User:Fastaro/Generalized Pythagorean tuning: Difference between revisions

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== Generalization of Ratios ==

The generalized Pythagorean ~~tuning~~ considers 'q'(2) as the octave term and 'p'(3) as the term usually associated with the fifth, such as in the traditional 3/2 ratio for a perfect fifth. The idea is to extend the Pythagorean tuning by generalizing the standard chain of fifths and fourths (using 3 and 2) method.

The generalized Pythagorean ratios considers 'q'(2) as the octave term and 'p'(3) as the term usually associated with the fifth, such as in the traditional 3/2 ratio for a perfect fifth. The idea is to extend the Pythagorean tuning by generalizing the standard chain of fifths and fourths (using 3 and 2) method.

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}\ = q]

== 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:

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 == Generalization of Ratios ==
-The generalized Pythagorean tuning considers 'q'(2) as the octave term and 'p'(3) as the term usually associated with the fifth, such as in the traditional 3/2 ratio for a perfect fifth. The idea is to extend the Pythagorean tuning by generalizing the standard chain of fifths and fourths (using 3 and 2) method.
+The generalized Pythagorean ratios considers 'q'(2) as the octave term and 'p'(3) as the term usually associated with the fifth, such as in the traditional 3/2 ratio for a perfect fifth. The idea is to extend the Pythagorean tuning by generalizing the standard chain of fifths and fourths (using 3 and 2) method.
+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}\ = q]
 == 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: