User:Fastaro/Generalized Pythagorean tuning: Difference between revisions
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Every pair of Pythagorean ratios with 3^x in the numerator and 3^x denominator always equals 2. In fact, Pythagorean tuning can be viewed as one particular case of the equation below where p=3 and q =2. | Every pair of Pythagorean ratios with 3^x in the numerator and 3^x denominator always equals 2. In fact, Pythagorean tuning can be viewed as one particular case of the equation below where p=3 and q =2. | ||
\[\frac{p^x}{q^n} \cdot \frac{q^n+1}{p^x} | \[\frac{p^x}{q^n} \cdot \frac{q^{n+1}}{p^x} = q\] | ||
== Derivation of 'n' == | == 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: | ||