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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="font-size: 105%; white-space: nowrap;" | Intervals in 34zpi
|-
|-
|+ style="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 47: Line 30:
! Cents
! Cents
! Ratios
! Ratios
! Ups and Downs Notation
! Ups and downs notation
|-
|-
| 0
| 0
Line 299: Line 282:
=== Interval mappings ===
=== 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''.  
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"
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| +9.610
| +9.610
| +9.629
| +9.629
|- style="background-color: #c0c0c0;"
|- style="background-color: #cccccc;"
| ''[[13/11]]''
| ''[[13/11]]''
| ''+10.212''
| ''+10.212''
Line 479: Line 462:
| '''+24.618'''
| '''+24.618'''
| '''+24.666'''
| '''+24.666'''
|- style="background-color: #c0c0c0;"
|- style="background-color: #cccccc;"
| ''[[13/7]]''
| ''[[13/7]]''
| ''+26.177''
| ''+26.177''
Line 491: Line 474:
| +27.923
| +27.923
| +27.977
| +27.977
|- style="background-color: #c0c0c0;"
|- style="background-color: #cccccc;"
| ''[[14/13]]''
| ''[[14/13]]''
| ''-28.491''
| ''-28.491''
Line 563: Line 546:
| -42.070
| -42.070
| -42.151
| -42.151
|- style="background-color: #c0c0c0;"
|- style="background-color: #cccccc;"
| ''[[13/5]]''
| ''[[13/5]]''
| ''+42.508''
| ''+42.508''
Line 579: Line 562:
| -44.384
| -44.384
| -44.470
| -44.470
|- style="background-color: #c0c0c0;"
|- style="background-color: #cccccc;"
| ''[[13/10]]''
| ''[[13/10]]''
| ''+44.822''
| ''+44.822''
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| +47.525
| +47.525
| +47.617
| +47.617
|- style="background-color: #c0c0c0;"
|- style="background-color: #cccccc;"
| ''[[11/9]]''
| ''[[11/9]]''
| ''-47.986''
| ''-47.986''
| ''-48.079''
| ''-48.079''
|- style="background-color: #c0c0c0;"
|- style="background-color: #cccccc;"
| ''[[15/13]]''
| ''[[15/13]]''
| ''-48.127''
| ''-48.127''
Line 611: Line 594:
| +48.516
| +48.516
| +48.610
| +48.610
|- style="background-color: #c0c0c0;"
|- style="background-color: #cccccc;"
| ''[[12/11]]''
| ''[[12/11]]''
| ''+48.977''
| ''+48.977''
Line 633: Line 616:
|-
|-
| [[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
|-
|-
| [[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'''
|-
|-
| [[7/2]]
| [[7/2]]
| +26.932 ¢
| +26.932
| +26.984 %
| +26.984
|-
|-
| [[14/3]]
| [[14/3]]
| +27.923 ¢
| +27.923
| +27.977 %
| +27.977
|-
|-
| [[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
|-
|-
| [[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
|-
|-
| [[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
|-
|-
| [[11/6]]
| [[11/6]]
| +48.516 ¢
| +48.516
| +48.610 %
| +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: #c0c0c0;"
|- style="background-color: #cccccc;"
| ''[[12/11]]''
| ''[[12/11]]''
| ''-50.830 ¢''
| ''-50.830''
| ''-50.928 %''
| ''-50.928''
|- style="background-color: #c0c0c0;"
|- style="background-color: #cccccc;"
| ''[[15/13]]''
| ''[[15/13]]''
| ''+51.680 ¢''
| ''+51.680''
| ''+51.780 %''
| ''+51.780''
|- style="background-color: #c0c0c0;"
|- style="background-color: #cccccc;"
| ''[[11/9]]''
| ''[[11/9]]''
| ''+51.821 ¢''
| ''+51.821''
| ''+51.921 %''
| ''+51.921''
|- style="background-color: #c0c0c0;"
|- style="background-color: #cccccc;"
| ''[[13/10]]''
| ''[[13/10]]''
| ''-54.985 ¢''
| ''-54.985''
| ''-55.091 %''
| ''-55.091''
|- style="background-color: #c0c0c0;"
|- style="background-color: #cccccc;"
| ''[[13/5]]''
| ''[[13/5]]''
| ''-57.299 ¢''
| ''-57.299''
| ''-57.410 %''
| ''-57.410''
|- style="background-color: #c0c0c0;"
|- style="background-color: #cccccc;"
| ''[[14/13]]''
| ''[[14/13]]''
| ''+71.316 ¢''
| ''+71.316''
| ''+71.454 %''
| ''+71.454''
|- style="background-color: #c0c0c0;"
|- style="background-color: #cccccc;"
| ''[[13/7]]''
| ''[[13/7]]''
| ''-73.630 ¢''
| ''-73.630''
| ''-73.772 %''
| ''-73.772''
|- style="background-color: #c0c0c0;"
|- style="background-color: #cccccc;"
| ''[[13/11]]''
| ''[[13/11]]''
| ''-89.595 ¢''
| ''-89.595''
| ''-89.768 %''
| ''-89.768''
|}
|}


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