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| '''[[16/1]]'''
| '''[[16/1]]'''
| P1 +4 oct
| P1 +4 oct
|}
== 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
|}
{| 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''
|}
|}


Revision as of 21:30, 12 August 2024

34 zeta peak index (abbreviated 34zpi), is the equal-step tuning system obtained from the 34th peak of the Riemann zeta function.

Tuning Strength Closest EDO Integer limit
ZPI Steps per octave Step size (cents) Height Integral Gap EDO Octave (cents) Consistent Distinct
34zpi 12.0231830072926 99.8071807833375 5.193290 1.269599 15.899282 12edo 1197.68616940005 10 6

Intervals

Intervals in 34zpi
JI ratios are comprised of 16-integer-limit ratios,
and are stylized as follows to indicate their accuracy:
  • Bold Underlined: relative error < 8.333 %
  • Bold: relative error < 16.667 %
  • Normal: relative error < 25 %
  • Small: relative error < 33.333 %
  • Small Small: relative error < 41.667 %
  • Small Small Small: relative error < 50 %
⟨12 19]

Whole tone = 2 steps
Limma = 1 step
Apotome = 1 step
Degree Cents Ratios Ups and Downs Notation
0 0.000 P1
1 99.807 16/15, 15/14, 14/13, 13/12 m2
2 199.614 12/11, 11/10, 10/9, 9/8, 8/7, 15/13 M2
3 299.422 7/6, 13/11, 6/5, 11/9 m3
4 399.229 16/13, 5/4, 14/11, 9/7 M3
5 499.036 13/10, 4/3, 15/11 P4
6 598.843 11/8, 7/5, 10/7, 13/9, 16/11 A4, d5
7 698.650 3/2 P5
8 798.457 14/9, 11/7, 8/5, 13/8 m6
9 898.265 5/3, 12/7 M6
10 998.072 7/4, 16/9, 9/5 m7
11 1097.879 11/6, 13/7, 15/8 M7
12 1197.686 2/1 P1 +1 oct
13 1297.493 15/7, 13/6 m2 +1 oct
14 1397.301 11/5, 9/4, 16/7 M2 +1 oct
15 1497.108 7/3, 12/5 m3 +1 oct
16 1596.915 5/2 M3 +1 oct
17 1696.722 13/5, 8/3 P4 +1 oct
18 1796.529 11/4, 14/5 A4 +1 oct, d5 +1 oct
19 1896.336 3/1 P5 +1 oct
20 1996.144 16/5, 13/4 m6 +1 oct
21 2095.951 10/3 M6 +1 oct
22 2195.758 7/2 m7 +1 oct
23 2295.565 11/3, 15/4 M7 +1 oct
24 2395.372 4/1 P1 +2 oct
25 2495.180 13/3 m2 +2 oct
26 2594.987 9/2 M2 +2 oct
27 2694.794 14/3 m3 +2 oct
28 2794.601 5/1 M3 +2 oct
29 2894.408 16/3 P4 +2 oct
30 2994.215 11/2 A4 +2 oct, d5 +2 oct
31 3094.023 6/1 P5 +2 oct
32 3193.830 13/2 m6 +2 oct
33 3293.637 M6 +2 oct
34 3393.444 7/1 m7 +2 oct
35 3493.251 15/2 M7 +2 oct
36 3593.059 8/1 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 10/1 M3 +3 oct
41 4092.094 P4 +3 oct
42 4191.902 11/1 A4 +3 oct, d5 +3 oct
43 4291.709 12/1 P5 +3 oct
44 4391.516 13/1 m6 +3 oct
45 4491.323 M6 +3 oct
46 4591.130 14/1 m7 +3 oct
47 4690.937 15/1 M7 +3 oct
48 4790.745 16/1 P1 +4 oct

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.

16-integer-limit intervals in 34zpi (by direct approximation)
Ratio Error (abs, ¢) Error (rel, %)
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
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
13/7 +26.177 +26.228
7/2 +26.932 +26.984
14/3 +27.923 +27.977
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
13/5 +42.508 +42.590
11/2 +42.897 +42.980
13/3 -43.393 -43.477
13/4 -44.384 -44.470
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
11/9 -47.986 -48.079
15/13 -48.127 -48.220
11/6 +48.516 +48.610
12/11 +48.977 +49.072
13/1 -49.012 -49.106
16/11 -49.839 -49.935
16-integer-limit intervals in 34zpi (by patent val mapping)
Ratio Error (abs, ¢) Error (rel, %)
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
12/11 -50.830 -50.928
15/13 +51.680 +51.780
11/9 +51.821 +51.921
13/10 -54.985 -55.091
13/5 -57.299 -57.410
14/13 +71.316 +71.454
13/7 -73.630 -73.772
13/11 -89.595 -89.768

See also


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