 center-4"
{| class="wikitable center-1 right-2 left-3 center-4"
|+ style="white-space: nowrap;" | 34zpi intervals and notation
|+ 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:
| 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 %
* '''<u>Bold Underlined:</u>''' relative error < 8.333 %
* '''Bold:''' relative error < 16.667 %
* '''Bold:''' relative error < 16.667 %
Line 42: Line 25:
* <small><small>Small Small:</small></small> relative error < 41.667 %
* <small><small>Small Small:</small></small> relative error < 41.667 %
* <small><small><small>Small Small Small:</small></small></small> relative error < 50 %
* <small><small><small>Small Small Small:</small></small></small> relative error < 50 %
| style="text-align:right;" | <center>'''12edo'''</center><br>[[9/8|Whole tone]] = 2 steps<br>[[256/243|Limma]] = 1 step<br>[[2187/2048|Apotome]] = 1 step
| 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
! Degree
! Cents
! Cents
! Ratios
! Ratios
! Ups and Downs Notation
! Ups and downs notation
|-
|-
| 0
| 0
Line 296: Line 279:


== Approximation to JI ==
== 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"
{| class="wikitable center-1 right-2 right-3 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Intervals by direct approximation (even if inconsistent)
|+ style="white-space: nowrap;" | 16-integer-limit intervals in 34zpi (by direct approximation)
|-
|-
! Ratio
! Ratio
Line 305: Line 292:
|-
|-
| [[4/3]]
| [[4/3]]
| -0.991
| +0.991
| -0.993
| +0.993
|-
|-
| [[8/3]]
| [[8/3]]
| +1.323
| -1.323
| +1.325
| -1.325
|-
|-
| [[16/9]]
| [[16/9]]
| -1.982
| +1.982
| -1.986
| +1.986
|-
|-
| '''[[2/1]]'''
| '''[[2/1]]'''
| '''+2.314'''
| '''-2.314'''
| '''+2.318'''
| '''-2.318'''
|-
|-
| [[15/1]]
| [[15/1]]
| -2.669
| +2.669
| -2.674
| +2.674
|-
|-
| [[3/2]]
| [[3/2]]
| +3.305
| -3.305
| +3.311
| -3.311
|-
|-
| [[16/3]]
| [[16/3]]
| +3.637
| -3.637
| +3.644
| -3.644
|-
|-
| [[9/8]]
| [[9/8]]
| +4.296
| -4.296
| +4.304
| -4.304
|-
|-
| [[4/1]]
| [[4/1]]
| +4.628
| -4.628
| +4.637
| -4.637
|-
|-
| [[15/2]]
| [[15/2]]
| -4.983
| +4.983
| -4.992
| +4.992
|-
|-
| '''[[3/1]]'''
| '''[[3/1]]'''
| '''+5.619'''
| '''-5.619'''
| '''+5.629'''
| '''-5.629'''
|-
|-
| [[10/1]]
| [[10/1]]
| -5.974
| +5.974
| -5.985
| +5.985
|-
|-
| [[9/4]]
| [[9/4]]
| +6.609
| -6.609
| +6.622
| -6.622
|-
|-
| [[8/1]]
| [[8/1]]
| +6.941
| -6.941
| +6.955
| -6.955
|-
|-
| [[15/4]]
| [[15/4]]
| -7.296
| +7.296
| -7.311
| +7.311
|-
|-
| [[6/1]]
| [[6/1]]
| +7.932
| -7.932
| +7.948
| -7.948
|-
|-
| '''[[5/1]]'''
| '''[[5/1]]'''
| '''-8.287'''
| '''+8.287'''
| '''-8.303'''
| '''+8.303'''
|-
|-
| [[9/2]]
| [[9/2]]
| +8.923
| -8.923
| +8.941
| -8.941
|-
|-
| [[16/1]]
| [[16/1]]
| +9.255
| -9.255
| +9.273
| -9.273
|-
|-
| [[15/8]]
| [[15/8]]
| -9.610
| +9.610
| -9.629
| +9.629
|-
|- style="background-color: #cccccc;"
| ''[[13/11]]''
| ''[[13/11]]''
| ''-10.212''
| ''+10.212''
| ''-10.232''
| ''+10.232''
|-
|-
| [[12/1]]
| [[12/1]]
| +10.246
| -10.246
| +10.266
| -10.266
|-
|-
| [[5/2]]
| [[5/2]]
| -10.601
| +10.601
| -10.622
| +10.622
|-
|-
| [[9/1]]
| [[9/1]]
| +11.237
| -11.237
| +11.259
| -11.259
|-
|-
| [[10/3]]
| [[10/3]]
| -11.592
| +11.592
| -11.614
| +11.614
|-
|-
| [[16/15]]
| [[16/15]]
| +11.924
| -11.924
| +11.947
| -11.947
|-
|-
| [[5/4]]
| [[5/4]]
| -12.915
| +12.915
| -12.940
| +12.940
|-
|-
| [[5/3]]
| [[5/3]]
| -13.906
| +13.906
| -13.933
| +13.933
|-
|-
| [[14/5]]
| [[14/5]]
| -14.017
| +14.017
| -14.044
| +14.044
|-
|-
| [[8/5]]
| [[8/5]]
| +15.229
| -15.229
| +15.258
| -15.258
|-
|-
| [[11/7]]
| [[11/7]]
| -15.965
| +15.965
| -15.996
| +15.996
|-
|-
| [[6/5]]
| [[6/5]]
| +16.220
| -16.220
| +16.251
| -16.251
|-
|-
| [[7/5]]
| [[7/5]]
| -16.331
| +16.331
| -16.362
| +16.362
|-
|-
| [[10/9]]
| [[10/9]]
| -17.211
| +17.211
| -17.244
| +17.244
|-
|-
| [[16/5]]
| [[16/5]]
| +17.543
| -17.543
| +17.577
| -17.577
|-
|-
| [[14/11]]
| [[14/11]]
| +18.279
| -18.279
| +18.315
| -18.315
|-
|-
| [[12/5]]
| [[12/5]]
| +18.534
| -18.534
| +18.569
| -18.569
|-
|-
| [[10/7]]
| [[10/7]]
| +18.645
| -18.645
| +18.681
| -18.681
|-
|-
| [[9/5]]
| [[9/5]]
| +19.524
| -19.524
| +19.562
| -19.562
|-
|-
| [[15/14]]
| [[15/14]]
| +19.636
| -19.636
| +19.674
| -19.674
|-
|-
| [[15/7]]
| [[15/7]]
| +21.949
| -21.949
| +21.992
| -21.992
|-
|-
| [[14/1]]
| [[14/1]]
| -22.304
| +22.304
| -22.347
| +22.347
|-
|-
| '''[[7/1]]'''
| '''[[7/1]]'''
| '''-24.618'''
| '''+24.618'''
| '''-24.666'''
| '''+24.666'''
|-
|- style="background-color: #cccccc;"
| ''[[13/7]]''
| ''[[13/7]]''
| ''-26.177''
| ''+26.177''
| ''-26.228''
| ''+26.228''
|-
|-
| [[7/2]]
| [[7/2]]
| -26.932
| +26.932
| -26.984
| +26.984
|-
|-
| [[14/3]]
| [[14/3]]
| -27.923
| +27.923
| -27.977
| +27.977
|-
|- style="background-color: #cccccc;"
| ''[[14/13]]''
| ''[[14/13]]''
| ''+28.491''
| ''-28.491''
| ''+28.546''
| ''-28.546''
|-
|-
| [[7/4]]
| [[7/4]]
| -29.246
| +29.246
| -29.302
| +29.302
|-
|-
| [[7/3]]
| [[7/3]]
| -30.237
| +30.237
| -30.295
| +30.295
|-
|-
| [[8/7]]
| [[8/7]]
| +31.560
| -31.560
| +31.621
| -31.621
|-
|-
| [[11/5]]
| [[11/5]]
| -32.296
| +32.296
| -32.359
| +32.359
|-
|-
| [[7/6]]
| [[7/6]]
| -32.551
| +32.551
| -32.614
| +32.614
|-
|-
| [[14/9]]
| [[14/9]]
| -33.542
| +33.542
| -33.606
| +33.606
|-
|-
| [[16/7]]
| [[16/7]]
| +33.874
| -33.874
| +33.939
| -33.939
|-
|-
| [[11/10]]
| [[11/10]]
| -34.610
| +34.610
| -34.677
| +34.677
|-
|-
| [[12/7]]
| [[12/7]]
| +34.864
| -34.864
| +34.932
| -34.932
|-
|-
| [[9/7]]
| [[9/7]]
| +35.855
| -35.855
| +35.925
| -35.925
|-
|-
| [[13/9]]
| [[13/9]]
| +37.775
| -37.775
| +37.848
| -37.848
|-
|-
| [[15/11]]
| [[15/11]]
| +37.915
| -37.915
| +37.988
| -37.988
|-
|-
| [[13/12]]
| [[13/12]]
| +38.765
| -38.765
| +38.840
| -38.840
|-
|-
| [[16/13]]
| [[16/13]]
| -39.756
| +39.756
| -39.833
| +39.833
|-
|-
| '''[[11/1]]'''
| '''[[11/1]]'''
| '''-40.584'''
| '''+40.584'''
| '''-40.662'''
| '''+40.662'''
|-
|-
| [[13/6]]
| [[13/6]]
| +41.079
| -41.079
| +41.159
| -41.159
|-
|-
| [[13/8]]
| [[13/8]]
| +42.070
| -42.070
| +42.151
| -42.151
|-
|- style="background-color: #cccccc;"
| ''[[13/5]]''
| ''[[13/5]]''
| ''-42.508''
| ''+42.508''
| ''-42.590''
| ''+42.590''
|-
|-
| [[11/2]]
| [[11/2]]
| -42.897
| +42.897
| -42.980
| +42.980
|-
|-
| [[13/3]]
| [[13/3]]
| +43.393
| -43.393
| +43.477
| -43.477
|-
|-
| [[13/4]]
| [[13/4]]
| +44.384
| -44.384
| +44.470
| -44.470
|-
|- style="background-color: #cccccc;"
| ''[[13/10]]''
| ''[[13/10]]''
| ''-44.822''
| ''+44.822''
| ''-44.909''
| ''+44.909''
|-
|-
| [[11/4]]
| [[11/4]]
| -45.211
| +45.211
| -45.299
| +45.299
|-
|-
| [[11/3]]
| [[11/3]]
| -46.202
| +46.202
| -46.291
| +46.291
|-
|-
| [[13/2]]
| [[13/2]]
| +46.698
| -46.698
| +46.788
| -46.788
|-
|-
| [[11/8]]
| [[11/8]]
| -47.525
| +47.525
| -47.617
| +47.617
|-
|- style="background-color: #cccccc;"
| ''[[11/9]]''
| ''[[11/9]]''
| ''+47.986''
| ''-47.986''
| ''+48.079''
| ''-48.079''
|-
|- style="background-color: #cccccc;"
| ''[[15/13]]''
| ''[[15/13]]''
| ''+48.127''
| ''-48.127''
| ''+48.220''
| ''-48.220''
|-
|-
| [[11/6]]
| [[11/6]]
| -48.516
| +48.516
| -48.610
| +48.610
|- style="background-color: #cccccc;"
| ''[[12/11]]''
| ''+48.977''
| ''+49.072''
|-
| '''[[13/1]]'''
| '''-49.012'''
| '''-49.106'''
|-
| [[16/11]]
| -49.839
| -49.935
|}
 
{| 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'''
|-
|-
| ''[[12/11]]''
| [[9/2]]
| ''-48.977''
| -8.923
| ''-49.072''
| -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]]'''
| '''[[13/1]]'''
| '''+49.012'''
| '''-49.012'''
| '''+49.106'''
| '''-49.106'''
|-
|-
| [[16/11]]
| [[16/11]]
| +49.839
| -49.839
| +49.935
| -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''
|}
|}


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