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| '''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>
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| == Theory ==
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| === The basics of Pythagorean tuning ===
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| 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.
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| === Generalization of ratios ===
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| 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.
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| 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.
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| \[\frac{p^x}{q^n} \cdot \frac{q^{n+1}}{p^x} = q\]
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| === Derivation of 'n' ===
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| 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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| 1. Start with the inequality that keeps the ratio within an octave:
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| \[ \frac{3^x}{2^n} \leq 2 \]
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| 2. To find when 'n' needs to increase, we set up the next inequality:
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| \[ \frac{3^x}{2^{n+1}} \leq 1 \]
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| 3. Solving for 'n', we take logarithms of both sides:
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| \[ 2^{n+1} \geq 3^x \]
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| \[ \ln(2^{n+1}) \geq \ln(3^x) \]
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| \[ (n + 1)\ln(2) \geq x\ln(3) \]
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| 4. Isolate 'n' and solve:
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| \[ n \geq \frac{x\ln(3)}{\ln(2)} - 1 \]
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| 5. Since 'n' must be an integer, we apply the ceiling function to get the largest integer less than or equal to the expression:
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| \[ n = \left\lceil \frac{x\ln(3)}{\ln(2)} - 1 \right\rceil \]
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| 6. Simplify:
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| \[ n = \left\lfloor \frac{x\ln(3)}{\ln(2)} \right\rfloor \]
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| === Generating tuple of ratios ===
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| <nowiki>Using the derived value of 'n':
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| \[ n = \left\lfloor \frac{x\ln(p)}{\ln(q)} \right\rfloor \]
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| 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>
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| \[ \prod_{x=0}^{k} R_{x_1} \cdot R_{x_2} = q^{k+1} \]
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| === Set notation for Pythagorean ratios ===
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| Given the definitions of \[ R_{x_1} \text { and } \ R_{x_2} \] (where p=3 and q=2) the set of pythagorean ratios for integer values of x from 0 to 6 is:
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| \[ S = \left\{ R_{x_1}, R_{x_2} \mid x \in \mathbb{Z}, 0 \leq x \leq 6 \right\} \]
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| <nowiki>This notation provides a compact and precise way to represent the set of all such tuples within the specified range of 'x'.</nowiki>
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| === Generating the ratios with Python code ===
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| <syntaxhighlight lang="python">
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| from math import log, floor
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|
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| # Define 'p' and 'q'(q is the octave term, to keep everything within an octave keep q = 2)
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| p = 3
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| q = 2
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|
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| # Define the function to calculate 'n' using the floor function
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| def calculate_n(x, p, q):
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| return floor(x * log(p) / log(q))
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|
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| # Define the range/edo for 'x'
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| limit = 12
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| # Print out the values of 'n' and the ratio for each 'x'
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| for x in range(limit):
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| n = calculate_n(x, p, q)
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| ratio1 = (p**x) / (q**n)
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| print(f'For x={x}, n={n}, the ratio p^x/q^n is: {ratio1}')
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| ratio2 = (q**(n+1)) / (p**x)
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| print(f'For x={x}, n={n}, the ratio q^n+1/p^x is: {ratio2}')
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| </syntaxhighlight>
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| # For x=0, n=0, the ratio p^x/q^n is: 1.0000000000000000
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| # For x=0, n=0, the ratio q^(n+1)/p^x is: 2.0000000000000000
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| # For x=1, n=1, the ratio p^x/q^n is: 1.5000000000000000
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| # For x=1, n=1, the ratio q^(n+1)/p^x is: 1.3333333333333333
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| # For x=2, n=3, the ratio p^x/q^n is: 1.1250000000000000
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| # For x=2, n=3, the ratio q^(n+1)/p^x is: 1.7777777777777777
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| # For x=3, n=4, the ratio p^x/q^n is: 1.6875000000000000
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| # For x=3, n=4, the ratio q^(n+1)/p^x is: 1.1851851851851851
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| # For x=4, n=6, the ratio p^x/q^n is: 1.2656250000000000
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| # For x=4, n=6, the ratio q^(n+1)/p^x is: 1.5802469135802468
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| # For x=5, n=7, the ratio p^x/q^n is: 1.8984375000000000
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| # For x=5, n=7, the ratio q^(n+1)/p^x is: 1.0534979423868314
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| # For x=6, n=9, the ratio p^x/q^n is: 1.4238281250000000
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| # For x=6, n=9, the ratio q^(n+1)/p^x is: 1.4046639231824416
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| # For x=7, n=11, the ratio p^x/q^n is: 1.0678710937500000
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| # For x=7, n=11, the ratio q^(n+1)/p^x is: 1.8728852309099222
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| # For x=8, n=12, the ratio p^x/q^n is: 1.6018066406250000
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| # For x=8, n=12, the ratio q^(n+1)/p^x is: 1.2485901539399482
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| # For x=9, n=14, the ratio p^x/q^n is: 1.2013549804687500
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| # For x=9, n=14, the ratio q^(n+1)/p^x is: 1.6647868719199308
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| # For x=10, n=15, the ratio p^x/q^n is: 1.8020324707031250
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| # For x=10, n=15, the ratio q^(n+1)/p^x is: 1.1098579146132872
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| # For x=11, n=17, the ratio p^x/q^n is: 1.3515243530273438
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| # For x=11, n=17, the ratio q^(n+1)/p^x is: 1.4798105528177163
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| == Implications and applications ==
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| 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.
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| === Chain of fifths / fourths ===
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| The original pythagorean tuning using the ratios from the chain of fifths and chain of fourths in a tuple. The generalized pythagorean ratios shows us how you can generate a scale of kth EDO with only the chain of fifths or it's octave complement, the chain of fourths.
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| <nowiki>
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| \[ S_{p_5} = \left\{ \frac{p^{x}}{q^n} \mid x \in \mathbb{Z}, 0 \leq x \leq k \right\} \]
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| \[ S_{p_4} = \left\{ \frac{q^{n+1}}{p^x} \mid x \in \mathbb{Z}, 0 \leq x \leq k \right\} \]
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| </nowiki>
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| == See also ==
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| * [[Harmonic lattice diagram]]
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| * [[Just intonation subgroup]]
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| * [[Chain of fifths]]
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| == References ==
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| <references/>
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| [[Category:Tuning]]
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| [[Category:Lattice]]
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| [[Category:Subgroup]]
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| {{Todo| review }}
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