Logarithmic intonation - Revision history

BudjarnLambeth: Mathematical interest

2025-08-10T22:27:44Z

Mathematical interest

← Older revision		Revision as of 22:27, 10 August 2025
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	{{~~niche~~}}		{{Mathematical interest}}

	'''Logarithmic intonation''' ('''LI'''){{idiosyncratic}} is a form of intonation that is similar to [[just intonation]] but rather than using primes as the basis elements, it uses the natural logarithms of integers (ln(2), ln(3), ln(4) and so on). Logarithmic intonation is a superset of just intonation, because the interval n/d can be expressed as ln(x<sup>n</sup>)/ln(x<sup>d</sup>) for any integer x, but the majority of it consists of irrational intervals. It can be viewed as using the e-[[logharmonic series]] instead of the [[harmonic series]].		'''Logarithmic intonation''' ('''LI'''){{idiosyncratic}} is a form of intonation that is similar to [[just intonation]] but rather than using primes as the basis elements, it uses the natural logarithms of integers (ln(2), ln(3), ln(4) and so on). Logarithmic intonation is a superset of just intonation, because the interval n/d can be expressed as ln(x<sup>n</sup>)/ln(x<sup>d</sup>) for any integer x, but the majority of it consists of irrational intervals. It can be viewed as using the e-[[logharmonic series]] instead of the [[harmonic series]].

BudjarnLambeth: Added category:math, removed todo:expand

2025-03-26T06:33:43Z

Added category:math, removed todo:expand

← Older revision		Revision as of 06:33, 26 March 2025
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	* [[Logharmonic series]]		* [[Logharmonic series]]

			[[Category:Math]]
	[[Category:Method]]		[[Category:Method]]
	[[Category:Irrational intervals]]		[[Category:Irrational intervals]]
	~~{{todo\|expand}}~~

BudjarnLambeth: Niche

2025-03-18T03:00:46Z

Niche

← Older revision		Revision as of 03:00, 18 March 2025
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	{{~~Novelty~~}}		{{niche}}

	'''Logarithmic intonation''' ('''LI'''){{idiosyncratic}} is a form of intonation that is similar to [[just intonation]] but rather than using primes as the basis elements, it uses the natural logarithms of integers (ln(2), ln(3), ln(4) and so on). Logarithmic intonation is a superset of just intonation, because the interval n/d can be expressed as ln(x<sup>n</sup>)/ln(x<sup>d</sup>) for any integer x, but the majority of it consists of irrational intervals. It can be viewed as using the e-[[logharmonic series]] instead of the [[harmonic series]].		'''Logarithmic intonation''' ('''LI'''){{idiosyncratic}} is a form of intonation that is similar to [[just intonation]] but rather than using primes as the basis elements, it uses the natural logarithms of integers (ln(2), ln(3), ln(4) and so on). Logarithmic intonation is a superset of just intonation, because the interval n/d can be expressed as ln(x<sup>n</sup>)/ln(x<sup>d</sup>) for any integer x, but the majority of it consists of irrational intervals. It can be viewed as using the e-[[logharmonic series]] instead of the [[harmonic series]].

BudjarnLambeth: -stub +todo:expand

2024-01-30T07:30:19Z

-stub +todo:expand

← Older revision		Revision as of 07:30, 30 January 2024
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	[[Category:Method]]		[[Category:Method]]
	[[Category:Irrational intervals]]		[[Category:Irrational intervals]]
			{{todo\|expand}}

	{{~~Stub~~}}

Fredg999: Categories

2023-12-12T20:15:31Z

Fredg999: Credit CompactStar for name proposal

2023-12-12T19:23:02Z

Credit CompactStar for name proposal

← Older revision		Revision as of 19:23, 12 December 2023
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	The simplest [[subgroup]] of logarithmic intonation is ln(2).ln(3), but this is a [[nonoctave]] system–if the octave is desired, the simplest subgroup is ln(2).ln(4) which contains the octave as ln(4)/ln(2) and can be rewritten as 2.ln(4). The subgroup ln(2).ln(3).ln(4) or equivalently 2.ln(3).ln(4) can be viewed as analogous to the [[5-limit]] of just intonation. [[Regular temperament]]s can be defined with these subgroups as they can with prime subgroups. The interval ln(2) itself is a descending wide tritone of -635 cents, with it's octave-equivalent ln(4) being a narrow tritone of 565 cents that can be approximated by [[25/18]] and [[18/13]].		The simplest [[subgroup]] of logarithmic intonation is ln(2).ln(3), but this is a [[nonoctave]] system–if the octave is desired, the simplest subgroup is ln(2).ln(4) which contains the octave as ln(4)/ln(2) and can be rewritten as 2.ln(4). The subgroup ln(2).ln(3).ln(4) or equivalently 2.ln(3).ln(4) can be viewed as analogous to the [[5-limit]] of just intonation. [[Regular temperament]]s can be defined with these subgroups as they can with prime subgroups. The interval ln(2) itself is a descending wide tritone of -635 cents, with it's octave-equivalent ln(4) being a narrow tritone of 565 cents that can be approximated by [[25/18]] and [[18/13]].

			The term "logarithmic intonation" was proposed by [[User:CompactStar\|CompactStar]].

	== See also ==		== See also ==

Fredg999: Mark term as idiosyncratic

2023-12-12T06:31:06Z

Mark term as idiosyncratic

← Older revision		Revision as of 06:31, 12 December 2023
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	{{Novelty}}		{{Novelty}}

	'''Logarithmic intonation''' ('''LI''') is a form of intonation that is similar to [[just intonation]] but rather than using primes as the basis elements, it uses the natural logarithms of integers (ln(2), ln(3), ln(4) and so on). Logarithmic intonation is a superset of just intonation, because the interval n/d can be expressed as ln(x<sup>n</sup>)/ln(x<sup>d</sup>) for any integer x, but the majority of it consists of irrational intervals. It can be viewed as using the e-[[logharmonic series]] instead of the [[harmonic series]].		'''Logarithmic intonation''' ('''LI'''){{idiosyncratic}} is a form of intonation that is similar to [[just intonation]] but rather than using primes as the basis elements, it uses the natural logarithms of integers (ln(2), ln(3), ln(4) and so on). Logarithmic intonation is a superset of just intonation, because the interval n/d can be expressed as ln(x<sup>n</sup>)/ln(x<sup>d</sup>) for any integer x, but the majority of it consists of irrational intervals. It can be viewed as using the e-[[logharmonic series]] instead of the [[harmonic series]].

	The simplest [[subgroup]] of logarithmic intonation is ln(2).ln(3), but this is a [[nonoctave]] system–if the octave is desired, the simplest subgroup is ln(2).ln(4) which contains the octave as ln(4)/ln(2) and can be rewritten as 2.ln(4). The subgroup ln(2).ln(3).ln(4) or equivalently 2.ln(3).ln(4) can be viewed as analogous to the [[5-limit]] of just intonation. [[Regular temperament]]s can be defined with these subgroups as they can with prime subgroups. The interval ln(2) itself is a descending wide tritone of -635 cents, with it's octave-equivalent ln(4) being a narrow tritone of 565 cents that can be approximated by [[25/18]] and [[18/13]].		The simplest [[subgroup]] of logarithmic intonation is ln(2).ln(3), but this is a [[nonoctave]] system–if the octave is desired, the simplest subgroup is ln(2).ln(4) which contains the octave as ln(4)/ln(2) and can be rewritten as 2.ln(4). The subgroup ln(2).ln(3).ln(4) or equivalently 2.ln(3).ln(4) can be viewed as analogous to the [[5-limit]] of just intonation. [[Regular temperament]]s can be defined with these subgroups as they can with prime subgroups. The interval ln(2) itself is a descending wide tritone of -635 cents, with it's octave-equivalent ln(4) being a narrow tritone of 565 cents that can be approximated by [[25/18]] and [[18/13]].

CompactStar at 10:34, 11 December 2023

2023-12-11T10:34:00Z

← Older revision		Revision as of 10:34, 11 December 2023
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	{{Novelty}}		{{Novelty}}

	'''Logarithmic intonation''' ('''LI''') is a form of intonation that is similar to [[just intonation]] but rather than using primes as the basis elements, it uses the natural logarithms of integers (ln(2), ln(3), ln(4) and so on). Logarithmic intonation is a superset of just intonation, because the interval n/d can be expressed as ln(x<sup>n</sup>)/ln(x<sup>d</sup>) for any integer x, but the majority of it consists of irrational intervals.		'''Logarithmic intonation''' ('''LI''') is a form of intonation that is similar to [[just intonation]] but rather than using primes as the basis elements, it uses the natural logarithms of integers (ln(2), ln(3), ln(4) and so on). Logarithmic intonation is a superset of just intonation, because the interval n/d can be expressed as ln(x<sup>n</sup>)/ln(x<sup>d</sup>) for any integer x, but the majority of it consists of irrational intervals. It can be viewed as using the e-[[logharmonic series]] instead of the [[harmonic series]].

	The simplest [[subgroup]] of logarithmic intonation is ln(2).ln(3), but this is a [[nonoctave]] system–if the octave is desired, the simplest subgroup is ln(2).ln(4) which contains the octave as ln(4)/ln(2) and can be rewritten as 2.ln(4). The subgroup ln(2).ln(3).ln(4) or equivalently 2.ln(3).ln(4) can be viewed as analogous to the [[5-limit]] of just intonation. [[Regular temperament]]s can be defined with these subgroups as they can with prime subgroups. The interval ln(2) itself is a descending wide tritone of -635 cents, with it's octave-equivalent ln(4) being a narrow tritone of 565 cents that can be approximated by [[25/18]] and [[18/13]].		The simplest [[subgroup]] of logarithmic intonation is ln(2).ln(3), but this is a [[nonoctave]] system–if the octave is desired, the simplest subgroup is ln(2).ln(4) which contains the octave as ln(4)/ln(2) and can be rewritten as 2.ln(4). The subgroup ln(2).ln(3).ln(4) or equivalently 2.ln(3).ln(4) can be viewed as analogous to the [[5-limit]] of just intonation. [[Regular temperament]]s can be defined with these subgroups as they can with prime subgroups. The interval ln(2) itself is a descending wide tritone of -635 cents, with it's octave-equivalent ln(4) being a narrow tritone of 565 cents that can be approximated by [[25/18]] and [[18/13]].

Fredg999: See also

2023-11-03T03:28:16Z

Fredg999: Add novelty template: this concept seems new and does not show clear xenharmonic value (i.e. the choice of natural logarithms appears arbitrary and does not seem supported by any desirable musical or psychoacoustic effect)

2023-11-02T05:14:33Z

Add novelty template: this concept seems new and does not show clear xenharmonic value (i.e. the choice of natural logarithms appears arbitrary and does not seem supported by any desirable musical or psychoacoustic effect)

← Older revision		Revision as of 05:14, 2 November 2023
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	{{~~Stub~~}}		{{Novelty}}

	'''Logarithmic intonation''' ('''LI''') is a form of intonation that is similar to [[just intonation]] but rather than using primes as the basis elements, it uses the natural logarithms of integers (ln(2), ln(3), ln(4) and so on). Logarithmic intonation is a superset of just intonation, because the interval n/d can be expressed as ln(x<sup>n</sup>)/ln(x<sup>d</sup>) for any integer x, but the majority of it consists of irrational intervals.		'''Logarithmic intonation''' ('''LI''') is a form of intonation that is similar to [[just intonation]] but rather than using primes as the basis elements, it uses the natural logarithms of integers (ln(2), ln(3), ln(4) and so on). Logarithmic intonation is a superset of just intonation, because the interval n/d can be expressed as ln(x<sup>n</sup>)/ln(x<sup>d</sup>) for any integer x, but the majority of it consists of irrational intervals.
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	[[Category:Terms]]		[[Category:Terms]]
	[[Category:Irrational intervals]]		[[Category:Irrational intervals]]


			{{Stub}}

← Older revision		Revision as of 20:15, 12 December 2023
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	[[Category:Method]]		[[Category:Method]]
	~~[[Category:Terms]]~~
	[[Category:Irrational intervals]]		[[Category:Irrational intervals]]


	{{Stub}}		{{Stub}}

← Older revision		Revision as of 03:28, 3 November 2023
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	The simplest [[subgroup]] of logarithmic intonation is ln(2).ln(3), but this is a [[nonoctave]] system–if the octave is desired, the simplest subgroup is ln(2).ln(4) which contains the octave as ln(4)/ln(2) and can be rewritten as 2.ln(4). The subgroup ln(2).ln(3).ln(4) or equivalently 2.ln(3).ln(4) can be viewed as analogous to the [[5-limit]] of just intonation. [[Regular temperament]]s can be defined with these subgroups as they can with prime subgroups. The interval ln(2) itself is a descending wide tritone of -635 cents, with it's octave-equivalent ln(4) being a narrow tritone of 565 cents that can be approximated by [[25/18]] and [[18/13]].		The simplest [[subgroup]] of logarithmic intonation is ln(2).ln(3), but this is a [[nonoctave]] system–if the octave is desired, the simplest subgroup is ln(2).ln(4) which contains the octave as ln(4)/ln(2) and can be rewritten as 2.ln(4). The subgroup ln(2).ln(3).ln(4) or equivalently 2.ln(3).ln(4) can be viewed as analogous to the [[5-limit]] of just intonation. [[Regular temperament]]s can be defined with these subgroups as they can with prime subgroups. The interval ln(2) itself is a descending wide tritone of -635 cents, with it's octave-equivalent ln(4) being a narrow tritone of 565 cents that can be approximated by [[25/18]] and [[18/13]].

			== See also ==
			* [[Logharmonic series]]

	[[Category:Method]]		[[Category:Method]]