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45zpi

Revision as of 22:54, 11 August 2024 by Contribution (talk | contribs) (→Intervals)

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

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