The Riemann zeta function and tuning: Difference between revisions
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<math>\displaystyle \xi_p(x) = \sum_{\substack{2 \leq q \leq p \\ q \text{ prime}}} \left(\frac{\rround{x \log_2 q}}{\log_2 q}\right)^2</math> | <math>\displaystyle \xi_p(x) = \sum_{\substack{2 \leq q \leq p \\ q \text{ prime}}} \left(\frac{\rround{x \log_2 q}}{\log_2 q}\right)^2</math> | ||
Here, the numerator represents the relative error on each prime, and the denominator represents a weight factor of the logarithm of each prime. | |||
This function has local minima, corresponding to associated generalized patent vals. The minima occur for values of ''x'' which are the [[Tenney-Euclidean_Tuning|Tenney-Euclidean tuning]]s of the octaves of the associated vals, while ξ<sub>''p''</sub> for these minima is the square of the [[Tenney–Euclidean_metrics|Tenney–Euclidean relative error]] of the val—equal to the TE error times the TE complexity, and sometimes known as "TE simple badness." | This function has local minima, corresponding to associated generalized patent vals. The minima occur for values of ''x'' which are the [[Tenney-Euclidean_Tuning|Tenney-Euclidean tuning]]s of the octaves of the associated vals, while ξ<sub>''p''</sub> for these minima is the square of the [[Tenney–Euclidean_metrics|Tenney–Euclidean relative error]] of the val—equal to the TE error times the TE complexity, and sometimes known as "TE simple badness." | ||