The Riemann zeta function and tuning: Difference between revisions

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=== Interpretation of results: "cosine relative error" ===

For every strictly positive rational ''n''/''d'', there is a cosine with period {{nowrap|2π log2({{frac|''n''|''d''}})}}. This cosine peaks at {{nowrap|''x'' {{=}} {{sfrac|''N''|log2(''n''/''d'')}}}} for all integers ''N'', or in other words, the Nth-equal division of the rational number {{frac|''n''|''d''}}, and hits troughs midway between.

For every strictly positive rational ''n''/''d'', there is a cosine with period {{nowrap|log2({{frac|''n''|''d''}})}}. This cosine peaks at {{nowrap|''x'' {{=}} {{sfrac|''N''|log2(''n''/''d'')}}}} for all integers ''N'', or in other words, the Nth-equal division of the rational number {{frac|''n''|''d''}}, and hits troughs midway between.

Our mysterious substitution above was chosen to set the units for this up nicely. The variable x now happens to be measured in divisions of the octave. (The original variable ''t'', which was the imaginary part of the zeta argument ''s'', can be thought of as the number of divisions of the interval {{nowrap|''e''2π ≈ 535.49}}, or what [[Keenan_Pepper|Keenan Pepper]] has called the "[[zetave|natural interval]].")

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 === Interpretation of results: "cosine relative error" ===
-For every strictly positive rational ''n''/''d'', there is a cosine with period {{nowrap|2π log<sub>2</sub>({{frac|''n''|''d''}})}}. This cosine peaks at {{nowrap|''x'' {{=}} {{sfrac|''N''|log<sub>2</sub>(''n''/''d'')}}}} for all integers ''N'', or in other words, the Nth-equal division of the rational number {{frac|''n''|''d''}}, and hits troughs midway between.
+For every strictly positive rational ''n''/''d'', there is a cosine with period {{nowrap|log<sub>2</sub>({{frac|''n''|''d''}})}}. This cosine peaks at {{nowrap|''x'' {{=}} {{sfrac|''N''|log<sub>2</sub>(''n''/''d'')}}}} for all integers ''N'', or in other words, the Nth-equal division of the rational number {{frac|''n''|''d''}}, and hits troughs midway between.
 Our mysterious substitution above was chosen to set the units for this up nicely. The variable x now happens to be measured in divisions of the octave. (The original variable ''t'', which was the imaginary part of the zeta argument ''s'', can be thought of as the number of divisions of the interval {{nowrap|''e''<sup>2π</sup> ≈ 535.49}}, or what [[Keenan_Pepper|Keenan Pepper]] has called the "[[zetave|natural interval]].")