71zpi

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71 zeta peak index (abbreviated 71zpi), is the equal-step tuning system obtained from the 71st 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
71zpi 20.2248393119540 59.3329806724710 3.531097 0.613581 12.986080 20edo 1186.65961344942 6 6
The Riemann zeta function around 71zpi

Theory

71zpi marks the most prominent zeta peak index in the vicinity of 20edo. While 70zpi is the nearest peak to 20edo and closely competes with 71zpi in terms of strength, 71zpi remains superior across all measures of strength. 71zpi may also be viewed as a tritave compression of 32edt, a no-2s zeta peak EDT (consistent in the no-2s 19-integer-limit), but with less extreme stretch than the no-2s peak at 59.271105 cents.

71zpi features a good 3:5:9:11:14:15:16:19:25:26:33 chord, which differs a lot from the harmonic characteristics of 20edo.

The nearest zeta peaks to 71zpi that surpass its strength are 65zpi and 75zpi.

71zpi is distinguished by its extensive EDO-deviation and substantial zeta strength, qualifying it as a strong candidate for no-octave tuning systems. It is noteworthy that only 19zpi exhibits both a greater octave error and stronger zeta height and integral than 71zpi, although 71zpi still has a more pronounced zeta gap. Other notable zeta peak indices in this category include 61zpi, 84zpi, 110zpi, 137zpi, 151zpi, 222zpi, and 273zpi, each demonstrating characteristics that make them suitable for similar applications.

Harmonic series

Approximation of harmonics in 71zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Error Absolute (¢) -13.3 -3.3 -26.7 +2.3 -16.6 +13.2 +19.3 -6.6 -11.0 +2.0 +29.4 +9.4 -0.2 -1.0 +6.0
Relative (%) -22.5 -5.6 -45.0 +3.9 -28.0 +22.2 +32.5 -11.1 -18.5 +3.4 +49.5 +15.9 -0.3 -1.6 +10.1
Steps 20 32 40 47 52 57 61 64 67 70 73 75 77 79 81
Approximation of harmonics in 71zpi
Harmonic 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
Error Absolute (¢) +19.7 -19.9 +5.1 -24.3 +9.9 -11.3 -29.0 +16.0 +4.7 -3.9 -9.9 -13.5 -14.9 -14.3 -11.7 -7.4 -1.3
Relative (%) +33.2 -33.6 +8.6 -41.0 +16.6 -19.1 -48.8 +27.0 +7.9 -6.6 -16.7 -22.8 -25.2 -24.1 -19.8 -12.4 -2.2
Steps 83 84 86 87 89 90 91 93 94 95 96 97 98 99 100 101 102

Intervals

There are multiple ways to approach notation. The simplest method is to use the notations from 20edo. However, this approach will not preserve octave compression when the audio is rendered by notation software. If maintaining accurate step compression in notation software is important, consider using the ups and downs notation from 182edo at every 9-degree step. With this method, the tonal difference will be less than 1 cent up to the 86th harmonic.

Intervals in 71zpi
JI ratios are comprised of 33-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 %
⟨182 288] at every 9 steps

Whole tone = 30 steps
Limma = 16 steps
Apotome = 14 steps
Degree Cents Ratios Ups and Downs Notation Step
0 0.000 P1 0
1 59.333 33/32, 32/31, 31/30, 30/29, 29/28, 28/27, 27/26, 26/25, 25/24, 24/23, 23/22, 22/21, 21/20, 20/19 v7m2 9
2 118.666 19/18, 18/17, 17/16, 33/31, 16/15, 31/29, 15/14, 29/27, 14/13, 27/25, 13/12, 25/23 ^^m2 18
3 177.999 12/11, 23/21, 11/10, 32/29, 21/19, 31/28, 10/9, 29/26, 19/17, 28/25, 9/8 vvvM2 27
4 237.332 26/23, 17/15, 25/22, 33/29, 8/7, 31/27, 23/20, 15/13, 22/19, 29/25, 7/6 ^6M2 36
5 296.665 27/23, 20/17, 33/28, 13/11, 32/27, 19/16, 25/21, 31/26, 6/5 vm3 45
6 355.998 29/24, 23/19, 17/14, 28/23, 11/9, 27/22, 16/13, 21/17, 26/21, 31/25 v6M3 54
7 415.331 5/4, 29/23, 24/19, 19/15, 33/26, 14/11, 23/18, 32/25, 9/7, 31/24 ^^^M3 63
8 474.664 22/17, 13/10, 30/23, 17/13, 21/16, 25/19, 29/22, 33/25, 4/3 v44 72
9 533.997 31/23, 27/20, 23/17, 19/14, 15/11, 26/19, 11/8, 29/21, 18/13 ^54 81
10 593.330 25/18, 32/23, 7/5, 31/22, 24/17, 17/12, 27/19, 10/7 A4 90
11 652.663 33/23, 23/16, 13/9, 29/20, 16/11, 19/13, 22/15, 25/17, 28/19, 31/21 ~5 99
12 711.996 3/2, 32/21, 29/19, 26/17, 23/15 ^^5 108
13 771.329 20/13, 17/11, 31/20, 14/9, 25/16, 11/7, 30/19, 19/12, 27/17 v5m6 117
14 830.662 8/5, 29/18, 21/13, 13/8, 31/19, 18/11, 23/14 ^4m6 126
15 889.995 28/17, 33/20, 5/3, 32/19, 27/16, 22/13, 17/10 vM6 135
16 949.328 29/17, 12/7, 31/18, 19/11, 26/15, 33/19, 7/4 v6A6, ^6d7 144
17 1008.661 30/17, 23/13, 16/9, 25/14, 9/5, 29/16, 20/11 ^m7 153
18 1067.994 31/17, 11/6, 24/13, 13/7, 28/15, 15/8, 32/17 v4M7 162
19 1127.327 17/9, 19/10, 21/11, 23/12, 25/13, 27/14, 29/15, 31/16, 33/17 ^5M7 171
20 1186.660 2/1 vv1 +1 oct 180
21 1245.993 33/16, 31/15, 29/14, 27/13, 25/12 ^71 +1 oct 189
22 1305.326 23/11, 21/10, 19/9, 17/8, 32/15, 15/7, 28/13 m2 +1 oct 198
23 1364.659 13/6, 24/11, 11/5, 31/14, 20/9, 29/13 v5M2 +1 oct 207
24 1423.992 9/4, 25/11, 16/7, 23/10, 30/13 ^4M2 +1 oct 216
25 1483.325 7/3, 33/14, 26/11, 19/8, 31/13 vvvm3 +1 oct 225
26 1542.657 12/5, 29/12, 17/7, 22/9, 27/11, 32/13 ^6m3 +1 oct 234
27 1601.990 5/2, 33/13, 28/11, 23/9 ^M3 +1 oct 243
28 1661.323 18/7, 31/12, 13/5, 21/8, 29/11 v64 +1 oct 252
29 1720.656 8/3, 27/10, 19/7, 30/11 ^^^4 +1 oct 261
30 1779.989 11/4, 25/9, 14/5, 31/11, 17/6 vvA4 +1 oct 270
31 1839.322 20/7, 23/8, 26/9, 29/10, 32/11 ^5d5 +1 oct 279
32 1898.655 3/1 P5 +1 oct 288
33 1957.988 31/10, 28/9, 25/8, 22/7 v7m6 +1 oct 297
34 2017.321 19/6, 16/5, 29/9, 13/4 ^^m6 +1 oct 306
35 2076.654 23/7, 33/10, 10/3, 27/8 vvvM6 +1 oct 315
36 2135.987 17/5, 24/7, 31/9 ^6M6 +1 oct 324
37 2195.320 7/2, 32/9, 25/7, 18/5 vm7 +1 oct 333
38 2254.653 29/8, 11/3, 26/7 v6M7 +1 oct 342
39 2313.986 15/4, 19/5, 23/6, 27/7 ^^^M7 +1 oct 351
40 2373.319 31/8, 4/1 v41 +2 oct 360
41 2432.652 33/8, 29/7 ^51 +2 oct 369
42 2491.985 25/6, 21/5, 17/4, 30/7 vvm2 +2 oct 378
43 2551.318 13/3, 22/5, 31/7 ~2 +2 oct 387
44 2610.651 9/2, 32/7 ^^M2 +2 oct 396
45 2669.984 23/5, 14/3, 33/7, 19/4 v5m3 +2 oct 405
46 2729.317 24/5, 29/6 ^4m3 +2 oct 414
47 2788.650 5/1 vM3 +2 oct 423
48 2847.983 31/6, 26/5, 21/4 v6A3 +2 oct, ^6d4 +2 oct 432
49 2907.316 16/3, 27/5 ^4 +2 oct 441
50 2966.649 11/2, 28/5 v4A4 +2 oct 450
51 3025.982 17/3, 23/4, 29/5 ^^^d5 +2 oct 459
52 3085.315 6/1 vv5 +2 oct 468
53 3144.648 31/5, 25/4 ^75 +2 oct 477
54 3203.981 19/3, 32/5 m6 +2 oct 486
55 3263.314 13/2, 33/5, 20/3 v5M6 +2 oct 495
56 3322.647 27/4 ^4M6 +2 oct 504
57 3381.980 7/1 vvvm7 +2 oct 513
58 3441.313 29/4, 22/3 ^6m7 +2 oct 522
59 3500.646 15/2, 23/3 ^M7 +2 oct 531
60 3559.979 31/4 v61 +3 oct 540
61 3619.312 8/1 ^^^1 +3 oct 549
62 3678.645 33/4, 25/3, 17/2 v4m2 +3 oct 558
63 3737.978 26/3 ^5m2 +3 oct 567
64 3797.311 9/1 M2 +3 oct 576
65 3856.644 28/3 v7m3 +3 oct 585
66 3915.977 19/2, 29/3 ^^m3 +3 oct 594
67 3975.310 10/1 vvvM3 +3 oct 603
68 4034.643 31/3 ^6M3 +3 oct 612
69 4093.976 21/2, 32/3 v4 +3 oct 621
70 4153.309 11/1 v6A4 +3 oct 630
71 4212.642 23/2 ^d5 +3 oct 639
72 4271.975 v45 +3 oct 648
73 4331.308 12/1 ^55 +3 oct 657
74 4390.641 25/2 vvm6 +3 oct 666
75 4449.974 13/1 ~6 +3 oct 675
76 4509.307 27/2 ^^M6 +3 oct 684
77 4568.640 14/1 v5m7 +3 oct 693
78 4627.972 29/2 ^4m7 +3 oct 702
79 4687.305 15/1 vM7 +3 oct 711
80 4746.638 31/2 v6A7 +3 oct, ^6d1 +4 oct 720
81 4805.971 16/1 ^1 +4 oct 729
82 4865.304 33/2 v6m2 +4 oct 738
83 4924.637 17/1 ^^^m2 +4 oct 747
84 4983.970 18/1 vvM2 +4 oct 756
85 5043.303 ^7M2 +4 oct 765
86 5102.636 19/1 m3 +4 oct 774
87 5161.969 20/1 v5M3 +4 oct 783
88 5221.302 ^4M3 +4 oct 792
89 5280.635 21/1 vvv4 +4 oct 801
90 5339.968 22/1 ^64 +4 oct 810
91 5399.301 23/1 ^A4 +4 oct, vd5 +4 oct 819
92 5458.634 v65 +4 oct 828
93 5517.967 24/1 ^^^5 +4 oct 837
94 5577.300 25/1 v4m6 +4 oct 846
95 5636.633 26/1 ^5m6 +4 oct 855
96 5695.966 27/1 M6 +4 oct 864
97 5755.299 28/1 v7m7 +4 oct 873
98 5814.632 29/1 ^^m7 +4 oct 882
99 5873.965 30/1 vvvM7 +4 oct 891
100 5933.298 31/1 ^6M7 +4 oct 900
101 5992.631 32/1 v1 +5 oct 909
102 6051.964 33/1 v6A1 +5 oct, ^6d2 +5 oct 918

Approximation to JI

Interval mappings

The following tables show how 33-integer-limit intervals are represented in 71zpi. Prime harmonics are in bold; inconsistent intervals are in italics.

33-integer-limit intervals in 71zpi (by direct approximation)
Ratio Error (abs, ¢) Error (rel, %)
14/1 -0.186 -0.314
11/5 -0.346 -0.583
17/8 +0.370 +0.624
31/22 -0.388 -0.654
21/13 +0.408 +0.688
25/19 -0.451 -0.759
26/3 -0.595 -1.003
30/29 +0.641 +1.081
31/10 -0.733 -1.236
32/9 -0.770 -1.297
15/14 -0.777 -1.309
19/16 -0.848 -1.429
15/1 -0.963 -1.623
23/12 +1.007 +1.698
27/10 +1.105 +1.863
33/14 -1.123 -1.892
25/16 -1.299 -2.189
33/1 -1.309 -2.206
29/28 -1.418 -2.390
27/22 +1.451 +2.445
31/2 +1.603 +2.702
29/2 -1.605 -2.705
29/6 +1.695 +2.857
31/28 +1.789 +3.016
31/27 -1.839 -3.099
11/1 +1.991 +3.355
14/11 -2.177 -3.669
23/4 -2.292 -3.864
5/1 +2.336 +3.938
14/5 -2.523 -4.252
32/27 +2.530 +4.264
31/30 +2.566 +4.325
33/26 +2.586 +4.358
25/11 +2.682 +4.520
26/9 +2.705 +4.559
19/5 +2.787 +4.697
24/7 +2.858 +4.817
26/15 -2.931 -4.940
15/11 -2.954 -4.979
14/3 +3.113 +5.247
19/11 +3.133 +5.280
31/29 +3.208 +5.406
3/1 -3.300 -5.561
27/2 +3.442 +5.800
16/13 -3.474 -5.856
29/22 -3.595 -6.060
28/27 -3.628 -6.115
16/5 +3.635 +6.127
33/5 -3.645 -6.144
24/17 -3.670 -6.185
13/7 -3.708 -6.250
21/16 +3.883 +6.544
26/1 -3.894 -6.564
29/10 -3.941 -6.642
16/11 +3.981 +6.709
32/3 -4.069 -6.858
19/13 -4.323 -7.285
32/31 +4.369 +7.363
10/9 -4.405 -7.424
23/20 -4.629 -7.801
25/1 +4.673 +7.875
21/19 +4.731 +7.974
22/9 -4.750 -8.006
25/13 -4.773 -8.045
25/14 +4.859 +8.190
31/6 +4.903 +8.263
29/18 +4.995 +8.418
29/27 -5.046 -8.505
19/1 +5.123 +8.635
31/9 -5.138 -8.660
25/21 -5.182 -8.733
11/3 +5.290 +8.916
19/14 +5.310 +8.949
5/3 +5.636 +9.499
26/11 -5.885 -9.919
16/1 +5.971 +10.064
33/25 -5.982 -10.082
27/26 -6.004 -10.120
33/32 +6.060 +10.214
19/15 +6.087 +10.258
8/7 +6.158 +10.378
26/5 -6.231 -10.502
32/15 -6.406 -10.796
14/9 +6.413 +10.808
33/19 -6.432 -10.841
17/7 +6.528 +11.002
24/13 +6.566 +11.067
9/1 -6.599 -11.122
9/2 +6.741 +11.362
28/9 -6.928 -11.676
16/15 +6.935 +11.688
13/5 +7.110 +11.982
16/7 -7.183 -12.106
33/16 -7.280 -12.270
32/1 -7.369 -12.420
13/11 +7.455 +12.565
21/5 +7.518 +12.671
32/29 +7.576 +12.769
10/3 -7.704 -12.985
31/26 -7.843 -13.219
21/11 +7.864 +13.253
25/3 +7.972 +13.437
19/7 -8.031 -13.535
22/3 -8.050 -13.568
31/18 +8.202 +13.824
29/9 -8.346 -14.066
19/3 +8.423 +14.196
31/3 -8.438 -14.221
25/7 -8.481 -14.294
26/25 -8.567 -14.439
11/9 +8.590 +14.478
9/5 -8.936 -15.060
26/19 -9.018 -15.199
23/18 -9.033 -15.225
16/3 +9.271 +15.625
32/11 -9.360 -15.775
29/20 +9.399 +15.842
13/1 +9.446 +15.920
21/8 -9.457 -15.940
14/13 -9.632 -16.234
17/12 -9.671 -16.299
33/10 +9.695 +16.340
32/5 -9.705 -16.357
27/14 -9.712 -16.369
21/17 -9.828 -16.563
21/1 +9.854 +16.609
13/8 -9.866 -16.628
27/1 -9.899 -16.684
3/2 +10.041 +16.923
28/3 -10.227 -17.237
17/13 +10.236 +17.252
22/15 -10.386 -17.505
15/13 -10.409 -17.544
33/31 +10.429 +17.576
23/17 +10.678 +17.997
33/13 -10.755 -18.126
31/15 -10.774 -18.159
7/5 +10.818 +18.232
24/19 +10.889 +18.352
10/1 -11.004 -18.546
23/8 +11.048 +18.620
29/26 -11.051 -18.625
11/7 -11.163 -18.815
25/9 +11.272 +18.998
25/24 -11.339 -19.112
22/1 -11.350 -19.129
31/14 -11.551 -19.468
29/3 -11.645 -19.627
19/9 +11.723 +19.757
29/4 +11.736 +19.779
31/1 -11.738 -19.782
27/11 -11.890 -20.039
33/2 +12.031 +20.278
32/25 -12.042 -20.295
33/28 +12.218 +20.592
27/5 -12.235 -20.621
23/6 -12.333 -20.786
15/2 +12.377 +20.860
32/19 -12.492 -21.055
28/15 -12.564 -21.175
16/9 +12.571 +21.187
31/20 +12.607 +21.248
13/3 +12.746 +21.481
17/4 -12.970 -21.860
11/10 +12.995 +21.901
7/1 +13.154 +22.170
2/1 -13.340 -22.484
28/1 -13.527 -22.798
33/29 +13.636 +22.982
24/5 +13.676 +23.049
22/5 -13.686 -23.067
17/16 +13.711 +23.108
31/11 -13.728 -23.138
26/21 -13.749 -23.172
29/15 -13.982 -23.565
24/11 +14.021 +23.632
29/23 +14.028 +23.643
31/5 -14.074 -23.720
15/7 -14.117 -23.793
19/8 -14.188 -23.913
30/1 -14.304 -24.107
24/23 -14.348 -24.182
27/20 +14.446 +24.347
33/7 -14.463 -24.376
19/17 -14.559 -24.537
27/25 -14.572 -24.559
25/8 -14.639 -24.673
30/23 +14.669 +24.724
29/14 -14.759 -24.874
31/4 +14.943 +25.185
29/1 -14.945 -25.189
25/17 -15.009 -25.297
27/19 -15.022 -25.318
29/12 +15.035 +25.341
20/17 +15.307 +25.798
11/2 +15.331 +25.839
28/23 +15.446 +26.033
28/11 -15.517 -26.153
23/2 -15.633 -26.347
5/2 +15.677 +26.422
28/5 -15.863 -26.736
27/16 -15.870 -26.748
24/1 +16.012 +26.987
25/22 +16.022 +27.004
13/9 +16.045 +27.043
19/10 +16.127 +27.181
12/7 +16.199 +27.301
30/11 -16.294 -27.463
31/25 -16.410 -27.658
7/3 +16.454 +27.731
22/19 -16.473 -27.764
6/1 -16.640 -28.045
27/4 +16.782 +28.284
32/13 -16.815 -28.340
31/19 -16.861 -28.417
29/11 -16.936 -28.544
8/5 +16.975 +28.610
26/7 -17.048 -28.734
23/7 +17.206 +28.999
32/21 -17.223 -29.028
31/23 +17.236 +29.049
29/5 -17.281 -29.126
11/8 -17.321 -29.193
17/5 +17.346 +29.234
23/22 -17.623 -29.703
17/11 +17.691 +29.817
31/16 -17.709 -29.847
20/9 -17.745 -29.908
23/10 -17.969 -30.285
25/2 +18.013 +30.359
28/25 -18.200 -30.674
31/12 +18.243 +30.747
19/2 +18.464 +31.119
11/6 +18.631 +31.400
28/19 -18.650 -31.433
6/5 -18.976 -31.983
27/23 +19.074 +32.148
8/1 +19.312 +32.548
27/13 -19.345 -32.604
30/19 -19.427 -32.742
7/4 -19.498 -32.862
29/25 -19.618 -33.064
17/1 +19.682 +33.172
18/17 +19.711 +33.222
9/7 -19.753 -33.292
17/14 +19.868 +33.486
13/12 -19.907 -33.551
18/1 -19.940 -33.606
29/19 -20.068 -33.823
9/4 +20.082 +33.845
15/8 -20.275 -34.172
13/10 +20.450 +34.466
23/21 +20.506 +34.560
32/7 -20.523 -34.589
33/8 -20.621 -34.754
17/15 +20.645 +34.796
22/13 -20.796 -35.049
21/10 +20.858 +35.155
23/13 +20.914 +35.249
29/16 -20.917 -35.253
33/17 -20.991 -35.378
20/3 -21.045 -35.469
31/13 -21.183 -35.703
22/21 -21.204 -35.737
25/6 +21.313 +35.921
31/21 -21.592 -36.391
32/23 +21.604 +36.412
19/6 +21.763 +36.680
20/7 +21.835 +36.800
18/11 -21.930 -36.961
18/5 -22.276 -37.544
23/9 -22.374 -37.709
8/3 +22.611 +38.109
13/2 +22.786 +38.404
21/4 -22.798 -38.424
28/13 -22.973 -38.718
17/3 +22.982 +38.733
17/6 -23.011 -38.783
33/20 +23.035 +38.824
27/7 -23.053 -38.853
21/2 +23.195 +39.093
13/4 -23.206 -39.112
4/3 -23.381 -39.407
26/17 -23.576 -39.736
30/13 -23.750 -40.028
10/7 -24.158 -40.716
19/12 -24.229 -40.836
20/1 -24.344 -41.030
23/16 +24.388 +41.104
29/13 -24.391 -41.109
22/7 -24.504 -41.299
25/18 +24.612 +41.482
25/12 -24.680 -41.595
29/17 +24.706 +41.639
29/21 -24.799 -41.797
31/7 -24.891 -41.952
19/18 +25.063 +42.241
29/8 +25.076 +42.263
26/23 +25.079 +42.268
21/20 -25.134 -42.361
23/19 +25.237 +42.534
30/17 +25.347 +42.721
33/4 +25.372 +42.762
20/13 +25.543 +43.050
23/3 -25.673 -43.270
25/23 -25.687 -43.293
15/4 +25.718 +43.344
9/8 -25.911 -43.671
13/6 +26.086 +43.965
28/17 +26.124 +44.030
18/7 +26.239 +44.224
17/9 +26.281 +44.294
17/2 -26.311 -44.344
20/11 -26.335 -44.385
7/2 +26.494 +44.654
4/1 -26.681 -44.968
12/5 +27.016 +45.533
32/17 -27.051 -45.592
12/11 +27.362 +46.116
30/7 -27.458 -46.277
19/4 -27.529 -46.397
33/23 +27.664 +46.625
31/24 -27.750 -46.769
31/17 +27.913 +47.045
25/4 -27.979 -47.157
23/15 -28.010 -47.208
23/5 +28.023 +47.231
29/7 -28.099 -47.358
31/8 +28.284 +47.669
22/17 +28.301 +47.699
23/11 +28.369 +47.813
29/24 +28.376 +47.824
17/10 -28.647 -48.282
11/4 +28.671 +48.323
23/14 -28.787 -48.517
23/1 -28.973 -48.831
5/4 +29.017 +48.906
27/8 -29.211 -49.232
12/1 +29.353 +49.471
18/13 -29.386 -49.526
20/19 -29.468 -49.665
7/6 -29.539 -49.785
27/17 -29.581 -49.856
33-integer-limit intervals in 71zpi (by patent val mapping)
Ratio Error (abs, ¢) Error (rel, %)
14/1 -0.186 -0.314
11/5 -0.346 -0.583
31/22 -0.388 -0.654
21/13 +0.408 +0.688
25/19 -0.451 -0.759
26/3 -0.595 -1.003
30/29 +0.641 +1.081
31/10 -0.733 -1.236
15/14 -0.777 -1.309
15/1 -0.963 -1.623
23/12 +1.007 +1.698
27/10 +1.105 +1.863
33/14 -1.123 -1.892
33/1 -1.309 -2.206
29/28 -1.418 -2.390
27/22 +1.451 +2.445
31/2 +1.603 +2.702
29/2 -1.605 -2.705
29/6 +1.695 +2.857
31/28 +1.789 +3.016
31/27 -1.839 -3.099
11/1 +1.991 +3.355
14/11 -2.177 -3.669
23/4 -2.292 -3.864
5/1 +2.336 +3.938
14/5 -2.523 -4.252
31/30 +2.566 +4.325
33/26 +2.586 +4.358
25/11 +2.682 +4.520
26/9 +2.705 +4.559
19/5 +2.787 +4.697
26/15 -2.931 -4.940
15/11 -2.954 -4.979
14/3 +3.113 +5.247
19/11 +3.133 +5.280
31/29 +3.208 +5.406
3/1 -3.300 -5.561
27/2 +3.442 +5.800
29/22 -3.595 -6.060
28/27 -3.628 -6.115
33/5 -3.645 -6.144
13/7 -3.708 -6.250
26/1 -3.894 -6.564
29/10 -3.941 -6.642
19/13 -4.323 -7.285
10/9 -4.405 -7.424
23/20 -4.629 -7.801
25/1 +4.673 +7.875
21/19 +4.731 +7.974
22/9 -4.750 -8.006
25/13 -4.773 -8.045
25/14 +4.859 +8.190
31/6 +4.903 +8.263
29/18 +4.995 +8.418
29/27 -5.046 -8.505
19/1 +5.123 +8.635
31/9 -5.138 -8.660
25/21 -5.182 -8.733
11/3 +5.290 +8.916
19/14 +5.310 +8.949
5/3 +5.636 +9.499
26/11 -5.885 -9.919
33/25 -5.982 -10.082
27/26 -6.004 -10.120
19/15 +6.087 +10.258
26/5 -6.231 -10.502
14/9 +6.413 +10.808
33/19 -6.432 -10.841
17/7 +6.528 +11.002
9/1 -6.599 -11.122
9/2 +6.741 +11.362
28/9 -6.928 -11.676
13/5 +7.110 +11.982
13/11 +7.455 +12.565
21/5 +7.518 +12.671
10/3 -7.704 -12.985
31/26 -7.843 -13.219
21/11 +7.864 +13.253
25/3 +7.972 +13.437
19/7 -8.031 -13.535
22/3 -8.050 -13.568
31/18 +8.202 +13.824
29/9 -8.346 -14.066
19/3 +8.423 +14.196
31/3 -8.438 -14.221
25/7 -8.481 -14.294
26/25 -8.567 -14.439
11/9 +8.590 +14.478
9/5 -8.936 -15.060
26/19 -9.018 -15.199
23/18 -9.033 -15.225
29/20 +9.399 +15.842
13/1 +9.446 +15.920
14/13 -9.632 -16.234
33/10 +9.695 +16.340
27/14 -9.712 -16.369
21/17 -9.828 -16.563
21/1 +9.854 +16.609
27/1 -9.899 -16.684
3/2 +10.041 +16.923
28/3 -10.227 -17.237
17/13 +10.236 +17.252
22/15 -10.386 -17.505
15/13 -10.409 -17.544
33/31 +10.429 +17.576
33/13 -10.755 -18.126
31/15 -10.774 -18.159
7/5 +10.818 +18.232
10/1 -11.004 -18.546
23/8 +11.048 +18.620
29/26 -11.051 -18.625
11/7 -11.163 -18.815
25/9 +11.272 +18.998
22/1 -11.350 -19.129
31/14 -11.551 -19.468
29/3 -11.645 -19.627
19/9 +11.723 +19.757
29/4 +11.736 +19.779
31/1 -11.738 -19.782
27/11 -11.890 -20.039
33/2 +12.031 +20.278
33/28 +12.218 +20.592
27/5 -12.235 -20.621
23/6 -12.333 -20.786
15/2 +12.377 +20.860
28/15 -12.564 -21.175
31/20 +12.607 +21.248
13/3 +12.746 +21.481
11/10 +12.995 +21.901
7/1 +13.154 +22.170
2/1 -13.340 -22.484
28/1 -13.527 -22.798
33/29 +13.636 +22.982
22/5 -13.686 -23.067
31/11 -13.728 -23.138
26/21 -13.749 -23.172
29/15 -13.982 -23.565
29/23 +14.028 +23.643
31/5 -14.074 -23.720
15/7 -14.117 -23.793
30/1 -14.304 -24.107
24/23 -14.348 -24.182
27/20 +14.446 +24.347
33/7 -14.463 -24.376
19/17 -14.559 -24.537
27/25 -14.572 -24.559
30/23 +14.669 +24.724
29/14 -14.759 -24.874
31/4 +14.943 +25.185
29/1 -14.945 -25.189
25/17 -15.009 -25.297
27/19 -15.022 -25.318
29/12 +15.035 +25.341
11/2 +15.331 +25.839
28/23 +15.446 +26.033
28/11 -15.517 -26.153
23/2 -15.633 -26.347
5/2 +15.677 +26.422
28/5 -15.863 -26.736
25/22 +16.022 +27.004
13/9 +16.045 +27.043
19/10 +16.127 +27.181
30/11 -16.294 -27.463
31/25 -16.410 -27.658
7/3 +16.454 +27.731
22/19 -16.473 -27.764
6/1 -16.640 -28.045
27/4 +16.782 +28.284
31/19 -16.861 -28.417
29/11 -16.936 -28.544
26/7 -17.048 -28.734
31/23 +17.236 +29.049
29/5 -17.281 -29.126
17/5 +17.346 +29.234
23/22 -17.623 -29.703
17/11 +17.691 +29.817
20/9 -17.745 -29.908
23/10 -17.969 -30.285
25/2 +18.013 +30.359
28/25 -18.200 -30.674
31/12 +18.243 +30.747
19/2 +18.464 +31.119
11/6 +18.631 +31.400
28/19 -18.650 -31.433
6/5 -18.976 -31.983
27/23 +19.074 +32.148
27/13 -19.345 -32.604
30/19 -19.427 -32.742
29/25 -19.618 -33.064
17/1 +19.682 +33.172
9/7 -19.753 -33.292
17/14 +19.868 +33.486
18/1 -19.940 -33.606
29/19 -20.068 -33.823
9/4 +20.082 +33.845
13/10 +20.450 +34.466
17/15 +20.645 +34.796
22/13 -20.796 -35.049
21/10 +20.858 +35.155
33/17 -20.991 -35.378
20/3 -21.045 -35.469
31/13 -21.183 -35.703
22/21 -21.204 -35.737
25/6 +21.313 +35.921
31/21 -21.592 -36.391
19/6 +21.763 +36.680
18/11 -21.930 -36.961
18/5 -22.276 -37.544
23/9 -22.374 -37.709
13/2 +22.786 +38.404
28/13 -22.973 -38.718
17/3 +22.982 +38.733
33/20 +23.035 +38.824
27/7 -23.053 -38.853
21/2 +23.195 +39.093
4/3 -23.381 -39.407
26/17 -23.576 -39.736
30/13 -23.750 -40.028
10/7 -24.158 -40.716
20/1 -24.344 -41.030
23/16 +24.388 +41.104
29/13 -24.391 -41.109
22/7 -24.504 -41.299
25/18 +24.612 +41.482
29/21 -24.799 -41.797
31/7 -24.891 -41.952
19/18 +25.063 +42.241
29/8 +25.076 +42.263
26/23 +25.079 +42.268
33/4 +25.372 +42.762
23/3 -25.673 -43.270
15/4 +25.718 +43.344
13/6 +26.086 +43.965
17/9 +26.281 +44.294
20/11 -26.335 -44.385
7/2 +26.494 +44.654
4/1 -26.681 -44.968
30/7 -27.458 -46.277
33/23 +27.664 +46.625
23/15 -28.010 -47.208
29/7 -28.099 -47.358
31/8 +28.284 +47.669
29/24 +28.376 +47.824
11/4 +28.671 +48.323
23/14 -28.787 -48.517
23/1 -28.973 -48.831
5/4 +29.017 +48.906
18/13 -29.386 -49.526
20/19 -29.468 -49.665
27/17 -29.581 -49.856
7/6 +29.794 +50.215
12/1 -29.980 -50.529
27/8 +30.122 +50.768
17/10 +30.686 +51.718
23/11 -30.964 -52.187
22/17 -31.032 -52.301
23/5 -31.309 -52.769
25/4 +31.354 +52.843
31/17 -31.419 -52.955
31/24 +31.583 +53.231
19/4 +31.804 +53.603
12/11 -31.971 -53.884
12/5 -32.317 -54.467
17/2 +33.022 +55.656
18/7 -33.094 -55.776
28/17 -33.209 -55.970
9/8 +33.422 +56.329
25/23 +33.646 +56.707
20/13 -33.790 -56.950
30/17 -33.986 -57.279
23/19 -34.096 -57.466
21/20 +34.199 +57.639
29/17 -34.627 -58.361
25/12 +34.653 +58.405
19/12 +35.104 +59.164
13/4 +36.127 +60.888
17/6 +36.322 +61.217
21/4 +36.535 +61.576
8/3 -36.722 -61.891
20/7 -37.498 -63.200
32/23 -37.729 -63.588
29/16 +38.416 +64.747
23/13 -38.419 -64.751
33/8 +38.712 +65.246
23/21 -38.827 -65.440
15/8 +39.058 +65.828
13/12 +39.426 +66.449
18/17 -39.622 -66.778
7/4 +39.835 +67.138
8/1 -40.021 -67.452
31/16 +41.624 +70.153
11/8 +42.012 +70.807
23/7 -42.127 -71.001
8/5 -42.358 -71.390
12/7 -43.134 -72.699
24/1 -43.321 -73.013
27/16 +43.463 +73.252
20/17 -44.026 -74.202
25/8 +44.694 +75.327
19/8 +45.144 +76.087
24/11 -45.311 -76.368
24/5 -45.657 -76.951
17/4 +46.363 +78.140
16/9 -46.762 -78.813
25/24 +47.994 +80.888
24/19 -48.444 -81.648
23/17 -48.655 -82.003
13/8 +49.467 +83.372
17/12 +49.662 +83.701
21/8 +49.876 +84.060
16/3 -50.062 -84.375
32/29 -51.757 -87.231
33/16 +52.053 +87.730
16/15 -52.398 -88.312
24/13 -52.767 -88.933
8/7 -53.175 -89.622
16/1 -53.362 -89.936
32/31 -54.964 -92.637
16/11 -55.352 -93.291
16/5 -55.698 -93.873
24/7 -56.475 -95.183
32/27 -56.803 -95.736
25/16 +58.034 +97.811
19/16 +58.485 +98.571
17/8 +59.703 +100.624
32/9 -60.103 -101.297
16/13 -62.807 -105.856
24/17 -63.003 -106.185
21/16 +63.216 +106.544
32/3 -63.402 -106.858
33/32 +65.393 +110.214
32/15 -65.739 -110.796
16/7 -66.516 -112.106
32/1 -66.702 -112.420
32/11 -68.693 -115.775
32/5 -69.038 -116.357
32/25 -71.375 -120.295
32/19 -71.825 -121.055
17/16 +73.044 +123.108
32/13 -76.148 -128.340
32/21 -76.556 -129.028
32/7 -79.856 -134.589
32/17 -86.384 -145.592

Record on the Riemann zeta function with prime 2 removed

71zpi sets a height record on the Riemann zeta function with prime 2 removed. The previous record is 53zpi and the next one is 93zpi. It is important to highlight that the optimal equal tunings obtained by excluding the prime number 2 from the Riemann zeta function differs slightly from the optimal equal tuning corresponding to the same peaks on the unmodified Riemann zeta function.

Unmodified Riemann zeta function Riemann zeta function with prime 2 removed
Tuning Strength Closest EDO Tuning Strength Closest EDO
ZPI Steps per octave Step size (cents) Height EDO Octave (cents) Steps per octave Step size (cents) Height EDO Octave (cents)
53zpi 16.3979501311478 73.1798786069366 2.518818 16edo 1170.87805771099 16.4044889390925 73.1507092025500 4.100909 16edo 1170.41134724080
71zpi 20.2248393119540 59.3329806724710 3.531097 20edo 1186.65961344942 20.2459529213541 59.2711049295348 4.137236 20edo 1185.42209859070
93zpi 24.5782550666850 48.8236449961234 2.810487 25edo 1220.59112490308 24.5738316304204 48.8324335434323 4.665720 25edo 1220.81083858581

Harmonic series in 71zpi with prime 2 removed

Approximation of harmonics in 71zpi with prime 2 removed
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Error Absolute (¢) -14.6 -5.3 -29.2 -0.6 -19.9 +9.6 +15.5 -10.6 -15.1 -2.3 +24.8 +4.8 -5.0 -5.9 +1.0
Relative (%) -24.6 -8.9 -49.2 -1.0 -33.5 +16.2 +26.2 -17.8 -25.6 -3.9 +41.9 +8.1 -8.4 -9.9 +1.6
Steps 20 32 40 47 52 57 61 64 67 70 73 75 77 79 81
Approximation of harmonics in 71zpi with prime 2 removed
Harmonic 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
Error Absolute (¢) +14.5 -25.1 -0.2 +29.5 +4.3 -16.9 +24.7 +10.3 -1.1 -9.8 -15.8 -19.5 -21.0 -20.4 -17.9 -13.6 -7.6 -0.0
Relative (%) +24.5 -42.4 -0.3 +49.8 +7.3 -28.5 +41.6 +17.3 -1.9 -16.5 -26.7 -32.9 -35.4 -34.5 -30.2 -23.0 -12.9 -0.1
Steps 83 84 86 88 89 90 92 93 94 95 96 97 98 99 100 101 102 103

Intervals in 71zpi with prime 2 removed

Intervals in 71zpi with prime 2 removed
JI ratios are comprised of 34-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 %
⟨81 128] at every 4 steps

Whole tone = 13 steps
Limma = 8 steps
Apotome = 5 steps
Degree Cents Ratios Ups and Downs Notation Step
0 0.000 P1 0
1 59.271 34/33, 33/32, 32/31, 31/30, 30/29, 29/28, 28/27, 27/26, 26/25, 25/24, 24/23, 23/22, 22/21, 21/20, 20/19 vA1, ^d2 4
2 118.542 19/18, 18/17, 17/16, 33/31, 16/15, 31/29, 15/14, 29/27, 14/13, 27/25, 13/12, 25/23 m2 8
3 177.813 12/11, 23/21, 34/31, 11/10, 32/29, 21/19, 31/28, 10/9, 29/26, 19/17, 28/25, 9/8 vM2 12
4 237.084 26/23, 17/15, 25/22, 33/29, 8/7, 31/27, 23/20, 15/13, 22/19, 29/25 vvA2 16
5 296.356 7/6, 34/29, 27/23, 20/17, 33/28, 13/11, 32/27, 19/16, 25/21, 31/26, 6/5 vm3 20
6 355.627 29/24, 23/19, 17/14, 28/23, 11/9, 27/22, 16/13, 21/17, 26/21, 31/25 vvM3 24
7 414.898 5/4, 34/27, 29/23, 24/19, 19/15, 33/26, 14/11, 23/18, 32/25, 9/7, 31/24 ^^M3 28
8 474.169 22/17, 13/10, 30/23, 17/13, 21/16, 25/19, 29/22, 33/25, 4/3 vv4 32
9 533.440 31/23, 27/20, 23/17, 19/14, 34/25, 15/11, 26/19, 11/8, 29/21 ^^4 36
10 592.711 18/13, 25/18, 32/23, 7/5, 31/22, 24/17, 17/12, 27/19, 10/7 ^A4 40
11 651.982 33/23, 23/16, 13/9, 29/20, 16/11, 19/13, 22/15, 25/17, 28/19, 31/21, 34/23 ^^d5 44
12 711.253 3/2, 32/21, 29/19, 26/17, 23/15 ^5 48
13 770.524 20/13, 17/11, 31/20, 14/9, 25/16, 11/7, 30/19, 19/12 ^^d6 52
14 829.795 27/17, 8/5, 29/18, 21/13, 34/21, 13/8, 31/19, 18/11 ^m6 56
15 889.067 23/14, 28/17, 33/20, 5/3, 32/19, 27/16, 22/13, 17/10 M6 60
16 948.338 29/17, 12/7, 31/18, 19/11, 26/15, 33/19, 7/4 vA6, ^d7 64
17 1007.609 30/17, 23/13, 16/9, 25/14, 34/19, 9/5, 29/16, 20/11 m7 68
18 1066.880 31/17, 11/6, 24/13, 13/7, 28/15, 15/8, 32/17 vM7 72
19 1126.151 17/9, 19/10, 21/11, 23/12, 25/13, 27/14, 29/15, 31/16, 33/17 vvA7 76
20 1185.422 2/1 v1 +1 oct 80
21 1244.693 33/16, 31/15, 29/14, 27/13, 25/12 vvA1 +1 oct 84
22 1303.964 23/11, 21/10, 19/9, 17/8, 32/15, 15/7, 28/13 vm2 +1 oct 88
23 1363.235 13/6, 24/11, 11/5, 31/14, 20/9, 29/13 vvM2 +1 oct 92
24 1422.507 9/4, 34/15, 25/11, 16/7, 23/10, 30/13 ^^M2 +1 oct 96
25 1481.778 7/3, 33/14, 26/11, 19/8, 31/13 vvm3 +1 oct 100
26 1541.049 12/5, 29/12, 17/7, 22/9, 27/11, 32/13 ^^m3 +1 oct 104
27 1600.320 5/2, 33/13, 28/11, 23/9 ^M3 +1 oct 108
28 1659.591 18/7, 31/12, 13/5, 34/13, 21/8, 29/11 ^^d4 +1 oct 112
29 1718.862 8/3, 27/10, 19/7, 30/11 ^4 +1 oct 116
30 1778.133 11/4, 25/9, 14/5, 31/11, 17/6 A4 +1 oct 120
31 1837.404 20/7, 23/8, 26/9, 29/10, 32/11 ^d5 +1 oct 124
32 1896.675 3/1 P5 +1 oct 128
33 1955.946 34/11, 31/10, 28/9, 25/8, 22/7 vA5 +1 oct, ^d6 +1 oct 132
34 2015.218 19/6, 16/5, 29/9, 13/4 m6 +1 oct 136
35 2074.489 23/7, 33/10, 10/3 vM6 +1 oct 140
36 2133.760 27/8, 17/5, 24/7, 31/9 vvA6 +1 oct 144
37 2193.031 7/2, 32/9, 25/7, 18/5 vm7 +1 oct 148
38 2252.302 29/8, 11/3, 26/7 vvM7 +1 oct 152
39 2311.573 15/4, 34/9, 19/5, 23/6, 27/7 ^^M7 +1 oct 156
40 2370.844 31/8, 4/1 vv1 +2 oct 160
41 2430.115 33/8 ^^1 +2 oct 164
42 2489.386 29/7, 25/6, 21/5, 17/4 vvm2 +2 oct 168
43 2548.658 30/7, 13/3, 22/5, 31/7 ^^m2 +2 oct 172
44 2607.929 9/2, 32/7 ^M2 +2 oct 176
45 2667.200 23/5, 14/3, 33/7 ^^d3 +2 oct 180
46 2726.471 19/4, 24/5, 29/6, 34/7 ^m3 +2 oct 184
47 2785.742 5/1 M3 +2 oct 188
48 2845.013 31/6, 26/5, 21/4 vA3 +2 oct, ^d4 +2 oct 192
49 2904.284 16/3, 27/5 P4 +2 oct 196
50 2963.555 11/2, 28/5 vA4 +2 oct 200
51 3022.826 17/3, 23/4, 29/5 d5 +2 oct 204
52 3082.097 6/1 v5 +2 oct 208
53 3141.369 31/5 vvA5 +2 oct 212
54 3200.640 25/4, 19/3, 32/5 vm6 +2 oct 216
55 3259.911 13/2, 33/5, 20/3 vvM6 +2 oct 220
56 3319.182 27/4, 34/5 ^^M6 +2 oct 224
57 3378.453 7/1 vvm7 +2 oct 228
58 3437.724 29/4, 22/3 ^^m7 +2 oct 232
59 3496.995 15/2, 23/3 ^M7 +2 oct 236
60 3556.266 31/4 ^^d1 +3 oct 240
61 3615.537 8/1 ^1 +3 oct 244
62 3674.809 33/4, 25/3 ^^d2 +3 oct 248
63 3734.080 17/2, 26/3 ^m2 +3 oct 252
64 3793.351 9/1 M2 +3 oct 256
65 3852.622 28/3 vA2 +3 oct, ^d3 +3 oct 260
66 3911.893 19/2, 29/3 m3 +3 oct 264
67 3971.164 10/1 vM3 +3 oct 268
68 4030.435 31/3 vvA3 +3 oct 272
69 4089.706 21/2, 32/3 v4 +3 oct 276
70 4148.977 11/1 vvA4 +3 oct 280
71 4208.248 34/3, 23/2 vd5 +3 oct 284
72 4267.520 vv5 +3 oct 288
73 4326.791 12/1 ^^5 +3 oct 292
74 4386.062 25/2 vvm6 +3 oct 296
75 4445.333 13/1 ^^m6 +3 oct 300
76 4504.604 27/2 ^M6 +3 oct 304
77 4563.875 14/1 ^^d7 +3 oct 308
78 4623.146 29/2 ^m7 +3 oct 312
79 4682.417 15/1 M7 +3 oct 316
80 4741.688 31/2 vA7 +3 oct, ^d1 +4 oct 320
81 4800.959 16/1 P1 +4 oct 324
82 4860.231 33/2 vA1 +4 oct, ^d2 +4 oct 328
83 4919.502 17/1 m2 +4 oct 332
84 4978.773 18/1 vM2 +4 oct 336
85 5038.044 vvA2 +4 oct 340
86 5097.315 19/1 vm3 +4 oct 344
87 5156.586 vvM3 +4 oct 348
88 5215.857 20/1 ^^M3 +4 oct 352
89 5275.128 21/1 vv4 +4 oct 356
90 5334.399 22/1 ^^4 +4 oct 360
91 5393.671 ^A4 +4 oct 364
92 5452.942 23/1 ^^d5 +4 oct 368
93 5512.213 24/1 ^5 +4 oct 372
94 5571.484 25/1 ^^d6 +4 oct 376
95 5630.755 26/1 ^m6 +4 oct 380
96 5690.026 27/1 M6 +4 oct 384
97 5749.297 28/1 vA6 +4 oct, ^d7 +4 oct 388
98 5808.568 29/1 m7 +4 oct 392
99 5867.839 30/1 vM7 +4 oct 396
100 5927.110 31/1 vvA7 +4 oct 400
101 5986.382 32/1 v1 +5 oct 404
102 6045.653 33/1 vvA1 +5 oct 408
103 6104.924 34/1 vm2 +5 oct 412

Approximation to JI in 71zpi with prime 2 removed

Interval mappings in 71zpi with prime 2 removed

The following tables show how 34-integer-limit intervals are represented in 71zpi with prime 2 removed. Prime harmonics are in bold; inconsistent intervals are in italics.

34-integer-limit intervals in 71zpi with prime 2 removed (by direct approximation)
Ratio Error (abs, ¢) Error (rel, %)
34/1 -0.032 -0.053
34/19 +0.166 +0.281
23/12 -0.168 -0.284
19/1 -0.198 -0.334
14/3 +0.329 +0.555
19/5 +0.374 +0.631
21/13 -0.458 -0.772
34/5 +0.540 +0.911
5/1 -0.572 -0.965
30/29 +0.580 +0.978
24/7 +0.631 +1.064
27/10 -0.689 -1.163
26/9 +0.787 +1.327
15/14 -0.901 -1.519
25/19 -0.946 -1.595
16/1 +0.959 +1.619
17/8 -0.991 -1.672
31/22 -1.007 -1.698
27/22 +1.080 +1.821
34/25 +1.112 +1.876
25/1 -1.144 -1.929
29/6 -1.151 -1.943
19/16 -1.157 -1.953
25/11 +1.197 +2.020
27/2 -1.261 -2.128
29/28 -1.480 -2.497
16/5 +1.531 +2.584
31/28 +1.604 +2.706
11/5 -1.769 -2.984
31/6 +1.932 +3.260
31/27 -2.086 -3.520
25/16 -2.103 -3.548
19/11 +2.143 +3.615
33/26 +2.152 +3.632
32/27 +2.221 +3.746
34/11 +2.309 +3.896
11/1 -2.341 -3.949
31/30 +2.504 +4.225
14/11 -2.610 -4.404
33/14 -2.669 -4.504
31/10 -2.775 -4.683
11/3 +2.939 +4.959
32/9 -3.059 -5.161
31/29 +3.084 +5.203
16/11 +3.300 +5.568
31/2 -3.347 -5.647
21/16 +3.388 +5.716
15/11 -3.511 -5.923
28/27 -3.690 -6.225
25/14 +3.807 +6.423
16/13 -3.846 -6.488
26/15 -3.921 -6.616
9/2 +4.019 +6.780
29/22 -4.090 -6.901
29/18 +4.128 +6.965
25/3 +4.136 +6.978
24/17 -4.289 -7.235
32/31 +4.307 +7.266
21/1 +4.347 +7.335
34/21 -4.379 -7.388
14/5 -4.379 -7.388
26/3 -4.493 -7.581
21/19 +4.545 +7.669
10/9 -4.590 -7.745
5/3 +4.708 +7.943
19/14 +4.753 +8.019
13/1 +4.805 +8.107
13/7 -4.822 -8.135
34/13 -4.837 -8.160
23/20 -4.876 -8.227
21/5 +4.919 +8.300
17/7 +4.919 +8.300
14/1 -4.951 -8.353
19/13 -5.003 -8.441
19/3 +5.082 +8.574
29/27 -5.170 -8.723
34/3 +5.248 +8.854
3/1 -5.280 -8.908
13/5 +5.377 +9.072
23/4 -5.448 -9.192
24/13 +5.453 +9.199
25/21 -5.491 -9.264
14/9 +5.608 +9.462
19/15 +5.653 +9.538
34/15 +5.820 +9.819
15/1 -5.851 -9.872
29/10 -5.859 -9.885
8/7 +5.910 +9.972
25/13 -5.949 -10.037
33/32 +5.998 +10.120
27/26 -6.066 -10.235
16/3 +6.239 +10.526
22/9 -6.359 -10.729
29/2 -6.431 -10.850
33/25 -6.477 -10.927
21/11 +6.688 +11.284
16/15 +6.811 +11.491
33/2 +6.958 +11.739
33/5 -7.048 -11.892
13/11 +7.146 +12.056
31/18 +7.212 +12.168
31/9 -7.366 -12.427
32/29 +7.391 +12.469
33/19 -7.422 -12.523
26/11 -7.432 -12.539
33/10 +7.529 +12.703
34/33 +7.589 +12.803
33/1 -7.620 -12.857
32/15 -7.767 -13.104
29/4 +8.147 +13.745
31/26 -8.152 -13.754
11/9 +8.219 +13.866
32/3 -8.339 -14.069
33/16 -8.580 -14.475
26/25 -8.629 -14.559
16/7 -8.668 -14.624
29/20 +8.719 +14.710
15/2 +8.726 +14.723
28/9 -8.969 -15.133
23/8 +9.130 +15.404
26/5 -9.201 -15.523
3/2 +9.298 +15.688
25/9 +9.416 +15.886
23/18 -9.467 -15.972
26/19 -9.575 -16.154
7/1 +9.627 +16.242
34/7 -9.659 -16.296
17/13 +9.741 +16.435
14/13 -9.756 -16.460
26/1 -9.773 -16.488
19/7 -9.825 -16.576
10/3 -9.870 -16.652
9/5 -9.988 -16.851
13/3 +10.085 +17.015
23/17 +10.121 +17.076
7/5 +10.199 +17.207
21/17 -10.199 -17.207
24/1 +10.258 +17.307
17/12 -10.289 -17.360
33/31 +10.305 +17.386
19/9 +10.361 +17.481
29/9 -10.450 -17.630
24/19 +10.456 +17.641
34/9 +10.528 +17.762
9/1 -10.559 -17.815
15/13 -10.657 -17.979
13/8 -10.732 -18.107
25/7 -10.771 -18.172
24/5 +10.830 +18.271
27/14 -10.888 -18.370
22/15 -11.067 -18.672
21/8 -11.190 -18.879
31/4 +11.231 +18.948
29/26 -11.236 -18.957
32/11 -11.278 -19.027
25/24 -11.401 -19.236
16/9 +11.519 +19.434
22/3 -11.639 -19.637
31/20 +11.803 +19.913
33/28 +11.908 +20.092
11/7 -11.968 -20.191
31/15 -12.074 -20.370
11/2 +12.237 +20.646
33/13 -12.425 -20.964
32/25 -12.475 -21.047
24/11 +12.598 +21.255
31/3 -12.645 -21.335
11/10 +12.809 +21.611
31/14 -12.974 -21.890
32/5 -13.047 -22.012
27/4 +13.317 +22.468
33/29 +13.389 +22.589
32/19 -13.420 -22.642
29/12 +13.427 +22.653
25/2 +13.434 +22.666
27/11 -13.498 -22.774
17/16 +13.587 +22.923
29/23 +13.595 +22.937
32/1 -13.618 -22.976
28/15 -13.677 -23.076
27/20 +13.889 +23.432
5/2 +14.006 +23.631
26/21 -14.120 -23.823
30/23 +14.174 +23.915
28/3 -14.249 -24.041
19/2 +14.380 +24.261
24/23 -14.410 -24.311
17/1 +14.546 +24.542
2/1 -14.578 -24.595
27/25 -14.695 -24.793
19/17 -14.744 -24.876
23/6 -14.746 -24.879
7/3 +14.907 +25.150
19/10 +14.952 +25.226
20/17 +14.997 +25.303
23/7 +15.040 +25.375
28/23 +15.075 +25.434
17/5 +15.118 +25.507
10/1 -15.150 -25.560
29/15 -15.157 -25.573
12/7 +15.209 +25.659
27/5 -15.267 -25.758
13/9 +15.364 +25.922
15/7 -15.478 -26.115
8/1 +15.537 +26.214
17/4 -15.569 -26.267
31/11 -15.584 -26.294
27/19 -15.641 -26.389
25/17 -15.690 -26.471
29/3 -15.729 -26.538
19/8 -15.735 -26.548
25/22 +15.775 +26.615
34/27 +15.807 +26.670
27/1 -15.839 -26.723
29/14 -16.058 -27.093
8/5 +16.109 +27.179
22/5 -16.347 -27.580
31/12 +16.510 +27.856
31/23 +16.679 +28.140
25/8 -16.681 -28.144
22/19 -16.721 -28.210
31/25 -16.782 -28.313
27/16 -16.798 -28.342
17/11 +16.887 +28.491
22/1 -16.918 -28.544
28/11 -17.188 -28.999
33/7 -17.247 -29.099
31/5 -17.353 -29.278
11/6 +17.517 +29.554
23/22 -17.685 -29.838
31/19 -17.727 -29.908
11/8 -17.878 -30.163
34/31 +17.893 +30.189
31/1 -17.925 -30.243
32/21 -17.966 -30.311
30/11 -18.089 -30.519
28/25 -18.385 -31.019
32/13 -18.424 -31.084
9/4 +18.597 +31.375
29/11 -18.668 -31.496
25/6 +18.714 +31.574
27/23 +18.765 +31.659
31/16 -18.885 -31.861
21/2 +18.925 +31.930
28/5 -18.957 -31.983
20/9 -19.168 -32.340
6/5 -19.286 -32.538
28/19 -19.331 -32.614
13/2 +19.383 +32.702
26/7 -19.400 -32.731
23/10 -19.454 -32.822
21/10 +19.497 +32.895
17/14 +19.497 +32.895
28/1 -19.529 -32.948
18/17 +19.588 +33.047
19/6 +19.660 +33.169
17/3 +19.826 +33.450
6/1 -19.858 -33.503
23/13 +19.862 +33.511
29/25 -19.865 -33.516
20/7 +19.916 +33.602
13/10 +19.955 +33.667
23/2 -20.026 -33.787
13/12 -20.030 -33.795
9/7 -20.186 -34.058
30/19 -20.231 -34.134
23/21 +20.320 +34.283
17/15 +20.398 +34.414
30/1 -20.429 -34.468
29/5 -20.437 -34.481
7/4 -20.488 -34.567
27/13 -20.644 -34.830
29/19 -20.811 -35.111
8/3 +20.817 +35.122
34/29 +20.977 +35.392
32/23 +20.985 +35.406
29/1 -21.009 -35.445
22/21 -21.266 -35.879
15/8 -21.389 -36.086
33/4 +21.536 +36.334
22/13 -21.724 -36.651
29/16 -21.968 -37.064
33/20 +22.107 +37.299
33/17 -22.167 -37.399
31/21 -22.273 -37.577
29/8 +22.725 +38.340
31/13 -22.730 -38.350
18/11 -22.797 -38.462
33/8 -23.158 -39.071
32/7 -23.245 -39.219
15/4 +23.304 +39.318
23/16 +23.708 +39.999
29/17 +23.716 +40.013
4/3 -23.876 -40.283
25/18 +23.994 +40.481
23/9 -24.045 -40.567
7/2 +24.205 +40.838
30/17 +24.295 +40.990
26/17 -24.319 -41.030
28/13 -24.334 -41.055
20/3 -24.448 -41.248
18/7 +24.507 +41.347
18/5 -24.565 -41.446
13/6 +24.663 +41.610
23/1 +24.667 +41.618
34/23 -24.699 -41.671
20/13 +24.738 +41.738
10/7 -24.777 -41.802
26/23 +24.831 +41.894
12/1 +24.836 +41.902
23/19 +24.865 +41.952
17/6 -24.867 -41.955
19/18 +24.939 +42.076
19/12 -25.034 -42.236
17/9 +25.106 +42.357
18/1 -25.137 -42.411
28/17 +25.196 +42.510
21/20 -25.196 -42.510
30/13 -25.235 -42.575
23/5 +25.239 +42.582
13/4 -25.310 -42.702
29/21 -25.356 -42.780
12/5 +25.407 +42.866
27/7 -25.466 -42.965
21/4 -25.768 -43.475
31/8 +25.809 +43.543
25/23 -25.811 -43.547
29/13 -25.814 -43.553
25/12 -25.979 -43.831
9/8 -26.097 -44.029
22/7 -26.546 -44.787
31/17 +26.800 +45.215
11/4 +26.815 +45.242
33/23 +26.984 +45.526
23/11 +27.008 +45.567
12/11 +27.176 +45.851
20/11 -27.387 -46.206
31/7 -27.552 -46.485
22/17 +27.806 +46.914
27/8 +27.895 +47.063
29/24 +28.004 +47.248
25/4 +28.012 +47.261
32/17 -28.165 -47.518
31/24 -28.183 -47.549
5/4 +28.584 +48.226
29/7 +28.635 +48.312
23/15 -28.752 -48.510
27/17 +28.886 +48.735
19/4 +28.958 +48.857
17/2 +29.124 +49.137
4/1 -29.156 -49.191
30/7 +29.215 +49.290
23/3 -29.324 -49.475
18/13 +29.329 +49.482
7/6 +29.485 +49.745
20/19 -29.530 -49.821
20/1 +29.544 +49.845
17/10 -29.575 -49.898
23/14 +29.618 +49.971
34-integer-limit intervals in 71zpi with prime 2 removed (by patent val mapping)
Ratio Error (abs, ¢) Error (rel, %)
34/1 -0.032 -0.053
34/19 +0.166 +0.281
19/1 -0.198 -0.334
14/3 +0.329 +0.555
19/5 +0.374 +0.631
21/13 -0.458 -0.772
34/5 +0.540 +0.911
5/1 -0.572 -0.965
30/29 +0.580 +0.978
27/10 -0.689 -1.163
26/9 +0.787 +1.327
15/14 -0.901 -1.519
25/19 -0.946 -1.595
31/22 -1.007 -1.698
27/22 +1.080 +1.821
34/25 +1.112 +1.876
25/1 -1.144 -1.929
29/6 -1.151 -1.943
25/11 +1.197 +2.020
27/2 -1.261 -2.128
29/28 -1.480 -2.497
31/28 +1.604 +2.706
11/5 -1.769 -2.984
31/6 +1.932 +3.260
31/27 -2.086 -3.520
19/11 +2.143 +3.615
33/26 +2.152 +3.632
34/11 +2.309 +3.896
11/1 -2.341 -3.949
31/30 +2.504 +4.225
14/11 -2.610 -4.404
33/14 -2.669 -4.504
31/10 -2.775 -4.683
11/3 +2.939 +4.959
31/29 +3.084 +5.203
31/2 -3.347 -5.647
15/11 -3.511 -5.923
28/27 -3.690 -6.225
25/14 +3.807 +6.423
26/15 -3.921 -6.616
9/2 +4.019 +6.780
29/22 -4.090 -6.901
29/18 +4.128 +6.965
25/3 +4.136 +6.978
21/1 +4.347 +7.335
34/21 -4.379 -7.388
14/5 -4.379 -7.388
26/3 -4.493 -7.581
21/19 +4.545 +7.669
10/9 -4.590 -7.745
5/3 +4.708 +7.943
19/14 +4.753 +8.019
13/1 +4.805 +8.107
13/7 -4.822 -8.135
34/13 -4.837 -8.160
21/5 +4.919 +8.300
17/7 +4.919 +8.300
14/1 -4.951 -8.353
19/13 -5.003 -8.441
19/3 +5.082 +8.574
29/27 -5.170 -8.723
34/3 +5.248 +8.854
3/1 -5.280 -8.908
13/5 +5.377 +9.072
25/21 -5.491 -9.264
14/9 +5.608 +9.462
19/15 +5.653 +9.538
34/15 +5.820 +9.819
15/1 -5.851 -9.872
29/10 -5.859 -9.885
25/13 -5.949 -10.037
27/26 -6.066 -10.235
22/9 -6.359 -10.729
29/2 -6.431 -10.850
33/25 -6.477 -10.927
21/11 +6.688 +11.284
33/2 +6.958 +11.739
33/5 -7.048 -11.892
13/11 +7.146 +12.056
31/18 +7.212 +12.168
31/9 -7.366 -12.427
33/19 -7.422 -12.523
26/11 -7.432 -12.539
33/10 +7.529 +12.703
34/33 +7.589 +12.803
33/1 -7.620 -12.857
29/4 +8.147 +13.745
31/26 -8.152 -13.754
11/9 +8.219 +13.866
26/25 -8.629 -14.559
29/20 +8.719 +14.710
15/2 +8.726 +14.723
28/9 -8.969 -15.133
26/5 -9.201 -15.523
3/2 +9.298 +15.688
25/9 +9.416 +15.886
26/19 -9.575 -16.154
7/1 +9.627 +16.242
34/7 -9.659 -16.296
17/13 +9.741 +16.435
14/13 -9.756 -16.460
26/1 -9.773 -16.488
19/7 -9.825 -16.576
10/3 -9.870 -16.652
9/5 -9.988 -16.851
13/3 +10.085 +17.015
23/17 +10.121 +17.076
7/5 +10.199 +17.207
21/17 -10.199 -17.207
33/31 +10.305 +17.386
19/9 +10.361 +17.481
29/9 -10.450 -17.630
34/9 +10.528 +17.762
9/1 -10.559 -17.815
15/13 -10.657 -17.979
25/7 -10.771 -18.172
27/14 -10.888 -18.370
22/15 -11.067 -18.672
31/4 +11.231 +18.948
29/26 -11.236 -18.957
22/3 -11.639 -19.637
31/20 +11.803 +19.913
33/28 +11.908 +20.092
11/7 -11.968 -20.191
31/15 -12.074 -20.370
11/2 +12.237 +20.646
33/13 -12.425 -20.964
31/3 -12.645 -21.335
11/10 +12.809 +21.611
31/14 -12.974 -21.890
27/4 +13.317 +22.468
33/29 +13.389 +22.589
29/12 +13.427 +22.653
25/2 +13.434 +22.666
27/11 -13.498 -22.774
28/15 -13.677 -23.076
27/20 +13.889 +23.432
5/2 +14.006 +23.631
26/21 -14.120 -23.823
28/3 -14.249 -24.041
19/2 +14.380 +24.261
17/1 +14.546 +24.542
2/1 -14.578 -24.595
27/25 -14.695 -24.793
19/17 -14.744 -24.876
7/3 +14.907 +25.150
19/10 +14.952 +25.226
23/7 +15.040 +25.375
17/5 +15.118 +25.507
10/1 -15.150 -25.560
29/15 -15.157 -25.573
27/5 -15.267 -25.758
13/9 +15.364 +25.922
15/7 -15.478 -26.115
31/11 -15.584 -26.294
27/19 -15.641 -26.389
25/17 -15.690 -26.471
29/3 -15.729 -26.538
25/22 +15.775 +26.615
34/27 +15.807 +26.670
27/1 -15.839 -26.723
29/14 -16.058 -27.093
22/5 -16.347 -27.580
31/12 +16.510 +27.856
22/19 -16.721 -28.210
31/25 -16.782 -28.313
17/11 +16.887 +28.491
22/1 -16.918 -28.544
28/11 -17.188 -28.999
33/7 -17.247 -29.099
31/5 -17.353 -29.278
11/6 +17.517 +29.554
31/19 -17.727 -29.908
34/31 +17.893 +30.189
31/1 -17.925 -30.243
30/11 -18.089 -30.519
28/25 -18.385 -31.019
9/4 +18.597 +31.375
29/11 -18.668 -31.496
25/6 +18.714 +31.574
21/2 +18.925 +31.930
28/5 -18.957 -31.983
20/9 -19.168 -32.340
6/5 -19.286 -32.538
28/19 -19.331 -32.614
13/2 +19.383 +32.702
26/7 -19.400 -32.731
21/10 +19.497 +32.895
17/14 +19.497 +32.895
28/1 -19.529 -32.948
19/6 +19.660 +33.169
17/3 +19.826 +33.450
6/1 -19.858 -33.503
23/13 +19.862 +33.511
29/25 -19.865 -33.516
13/10 +19.955 +33.667
9/7 -20.186 -34.058
30/19 -20.231 -34.134
23/21 +20.320 +34.283
17/15 +20.398 +34.414
30/1 -20.429 -34.468
29/5 -20.437 -34.481
27/13 -20.644 -34.830
29/19 -20.811 -35.111
34/29 +20.977 +35.392
29/1 -21.009 -35.445
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