Operations on intervals - Revision history

Inthar: /* For JI ratios */

2026-02-09T01:54:33Z

For JI ratios

← Older revision		Revision as of 01:54, 9 February 2026
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	=== For JI ratios ===		=== For JI ratios ===
	~~Example~~:		Examples:
	# <math>\frac{1}{1} \oplus \frac{1}{0} = \frac{1+1}{1+0} = \frac{2}{1}</math>		# <math>\frac{1}{1} \oplus \frac{1}{0} = \frac{1+1}{1+0} = \frac{2}{1}</math>
	# <math>\frac{1}{1} \oplus \frac{2}{1} = \frac{1+2}{1+1} = \frac{3}{2}</math>		# <math>\frac{1}{1} \oplus \frac{2}{1} = \frac{1+2}{1+1} = \frac{3}{2}</math>

Inthar: /* Linear stretching */

2026-01-07T22:13:46Z

Linear stretching

← Older revision		Revision as of 22:13, 7 January 2026
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	* 13:16:19 {{=}} (4+1/3):(5+1/3):(6+1/3) is a linear compression of 4:5:6 (approximately 0-6-11\20)		* 13:16:19 {{=}} (4+1/3):(5+1/3):(6+1/3) is a linear compression of 4:5:6 (approximately 0-6-11\20)

	{{Adv\|Linearly stretching a chord preserves ~~the~~ [[delta signature]], unlike logarithmic stretching, which preserves the logarithmic ratios between intervals (ratios between cent values) in a chord. Note that all the chords in the examples above are isodifferential (+1+1).}}		{{Adv\|Linearly stretching a chord preserves its [[delta signature]], unlike logarithmic stretching, which preserves the logarithmic ratios between intervals (ratios between cent values) in a chord. Note that all the chords in the examples above are isodifferential (+1+1).}}

	{{adv\|A small logarithmic stretch nevertheless ''approximates'' a small linear stretch. This can be useful when hunting for approximate DR chords in equal divisions: for example, the 19edo major triad 0-6-11\19 can be logarithmically stretched to 0-6-11\18 and logarithmically compressed to 0-6-11\20. All of these chords are roughly +1+1.}}		{{adv\|A small logarithmic stretch nevertheless ''approximates'' a small linear stretch. This can be useful when hunting for approximate DR chords in equal divisions: for example, the 19edo major triad 0-6-11\19 can be logarithmically stretched to 0-6-11\18 and logarithmically compressed to 0-6-11\20. All of these chords are roughly +1+1.}}

	{{cat\|Core knowledge}}		{{cat\|Core knowledge}}

Inthar: /* Stacking */

2026-01-07T19:17:59Z

Stacking

← Older revision		Revision as of 19:17, 7 January 2026
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	This also means that going up (positive cents value) corresponds to a ratio greater than 1, and going down (negative cents value) corresponds to a ratio less than 1.		This also means that going up (positive cents value) corresponds to a ratio greater than 1, and going down (negative cents value) corresponds to a ratio less than 1.

	When applied to chords, stacking in the sense of using the top note of the first chord as the root note of the second may be called ''concatenation''. For example, the ~~concatenstion~~ of two 4:5:6 triads is 8:10:12:15:18.		When applied to chords, stacking in the sense of using the top note of the first chord as the root note of the second may be called ''concatenation''. For example, the concatenation of two 4:5:6 triads is 8:10:12:15:18.
	=== Examples ===		=== Examples ===
	# <math>1200 \log_2 (3/2) - 1200 \log_2 (5/4) = 1200 \log_2 (6/5)</math>		# <math>1200 \log_2 (3/2) - 1200 \log_2 (5/4) = 1200 \log_2 (6/5)</math>

Inthar: /* Stacking */

2026-01-07T19:17:44Z

Stacking

← Older revision		Revision as of 19:17, 7 January 2026
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	This also means that going up (positive cents value) corresponds to a ratio greater than 1, and going down (negative cents value) corresponds to a ratio less than 1.		This also means that going up (positive cents value) corresponds to a ratio greater than 1, and going down (negative cents value) corresponds to a ratio less than 1.

	When applied to chords, stacking in the sense of using the top note of the first chord as the root note of the second may be called ''concatenation''.		When applied to chords, stacking in the sense of using the top note of the first chord as the root note of the second may be called ''concatenation''. For example, the concatenstion of two 4:5:6 triads is 8:10:12:15:18.
	=== Examples ===		=== Examples ===
	# <math>1200 \log_2 (3/2) - 1200 \log_2 (5/4) = 1200 \log_2 (6/5)</math>		# <math>1200 \log_2 (3/2) - 1200 \log_2 (5/4) = 1200 \log_2 (6/5)</math>

Inthar: /* Stacking */

2026-01-07T19:10:57Z

Stacking

← Older revision		Revision as of 19:10, 7 January 2026
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	This also means that going up (positive cents value) corresponds to a ratio greater than 1, and going down (negative cents value) corresponds to a ratio less than 1.		This also means that going up (positive cents value) corresponds to a ratio greater than 1, and going down (negative cents value) corresponds to a ratio less than 1.

			When applied to chords, stacking in the sense of using the top note of the first chord as the root note of the second may be called ''concatenation''.
	=== Examples ===		=== Examples ===
	# <math>1200 \log_2 (3/2) - 1200 \log_2 (5/4) = 1200 \log_2 (6/5)</math>		# <math>1200 \log_2 (3/2) - 1200 \log_2 (5/4) = 1200 \log_2 (6/5)</math>

Inthar: /* Conversion between frequency ratios and cent values */

2026-01-01T16:03:56Z

Conversion between frequency ratios and cent values

← Older revision		Revision as of 16:03, 1 January 2026
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	To convert a cent value to a frequency ratio:		To convert a cent value to a frequency ratio:

	<math>(\mathsf{ratio}) = 2^{(\mathsf{cents})/1200}</math>		<math>\mathsf{ratio} = 2^{\mathsf{cents}/1200}</math>

	== Stacking ==		== Stacking ==

Inthar: /* Conversion between frequency ratios and cent values */

2026-01-01T16:03:29Z

Conversion between frequency ratios and cent values

← Older revision		Revision as of 16:03, 1 January 2026
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	<math>\mathsf{cents} = 1200 \log_2(\mathsf{ratio}) = 1200 \frac{\log(\mathsf{ratio})}{\log 2}</math>		<math>\mathsf{cents} = 1200 \log_2(\mathsf{ratio}) = 1200 \frac{\log(\mathsf{ratio})}{\log 2}</math>

	To convert a cent value ~~''c''~~ to a frequency ratio ~~''r''~~:		To convert a cent value to a frequency ratio:

	<math>(\mathsf{ratio}) = 2^{(\mathsf{cents})/1200}</math>		<math>(\mathsf{ratio}) = 2^{(\mathsf{cents})/1200}</math>

Inthar: /* Conversion between frequency ratios and cent values */

2026-01-01T16:03:16Z

Conversion between frequency ratios and cent values

← Older revision		Revision as of 16:03, 1 January 2026
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	== Conversion between frequency ratios and cent values ==		== Conversion between frequency ratios and cent values ==
	To convert a ratio ~~''r''~~ to a cent value ~~''c''~~:		To convert a ratio to a cent value:

	<math>c = 1200 \log_2 r = 1200 \frac{\log r}{\log 2}</math>		<math>\mathsf{cents} = 1200 \log_2(\mathsf{ratio}) = 1200 \frac{\log(\mathsf{ratio})}{\log 2}</math>

	To convert a cent value ''c'' to a frequency ratio ''r'':		To convert a cent value ''c'' to a frequency ratio ''r'':

	<math>r = 2^{c/1200}</math>		<math>(\mathsf{ratio}) = 2^{(\mathsf{cents})/1200}</math>

	== Stacking ==		== Stacking ==

Inthar: Undo revision 2103 by Inthar (talk)

2025-12-30T04:52:53Z

Undo revision 2103 by Inthar (talk)

← Older revision		Revision as of 04:52, 30 December 2025
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	For example, the mediant of 4/3 and 5/4 is 9/7. The mediant operation is common in xenharmonic theory, as the mediant of two unequal ratios is always strictly between the two ratios, e.g. 3/5 < 7/12 < 4/7.		For example, the mediant of 4/3 and 5/4 is 9/7. The mediant operation is common in xenharmonic theory, as the mediant of two unequal ratios is always strictly between the two ratios, e.g. 3/5 < 7/12 < 4/7.
	=== For intervals in equal divisions ===		=== For intervals in equal divisions ===
	Mediants can be used to obtain generators of [[MOS]] scales in edos (the procedure for any other equal division, given the ~~formal octave~~, is the same). For example, if a\m and b\n are both diatonic fifths (including equalized and collapsed diatonic tunings), then (a+b)\(m+n) is also a diatonic fifth; for example, we obtain 10\17 by taking the mediant of 7\12 and 3\5, thus sharpening 7\12 into 10\17. On the other hand, we can flatten fifth generators by taking the mediant of the diatonic generator with the equalized diatonic generator 4\7: 7\12 ⊕ 4\7 = 11\19.		Mediants can be used to obtain generators of [[MOS]] scales in edos (the procedure for any other equal division, given the equave, is the same). For example, if a\m and b\n are both diatonic fifths (including equalized and collapsed diatonic tunings), then (a+b)\(m+n) is also a diatonic fifth; for example, we obtain 10\17 by taking the mediant of 7\12 and 3\5, thus sharpening 7\12 into 10\17. On the other hand, we can flatten fifth generators by taking the mediant of the diatonic generator with the equalized diatonic generator 4\7: 7\12 ⊕ 4\7 = 11\19.

	=== For JI ratios ===		=== For JI ratios ===

Inthar at 04:31, 30 December 2025

2025-12-30T04:31:51Z

← Older revision		Revision as of 04:31, 30 December 2025
Line 86:		Line 86:
	For example, the mediant of 4/3 and 5/4 is 9/7. The mediant operation is common in xenharmonic theory, as the mediant of two unequal ratios is always strictly between the two ratios, e.g. 3/5 < 7/12 < 4/7.		For example, the mediant of 4/3 and 5/4 is 9/7. The mediant operation is common in xenharmonic theory, as the mediant of two unequal ratios is always strictly between the two ratios, e.g. 3/5 < 7/12 < 4/7.
	=== For intervals in equal divisions ===		=== For intervals in equal divisions ===
	Mediants can be used to obtain generators of [[MOS]] scales in edos (the procedure for any other equal division, given the ~~equave~~, is the same). For example, if a\m and b\n are both diatonic fifths (including equalized and collapsed diatonic tunings), then (a+b)\(m+n) is also a diatonic fifth; for example, we obtain 10\17 by taking the mediant of 7\12 and 3\5, thus sharpening 7\12 into 10\17. On the other hand, we can flatten fifth generators by taking the mediant of the diatonic generator with the equalized diatonic generator 4\7: 7\12 ⊕ 4\7 = 11\19.		Mediants can be used to obtain generators of [[MOS]] scales in edos (the procedure for any other equal division, given the formal octave, is the same). For example, if a\m and b\n are both diatonic fifths (including equalized and collapsed diatonic tunings), then (a+b)\(m+n) is also a diatonic fifth; for example, we obtain 10\17 by taking the mediant of 7\12 and 3\5, thus sharpening 7\12 into 10\17. On the other hand, we can flatten fifth generators by taking the mediant of the diatonic generator with the equalized diatonic generator 4\7: 7\12 ⊕ 4\7 = 11\19.

	=== For JI ratios ===		=== For JI ratios ===