45zpi
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
| 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 | |
| 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
| JI ratios are comprised of 32-integer limit ratios, and are stylized as follows to indicate their accuracy:
|
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
|
This page is a stub. You can help the Xenharmonic Wiki by expanding it. |