The Riemann zeta function and tuning: Difference between revisions

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== Mike Battaglia's expanded results ==

=== Zeta yields "relative error" over all rationals ===

Above, Gene proves that the zeta function measures the [[Tenney-Euclidean_metrics|Tenney-Euclidean relative error]], sometimes called "Tenney-Euclidean Simple Badness," of any EDO, taken over all 'prime powers'. The relative error is simply equal to the tuning error times the size of the EDO, so we can easily get the raw "non-relative" tuning error from this as well by simply dividing by the size of the EDO.

Above, Gene proves that the zeta function measures the [[Tenney-Euclidean_metrics|Tenney–Euclidean relative error]], sometimes called "Tenney–Euclidean Simple Badness," of any EDO, taken over all "prime powers". The relative error is simply equal to the tuning error times the size of the EDO, so we can easily get the raw "non-relative" tuning error from this as well by simply dividing by the size of the EDO.

Here, we strengthen that result to show that the zeta function additionally measures weighted relative error over all rational numbers, relative to the size of the EDO.

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\zeta(s) = \sum_n n^{-s}</math>

Now let's do two things: we're going to expand {{nowrap|''s'' {{=}} σ + ''it''}}, and we're going to multiply ζ(s) by its conjugate ζ(''s''){{'}}, noting that {{nowrap|ζ(''s'')' {{=}} ζ(''s''{{'}})}} and {{nowrap|ζ(''s'')·ζ(''s''){{'}} {{=}} ζ(''s'')<sup>2</sup>}}. We get:

Now let's do two things: we're going to expand {{nowrap|''s'' {{=}} σ + ''it''}}, and we're going to multiply ζ(s) by its conjugate ζ(''s''){{'}}, noting that {{nowrap|ζ(''s''){{'}} {{=}} ζ(''s''{{'}})}} and {{nowrap|ζ(''s'')·ζ(''s''){{'}} {{=}} ζ(''s'')<sup>2</sup>}}. We get:

<math> \displaystyle

@@ Line 104: / Line 104: @@
 == Mike Battaglia's expanded results ==
 === Zeta yields "relative error" over all rationals ===
-Above, Gene proves that the zeta function measures the [[Tenney-Euclidean_metrics|Tenney-Euclidean relative error]], sometimes called "Tenney-Euclidean Simple Badness," of any EDO, taken over all 'prime powers'. The relative error is simply equal to the tuning error times the size of the EDO, so we can easily get the raw "non-relative" tuning error from this as well by simply dividing by the size of the EDO.
+Above, Gene proves that the zeta function measures the [[Tenney-Euclidean_metrics|Tenney&ndash;Euclidean relative error]], sometimes called "Tenney&ndash;Euclidean Simple Badness," of any EDO, taken over all "prime powers". The relative error is simply equal to the tuning error times the size of the EDO, so we can easily get the raw "non-relative" tuning error from this as well by simply dividing by the size of the EDO.
 Here, we strengthen that result to show that the zeta function additionally measures weighted relative error over all rational numbers, relative to the size of the EDO.
@@ Line 115: / Line 115: @@
 \zeta(s) = \sum_n n^{-s}</math>
-Now let's do two things: we're going to expand {{nowrap|''s'' {{=}} &sigma; + ''it''}}, and we're going to multiply &zeta;(s) by its conjugate &zeta;(''s''){{'}}, noting that {{nowrap|&zeta;(''s'')' {{=}} &zeta;(''s''{{'}})}} and {{nowrap|&zeta;(''s'')·&zeta;(''s''){{'}} {{=}} &zeta;(''s'')<sup>2</sup>}}. We get:
+Now let's do two things: we're going to expand {{nowrap|''s'' {{=}} &sigma; + ''it''}}, and we're going to multiply &zeta;(s) by its conjugate &zeta;(''s''){{'}}, noting that {{nowrap|&zeta;(''s''){{'}} {{=}} &zeta;(''s''{{'}})}} and {{nowrap|&zeta;(''s'')·&zeta;(''s''){{'}} {{=}} &zeta;(''s'')<sup>2</sup>}}. We get:
 <math> \displaystyle