The Riemann zeta function and tuning: Difference between revisions
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m black magic formulas do not have to do with removing primes |
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Removing 2 leads to increasing adjusted peak values corresponding to edts (division of the [[3/1|{{ordinal|3}}]] harmonic, or "tritave") into {{EDTs| 4, 7, 9, 13, 15, 17, 26, 32, 39, 45, 52, 56, 71, 75, 88, 131, 245, 316,}} … parts. We can also compare zeta peak EDTs with pure and tempered tritaves just like [[#zeta peak edos|zeta peak]] edos. A striking feature of this list is the appearance not only of [[13edt]], the [[Bohlen–Pierce]] division of the tritave, but the multiples 26, 39, and 52 also. | Removing 2 leads to increasing adjusted peak values corresponding to edts (division of the [[3/1|{{ordinal|3}}]] harmonic, or "tritave") into {{EDTs| 4, 7, 9, 13, 15, 17, 26, 32, 39, 45, 52, 56, 71, 75, 88, 131, 245, 316,}} … parts. We can also compare zeta peak EDTs with pure and tempered tritaves just like [[#zeta peak edos|zeta peak]] edos. A striking feature of this list is the appearance not only of [[13edt]], the [[Bohlen–Pierce]] division of the tritave, but the multiples 26, 39, and 52 also. | ||
== Black magic formulas == | |||
When [[Gene Ward Smith|Gene Smith]] discovered these formulas in the 70s, he thought of them as "black magic" formulas not because of any aura of evil, but because they seemed mysteriously to give you something for next to nothing. They are based on Gram points and the Riemann–Siegel theta function θ(''t''). Recall that a Gram point is a point on the critical line where {{nowrap|ζ({{frac|1|2}} + ''ig'')}} is real. This implies that exp(''i''θ(''g'')) is real, so that {{frac|θ(''g'')|π}} is an integer. Theta has an {{w|asymptotic expansion}} | When [[Gene Ward Smith|Gene Smith]] discovered these formulas in the 70s, he thought of them as "black magic" formulas not because of any aura of evil, but because they seemed mysteriously to give you something for next to nothing. They are based on Gram points and the Riemann–Siegel theta function θ(''t''). Recall that a Gram point is a point on the critical line where {{nowrap|ζ({{frac|1|2}} + ''ig'')}} is real. This implies that exp(''i''θ(''g'')) is real, so that {{frac|θ(''g'')|π}} is an integer. Theta has an {{w|asymptotic expansion}} | ||