User:Sintel/Zeta working page: Difference between revisions
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If <math>x \log_2(n)</math> is very close to an integer, it means our equal temperament system can closely approximate that harmonic. If it's far from an integer, the approximation will be poor. | If <math>x \log_2(n)</math> is very close to an integer, it means our equal temperament system can closely approximate that harmonic. If it's far from an integer, the approximation will be poor. | ||
To quantify this proximity to integers, we can use the cosine function <math>f(x) = \cos(2 \pi x)</math>, which peaks at integer values. This provides a simple metric where values close to 1 indicate good approximations and values close to -1 indicate poor ones. | To quantify this proximity to integers, we can use the cosine function <math>f(x) = \cos(2 \pi x)</math>, which peaks at integer values. This provides a simple metric for the [[Relative interval error|relative error]], where values close to 1 indicate good approximations and values close to -1 indicate poor ones. | ||
Let's begin by considering only the first few harmonics and look at <math>f(x \log_2 2)</math>, <math>f(x \log_2 3)</math>, <math>f(x \log_2 4)</math>, <math>f(x \log_2 5)</math>. | Let's begin by considering only the first few harmonics and look at <math>f(x \log_2 2)</math>, <math>f(x \log_2 3)</math>, <math>f(x \log_2 4)</math>, <math>f(x \log_2 5)</math>. | ||
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== Choice of sigma == | == Choice of sigma == | ||
{{todo|inline=1|text=This section is incomplete! Discuss symmetry and add some conclusion}} | {{todo|complete section|inline=1|text=This section is incomplete! Discuss symmetry and add some conclusion}} | ||
The infinite sum derived above converges only when <math>\mathrm{Re}(s) = \sigma > 1</math>. However, mathematicians found that the zeta function can be "extended" or "continued" to other values of s where the original sum doesn't converge. This mathematical technique is called {{w|analytic continuation}}. | The infinite sum derived above converges only when <math>\mathrm{Re}(s) = \sigma > 1</math>. However, mathematicians found that the zeta function can be "extended" or "continued" to other values of s where the original sum doesn't converge. This mathematical technique is called {{w|analytic continuation}}. | ||
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== Zeta lists == | == Zeta lists == | ||
{{todo|inline=1|text=add the original lists back here, and discuss their relevance.}} | {{todo|complete list|inline=1|text=add the original lists back here, and discuss their relevance.}} | ||
== Optimal octave stretch == | == Optimal octave stretch == | ||
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[[Category:Number theory]] | [[Category:Number theory]] | ||
[[Category:Pages with proofs]] | [[Category:Pages with proofs]] | ||