User:Fastaro/Generalized Pythagorean tuning: Difference between revisions
m Fredg999 moved page Generalized Pythagorean Tuning to Generalized Pythagorean tuning: WP:NCCAPS |
Cleanup |
||
| Line 1: | Line 1: | ||
'''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]].<ref>Fustero, Robert. [https://robertfustero.medium.com/fourier-series-of-music-902c9dd57629 "The Fourier Series of Music"]</ref> | |||
== | == Theory == | ||
=== 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. | |||
== | === Generalization of ratios === | ||
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. | 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. | ||
| Line 14: | Line 12: | ||
\[\frac{p^x}{q^n} \cdot \frac{q^{n+1}}{p^x} = q\] | \[\frac{p^x}{q^n} \cdot \frac{q^{n+1}}{p^x} = q\] | ||
== 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: | ||
| Line 37: | Line 35: | ||
\[ n = \left\lfloor \frac{x\ln(3)}{\ln(2)} \right\rfloor \] | \[ n = \left\lfloor \frac{x\ln(3)}{\ln(2)} \right\rfloor \] | ||
== Generating | === Generating tuple of ratios === | ||
<nowiki>Using the derived value of 'n', 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> | <nowiki>Using the derived value of 'n', 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> | ||
\[ \prod_{x=0}^{k} R_{x_1} \cdot R_{x_2} = q^{k+1} \] | \[ \prod_{x=0}^{k} R_{x_1} \cdot R_{x_2} = q^{k+1} \] | ||
== Generating the | === Generating the ratios with Python code === | ||
< | <syntaxhighlight lang="python"> | ||
from math import log, floor | from math import log, floor | ||
| Line 64: | Line 62: | ||
ratio2 = (q**(n+1)) / (p**x) | ratio2 = (q**(n+1)) / (p**x) | ||
print(f'For x={x}, n={n}, the ratio q^n+1/p^x is: {ratio2}') | print(f'For x={x}, n={n}, the ratio q^n+1/p^x is: {ratio2}') | ||
</ | </syntaxhighlight> | ||
# For x=0, n=0, the ratio p^x/q^n is: 1.0000000000000000 | # For x=0, n=0, the ratio p^x/q^n is: 1.0000000000000000 | ||
| Line 91: | Line 89: | ||
# For x=11, n=17, the ratio q^(n+1)/p^x is: 1.4798105528177163 | # For x=11, n=17, the ratio q^(n+1)/p^x is: 1.4798105528177163 | ||
== Implications and | == 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. | 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. | ||
== See also == | |||
* [[Harmonic lattice diagram]] | |||
* [[Just intonation subgroup]] | |||
== References == | |||
<references/> | |||
[[Category:Tuning]] | |||
[[Category:Lattice]] | |||
[[Category:Subgroup]] | |||