User:Fastaro/Generalized Pythagorean tuning: Difference between revisions

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 == Derivation of 'n' ==
-In generalized Pythagorean tuning, the goal is to find values of 'n' that keep the ratio \( \frac{p^x}{q^n} \) within an octave. This is achieved by ensuring that the ratio does not exceed 2 (the frequency doubling that marks the octave). When the ratio \( \frac{3^x}{2^n} \) is greater than 2, we add 1 to 'n' to bring the ratio back within the octave range. To avoid using an 'if' statement and make the function linear, we derive 'n' as follows:
+In generalized Pythagorean tuning, the goal is to find values of 'n' that keep the ratio \[ \frac{p^x}{q^n} \] within an octave. This is achieved by ensuring that the ratio does not exceed 2 (the frequency doubling that marks the octave). When the ratio \[ \frac{3^x}{2^n} \] is greater than 2, we add 1 to 'n' to bring the ratio back within the octave range. To avoid using an 'if' statement and make the function linear, we derive 'n' as follows:
 . Start with the inequality that keeps the ratio within an octave: