34zpi: Difference between revisions
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{| class="wikitable center-1 right-2 left-3 center-4" | {| class="wikitable center-1 right-2 left-3 center-4" | ||
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|+ style="white-space:nowrap" | <big> | |+ style="white-space:nowrap" | <big>Intervals in 34zpi</big> | ||
| colspan="3" style="text-align:left;" | JI ratios are comprised of 16-integer limit ratios,<br>and are stylized as follows to indicate their accuracy: | | colspan="3" style="text-align:left;" | JI ratios are comprised of 16-integer limit ratios,<br>and are stylized as follows to indicate their accuracy: | ||
* '''<u>Bold Underlined:</u>''' relative error < 8.333 % | * '''<u>Bold Underlined:</u>''' relative error < 8.333 % | ||
Revision as of 15:49, 11 August 2024
34 zeta peak index (abbreviated 34zpi), is the equal-step tuning system obtained from the 34th 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 |
| 34zpi | 12.0231830072926 | 99.8071807833375 | 5.193290 | 1.269599 | 15.899282 | 12edo | 1197.68616940005 | 10 | 6 |
Intervals
| JI ratios are comprised of 16-integer limit ratios, and are stylized as follows to indicate their accuracy:
|
Whole tone = 2 steps Limma = 1 step Apotome = 1 step | ||
| Degree | Cents | Ratios | Ups and Downs Notation |
|---|---|---|---|
| 0 | 0.000 | P1 | |
| 1 | 99.807 | 16/15, 15/14, 14/13, 13/12 | m2 |
| 2 | 199.614 | 12/11, 11/10, 10/9, 9/8, 8/7, 15/13 | M2 |
| 3 | 299.422 | 7/6, 13/11, 6/5, 11/9 | m3 |
| 4 | 399.229 | 16/13, 5/4, 14/11, 9/7 | M3 |
| 5 | 499.036 | 13/10, 4/3, 15/11 | P4 |
| 6 | 598.843 | 11/8, 7/5, 10/7, 13/9, 16/11 | A4, d5 |
| 7 | 698.650 | 3/2 | P5 |
| 8 | 798.457 | 14/9, 11/7, 8/5, 13/8 | m6 |
| 9 | 898.265 | 5/3, 12/7 | M6 |
| 10 | 998.072 | 7/4, 16/9, 9/5 | m7 |
| 11 | 1097.879 | 11/6, 13/7, 15/8 | M7 |
| 12 | 1197.686 | 2/1 | P1 +1 oct |
| 13 | 1297.493 | 15/7, 13/6 | m2 +1 oct |
| 14 | 1397.301 | 11/5, 9/4, 16/7 | M2 +1 oct |
| 15 | 1497.108 | 7/3, 12/5 | m3 +1 oct |
| 16 | 1596.915 | 5/2 | M3 +1 oct |
| 17 | 1696.722 | 13/5, 8/3 | P4 +1 oct |
| 18 | 1796.529 | 11/4, 14/5 | A4 +1 oct, d5 +1 oct |
| 19 | 1896.336 | 3/1 | P5 +1 oct |
| 20 | 1996.144 | 16/5, 13/4 | m6 +1 oct |
| 21 | 2095.951 | 10/3 | M6 +1 oct |
| 22 | 2195.758 | 7/2 | m7 +1 oct |
| 23 | 2295.565 | 11/3, 15/4 | M7 +1 oct |
| 24 | 2395.372 | 4/1 | P1 +2 oct |
| 25 | 2495.180 | 13/3 | m2 +2 oct |
| 26 | 2594.987 | 9/2 | M2 +2 oct |
| 27 | 2694.794 | 14/3 | m3 +2 oct |
| 28 | 2794.601 | 5/1 | M3 +2 oct |
| 29 | 2894.408 | 16/3 | P4 +2 oct |
| 30 | 2994.215 | 11/2 | A4 +2 oct, d5 +2 oct |
| 31 | 3094.023 | 6/1 | P5 +2 oct |
| 32 | 3193.830 | 13/2 | m6 +2 oct |
| 33 | 3293.637 | M6 +2 oct | |
| 34 | 3393.444 | 7/1 | m7 +2 oct |
| 35 | 3493.251 | 15/2 | M7 +2 oct |
| 36 | 3593.059 | 8/1 | P1 +3 oct |
| 37 | 3692.866 | m2 +3 oct | |
| 38 | 3792.673 | 9/1 | M2 +3 oct |
| 39 | 3892.480 | m3 +3 oct | |
| 40 | 3992.287 | 10/1 | M3 +3 oct |
| 41 | 4092.094 | P4 +3 oct | |
| 42 | 4191.902 | 11/1 | A4 +3 oct, d5 +3 oct |
| 43 | 4291.709 | 12/1 | P5 +3 oct |
| 44 | 4391.516 | 13/1 | m6 +3 oct |
| 45 | 4491.323 | M6 +3 oct | |
| 46 | 4591.130 | 14/1 | m7 +3 oct |
| 47 | 4690.937 | 15/1 | M7 +3 oct |
| 48 | 4790.745 | 16/1 | P1 +4 oct |
Approximation to JI
| Ratio | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 4/3 | -0.991 | -0.993 |
| 8/3 | +1.323 | +1.325 |
| 16/9 | -1.982 | -1.986 |
| 2/1 | +2.314 | +2.318 |
| 15/1 | -2.669 | -2.674 |
| 3/2 | +3.305 | +3.311 |
| 16/3 | +3.637 | +3.644 |
| 9/8 | +4.296 | +4.304 |
| 4/1 | +4.628 | +4.637 |
| 15/2 | -4.983 | -4.992 |
| 3/1 | +5.619 | +5.629 |
| 10/1 | -5.974 | -5.985 |
| 9/4 | +6.609 | +6.622 |
| 8/1 | +6.941 | +6.955 |
| 15/4 | -7.296 | -7.311 |
| 6/1 | +7.932 | +7.948 |
| 5/1 | -8.287 | -8.303 |
| 9/2 | +8.923 | +8.941 |
| 16/1 | +9.255 | +9.273 |
| 15/8 | -9.610 | -9.629 |
| 13/11 | -10.212 | -10.232 |
| 12/1 | +10.246 | +10.266 |
| 5/2 | -10.601 | -10.622 |
| 9/1 | +11.237 | +11.259 |
| 10/3 | -11.592 | -11.614 |
| 16/15 | +11.924 | +11.947 |
| 5/4 | -12.915 | -12.940 |
| 5/3 | -13.906 | -13.933 |
| 14/5 | -14.017 | -14.044 |
| 8/5 | +15.229 | +15.258 |
| 11/7 | -15.965 | -15.996 |
| 6/5 | +16.220 | +16.251 |
| 7/5 | -16.331 | -16.362 |
| 10/9 | -17.211 | -17.244 |
| 16/5 | +17.543 | +17.577 |
| 14/11 | +18.279 | +18.315 |
| 12/5 | +18.534 | +18.569 |
| 10/7 | +18.645 | +18.681 |
| 9/5 | +19.524 | +19.562 |
| 15/14 | +19.636 | +19.674 |
| 15/7 | +21.949 | +21.992 |
| 14/1 | -22.304 | -22.347 |
| 7/1 | -24.618 | -24.666 |
| 13/7 | -26.177 | -26.228 |
| 7/2 | -26.932 | -26.984 |
| 14/3 | -27.923 | -27.977 |
| 14/13 | +28.491 | +28.546 |
| 7/4 | -29.246 | -29.302 |
| 7/3 | -30.237 | -30.295 |
| 8/7 | +31.560 | +31.621 |
| 11/5 | -32.296 | -32.359 |
| 7/6 | -32.551 | -32.614 |
| 14/9 | -33.542 | -33.606 |
| 16/7 | +33.874 | +33.939 |
| 11/10 | -34.610 | -34.677 |
| 12/7 | +34.864 | +34.932 |
| 9/7 | +35.855 | +35.925 |
| 13/9 | +37.775 | +37.848 |
| 15/11 | +37.915 | +37.988 |
| 13/12 | +38.765 | +38.840 |
| 16/13 | -39.756 | -39.833 |
| 11/1 | -40.584 | -40.662 |
| 13/6 | +41.079 | +41.159 |
| 13/8 | +42.070 | +42.151 |
| 13/5 | -42.508 | -42.590 |
| 11/2 | -42.897 | -42.980 |
| 13/3 | +43.393 | +43.477 |
| 13/4 | +44.384 | +44.470 |
| 13/10 | -44.822 | -44.909 |
| 11/4 | -45.211 | -45.299 |
| 11/3 | -46.202 | -46.291 |
| 13/2 | +46.698 | +46.788 |
| 11/8 | -47.525 | -47.617 |
| 11/9 | +47.986 | +48.079 |
| 15/13 | +48.127 | +48.220 |
| 11/6 | -48.516 | -48.610 |
| 12/11 | -48.977 | -49.072 |
| 13/1 | +49.012 | +49.106 |
| 16/11 | +49.839 | +49.935 |
See also
|
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