45zpi

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45 zeta peak index (abbreviated 45zpi), is the equal-step tuning system obtained from the 45th 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
45zpi 14.5944346577250 82.2231232756126 2.097730 0.344839 10.594800 15edo 1233.34684913419 2 2

Theory

45zpi is characterized by a very broad octave error, yet it maintains a quite decent zeta strength. This combination makes it an ideal candidate for no-octave tuning applications.

No other zeta peak indexes exhibit both a larger octave error and greater zeta height than 45zpi.

45zpi supports a complex chord structure with ratios of 1:3:4:5:7:9:13:15:18:19:20:21:22:23:24:25, which further exemplifies its capabilities.

The closest zeta peak indexes to 45zpi that exceed its strength are 42zpi and 47zpi, though 43zpi is nearly as strong as 45zpi.

Harmonic series

Approximation of harmonics in 45zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Error Absolute (¢) +33.3 -10.8 -15.5 +9.3 +22.5 +2.3 +17.8 -21.6 -39.6 -40.2 -26.4 -0.5 +35.7 -1.6 -31.1
Relative (%) +40.6 -13.2 -18.9 +11.3 +27.4 +2.8 +21.7 -26.3 -48.2 -48.8 -32.1 -0.6 +43.4 -1.9 -37.8
Step 15 23 29 34 38 41 44 46 48 50 52 54 56 57 58
Approximation of harmonics in 45zpi
Harmonic 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
Error Absolute (¢) +28.4 +11.7 +0.3 -6.3 -8.5 -6.8 -1.5 +7.0 +18.5 +32.9 -32.5 -13.2 +8.3 +31.8 -25.0 +2.3
Relative (%) +34.6 +14.2 +0.4 -7.6 -10.3 -8.3 -1.9 +8.5 +22.6 +40.0 -39.5 -16.1 +10.1 +38.7 -30.4 +2.8
Step 60 61 62 63 64 65 66 67 68 69 69 70 71 72 72 73

Intervals

Intervals in 45zpi
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 %
⟨73 116] at every 5 steps

Whole tone = 13 steps
Limma = 4 steps
Apotome = 9 steps
Degree Cents Ratios Ups and Downs Notation Step
0 0.000 P1 0
1 82.223 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, 19/18, 18/17, 17/16, 16/15, 31/29, 15/14 ^m2 5
2 164.446 29/27, 14/13, 27/25, 13/12, 25/23, 12/11, 23/21, 11/10, 32/29, 21/19, 31/28, 10/9, 29/26, 19/17, 28/25, 9/8 vvvM2 10
3 246.669 26/23, 17/15, 25/22, 8/7, 31/27, 23/20, 15/13, 22/19, 29/25, 7/6, 27/23, 20/17 ^^M2, vvm3 15
4 328.892 13/11, 32/27, 19/16, 25/21, 31/26, 6/5, 29/24, 23/19, 17/14, 28/23, 11/9, 27/22, 16/13, 21/17, 26/21 ^^^m3 20
5 411.116 31/25, 5/4, 29/23, 24/19, 19/15, 14/11, 23/18, 32/25, 9/7, 31/24, 22/17 vM3 25
6 493.339 13/10, 30/23, 17/13, 21/16, 25/19, 29/22, 4/3, 31/23, 27/20, 23/17, 19/14 P4 30
7 575.562 15/11, 26/19, 11/8, 29/21, 18/13, 25/18, 32/23, 7/5, 31/22, 24/17, 17/12, 27/19 v4A4 35
8 657.785 10/7, 23/16, 13/9, 29/20, 16/11, 19/13, 22/15, 25/17, 28/19, 31/21 vvv5 40
9 740.008 3/2, 32/21, 29/19, 26/17, 23/15, 20/13, 17/11, 31/20, 14/9, 25/16 ^^5, vvm6 45
10 822.231 11/7, 30/19, 19/12, 27/17, 8/5, 29/18, 21/13, 13/8, 31/19, 18/11, 23/14 ^^^m6 50
11 904.454 28/17, 5/3, 32/19, 27/16, 22/13, 17/10, 29/17, 12/7, 31/18 vM6 55
12 986.677 19/11, 26/15, 7/4, 30/17, 23/13, 16/9, 25/14, 9/5 m7 60
13 1068.901 29/16, 20/11, 31/17, 11/6, 24/13, 13/7, 28/15, 15/8, 32/17, 17/9 v4M7 65
14 1151.124 19/10, 21/11, 23/12, 25/13, 27/14, 29/15, 31/16 ^M7 70
15 1233.347 2/1, 31/15, 29/14, 27/13, 25/12 ^^1 +1 oct, vvm2 +1 oct 75
16 1315.570 23/11, 21/10, 19/9, 17/8, 32/15, 15/7, 28/13, 13/6, 24/11 ^^^m2 +1 oct 80
17 1397.793 11/5, 31/14, 20/9, 29/13, 9/4, 25/11, 16/7 vM2 +1 oct 85
18 1480.016 23/10, 30/13, 7/3, 26/11, 19/8, 31/13, 12/5 m3 +1 oct 90
19 1562.239 29/12, 17/7, 22/9, 27/11, 32/13, 5/2 v4M3 +1 oct 95
20 1644.462 28/11, 23/9, 18/7, 31/12, 13/5, 21/8, 29/11 ^M3 +1 oct 100
21 1726.686 8/3, 27/10, 19/7, 30/11, 11/4 ^^4 +1 oct 105
22 1808.909 25/9, 14/5, 31/11, 17/6, 20/7, 23/8, 26/9, 29/10, 32/11 ^^^d5 +1 oct 110
23 1891.132 3/1 v5 +1 oct 115
24 1973.355 31/10, 28/9, 25/8, 22/7, 19/6, 16/5 m6 +1 oct 120
25 2055.578 29/9, 13/4, 23/7, 10/3 v4M6 +1 oct 125
26 2137.801 27/8, 17/5, 24/7, 31/9, 7/2 ^M6 +1 oct 130
27 2220.024 32/9, 25/7, 18/5, 29/8, 11/3 ^^m7 +1 oct 135
28 2302.247 26/7, 15/4, 19/5, 23/6, 27/7 vvM7 +1 oct 140
29 2384.471 31/8, 4/1 v1 +2 oct 145
30 2466.694 29/7, 25/6, 21/5, 17/4 m2 +2 oct 150
31 2548.917 30/7, 13/3, 22/5, 31/7 v4M2 +2 oct 155
32 2631.140 9/2, 32/7, 23/5, 14/3 ^M2 +2 oct 160
33 2713.363 19/4, 24/5, 29/6 ^^m3 +2 oct 165
34 2795.586 5/1 vvM3 +2 oct 170
35 2877.809 31/6, 26/5, 21/4, 16/3 v4 +2 oct 175
36 2960.032 27/5, 11/2, 28/5 ^44 +2 oct 180
37 3042.256 17/3, 23/4, 29/5 v45 +2 oct 185
38 3124.479 6/1, 31/5 ^5 +2 oct 190
39 3206.702 25/4, 19/3, 32/5, 13/2 ^^m6 +2 oct 195
40 3288.925 20/3, 27/4 vvM6 +2 oct 200
41 3371.148 7/1 vm7 +2 oct 205
42 3453.371 29/4, 22/3, 15/2 ^4m7 +2 oct 210
43 3535.594 23/3, 31/4 M7 +2 oct 215
44 3617.817 8/1 ^1 +3 oct 220
45 3700.041 25/3, 17/2, 26/3 ^^m2 +3 oct 225
46 3782.264 9/1 vvM2 +3 oct 230
47 3864.487 28/3, 19/2 vm3 +3 oct 235
48 3946.710 29/3, 10/1 ^4m3 +3 oct 240
49 4028.933 31/3 M3 +3 oct 245
50 4111.156 21/2, 32/3, 11/1 ^4 +3 oct 250
51 4193.379 23/2 vvvA4 +3 oct 255
52 4275.602 12/1 vv5 +3 oct 260
53 4357.826 25/2 vm6 +3 oct 265
54 4440.049 13/1 ^4m6 +3 oct 270
55 4522.272 27/2 M6 +3 oct 275
56 4604.495 14/1, 29/2 ^m7 +3 oct 280
57 4686.718 15/1 vvvM7 +3 oct 285
58 4768.941 31/2, 16/1 ^^M7 +3 oct, vv1 +4 oct 290
59 4851.164 vm2 +4 oct 295
60 4933.387 17/1 ^4m2 +4 oct 300
61 5015.611 18/1 M2 +4 oct 305
62 5097.834 19/1 ^m3 +4 oct 310
63 5180.057 20/1 vvvM3 +4 oct 315
64 5262.280 21/1 ^^M3 +4 oct, vv4 +4 oct 320
65 5344.503 22/1 ^^^4 +4 oct 325
66 5426.726 23/1 ^4d5 +4 oct 330
67 5508.949 24/1 P5 +4 oct 335
68 5591.172 25/1 ^m6 +4 oct 340
69 5673.396 26/1, 27/1 vvvM6 +4 oct 345
70 5755.619 28/1 ^^M6 +4 oct, vvm7 +4 oct 350
71 5837.842 29/1 ^^^m7 +4 oct 355
72 5920.065 30/1, 31/1 vM7 +4 oct 360
73 6002.288 32/1 P1 +5 oct 365

Approximation to JI

Intervals by direct approximation (even if inconsistent)
Ratio Error (abs, ¢) Error (rel, %)
23/15 -0.002 -0.003
32/7 +0.034 +0.042
19/1 -0.321 -0.390
13/1 +0.479 +0.583
11/10 +0.558 +0.679
25/8 -0.728 -0.885
19/13 -0.800 -0.973
29/5 +1.008 +1.226
23/13 +1.069 +1.300
15/13 +1.072 +1.303
29/24 -1.270 -1.545
31/12 -1.382 -1.681
27/16 +1.411 +1.716
23/1 +1.548 +1.883
15/1 +1.551 +1.886
22/21 -1.686 -2.051
23/19 +1.869 +2.273
19/15 -1.871 -2.276
32/19 -1.967 -2.393
19/7 +2.002 +2.434
21/20 +2.244 +2.729
24/5 +2.278 +2.771
32/1 -2.288 -2.783
7/1 -2.322 -2.824
28/3 +2.384 +2.900
18/5 -2.428 -2.953
32/13 -2.767 -3.365
13/7 +2.801 +3.407
31/9 +3.324 +4.043
30/17 -3.364 -4.092
29/18 +3.436 +4.179
32/23 -3.836 -4.666
32/15 -3.839 -4.669
23/7 +3.870 +4.707
15/7 +3.873 +4.710
25/6 +3.979 +4.839
22/3 -4.008 -4.875
26/17 -4.436 -5.395
20/3 -4.566 -5.553
24/7 -4.672 -5.682
4/3 +4.706 +5.724
23/20 -4.709 -5.727
17/2 +4.915 +5.977
22/15 +5.264 +6.402
23/22 -5.267 -6.405
20/13 +5.778 +7.027
17/6 -5.908 -7.186
29/7 -5.942 -7.227
32/29 +5.977 +7.269
31/16 -6.088 -7.404
9/4 +6.117 +7.439
20/1 +6.257 +7.610
22/13 +6.336 +7.706
14/11 +6.392 +7.774
20/19 +6.578 +8.000
24/19 -6.674 -8.116
22/1 +6.815 +8.288
25/18 -6.844 -8.324
7/5 +6.950 +8.453
23/21 -6.953 -8.456
32/5 +6.984 +8.495
24/1 -6.994 -8.506
21/4 -7.028 -8.548
27/10 -7.134 -8.677
22/19 +7.136 +8.678
17/14 +7.237 +8.802
24/13 -7.473 -9.089
31/27 -7.499 -9.120
27/11 -7.692 -9.355
29/19 -7.944 -9.661
21/13 +8.022 +9.756
29/1 -8.265 -10.051
28/9 -8.439 -10.264
21/1 +8.501 +10.339
24/23 -8.542 -10.389
8/5 -8.545 -10.392
20/7 +8.579 +10.434
11/2 -8.714 -10.599
29/13 -8.744 -10.634
21/19 +8.822 +10.729
19/5 +8.952 +10.887
16/11 -9.103 -11.071
22/7 +9.137 +11.113
26/11 +9.194 +11.181
5/1 -9.272 -11.277
23/3 -9.275 -11.280
18/7 -9.378 -11.406
16/9 +9.413 +11.448
31/4 +9.441 +11.483
29/8 +9.553 +11.618
13/5 +9.751 +11.860
29/23 -9.813 -11.934
29/15 -9.815 -11.937
25/17 +9.887 +12.025
30/11 +10.265 +12.485
29/25 +10.280 +12.503
13/3 -10.344 -12.581
17/8 -10.615 -12.909
32/21 -10.789 -13.122
23/5 +10.821 +13.160
3/1 +10.823 +13.163
19/3 -11.144 -13.553
19/18 +11.380 +13.840
25/24 -11.551 -14.048
28/15 +11.657 +14.177
28/23 +11.659 +14.180
18/1 -11.701 -14.230
31/28 +11.763 +14.307
18/13 -12.180 -14.813
28/13 +12.728 +15.480
32/3 -13.111 -15.946
7/3 -13.145 -15.987
28/1 +13.207 +16.063
23/18 +13.249 +16.113
6/5 -13.251 -16.116
28/19 +13.528 +16.453
17/11 +13.629 +16.576
12/11 -13.809 -16.795
15/4 -13.979 -17.001
23/4 -13.981 -17.004
27/14 -14.085 -17.130
31/3 +14.148 +17.206
17/10 +14.187 +17.255
29/6 +14.259 +17.342
26/25 -14.323 -17.420
29/20 -14.522 -17.661
31/10 -14.633 -17.797
25/2 +14.802 +18.002
22/9 -14.831 -18.038
13/4 -15.050 -18.304
29/22 -15.079 -18.340
31/11 -15.191 -18.475
20/9 -15.389 -18.717
8/7 -15.495 -18.845
4/1 +15.529 +18.887
19/4 -15.850 -19.277
22/5 +16.087 +19.566
25/7 -16.223 -19.730
32/25 +16.257 +19.772
27/2 -16.407 -19.954
31/21 +16.470 +20.030
18/17 +16.731 +20.349
29/21 -16.766 -20.390
27/26 -16.886 -20.537
27/4 +16.940 +20.603
25/14 +17.124 +20.826
19/8 +17.497 +21.280
21/5 +17.773 +21.616
8/1 -17.817 -21.670
7/4 -17.852 -21.711
10/9 +17.957 +21.840
31/22 +18.156 +22.081
25/19 -18.224 -22.164
13/8 +18.296 +22.252
11/9 +18.515 +22.519
25/1 -18.545 -22.554
31/20 +18.714 +22.760
25/13 -19.024 -23.137
29/3 -19.088 -23.215
17/5 -19.160 -23.302
28/27 -19.262 -23.427
23/8 +19.366 +23.553
15/8 +19.368 +23.556
11/6 -19.538 -23.762
25/23 -20.093 -24.437
5/3 -20.096 -24.440
23/9 -20.098 -24.443
29/17 +20.167 +24.528
7/6 +20.202 +24.569
16/3 +20.236 +24.611
13/9 -21.167 -25.744
27/17 -21.322 -25.931
24/17 +21.438 +26.073
29/28 -21.472 -26.114
31/14 -21.583 -26.250
9/1 +21.646 +26.326
19/9 -21.967 -26.716
19/6 +22.203 +27.003
28/5 +22.480 +27.340
6/1 -22.524 -27.393
21/16 -22.558 -27.435
17/16 +22.732 +27.647
13/6 +23.003 +27.976
31/15 +23.420 +28.483
31/23 +23.422 +28.486
25/11 +23.516 +28.601
30/29 -23.532 -28.619
26/5 -23.595 -28.697
29/4 -23.794 -28.938
31/2 -23.906 -29.074
32/9 -23.934 -29.109
9/7 +23.968 +29.151
23/6 +24.072 +29.276
5/2 +24.074 +29.279
11/8 -24.244 -29.486
31/26 -24.385 -29.657
31/13 +24.492 +29.787
29/26 +24.603 +29.923
11/4 +24.632 +29.958
5/4 -24.802 -30.164
23/12 -24.804 -30.167
14/9 +24.908 +30.293
31/1 +24.971 +30.369
29/2 +25.082 +30.505
31/19 +25.291 +30.759
25/22 -25.360 -30.843
31/30 -25.456 -30.960
27/22 +25.655 +31.201
13/12 -25.874 -31.468
17/7 -26.110 -31.755
32/17 +26.144 +31.796
27/20 +26.213 +31.880
21/8 +26.318 +32.009
12/1 +26.353 +32.050
14/5 -26.397 -32.104
19/12 -26.673 -32.440
28/11 -26.955 -32.782
25/21 -27.046 -32.893
9/2 -27.230 -33.117
32/31 -27.259 -33.152
31/7 +27.293 +33.194
29/14 +27.404 +33.329
17/12 +27.439 +33.371
26/9 +27.709 +33.700
19/17 +28.111 +34.189
17/1 -28.432 -34.579
8/3 -28.641 -34.833
12/7 +28.675 +34.874
10/3 +28.781 +35.003
31/17 -28.820 -35.052
17/13 -28.911 -35.162
11/3 +29.339 +35.682
25/3 -29.368 -35.718
30/7 -29.474 -35.846
16/15 +29.508 +35.888
23/16 -29.511 -35.891
29/9 -29.911 -36.378
23/17 +29.980 +36.462
17/15 -29.983 -36.465
18/11 +30.361 +36.925
26/7 -30.546 -37.150
16/13 +30.580 +37.191
9/5 +30.919 +37.604
27/23 +30.921 +37.607
7/2 +31.025 +37.732
16/1 +31.059 +37.774
21/10 -31.103 -37.827
27/25 -31.209 -37.956
19/16 -31.379 -38.164
30/19 -31.476 -38.281
21/11 -31.661 -38.506
28/25 +31.752 +38.617
30/1 -31.796 -38.671
27/8 -31.936 -38.841
31/24 +31.965 +38.876
27/13 +31.990 +38.907
17/9 +32.145 +39.095
30/13 -32.275 -39.253
27/1 +32.469 +39.489
26/19 -32.547 -39.584
25/16 +32.619 +39.672
11/5 -32.789 -39.878
27/19 +32.790 +39.879
26/1 -32.868 -39.974
19/2 +33.026 +40.167
31/29 +33.235 +40.421
30/23 -33.344 -40.554
2/1 -33.347 -40.557
16/7 +33.381 +40.598
29/11 +33.797 +41.104
13/2 +33.826 +41.139
20/11 -33.905 -41.235
25/4 -34.074 -41.441
31/5 +34.243 +41.647
29/10 +34.355 +41.782
26/23 -34.416 -41.857
26/15 -34.419 -41.860
29/12 -34.617 -42.101
20/17 +34.689 +42.189
31/6 -34.729 -42.237
32/27 -34.757 -42.272
27/7 +34.792 +42.314
23/2 +34.895 +42.439
15/2 +34.898 +42.442
24/11 +35.067 +42.649
22/17 +35.247 +42.867
19/14 +35.348 +42.991
12/5 +35.625 +43.327
14/1 -35.669 -43.381
14/3 +35.731 +43.456
14/13 -36.148 -43.963
31/18 +36.671 +44.600
21/17 +36.933 +44.918
23/14 +37.217 +45.264
15/14 +37.220 +45.267
25/12 +37.326 +45.395
3/2 -38.053 -46.280
23/10 -38.056 -46.283
17/4 +38.262 +46.534
26/3 +38.532 +46.863
15/11 -38.611 -46.959
23/11 -38.614 -46.962
31/25 -38.707 -47.076
13/10 -39.125 -47.584
17/3 -39.255 -47.742
29/16 -39.323 -47.825
31/8 -39.435 -47.961
9/8 +39.464 +47.996
10/1 +39.604 +48.166
13/11 -39.683 -48.262
11/7 -39.739 -48.331
32/11 +39.773 +48.372
19/10 -39.924 -48.556
11/1 +40.162 +48.845
25/9 -40.191 -48.881
10/7 -40.297 -49.010
16/5 +40.331 +49.051
21/2 -40.375 -49.105
27/5 -40.481 -49.233
19/11 -40.482 -49.235
28/17 -40.584 -49.358
29/27 -40.734 -49.541
26/21 +40.854 +49.687
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