186zpi
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 previous record is 125zpi and the next one 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
As a non-octave, non-tritave scale, 186zpi features a well-balanced harmonic series segment from 5 to 9, and performs exceptionally well across all prime harmonics from 5 to 23, with the exception of 19.
| 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 | |
Approximation of EDONOIs
Based on harmonics with less than 1 cent of error, 186zpi can be approximated by 96ed5, 124ed8 (or every 3 steps of 124edo), 143ed11, 153ed13, 169ed17, 187ed23, and 192ed25.
Intervals and notation
There are multiple ways to approach notation. The simplest method is to use the notations from 41edo. However, this approach will not preserve octave compression when the audio is rendered by notation software. To address this, consider using the ups and downs notation from 124edo at every 3-degree step (i.e., the edonoi 124ed8).