User:Sintel/Zeta working page: Difference between revisions
Created page with "== Derivation == Our goal is to derive a function that quantifies how effectively an equal division of the octave (EDO) approximates just intonation (JI). We will use the variable ''x'' to denote an EDO where x=12 represents 12edo. Importantly, we allow x to take continuous values, enabling it to represent any equal-step tuning where the step size is 1200/x cents. For example, x=8.202 corresponds to 13 equal divisions of the tritave (3/1). To a..." |
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== Derivation == | == Derivation == | ||
Our goal is to derive a function that quantifies how effectively an [[equal division of the octave]] (EDO) approximates [[just intonation]] (JI). We will use the variable ''x'' to denote an EDO where x=12 represents [[12edo]]. | Our goal is to derive a function that quantifies how effectively an [[equal division of the octave]] (EDO) approximates [[just intonation]] (JI). We will use the variable ''x'' to denote an EDO where x=12 represents [[12edo]]. | ||
Importantly, we allow x to take continuous values, enabling it to represent any [[equal-step tuning]] where the step size is 1200/x cents. For example, x=8.202 corresponds to [[13edt|13 equal divisions of the tritave]] | Importantly, we allow x to take continuous values, enabling it to represent any [[equal-step tuning]] where the step size is 1200/x cents. For example, x=8.202 corresponds to [[13edt|13 equal divisions of the tritave]]. | ||
To assess how well an equal temperament approximates the [[harmonic series]], we need a function that measures the accuracy of approximation for each overtone. | To assess how well an equal temperament approximates the [[harmonic series]], we need a function that measures the accuracy of approximation for each overtone. | ||
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</math> | </math> | ||
{{todo|add a figure here|inline= | {{todo|text=add a figure here|inline=1}} | ||
Ideally, we would like to extend this function to sum over the entire harmonic series: | Ideally, we would like to extend this function to sum over the entire harmonic series: | ||
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F(x) = \mathrm{Re} \, \zeta \left( \sigma + \frac{2 \pi i}{\ln(2)} x \right) | F(x) = \mathrm{Re} \, \zeta \left( \sigma + \frac{2 \pi i}{\ln(2)} x \right) | ||
</math> | </math> | ||
{{todo|text=add figure of zeta here along s=1 or something|inline=1}} | |||