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 |
71zpi marks the most prominent zeta peak index in the vicinity of 20edo, ranging between 19.5 EDO and 20.5 EDO. 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.
| 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 (%) | -22.5 | -5.6 | -45.0 | +3.9 | -28.0 | +22.2 | +32.5 | -11.1 | -18.5 | +3.4 | +49.5 | +15.9 | -0.3 | -1.6 | +10.1 | |
| Step | 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 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 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 |
| Relative (%) | +33.2 | -33.6 | +8.6 | -41.0 | +16.6 | -19.1 | -48.8 | +27.0 | +7.9 | -6.6 | -16.7 | -22.8 | -25.2 | -24.1 | -19.8 | -12.4 | |
| Step | 83 | 84 | 86 | 87 | 89 | 90 | 91 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 | 101 | |
| Step | Cents | Ratios |
|---|---|---|
| 1 | 59.333 | 30/29, 29/28 |
| 2 | 118.666 | 15/14 |
| 3 | 177.999 | 10/9 |
| 4 | 237.332 | 8/7 |
| 5 | 296.665 | 13/11, 19/16, 6/5 |
| 6 | 355.998 | 11/9, 27/22, 16/13 |
| 7 | 415.331 | 5/4, 14/11 |
| 8 | 474.664 | 25/19, 4/3 |
| 9 | 533.997 | 15/11 |
| 10 | 593.330 | 7/5, 31/22 |
| 11 | 652.663 | 16/11, 19/13 |
| 12 | 711.996 | 3/2 |
| 13 | 771.329 | 14/9, 25/16, 11/7 |
| 14 | 830.662 | 8/5, 21/13, 13/8 |
| 15 | 889.995 | 5/3 |
| 16 | 949.328 | 19/11, 26/15, 7/4 |
| 17 | 1008.661 | 9/5 |
| 18 | 1067.994 | 13/7 |
| 19 | 1127.327 | 23/12 |
| 20 | 1186.660 | 2/1 |
| 22 | 1305.326 | 17/8 |
| 23 | 1364.659 | 11/5 |
| 24 | 1423.992 | 9/4, 25/11, 16/7 |
| 25 | 1483.325 | 7/3 |
| 26 | 1542.657 | 17/7, 22/9 |
| 27 | 1601.990 | 5/2 |
| 28 | 1661.323 | 13/5 |
| 29 | 1720.656 | 8/3, 27/10, 19/7 |
| 30 | 1779.989 | 14/5 |
| 31 | 1839.322 | 26/9, 29/10 |
| 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 |
| 42 | 2491.985 | 21/5, 17/4 |
| 43 | 2551.318 | 13/3 |
| 44 | 2610.651 | 9/2 |
| 45 | 2669.984 | 14/3 |
| 46 | 2729.317 | 29/6 |
| 47 | 2788.650 | 5/1 |
| 48 | 2847.983 | 31/6, 26/5 |
| 49 | 2907.316 | 16/3 |
| 50 | 2966.649 | 11/2 |
| 51 | 3025.982 | 17/3, 23/4 |
| 52 | 3085.315 | 6/1 |
| 54 | 3203.981 | 19/3 |
| 55 | 3263.314 | 13/2 |
| 57 | 3381.980 | 7/1 |
| 58 | 3441.313 | 22/3 |
| 59 | 3500.646 | 15/2 |
| 61 | 3619.312 | 8/1 |
| 62 | 3678.645 | 25/3, 17/2 |
| 63 | 3737.978 | 26/3 |
| 64 | 3797.311 | 9/1 |
| 65 | 3856.644 | 28/3 |
| 66 | 3915.977 | 19/2, 29/3 |
| 67 | 3975.310 | 10/1 |
| 68 | 4034.643 | 31/3 |
| 69 | 4093.976 | 21/2, 32/3 |
| 70 | 4153.309 | 11/1 |
| 71 | 4212.642 | 23/2 |
| 73 | 4331.308 | 12/1 |
| 74 | 4390.641 | 25/2 |
| 75 | 4449.974 | 13/1 |
| 76 | 4509.307 | 27/2 |
| 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 |
| 83 | 4924.637 | 17/1 |
| 84 | 4983.970 | 18/1 |
| 86 | 5102.636 | 19/1 |
| 87 | 5161.969 | 20/1 |
| 89 | 5280.635 | 21/1 |
| 90 | 5339.968 | 22/1 |
| 91 | 5399.301 | 23/1 |
| 93 | 5517.967 | 24/1 |
| 94 | 5577.300 | 25/1 |
| 95 | 5636.633 | 26/1 |
| 96 | 5695.966 | 27/1 |
| 97 | 5755.299 | 28/1 |
| 98 | 5814.632 | 29/1 |
| 99 | 5873.965 | 30/1 |
| 100 | 5933.298 | 31/1 |
| 101 | 5992.631 | 32/1 |
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