User:Fastaro/Generalized Pythagorean tuning: Difference between revisions
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=== Generating tuple of ratios === | === Generating tuple of ratios === | ||
<nowiki>Using the derived value of 'n' | <nowiki>Using the derived value of 'n': | ||
\[ n = \left\lfloor \frac{x\ln(p)}{\ln(q)} \right\rfloor \] | |||
we can generate a tuple of ratios \[ R_{x_1} \text {and}\ R_{x_2} \text{ , where } R_{x_1} = \frac{p^x}{q^n} \text { and}\ R_{x_2} = \frac{q^{n+1}}{p^x} \]. This pair of ratios represents the upper and lower bounds of a frequency range for a given 'x'. The product of \[ R_{x_1} \cdot R_{x_2} \] for all 'x' from 0 to 'k' yields the result:</nowiki> | |||
\[ \prod_{x=0}^{k} R_{x_1} \cdot R_{x_2} = q^{k+1} \] | \[ \prod_{x=0}^{k} R_{x_1} \cdot R_{x_2} = q^{k+1} \] | ||