186zpi: Difference between revisions
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== Theory == | == Theory == | ||
'''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. | '''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. | ||
{| class="wikitable" | |||
! colspan="5" |Unmodified Riemann zeta function | |||
|- | |||
! colspan="3" | Tuning | |||
! colspan="2" | Closest EDO | |||
|- | |||
! ZPI | |||
! Steps per octave | |||
! Step size (cents) | |||
! EDO | |||
! Octave (cents) | |||
|- | |||
|[[186zpi]] | |||
| 41.3438354846780 | |||
| 29.0248832971658 | |||
|[[41edo]] | |||
| 1190.02021518380 | |||
|} | |||
Revision as of 14:32, 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
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 | ||||
|---|---|---|---|---|
| Tuning | Closest EDO | |||
| ZPI | Steps per octave | Step size (cents) | EDO | Octave (cents) |
| 186zpi | 41.3438354846780 | 29.0248832971658 | 41edo | 1190.02021518380 |