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 |
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 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 indexes in this category include 61zpi, 84zpi, 110zpi, 137zpi, 151zpi, 222zpi, and 273zpi, each demonstrating characteristics that make them suitable for similar applications.
Harmonic series
| 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 | -6 | -45 | +4 | -28 | +22 | +33 | -11 | -19 | +3 | +49 | +16 | -0 | -2 | +10 | |
| Steps | 20 | 32 | 40 | 47 | 52 | 57 | 61 | 64 | 67 | 70 | 73 | 75 | 77 | 79 | 81 | |
| 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 | -34 | +9 | -41 | +17 | -19 | -49 | +27 | +8 | -7 | -17 | -23 | -25 | -24 | -20 | -12 | -2 | |
| Steps | 83 | 84 | 86 | 87 | 89 | 90 | 91 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 | 101 | 102 | |
Intervals
The table bellow presents 32 integer-limit ratios by direct mapping to 71zpi steps, with an allowable error determined by the formula: abs (cents_error) < (1200 / (n * d)), where n/d represents the octave-reduced ratio. If you have an alternative or more effective formula, please feel free to suggest it.
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. With this method, the tonal difference will be less than 1 cent up to the 86th harmonic.
Todo: complete table
Fill in the blank sections with the ups and downs notation from 20-EDO and 182-EDO. Additionally, incorporate a new column adjacent to each notation to indicate the corresponding octave.
| Step | Cents | Ratios | Ups and Downs Notation from 20EDO | Ups and Downs Notation from 182EDO | ||||
|---|---|---|---|---|---|---|---|---|
| 0 | 0.000 | 1/1 | unison | P1 | D | unison | P1 | D |
| 1 | 59.333 | 30/29, 29/28 | up unison, upminor 2nd | ^1, ^m2 | ^D, ^Eb | |||
| 2 | 118.666 | 15/14 | dup unison, mid 2nd | ^^1, ~2 | ^^D, vvE | |||
| 3 | 177.999 | 10/9 | downmajor 2nd | vM2 | vE | |||
| 4 | 237.332 | 8/7 | major 2nd, minor 3rd | M2, m3 | E, F | |||
| 5 | 296.665 | 13/11, 19/16, 6/5 | upminor 3rd | ^m3 | ^F | |||
| 6 | 355.998 | 11/9, 27/22, 16/13 | mid 3rd | ~3 | ^^F, vvF# | |||
| 7 | 415.331 | 5/4, 14/11 | downmajor 3rd | vM3 | vF# | |||
| 8 | 474.664 | 25/19, 4/3 | major 3rd, perfect fourth | M3, P4 | F#, G | |||
| 9 | 533.997 | 15/11 | up-fourth | ^4 | ^G | |||
| 10 | 593.330 | 7/5, 31/22 | mid fourth, mid fifth | ~4, ~5 | ^^G, vvA | |||
| 11 | 652.663 | 16/11, 19/13 | down-fifth | v5 | vA | |||
| 12 | 711.996 | 3/2 | fifth | P5, m6 | A | |||
| 13 | 771.329 | 14/9, 25/16, 11/7 | upfifth, upminor 6th | ^5, ^m6 | ^A, ^Bb | |||
| 14 | 830.662 | 8/5, 21/13, 13/8 | mid 6th | ~6 | ^^A, vvB | |||
| 15 | 889.995 | 5/3 | downmajor 6th | vM6 | vB | |||
| 16 | 949.328 | 19/11, 26/15, 7/4 | major 6th, minor 7th | M6, m7 | B, C | |||
| 17 | 1008.661 | 9/5 | upminor 7th | ^m7 | ^C | |||
| 18 | 1067.994 | 13/7 | mid 7th | ~7 | ^^C, vvD | |||
| 19 | 1127.327 | 23/12 | downmajor 7th | vM7 | vD | |||
| 20 | 1186.660 | 2/1 | octave | P8 | D | |||
| 22 | 1305.326 | 17/8 | ||||||
| 23 | 1364.659 | 11/5 | ||||||
| 25 | 1483.325 | 7/3 | ||||||
| 27 | 1601.990 | 5/2 | ||||||
| 28 | 1661.323 | 13/5 | ||||||
| 29 | 1720.656 | 8/3, 27/10 | ||||||
| 30 | 1779.989 | 14/5 | ||||||
| 32 | 1898.655 | 3/1 | ||||||
| 33 | 1957.988 | 31/10 | ||||||
| 34 | 2017.321 | 16/5 | ||||||
| 35 | 2076.654 | 10/3 | ||||||
| 36 | 2135.987 | 24/7 | ||||||
| 37 | 2195.320 | 7/2, 32/9 | ||||||
| 38 | 2254.653 | 11/3 | ||||||
| 39 | 2313.986 | 19/5 | ||||||
| 40 | 2373.319 | 4/1 | ||||||
| 44 | 2610.651 | 9/2 | ||||||
| 45 | 2669.984 | 14/3 | ||||||
| 46 | 2729.317 | 29/6 | ||||||
| 47 | 2788.650 | 5/1 | ||||||
| 51 | 3025.982 | 23/4 | ||||||
| 52 | 3085.315 | 6/1 | ||||||
| 57 | 3381.980 | 7/1 | ||||||
| 61 | 3619.312 | 8/1 | ||||||
| 63 | 3737.978 | 26/3 | ||||||
| 64 | 3797.311 | 9/1 | ||||||
| 67 | 3975.310 | 10/1 | ||||||
| 70 | 4153.309 | 11/1 | ||||||
| 75 | 4449.974 | 13/1 | ||||||
| 77 | 4568.640 | 14/1 | ||||||
| 78 | 4627.972 | 29/2 | ||||||
| 79 | 4687.305 | 15/1 | ||||||
| 80 | 4746.638 | 31/2 | ||||||
| 81 | 4805.971 | 16/1 | ||||||
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.
The threshold is calculated using the formula: abs (cents_error) < (1200 / (n * d)), where n/d denotes the octave-reduced ratio.
| 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 |
| 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 |
| 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 |
| 3/1 | 3.300 | 5.561 |
| 16/13 | 3.474 | 5.856 |
| 16/5 | 3.635 | 6.127 |
| 13/7 | 3.708 | 6.250 |
| 16/11 | 3.981 | 6.709 |
| 19/13 | 4.323 | 7.285 |
| 10/9 | 4.405 | 7.424 |
| 11/3 | 5.290 | 8.916 |
| 5/3 | 5.636 | 9.499 |
| 16/1 | 5.971 | 10.064 |
| 8/7 | 6.158 | 10.378 |
| 14/9 | 6.413 | 10.808 |
| 9/1 | 6.599 | 11.122 |
| 9/2 | 6.741 | 11.362 |
| 13/5 | 7.110 | 11.982 |
| 13/11 | 7.455 | 12.565 |
| 10/3 | 7.704 | 12.985 |
| 11/9 | 8.590 | 14.478 |
| 9/5 | 8.936 | 15.060 |
| 13/1 | 9.446 | 15.920 |
| 13/8 | 9.866 | 16.628 |
| 3/2 | 10.041 | 16.923 |
| 7/5 | 10.818 | 18.232 |
| 10/1 | 11.004 | 18.546 |
| 11/7 | 11.163 | 18.815 |
| 7/1 | 13.154 | 22.170 |
| 2/1 | 13.340 | 22.484 |
| 5/2 | 15.677 | 26.422 |
| 7/3 | 16.454 | 27.731 |
| 6/1 | 16.640 | 28.045 |
| 8/5 | 16.975 | 28.610 |
| 6/5 | 18.976 | 31.983 |
| 8/1 | 19.312 | 32.548 |
| 7/4 | 19.498 | 32.862 |
| 8/3 | 22.611 | 38.109 |
| 4/3 | 23.381 | 39.407 |
| 7/2 | 26.494 | 44.654 |
| 4/1 | 26.681 | 44.968 |
| 5/4 | 29.017 | 48.906 |