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>. | ||