User:Fastaro/Generalized Pythagorean tuning - Revision history

Fastaro: Blanked the page

2024-02-02T05:55:49Z

Blanked the page

Fastaro: /* Generating the ratios with Python code */ - I changed the scale work shop code a little so the limit will align with the MOS values

2024-01-08T08:13:39Z

Generating the ratios with Python code: - I changed the scale work shop code a little so the limit will align with the MOS values

← Older revision		Revision as of 08:13, 8 January 2024
Line 76:		Line 76:


	# Define the range/edo for 'x'		# Define the range/edo for 'x' - NOTE: this will generate twice the amount ratios as the limit
	limit = 7		limit = 6

	# Initialize lists to store the results		# Initialize lists to store the results
	ratios1 = []		ratios1 = []
	ratios2 = []		ratios2 = []



	# Calculate and store the values of 'n' and the ratios for each 'x'		# Calculate and store the values of 'n' and the ratios for each 'x'
	for x in range(limit):		for x in range(limit+1):

	n = calculate_n(x, p, q)		n = calculate_n(x, p, q)
Line 92:		Line 93:
	ratios1.append(ratio1)		ratios1.append(ratio1)
	# Must have this check so an extra note is not placed in for specific limit		# Must have this check so an extra note is not placed in for specific limit
	if (x != limit-1):		if (x != limit):
	ratio2 = Fraction(q(n+1), px)		ratio2 = Fraction(q(n+1), px)
	ratios2.append(ratio2)		ratios2.append(ratio2)
Line 98:		Line 99:

	combined_ratios = ratios1 + ratios2		combined_ratios = ratios1 + ratios2


	# Sort the combined list from lowest to highest		# Sort the combined list from lowest to highest

Fastaro: /* Finding Potential MOS */

2024-01-08T07:58:29Z

Finding Potential MOS

@@ Line 175: / Line 175: @@
 print('\n'.join(formatted_mos_values))
-</syntaxhighlight>
+</syntaxhighlight>MOS val: 53 Error: 0.0062622215
 == Implications and applications ==

Fastaro: /* Generating the ratios with Python code */

2024-01-08T07:58:08Z

Generating the ratios with Python code

← Older revision		Revision as of 07:58, 8 January 2024
Line 132:		Line 132:

	2/1		2/1

			=== Finding Potential MOS ===
			<syntaxhighlight lang="python">
			from math import log, floor
			from fractions import Fraction

			p = 3
			q = 2

			# Define the function to calculate 'n' using the floor function
			def calculate_n(x, p, q):
			return floor(x * log(p) / log(q))

			mos_limit = 100 # This can be adjusted as needed

			# Initialize lists to store the results
			ratios = []

			# Calculate and store the values of the ratios for each 'x'
			for x in range(mos_limit + 1):
			n = calculate_n(x, p, q)
			ratio1 = Fraction(px, qn)
			ratio2 = Fraction(q(n+1), px)
			ratios.append((ratio1, ratio2))

			# Find Moments of Symmetry (MOS)
			threshold = 0.1 # Define how close the absolute difference must be to 0 or 1 to qualify as MOS
			mos_values = []

			for i, (ratio1, ratio2) in enumerate(ratios):
			abs_diff = abs(float(ratio1) - float(ratio2))
			if abs_diff < threshold:
			mos_values.append((i, abs_diff))
			elif abs(1 - abs_diff) < threshold:
			mos_values.append((i, abs(1 - abs_diff)))

			# Sort MOS values by the error term
			mos_values_sorted = sorted(mos_values, key=lambda x: x[1])[1:]

			# Format the sorted MOS values for printing
			formatted_mos_values = ['MOS val: {} Error: {:.10f}'.format(mos_val, error) for mos_val, error in mos_values_sorted]

			print('\n'.join(formatted_mos_values))
			</syntaxhighlight>

	== Implications and applications ==		== Implications and applications ==

Fastaro: /* Generating the ratios with Python code */ - changed to decimal format

2024-01-08T07:24:07Z

Generating the ratios with Python code: - changed to decimal format

← Older revision		Revision as of 07:24, 8 January 2024
Line 56:		Line 56:
	<syntaxhighlight lang="python">		<syntaxhighlight lang="python">
	from math import log, floor		from math import log, floor
			from fractions import Fraction


	# Define 'p' and 'q'(q is the octave term, to keep everything within an octave keep q = 2)		# Define 'p' and 'q'(q is the octave term, to keep everything within an octave keep q = 2)
	p = 3		p = 3
	q = 2		q = 2

			# Format fractions so 1 can be in denominator
			def format_fraction(frac):
			if frac.denominator == 1:
			return f"{frac.numerator}/1"
			else:
			return str(frac)


	# Define the function to calculate 'n' using the floor function		# Define the function to calculate 'n' using the floor function
	def calculate_n(x, p, q):		def calculate_n(x, p, q):
	return floor(x * log(p) / log(q))		return floor(x * log(p) / log(q))


	# Define the range/edo for 'x'		# Define the range/edo for 'x'
Line 71:		Line 82:
	ratios1 = []		ratios1 = []
	ratios2 = []		ratios2 = []


	# Calculate and store the values of 'n' and the ratios for each 'x'		# Calculate and store the values of 'n' and the ratios for each 'x'
	for x in range(limit):		for x in range(limit):

	n = calculate_n(x, p, q)		n = calculate_n(x, p, q)
	ratio1 = (px~~) / (~~qn)
			ratio1 = Fraction(px, qn)
	ratios1.append(ratio1)		ratios1.append(ratio1)
			# Must have this check so an extra note is not placed in for specific limit
	if (x != limit-1):		if (x != limit-1):
	ratio2 = (q(n+1)~~) / (~~px)		ratio2 = Fraction(q(n+1), px)
	ratios2.append(ratio2)		ratios2.append(ratio2)


	combined_ratios = ratios1 + ratios2		combined_ratios = ratios1 + ratios2

	# Sort the combined list from lowest to highest		# Sort the combined list from lowest to highest
	sorted_combined_ratios = sorted(combined_ratios)		sorted_combined_ratios = sorted(combined_ratios, key=lambda x: float(x))
	~~formatted_ratios~~ = '\n'.join([~~f'{~~ratio~~:.16f}'.replace('.', ',').rstrip('0').rstrip(','~~) for ratio in sorted_combined_ratios[1:]])

			# Convert the sorted ratios to strings, remove the first item (1/1), and put it in scaleworkshop format
			formatted_ratios_as_fractions = '\n'.join([format_fraction(ratio) for ratio in sorted_combined_ratios][1:])


	print(~~formatted_ratios~~)		print(formatted_ratios_as_fractions)

	</syntaxhighlight>~~1,0534979423868314~~		</syntaxhighlight>256/243

	~~1,125~~		9/8

	~~1,1851851851851851~~		32/27

	~~1,265625~~		81/64

	~~1,3333333333333333~~		4/3

	~~1,423828125~~		729/512

	~~1,5~~		3/2

	~~1,5802469135802468~~		128/81

	~~1,6875~~		27/16

	~~1,7777777777777777~~		16/9

	~~1,8984375~~		243/128

	2		2/1

	== Implications and applications ==		== Implications and applications ==

Fastaro: /* Generating the ratios with Python code */ - I changed the python code so it can be used in scale work shop

2024-01-08T06:39:57Z

Generating the ratios with Python code: - I changed the python code so it can be used in scale work shop

@@ Line 89: / Line 89: @@
 print(formatted_ratios)
-</syntaxhighlight>
+</syntaxhighlight>1,0534979423868314
 == Implications and applications ==

Fastaro: /* Generating the ratios with Python code */

2024-01-08T06:38:34Z

Generating the ratios with Python code

@@ Line 66: / Line 66: @@
 # Define the range/edo for 'x'
-limit = 12
+limit = 7
-# Print out the values of 'n' and the ratio for each 'x'
+# Initialize lists to store the results
 for x in range(limit):
      n = calculate_n(x, p, q)
      ratio1 = (p**x) / (q**n)
-     print(f'For x={x}, n={n}, the ratio p^x/q^n is: {ratio1}')
+     ratios1.append(ratio1)
-    ratio2 = (q**(n+1)) / (p**x)
+    if (x != limit-1):
-    print(f'For x={x}, n={n}, the ratio q^n+1/p^x is: {ratio2}')
+        ratio2 = (q**(n+1)) / (p**x)
 </syntaxhighlight>
 == Implications and applications ==

Fastaro: /* Generalization of ratios */

2024-01-08T06:13:10Z

Generalization of ratios

← Older revision		Revision as of 06:13, 8 January 2024
Line 9:		Line 9:

	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.

			\[ \frac{p^x}{q^n} \cdot \frac{q^{n+1}}{p^x} = q\]

	\[ S = \left\{ \frac{p^x}{q^n},\frac{q^{n+1}}{p^x} \mid x \in \mathbb{Z}, 0 \leq x \leq 6 \right\} \]		\[ S = \left\{ \frac{p^x}{q^n},\frac{q^{n+1}}{p^x} \mid x \in \mathbb{Z}, 0 \leq x \leq 6 \right\} \]

Fastaro: /* Generalization of ratios */

2024-01-08T06:11:16Z

Generalization of ratios

← Older revision		Revision as of 06:11, 8 January 2024
Line 8:		Line 8:
	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.

	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.		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.

	\[\frac{p^x}{q^n} ~~\cdot~~ \frac{q^{n+1}}{p^x} ~~= q~~\]		\[ S = \left\{ \frac{p^x}{q^n},\frac{q^{n+1}}{p^x} \mid x \in \mathbb{Z}, 0 \leq x \leq 6 \right\} \]

	=== Derivation of 'n' ===		=== Derivation of 'n' ===

Fredg999: Fredg999 moved page Generalized Pythagorean tuning to User:Fastaro/Generalized Pythagorean tuning: Page not yet ready for main namespace (after discussion on Discord)

2024-01-08T03:47:55Z

Fredg999 moved page Generalized Pythagorean tuning to User:Fastaro/Generalized Pythagorean tuning: Page not yet ready for main namespace (after discussion on Discord)

← Older revision	Revision as of 03:47, 8 January 2024
(No difference)