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 (its octave complement). 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 (its octave complement). In fact, Pythagorean tuning can be viewed as one particular case of the equation below where p=3 and q =2. | ||
\[ \frac{p^x}{q^n} \cdot \frac{q^{n+1}}{p^x} = q\] | |||
\[ S = \left\{ \frac{p^x}{q^n},\frac{q^{n+1}}{p^x} \mid x \in \mathbb{Z}, 0 \leq x \leq 6 \right\} \] | \[ S = \left\{ \frac{p^x}{q^n},\frac{q^{n+1}}{p^x} \mid x \in \mathbb{Z}, 0 \leq x \leq 6 \right\} \] | ||