186zpi: Difference between revisions
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{| class="wikitable" | {| class="wikitable" | ||
! colspan="5" |Unmodified Riemann zeta function | ! colspan="5" |Unmodified Riemann zeta function | ||
! colspan=" | ! colspan="4" |Riemann zeta function with primes 2 and 3 removed | ||
|- | |- | ||
! colspan="3" | Tuning | ! colspan="3" | Tuning | ||
! colspan="2" | Closest EDO | ! colspan="2" | Closest EDO | ||
! colspan=" | ! colspan="2" | Tuning | ||
! colspan="2" | Closest EDO | ! colspan="2" | Closest EDO | ||
|- | |- | ||
| Line 47: | Line 47: | ||
! EDO | ! EDO | ||
! Octave (cents) | ! Octave (cents) | ||
! Steps per octave | ! Steps per octave | ||
! Step size (cents) | ! Step size (cents) | ||
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| [[41edo]] | | [[41edo]] | ||
| 1190.02021518380 | | 1190.02021518380 | ||
| 41.3438354846780 | | 41.3438354846780 | ||
| 29.0248832971658 | | 29.0248832971658 | ||
Revision as of 14:34, 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 | Riemann zeta function with primes 2 and 3 removed | |||||||
|---|---|---|---|---|---|---|---|---|
| Tuning | Closest EDO | Tuning | Closest EDO | |||||
| ZPI | Steps per octave | Step size (cents) | EDO | Octave (cents) | Steps per octave | Step size (cents) | EDO | Octave (cents) |
| 186zpi | 41.3438354846780 | 29.0248832971658 | 41edo | 1190.02021518380 | 41.3438354846780 | 29.0248832971658 | 41edo | 1190.02021518380 |