186zpi: Difference between revisions
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=== Harmonic series === | === Harmonic series === | ||
{{Harmonics in cet|29.0248832971658|columns=15|title=Approximation of harmonics in 186zpi}} | |||
{{Harmonics in cet|29.0248832971658|columns=16|start=16|title=Approximation of harmonics in 186zpi}} | |||
Revision as of 15:08, 28 June 2024
186 zeta peak index (abbreviated 186zpi), is the equal-step tuning system obtained from the 186st 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 |
| 186zpi | 41.3438354846780 | 29.0248832971658 | 1.876590 | 0.241233 | 11.567493 | 41edo | 1190.02021518380 | 2 | 2 |
Theory
Record on the Riemann zeta function with primes 2 and 3 removed
186zpi sets a height record on the Riemann zeta function with primes 2 and 3 removed. The last record is 125zpi and the next is 565zpi. It is important to highlight that the optimal equal tunings obtained by excluding the prime numbers 2 and 3 from the Riemann zeta function differs very slightly from the optimal equal tuning corresponding to the same peaks on the unmodified Riemann zeta function.
| Unmodified Riemann zeta function | Riemann zeta function with primes 2 and 3 removed | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Tuning | Strength | Closest EDO | Tuning | Strength | Closest EDO | |||||
| ZPI | Steps per octave | Step size (cents) | Height | EDO | Octave (cents) | Steps per octave | Step size (cents) | Height | EDO | Octave (cents) |
| 125zpi | 30.6006474885974 | 39.2148564976330 | 1.468164 | 31edo | 1215.66055142662 | 30.5974484926723 | 39.2189564527704 | 3.769318 | 31edo | 1215.78765003588 |
| 186zpi | 41.3438354846780 | 29.0248832971658 | 1.876590 | 41edo | 1190.02021518380 | 41.3477989230936 | 29.0221010852836 | 4.469823 | 41edo | 1189.90614449663 |
| 565zpi | 98.6209462564991 | 12.1678005084130 | 2.305330 | 99edo | 1204.61225033289 | 98.6257548378926 | 12.1672072570942 | 4.883729 | 99edo | 1204.55351845233 |
Harmonic series
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -10.0 | +13.7 | +9.1 | +0.1 | +3.7 | -1.9 | -0.9 | -1.7 | -9.9 | -0.8 | -6.3 | +0.3 | -11.9 | +13.8 | -10.9 |
| Relative (%) | -34.4 | +47.2 | +31.2 | +0.3 | +12.8 | -6.7 | -3.2 | -5.7 | -34.1 | -2.6 | -21.6 | +1.0 | -41.1 | +47.4 | -37.5 | |
| Step | 41 | 66 | 83 | 96 | 107 | 116 | 124 | 131 | 137 | 143 | 148 | 153 | 157 | 162 | 165 | |
| Harmonic | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.2 | -11.6 | +10.9 | +9.1 | +11.7 | -10.7 | -0.6 | +12.8 | +0.2 | -9.7 | +12.0 | +7.1 | +4.4 | +3.8 | +5.1 | +8.2 |
| Relative (%) | +0.9 | -40.1 | +37.4 | +31.5 | +40.5 | -37.0 | -2.1 | +44.0 | +0.5 | -33.4 | +41.5 | +24.6 | +15.2 | +13.0 | +17.5 | +28.1 | |
| Step | 169 | 172 | 176 | 179 | 182 | 184 | 187 | 190 | 192 | 194 | 197 | 199 | 201 | 203 | 205 | 207 | |