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'''186 zeta peak index''' (abbreviated '''186zpi'''), is the [[Equal-step tuning|equal-step]] [[tuning system]] obtained from the 186st [[Zeta peak index|peak]] of the [[The Riemann zeta function and tuning|Riemann zeta function]]. | '''186 zeta peak index''' (abbreviated '''186zpi'''), is the [[Equal-step tuning|equal-step]] [[tuning system]] obtained from the 186st [[Zeta peak index|peak]] of the [[The Riemann zeta function and tuning|Riemann zeta function]]. | ||
{ | {{ZPI | ||
| zpi = 186 | |||
| steps = 41.3438354846780 | |||
| step size = 29.0248832971658 | |||
| height = 1.876590 | |||
| integral = 0.241233 | |||
| gap = 11.567493 | |||
| edo = 41edo | |||
| octave = 1190.02021518380 | |||
| consistent = 2 | |||
| distinct = 2 | |||
}} | |||
| 29.0248832971658 | |||
| 1.876590 | |||
| 0.241233 | |||
| 11.567493 | |||
| | |||
| 1190.02021518380 | |||
| 2 | |||
| 2 | |||
== Theory == | == Theory == | ||
=== Record on the Riemann zeta function with primes 2 and 3 removed === | === 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. | '''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. | ||
{| class="wikitable" | {| class="wikitable" | ||
! colspan="6" |Unmodified Riemann zeta function | |- | ||
! colspan="5" |Riemann zeta function with primes 2 and 3 removed | ! colspan="6" | Unmodified Riemann zeta function | ||
! colspan="5" | Riemann zeta function with primes 2 and 3 removed | |||
|- | |- | ||
! colspan="3" | Tuning | ! colspan="3" | Tuning | ||
| Line 61: | Line 45: | ||
| 30.6006474885974 | | 30.6006474885974 | ||
| 39.2148564976330 | | 39.2148564976330 | ||
|1.468164 | | 1.468164 | ||
| [[31edo]] | | [[31edo]] | ||
| 1215.66055142662 | | 1215.66055142662 | ||
| 30.5974484926723 | | 30.5974484926723 | ||
| 39.2189564527704 | | 39.2189564527704 | ||
|3.769318 | | 3.769318 | ||
| [[31edo]] | | [[31edo]] | ||
| 1215.78765003588 | | 1215.78765003588 | ||
|- | |- | ||
|[[186zpi]] | | [[186zpi]] | ||
|41.3438354846780 | | 41.3438354846780 | ||
|29.0248832971658 | | 29.0248832971658 | ||
|1.876590 | | 1.876590 | ||
|[[41edo]] | | [[41edo]] | ||
|1190.02021518380 | | 1190.02021518380 | ||
|41.3477989230936 | | 41.3477989230936 | ||
|29.0221010852836 | | 29.0221010852836 | ||
|4.469823 | | 4.469823 | ||
|[[41edo]] | | [[41edo]] | ||
|1189.90614449663 | | 1189.90614449663 | ||
|- | |- | ||
|[[565zpi]] | | [[565zpi]] | ||
|98.6209462564991 | | 98.6209462564991 | ||
|12.1678005084130 | | 12.1678005084130 | ||
|2.305330 | | 2.305330 | ||
|[[99edo]] | | [[99edo]] | ||
|1204.61225033289 | | 1204.61225033289 | ||
|98.6257548378926 | | 98.6257548378926 | ||
|12.1672072570942 | | 12.1672072570942 | ||
|4.883729 | | 4.883729 | ||
|[[99edo]] | | [[99edo]] | ||
|1204.55351845233 | | 1204.55351845233 | ||
|} | |} | ||
| Line 108: | Line 92: | ||
{| class="wikitable center-1 right-2 left-3 center-4 center-5" | {| class="wikitable center-1 right-2 left-3 center-4 center-5" | ||
|+ style="white-space:nowrap" | Intervals in 186zpi | |+ style="font-size: 105%; white-space: nowrap;" | Intervals in 186zpi | ||
|- | |- | ||
| colspan="3" style="text-align:left;" | JI ratios are comprised of [[32-integer-limit]] ratios,<br>and are stylized as follows to indicate their accuracy: | | colspan="3" style="text-align:left;" | JI ratios are comprised of [[32-integer-limit]] ratios,<br>and are stylized as follows to indicate their accuracy: | ||
| Line 122: | Line 106: | ||
! Cents | ! Cents | ||
! Ratios | ! Ratios | ||
! Ups and | ! Ups and downs notation | ||
! Step | ! Step | ||
|- | |- | ||