Riemann zeta function: Difference between revisions

Line 712:

Multiplying the Z-function by this factor of adjustment gives a Z-function with the prime ''p'' removed from consideration. Zeta peak and zeta integral tunings may then be found as before.

~~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, 56, 69, 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.

For example, the function used to find peak values of EDTs (division of the [[3/1|{{ordinal|3}}]] harmonic, or "tritave") is

<math>

\displaystyle Z\left(\frac{2\pi}{\ln(3)}t\right)\sqrt{\frac{3}{2}-\sqrt{2}\cos\left(\frac{2\pi\ln(2)}{\ln(3)}t\right)}

</math>

Removing 2 leads to increasing adjusted peak values corresponding to edts into {{EDTs| 4, 7, 9, 13, 15, 17, 26, 32, 39, 56, 69, 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 ==

@@ Line 712: / Line 712: @@
 Multiplying the Z-function by this factor of adjustment gives a Z-function with the prime ''p'' removed from consideration. Zeta peak and zeta integral tunings may then be found as before.
-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, 56, 69, 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.
+For example, the function used to find peak values of EDTs (division of the [[3/1|{{ordinal|3}}]] harmonic, or "tritave") is
+<math>
+\displaystyle Z\left(\frac{2\pi}{\ln(3)}t\right)\sqrt{\frac{3}{2}-\sqrt{2}\cos\left(\frac{2\pi\ln(2)}{\ln(3)}t\right)}
+</math>
+Removing 2 leads to increasing adjusted peak values corresponding to edts into {{EDTs| 4, 7, 9, 13, 15, 17, 26, 32, 39, 56, 69, 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 ==