Xenharmonic Wiki

71zpi

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 21-throdd-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
Step 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
Step 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 32-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 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, 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, 8/7, 31/27, 23/20, 15/13, 22/19, 29/25, 7/6 ^6M2 36
5 296.665 27/23, 20/17, 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, 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, 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 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, 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, 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 ^5M7 171
20 1186.660 2/1 vv1 +1 oct 180
21 1245.993 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, 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, 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, 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 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, 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, 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 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 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

Approximation to JI

The following table illustrates the representation of the 32-integer limit intervals in 71zpi. Prime harmonics are in bold; inconsistent intervals are in italic.

Intervals by direct approximation (even if inconsistent)
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
25/16 +1.299 +2.189
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
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
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
27/26 +6.004 +10.120
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
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
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
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
23/17 -10.678 -17.997
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
32/25 +12.042 +20.295
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
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
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
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
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
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
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
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

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 numbers 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

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
Step 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
Step 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
JI ratios are comprised of 32-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 59.333 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
2 118.666 19/18, 18/17, 17/16, 16/15, 31/29, 15/14, 29/27, 14/13, 27/25, 13/12, 25/23 m2 8
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 vM2 12
4 237.332 26/23, 17/15, 25/22, 8/7, 31/27, 23/20, 15/13, 22/19, 29/25, 7/6 vvA2 16
5 296.665 27/23, 20/17, 13/11, 32/27, 19/16, 25/21, 31/26, 6/5 vm3 20
6 355.998 29/24, 23/19, 17/14, 28/23, 11/9, 27/22, 16/13, 21/17, 26/21, 31/25 vvM3 24
7 415.331 5/4, 29/23, 24/19, 19/15, 14/11, 23/18, 32/25, 9/7, 31/24 ^^M3 28
8 474.664 22/17, 13/10, 30/23, 17/13, 21/16, 25/19, 29/22, 4/3 vv4 32
9 533.997 31/23, 27/20, 23/17, 19/14, 15/11, 26/19, 11/8, 29/21, 18/13 ^^4 36
10 593.330 25/18, 32/23, 7/5, 31/22, 24/17, 17/12, 27/19, 10/7 ^A4 40
11 652.663 23/16, 13/9, 29/20, 16/11, 19/13, 22/15, 25/17, 28/19, 31/21 ^^d5 44
12 711.996 3/2, 32/21, 29/19, 26/17, 23/15 ^5 48
13 771.329 20/13, 17/11, 31/20, 14/9, 25/16, 11/7, 30/19, 19/12, 27/17 ^^d6 52
14 830.662 8/5, 29/18, 21/13, 13/8, 31/19, 18/11, 23/14 ^m6 56
15 889.995 28/17, 5/3, 32/19, 27/16, 22/13, 17/10 M6 60
16 949.328 29/17, 12/7, 31/18, 19/11, 26/15, 7/4 vA6, ^d7 64
17 1008.661 30/17, 23/13, 16/9, 25/14, 9/5, 29/16, 20/11 m7 68
18 1067.994 31/17, 11/6, 24/13, 13/7, 28/15, 15/8, 32/17 vM7 72
19 1127.327 17/9, 19/10, 21/11, 23/12, 25/13, 27/14, 29/15, 31/16 vvA7 76
20 1186.660 2/1 v1 +1 oct 80
21 1245.993 31/15, 29/14, 27/13, 25/12 vvA1 +1 oct 84
22 1305.326 23/11, 21/10, 19/9, 17/8, 32/15, 15/7, 28/13 vm2 +1 oct 88
23 1364.659 13/6, 24/11, 11/5, 31/14, 20/9, 29/13 vvM2 +1 oct 92
24 1423.992 9/4, 25/11, 16/7, 23/10, 30/13 ^^M2 +1 oct 96
25 1483.325 7/3, 26/11, 19/8, 31/13 vvm3 +1 oct 100
26 1542.657 12/5, 29/12, 17/7, 22/9, 27/11, 32/13 ^^m3 +1 oct 104
27 1601.990 5/2, 28/11, 23/9 ^M3 +1 oct 108
28 1661.323 18/7, 31/12, 13/5, 21/8, 29/11 ^^d4 +1 oct 112
29 1720.656 8/3, 27/10, 19/7, 30/11 ^4 +1 oct 116
30 1779.989 11/4, 25/9, 14/5, 31/11, 17/6 A4 +1 oct 120
31 1839.322 20/7, 23/8, 26/9, 29/10, 32/11 ^d5 +1 oct 124
32 1898.655 3/1 P5 +1 oct 128
33 1957.988 31/10, 28/9, 25/8, 22/7 vA5 +1 oct, ^d6 +1 oct 132
34 2017.321 19/6, 16/5, 29/9, 13/4 m6 +1 oct 136
35 2076.654 23/7, 10/3, 27/8 vM6 +1 oct 140
36 2135.987 17/5, 24/7, 31/9 vvA6 +1 oct 144
37 2195.320 7/2, 32/9, 25/7, 18/5 vm7 +1 oct 148
38 2254.653 29/8, 11/3, 26/7 vvM7 +1 oct 152
39 2313.986 15/4, 19/5, 23/6, 27/7 ^^M7 +1 oct 156
40 2373.319 31/8, 4/1 vv1 +2 oct 160
41 2432.652 29/7 ^^1 +2 oct 164
42 2491.985 25/6, 21/5, 17/4, 30/7 vvm2 +2 oct 168
43 2551.318 13/3, 22/5, 31/7 ^^m2 +2 oct 172
44 2610.651 9/2, 32/7 ^M2 +2 oct 176
45 2669.984 23/5, 14/3, 19/4 ^^d3 +2 oct 180
46 2729.317 24/5, 29/6 ^m3 +2 oct 184
47 2788.650 5/1 M3 +2 oct 188
48 2847.983 31/6, 26/5, 21/4 vA3 +2 oct, ^d4 +2 oct 192
49 2907.316 16/3, 27/5 P4 +2 oct 196
50 2966.649 11/2, 28/5 vA4 +2 oct 200
51 3025.982 17/3, 23/4, 29/5 d5 +2 oct 204
52 3085.315 6/1 v5 +2 oct 208
53 3144.648 31/5, 25/4 vvA5 +2 oct 212
54 3203.981 19/3, 32/5 vm6 +2 oct 216
55 3263.314 13/2, 20/3 vvM6 +2 oct 220
56 3322.647 27/4 ^^M6 +2 oct 224
57 3381.980 7/1 vvm7 +2 oct 228
58 3441.313 29/4, 22/3 ^^m7 +2 oct 232
59 3500.646 15/2, 23/3 ^M7 +2 oct 236
60 3559.979 31/4 ^^d1 +3 oct 240
61 3619.312 8/1 ^1 +3 oct 244
62 3678.645 25/3, 17/2 ^^d2 +3 oct 248
63 3737.978 26/3 ^m2 +3 oct 252
64 3797.311 9/1 M2 +3 oct 256
65 3856.644 28/3 vA2 +3 oct, ^d3 +3 oct 260
66 3915.977 19/2, 29/3 m3 +3 oct 264
67 3975.310 10/1 vM3 +3 oct 268
68 4034.643 31/3 vvA3 +3 oct 272
69 4093.976 21/2, 32/3 v4 +3 oct 276
70 4153.309 11/1 vvA4 +3 oct 280
71 4212.642 23/2 vd5 +3 oct 284
72 4271.975 vv5 +3 oct 288
73 4331.308 12/1 ^^5 +3 oct 292
74 4390.641 25/2 vvm6 +3 oct 296
75 4449.974 13/1 ^^m6 +3 oct 300
76 4509.307 27/2 ^M6 +3 oct 304
77 4568.640 14/1 ^^d7 +3 oct 308
78 4627.972 29/2 ^m7 +3 oct 312
79 4687.305 15/1 M7 +3 oct 316
80 4746.638 31/2 vA7 +3 oct, ^d1 +4 oct 320
81 4805.971 16/1 P1 +4 oct 324
82 4865.304 vA1 +4 oct, ^d2 +4 oct 328
83 4924.637 17/1 m2 +4 oct 332
84 4983.970 18/1 vM2 +4 oct 336
85 5043.303 vvA2 +4 oct 340
86 5102.636 19/1 vm3 +4 oct 344
87 5161.969 20/1 vvM3 +4 oct 348
88 5221.302 ^^M3 +4 oct 352
89 5280.635 21/1 vv4 +4 oct 356
90 5339.968 22/1 ^^4 +4 oct 360
91 5399.301 23/1 ^A4 +4 oct 364
92 5458.634 ^^d5 +4 oct 368
93 5517.967 24/1 ^5 +4 oct 372
94 5577.300 25/1 ^^d6 +4 oct 376
95 5636.633 26/1 ^m6 +4 oct 380
96 5695.966 27/1 M6 +4 oct 384
97 5755.299 28/1 vA6 +4 oct, ^d7 +4 oct 388
98 5814.632 29/1 m7 +4 oct 392
99 5873.965 30/1 vM7 +4 oct 396
100 5933.298 31/1 vvA7 +4 oct 400
101 5992.631 32/1 v1 +5 oct 404
Intervals by direct approximation (even if inconsistent)
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
25/16 +1.299 +2.189
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
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
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
27/26 +6.004 +10.120
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
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
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
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
23/17 -10.678 -17.997
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
32/25 +12.042 +20.295
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
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
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
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
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
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
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
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
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