45zpi

From Xenharmonic Wiki
Jump to navigation Jump to search

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
Steps 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
Steps 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 25-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 25/24, 24/23, 23/22, 22/21, 21/20, 20/19, 19/18, 18/17, 17/16, 16/15, 15/14 ^m2 5
2 164.446 14/13, 13/12, 25/23, 12/11, 23/21, 11/10, 21/19, 10/9, 19/17, 9/8 vvvM2 10
3 246.669 17/15, 25/22, 8/7, 23/20, 15/13, 22/19, 7/6, 20/17 ^^M2, vvm3 15
4 328.892 13/11, 19/16, 25/21, 6/5, 23/19, 17/14, 11/9, 16/13, 21/17 ^^^m3 20
5 411.116 5/4, 24/19, 19/15, 14/11, 23/18, 9/7, 22/17 vM3 25
6 493.339 13/10, 17/13, 21/16, 25/19, 4/3, 23/17, 19/14 P4 30
7 575.562 15/11, 11/8, 18/13, 25/18, 7/5, 24/17, 17/12 v4A4 35
8 657.785 10/7, 23/16, 13/9, 16/11, 19/13, 22/15, 25/17 vvv5 40
9 740.008 3/2, 23/15, 20/13, 17/11, 14/9, 25/16 ^^5, vvm6 45
10 822.231 11/7, 19/12, 8/5, 21/13, 13/8, 18/11, 23/14 ^^^m6 50
11 904.454 5/3, 22/13, 17/10, 12/7 vM6 55
12 986.677 19/11, 7/4, 23/13, 16/9, 25/14, 9/5 m7 60
13 1068.901 20/11, 11/6, 24/13, 13/7, 15/8, 17/9 v4M7 65
14 1151.124 19/10, 21/11, 23/12, 25/13 ^M7 70
15 1233.347 2/1, 25/12 ^^1 +1 oct, vvm2 +1 oct 75
16 1315.570 23/11, 21/10, 19/9, 17/8, 15/7, 13/6, 24/11 ^^^m2 +1 oct 80
17 1397.793 11/5, 20/9, 9/4, 25/11, 16/7 vM2 +1 oct 85
18 1480.016 23/10, 7/3, 19/8, 12/5 m3 +1 oct 90
19 1562.239 17/7, 22/9, 5/2 v4M3 +1 oct 95
20 1644.462 23/9, 18/7, 13/5, 21/8 ^M3 +1 oct 100
21 1726.686 8/3, 19/7, 11/4 ^^4 +1 oct 105
22 1808.909 25/9, 14/5, 17/6, 20/7, 23/8 ^^^d5 +1 oct 110
23 1891.132 3/1 v5 +1 oct 115
24 1973.355 25/8, 22/7, 19/6, 16/5 m6 +1 oct 120
25 2055.578 13/4, 23/7, 10/3 v4M6 +1 oct 125
26 2137.801 17/5, 24/7, 7/2 ^M6 +1 oct 130
27 2220.024 25/7, 18/5, 11/3 ^^m7 +1 oct 135
28 2302.247 15/4, 19/5, 23/6 vvM7 +1 oct 140
29 2384.471 4/1 v1 +2 oct 145
30 2466.694 25/6, 21/5, 17/4 m2 +2 oct 150
31 2548.917 13/3, 22/5 v4M2 +2 oct 155
32 2631.140 9/2, 23/5, 14/3 ^M2 +2 oct 160
33 2713.363 19/4, 24/5 ^^m3 +2 oct 165
34 2795.586 5/1 vvM3 +2 oct 170
35 2877.809 21/4, 16/3 v4 +2 oct 175
36 2960.032 11/2 ^44 +2 oct 180
37 3042.256 17/3, 23/4 v45 +2 oct 185
38 3124.479 6/1 ^5 +2 oct 190
39 3206.702 25/4, 19/3, 13/2 ^^m6 +2 oct 195
40 3288.925 20/3 vvM6 +2 oct 200
41 3371.148 7/1 vm7 +2 oct 205
42 3453.371 22/3, 15/2 ^4m7 +2 oct 210
43 3535.594 23/3 M7 +2 oct 215
44 3617.817 8/1 ^1 +3 oct 220
45 3700.041 25/3, 17/2 ^^m2 +3 oct 225
46 3782.264 9/1 vvM2 +3 oct 230
47 3864.487 19/2 vm3 +3 oct 235
48 3946.710 10/1 ^4m3 +3 oct 240
49 4028.933 M3 +3 oct 245
50 4111.156 21/2, 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 M6 +3 oct 275
56 4604.495 14/1 ^m7 +3 oct 280
57 4686.718 15/1 vvvM7 +3 oct 285
58 4768.941 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

Approximation to JI

Interval mappings

The following tables show how 25-integer-limit intervals are represented in 45zpi. Prime harmonics are in bold; inconsistent intervals are in italics.

25-integer-limit intervals in 45zpi (by direct approximation)
Ratio Error (abs, ¢) Error (rel, %)
23/15 +0.002 +0.003
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
23/13 -1.069 -1.300
15/13 -1.072 -1.303
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
19/7 -2.002 -2.434
21/20 -2.244 -2.729
24/5 -2.278 -2.771
7/1 +2.322 +2.824
18/5 +2.428 +2.953
13/7 -2.801 -3.407
23/7 -3.870 -4.707
15/7 -3.873 -4.710
25/6 -3.979 -4.839
22/3 +4.008 +4.875
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
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
24/1 +6.994 +8.506
21/4 +7.028 +8.548
22/19 -7.136 -8.678
17/14 -7.237 -8.802
24/13 +7.473 +9.089
21/13 -8.022 -9.756
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
21/19 -8.822 -10.729
19/5 -8.952 -10.887
16/11 +9.103 +11.071
22/7 -9.137 -11.113
5/1 +9.272 +11.277
23/3 +9.275 +11.280
18/7 +9.378 +11.406
16/9 -9.413 -11.448
13/5 -9.751 -11.860
25/17 -9.887 -12.025
13/3 +10.344 +12.581
17/8 +10.615 +12.909
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
18/1 +11.701 +14.230
18/13 +12.180 +14.813
7/3 +13.145 +15.987
23/18 -13.249 -16.113
6/5 +13.251 +16.116
17/11 -13.629 -16.576
12/11 +13.809 +16.795
15/4 +13.979 +17.001
23/4 +13.981 +17.004
17/10 -14.187 -17.255
25/2 -14.802 -18.002
22/9 +14.831 +18.038
13/4 +15.050 +18.304
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
18/17 -16.731 -20.349
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
25/19 +18.224 +22.164
13/8 -18.296 -22.252
11/9 -18.515 -22.519
25/1 +18.545 +22.554
25/13 +19.024 +23.137
17/5 +19.160 +23.302
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
7/6 -20.202 -24.569
16/3 -20.236 -24.611
13/9 +21.167 +25.744
24/17 -21.438 -26.073
9/1 -21.646 -26.326
19/9 +21.967 +26.716
19/6 -22.203 -27.003
6/1 +22.524 +27.393
21/16 +22.558 +27.435
17/16 -22.732 -27.647
13/6 -23.003 -27.976
25/11 -23.516 -28.601
9/7 -23.968 -29.151
23/6 -24.072 -29.276
5/2 -24.074 -29.279
11/8 +24.244 +29.486
11/4 -24.632 -29.958
5/4 +24.802 +30.164
23/12 +24.804 +30.167
14/9 -24.908 -30.293
25/22 +25.360 +30.843
13/12 +25.874 +31.468
17/7 +26.110 +31.755
21/8 -26.318 -32.009
12/1 -26.353 -32.050
14/5 +26.397 +32.104
19/12 +26.673 +32.440
25/21 +27.046 +32.893
9/2 +27.230 +33.117
17/12 -27.439 -33.371
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
17/13 +28.911 +35.162
11/3 -29.339 -35.682
25/3 +29.368 +35.718
16/15 -29.508 -35.888
23/16 +29.511 +35.891
23/17 -29.980 -36.462
17/15 +29.983 +36.465
18/11 -30.361 -36.925
16/13 -30.580 -37.191
9/5 -30.919 -37.604
7/2 -31.025 -37.732
16/1 -31.059 -37.774
21/10 +31.103 +37.827
19/16 +31.379 +38.164
21/11 +31.661 +38.506
17/9 -32.145 -39.095
25/16 -32.619 -39.672
11/5 +32.789 +39.878
19/2 -33.026 -40.167
2/1 +33.347 +40.557
16/7 -33.381 -40.598
13/2 -33.826 -41.139
20/11 +33.905 +41.235
25/4 +34.074 +41.441
20/17 -34.689 -42.189
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
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
15/11 +38.611 +46.959
23/11 +38.614 +46.962
13/10 +39.125 +47.584
17/3 +39.255 +47.742
9/8 -39.464 -47.996
10/1 -39.604 -48.166
13/11 +39.683 +48.262
11/7 +39.739 +48.331
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
19/11 +40.482 +49.235
25-integer-limit intervals in 45zpi (by patent val mapping)
Ratio Error (abs, ¢) Error (rel, %)
23/15 +0.002 +0.003
19/1 +0.321 +0.390
13/1 -0.479 -0.583
19/13 +0.800 +0.973
23/13 -1.069 -1.300
15/13 -1.072 -1.303
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
19/7 -2.002 -2.434
7/1 +2.322 +2.824
18/5 +2.428 +2.953
13/7 -2.801 -3.407
23/7 -3.870 -4.707
15/7 -3.873 -4.710
25/6 -3.979 -4.839
22/3 +4.008 +4.875
17/2 -4.915 -5.977
22/15 -5.264 -6.402
23/22 +5.267 +6.405
17/6 +5.908 +7.186
22/13 -6.336 -7.706
22/1 -6.815 -8.288
25/18 +6.844 +8.324
7/5 -6.950 -8.453
23/21 +6.953 +8.456
22/19 -7.136 -8.678
17/14 -7.237 -8.802
21/13 -8.022 -9.756
21/1 -8.501 -10.339
21/19 -8.822 -10.729
19/5 -8.952 -10.887
22/7 -9.137 -11.113
5/1 +9.272 +11.277
23/3 +9.275 +11.280
18/7 +9.378 +11.406
13/5 -9.751 -11.860
25/17 -9.887 -12.025
13/3 +10.344 +12.581
23/5 -10.821 -13.160
3/1 -10.823 -13.163
19/3 +11.144 +13.553
19/18 -11.380 -13.840
18/1 +11.701 +14.230
18/13 +12.180 +14.813
7/3 +13.145 +15.987
23/18 -13.249 -16.113
6/5 +13.251 +16.116
17/10 -14.187 -17.255
25/2 -14.802 -18.002
22/9 +14.831 +18.038
22/5 -16.087 -19.566
25/7 +16.223 +19.730
18/17 -16.731 -20.349
25/14 -17.124 -20.826
21/5 -17.773 -21.616
25/19 +18.224 +22.164
11/9 -18.515 -22.519
25/1 +18.545 +22.554
25/13 +19.024 +23.137
17/5 +19.160 +23.302
25/23 +20.093 +24.437
5/3 +20.096 +24.440
23/9 +20.098 +24.443
7/6 -20.202 -24.569
13/9 +21.167 +25.744
9/1 -21.646 -26.326
19/9 +21.967 +26.716
19/6 -22.203 -27.003
6/1 +22.524 +27.393
13/6 -23.003 -27.976
9/7 -23.968 -29.151
23/6 -24.072 -29.276
5/2 -24.074 -29.279
25/22 +25.360 +30.843
17/7 +26.110 +31.755
14/5 +26.397 +32.104
25/21 +27.046 +32.893
17/12 -27.439 -33.371
19/17 -28.111 -34.189
17/1 +28.432 +34.579
17/13 +28.911 +35.162
11/3 -29.339 -35.682
25/3 +29.368 +35.718
23/17 -29.980 -36.462
17/15 +29.983 +36.465
9/5 -30.919 -37.604
7/2 -31.025 -37.732
21/11 +31.661 +38.506
19/2 -33.026 -40.167
2/1 +33.347 +40.557
13/2 -33.826 -41.139
23/2 -34.895 -42.439
15/2 -34.898 -42.442
22/17 -35.247 -42.867
19/14 -35.348 -42.991
14/1 +35.669 +43.381
14/13 +36.148 +43.963
21/17 -36.933 -44.918
23/14 -37.217 -45.264
15/14 -37.220 -45.267
25/12 -37.326 -45.395
17/4 -38.262 -46.534
15/11 +38.611 +46.959
23/11 +38.614 +46.962
17/3 +39.255 +47.742
13/11 +39.683 +48.262
11/1 -40.162 -48.845
25/9 +40.191 +48.881
10/7 +40.297 +49.010
19/11 +40.482 +49.235
21/2 -41.848 -50.895
19/10 -42.299 -51.444
11/7 -42.484 -51.669
10/1 +42.619 +51.834
13/10 -43.098 -52.416
23/10 -44.168 -53.717
3/2 -44.170 -53.720
14/3 +46.492 +56.544
12/5 +46.598 +56.673
20/17 +47.534 +57.811
25/4 -48.149 -58.559
11/5 -49.434 -60.122
17/9 +50.078 +60.905
21/10 -51.120 -62.173
18/11 +51.862 +63.075
10/3 +53.442 +64.997
12/7 +53.548 +65.126
9/2 -54.993 -66.883
19/12 -55.550 -67.560
12/1 +55.871 +67.950
13/12 -56.350 -68.532
14/9 +57.315 +69.707
23/12 -57.419 -69.833
5/4 -57.421 -69.836
25/11 +58.707 +71.399
24/17 +60.785 +73.927
11/6 -62.685 -76.238
10/9 +64.266 +78.160
7/4 -64.372 -78.289
19/4 -66.373 -80.723
4/1 +66.694 +81.113
13/4 -67.173 -81.696
23/4 -68.242 -82.996
15/4 -68.244 -82.999
17/11 +68.594 +83.424
25/24 -70.672 -85.952
17/8 -71.609 -87.091
11/2 -73.509 -89.401
20/7 +73.644 +89.566
21/4 -75.195 -91.452
20/19 +75.646 +92.000
14/11 +75.831 +92.226
20/1 +75.966 +92.390
20/13 +76.445 +92.973
23/20 -77.514 -94.273
4/3 +77.517 +94.276
24/5 +79.945 +97.229
25/8 -81.496 -99.115
11/10 -82.781 -100.679
21/20 -84.467 -102.729
20/3 +86.789 +105.553
24/7 +86.895 +105.682
9/4 -88.340 -107.439
24/19 +88.897 +108.116
24/1 +89.217 +108.506
24/13 +89.696 +109.089
24/23 +90.766 +110.389
8/5 +90.768 +110.392
12/11 +96.032 +116.795
20/9 +97.613 +118.717
8/7 +97.718 +118.845
19/8 -99.720 -121.280
8/1 +100.041 +121.670
13/8 -100.520 -122.252
23/8 -101.589 -123.553
15/8 -101.591 -123.556
17/16 -104.955 -127.647
11/4 -106.855 -129.958
21/8 -108.542 -132.009
8/3 +110.864 +134.833
25/16 -114.842 -139.672
20/11 +116.128 +141.235
9/8 -121.687 -147.996
16/5 +124.115 +150.949
24/11 +129.379 +157.351
16/7 +131.065 +159.402
19/16 -133.067 -161.836
16/1 +133.387 +162.226
16/13 +133.866 +162.809
23/16 -134.936 -164.109
16/15 +134.938 +164.112
11/8 -140.202 -170.514
21/16 -141.888 -172.565
16/3 +144.211 +175.389
16/9 +155.034 +188.552
16/11 +173.549 +211.071
Icon-Stub.png This page is a stub. You can help the Xenharmonic Wiki by expanding it.
Retrieved from "https://en.xen.wiki/index.php?title=45zpi&oldid=150497"