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Generalized Pythagorean Tuning

Introduction

Generalized Pythagorean Tuning is an extension of the traditional Pythagorean tuning method, which is based on chains of perfect fifths and fourths. This method extends the Pythagorean ratios to any two numbers, not just 3 and 2, allowing for a more versatile approach to musical tuning.

The Basics of Pythagorean Tuning

Pythagorean tuning is a system based on the ratio of 3/2, known as a perfect fifth. The method involves generating scales through a chain of fifths, multiplying the frequency by 3/2 until passing an octave. This system is limited by the specific ratios it employs and does not return to the unison ratio of 1/1, leading to the need for more generalized methods.

Generalization of Ratios

The generalized Pythagorean tuning considers 'q' as the octave term and 'p' 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 beyond the fixed intervals of 3/2 and 4/3, allowing any two numbers to define the tuning system.

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:

1. Start with the inequality that keeps the ratio within an octave: \[ \frac{3^x}{2^n} \leq 2 \]

2. To find when 'n' needs to increase, we set up the next inequality: \[ \frac{3^x}{2^{n+1}} \leq 1 \]

3. Solving for 'n', we take logarithms of both sides: \[ 2^{n+1} \geq 3^x \] \[ \ln(2^{n+1}) \geq \ln(3^x) \] \[ (n + 1)\ln(2) \geq x\ln(3) \]

4. Isolate 'n' and solve: \[ n \geq \frac{x\ln(3)}{\ln(2)} - 1 \]

5. Since 'n' must be an integer, we apply the floor function to get the largest integer less than or equal to the expression: \[ n = \left\lfloor \frac{x\ln(3)}{\ln(2)} - 1 \right\rfloor \]

Generating Tuple of Ratios

Using the derived value of 'n', we can generate a tuple of ratios \[ R_{x_1} \] and \[ R_{x_2} \], where \[ R_{x_1} = \frac{p^x}{q^n} \] 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 \[ R_{x_1} \text and\ R_{x_2} \] for all 'x' from 0 to 'k' yields the series:

\[ \prod_{x=0}^{k} R_{x_1} \cdot R_{x_2} = q^{k+1} \]

This series of ratios describes the continuum of the Pythagorean Ratios from the smallest numerator/denominator to the largest, encompassing all the Pythagorean Ratios within the range of 'k' octaves.

Implications and Applications

The generalized Pythagorean tuning provides a more versatile framework for musical tuning, allowing composers and musicians to explore scales and harmonies beyond the traditional limits. This approach can lead to new musical expressions and better alignment with various musical traditions and instruments.