Golden ratio - Revision history

Pailiaq at 09:08, 26 November 2025

2025-11-26T09:08:01Z

← Older revision		Revision as of 09:08, 26 November 2025
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	== Musical applications ==		== Musical applications ==

	The golden ratio can be used as a frequency multiplier or as a pitch fraction; in the former case it is known as [[acoustic phi]] and in the latter case it is known as [[logarithmic phi]]. These two versions of phi have completely different musical applications which can be read about in detail on their separate pages. [[~~Lemba~~]] ~~is a notable~~ [[~~regular temperament~~]] ~~for approximating both versions of phi simultaneously~~, ~~requiring only~~ two ~~of its [[generators]] for logarithmic~~ phi, ~~and only one each of its generator and [[period]] for~~ acoustic phi.		The golden ratio can be used as a frequency multiplier or as a pitch fraction; in the former case it is known as [[acoustic phi]] and in the latter case it is known as [[logarithmic phi]]. These two versions of phi have completely different musical applications which can be read about in detail on their separate pages. A third interval, the [[phith root of phi]] ([math]\displaystyle{ \sqrt[\varphi]{\varphi} }[/math]), acts as a bridge between the two: it divides acoustic phi logarithmically by phi, enabling golden MOS scales with acoustic phi as the equave.

	~~==Phith root of Phi (<math>\sqrt~~[~~\varphi~~]~~{\varphi}</math>) ==~~		[[Lemba]] is a notable [[regular temperament]] for approximating both acoustic and logarithmic phi simultaneously, requiring only two of its [[generators]] for logarithmic phi, and only one each of its generator and [[period]] for acoustic phi.

	~~The '''phith root of phi''' (<math>\varphi^{1/\varphi}</math>, <math>\sqrt[\varphi~~]~~{\varphi}</math>, or approximately 514.878 cents)~~ is ~~another useful interval for generating golden scales. Similarly to~~ [[~~logarithmic phi~~]] ~~(<math>2^{\varphi}</math>), <math>\sqrt[\varphi]{\varphi}</math> can be used as a generator interval to produce MOS scales whose sizes are Fibonacci numbers, where the equave is the~~ acoustic phi ~~instead~~ of ~~the octave.~~

	~~<math>\sqrt[\varphi]{\varphi}</math> divides~~ [[~~acoustic phi~~]] ~~logarithmically by~~ phi, ~~just as [[logarithmic phi]] divides the octave logarithmically by phi:~~
	~~: 833.09¢ ÷ φ = 514.878¢~~

	~~When using acoustic phi (~833.09¢) as the~~ [[~~equave~~]] instead of the octave, <math>\varphi^{1/\varphi}</math> generates MOS scales at Fibonacci cardinalities (3, 5, 8, 13...). This creates fractal, self-similar scales where every interval relationship exhibits the golden ratio, which is useful for ~~compositions that maximize golden properties while avoiding octave equivalence entirely.~~

	~~[[49edo]] provides an exceptionally accurate approximation. Because 34 steps ≈ 832.65¢ (~~acoustic phi) and because 34 is a fibonacci number, it naturally follows that 21 steps, its previous fibonacci number, ≈ 514.29¢ (<math>\varphi^{1/\varphi}</math>). This pattern continues, creating a highly accurate golden scale of self-similar frequency relationships at step sizes 2, 3, 5, 8, 13, 21, 34.

	~~[[7edo]] approximates this interval extremely well (0.59c off), as does all edos that are multiples of 7~~.

	~~{{Interval Edo Approximation \| interval = 134636/100000 \| interval_name = <math>\varphi^{1/\varphi}</math>}}~~

	== Compositions based on the golden ratio ==		== Compositions based on the golden ratio ==

Pailiaq: restored and updated sevish's contributions about the phith root of phi

2025-11-26T08:51:10Z

restored and updated sevish's contributions about the phith root of phi

← Older revision		Revision as of 08:51, 26 November 2025
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	The golden ratio can be used as a frequency multiplier or as a pitch fraction; in the former case it is known as [[acoustic phi]] and in the latter case it is known as [[logarithmic phi]]. These two versions of phi have completely different musical applications which can be read about in detail on their separate pages. [[Lemba]] is a notable [[regular temperament]] for approximating both versions of phi simultaneously, requiring only two of its [[generators]] for logarithmic phi, and only one each of its generator and [[period]] for acoustic phi.		The golden ratio can be used as a frequency multiplier or as a pitch fraction; in the former case it is known as [[acoustic phi]] and in the latter case it is known as [[logarithmic phi]]. These two versions of phi have completely different musical applications which can be read about in detail on their separate pages. [[Lemba]] is a notable [[regular temperament]] for approximating both versions of phi simultaneously, requiring only two of its [[generators]] for logarithmic phi, and only one each of its generator and [[period]] for acoustic phi.

			==Phith root of Phi (<math>\sqrt[\varphi]{\varphi}</math>) ==

			The '''phith root of phi''' (<math>\varphi^{1/\varphi}</math>, <math>\sqrt[\varphi]{\varphi}</math>, or approximately 514.878 cents) is another useful interval for generating golden scales. Similarly to [[logarithmic phi]] (<math>2^{\varphi}</math>), <math>\sqrt[\varphi]{\varphi}</math> can be used as a generator interval to produce MOS scales whose sizes are Fibonacci numbers, where the equave is the acoustic phi instead of the octave.

			<math>\sqrt[\varphi]{\varphi}</math> divides [[acoustic phi]] logarithmically by phi, just as [[logarithmic phi]] divides the octave logarithmically by phi:
			: 833.09¢ ÷ φ = 514.878¢

			When using acoustic phi (~833.09¢) as the [[equave]] instead of the octave, <math>\varphi^{1/\varphi}</math> generates MOS scales at Fibonacci cardinalities (3, 5, 8, 13...). This creates fractal, self-similar scales where every interval relationship exhibits the golden ratio, which is useful for compositions that maximize golden properties while avoiding octave equivalence entirely.

			[[49edo]] provides an exceptionally accurate approximation. Because 34 steps ≈ 832.65¢ (acoustic phi) and because 34 is a fibonacci number, it naturally follows that 21 steps, its previous fibonacci number, ≈ 514.29¢ (<math>\varphi^{1/\varphi}</math>). This pattern continues, creating a highly accurate golden scale of self-similar frequency relationships at step sizes 2, 3, 5, 8, 13, 21, 34.

			[[7edo]] approximates this interval extremely well (0.59c off), as does all edos that are multiples of 7.

			{{Interval Edo Approximation \| interval = 134636/100000 \| interval_name = <math>\varphi^{1/\varphi}</math>}}

	== Compositions based on the golden ratio ==		== Compositions based on the golden ratio ==
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	== External links ==		== External links ==
			* [https://sevish.com/2017/golden-ratio-music-interval/ The Golden Ratio as a musical interval] by [[Sevish]]
	* [http://tonalsoft.com/enc/p/phi.aspx Phi Φ / phi φ] on [[Tonalsoft Encyclopedia]]		* [http://tonalsoft.com/enc/p/phi.aspx Phi Φ / phi φ] on [[Tonalsoft Encyclopedia]]

	[[Category:Golden ratio]]		[[Category:Golden ratio]]
	[[Category:Irrational intervals]]		[[Category:Irrational intervals]]

BudjarnLambeth: Add section including link to orphaned page

2024-12-02T00:31:27Z

Add section including link to orphaned page

← Older revision		Revision as of 00:31, 2 December 2024
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	The golden ratio can be used as a frequency multiplier or as a pitch fraction; in the former case it is known as [[acoustic phi]] and in the latter case it is known as [[logarithmic phi]]. These two versions of phi have completely different musical applications which can be read about in detail on their separate pages. [[Lemba]] is a notable [[regular temperament]] for approximating both versions of phi simultaneously, requiring only two of its [[generators]] for logarithmic phi, and only one each of its generator and [[period]] for acoustic phi.		The golden ratio can be used as a frequency multiplier or as a pitch fraction; in the former case it is known as [[acoustic phi]] and in the latter case it is known as [[logarithmic phi]]. These two versions of phi have completely different musical applications which can be read about in detail on their separate pages. [[Lemba]] is a notable [[regular temperament]] for approximating both versions of phi simultaneously, requiring only two of its [[generators]] for logarithmic phi, and only one each of its generator and [[period]] for acoustic phi.

			== Compositions based on the golden ratio ==
			* ''[[Star Nursery]]'' - [[Sean Archibald]] (2021)
			* ''[[Abyss]]'' - [[T.C. Edwards]] (2024)

	== External links ==		== External links ==

Cmloegcmluin: clean up, clarity

2024-07-01T08:51:12Z

clean up, clarity

← Older revision		Revision as of 08:51, 1 July 2024
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	{{Wikipedia}}		{{Wikipedia}}
	The '''golden ratio''' or '''phi''' (Greek letter <math>\~~phi~~</math> / <math>\~~varphi~~</math>) ~~may be~~ defined by <math>\frac{a}{b}</math> such that <math>\frac{a}{b} = \frac{a+b}{a}</math>. It follows that <math>\varphi - 1 = ~~1 /~~ \varphi</math>, and also that <math>\varphi = \frac{1+\sqrt{5}}{2}</math>, or approximately 1.6180339887..~~. <math>\varphi</math> is an irrational number that appears in many branches of mathematics~~.		The '''golden ratio''' or '''phi''' (Greek letter <math>\varphi</math> or <math>\phi</math>) is an irrational number that appears in many branches of mathematics, defined as the <math>\frac{a}{b}</math> such that <math>\frac{a}{b} = \frac{a+b}{a}</math>. It follows that <math>\varphi - 1 = \frac1{\varphi}</math>, and also that <math>\varphi = \frac{1+\sqrt{5}}{2}</math>, or approximately 1.6180339887...

	== Musical applications ==		== Musical applications ==
	The golden ratio can be used as a frequency multiplier or as a pitch fraction; in the former case it is known as [[acoustic phi]] and in the latter case it is known as [[logarithmic phi]]. [[Lemba]] is particularly notable for approximating both simply and accurately simultaneously, at a generator + a period for acoustic and 2 generators for logarithmic, making it an excellent choice for experimenting with phi based composition. [[Triforce]] is also essentially based on dividing the 1/3 octave period into logarithmic phi sized fractions.

	The ~~phith root of~~ phi ~~(<math>\sqrt[\varphi~~]~~{\varphi}</math> or approximately 514.878 cents)~~ is ~~another useful interval for generating golden scales. Similarly to~~ [[logarithmic phi]]~~, <math>\sqrt~~[~~\varphi~~]~~{\varphi}</math> can be used as~~ a ~~generator interval to produce MOS scales whose sizes are Fibonacci numbers~~, ~~where the equave is the acoustic~~ phi ~~instead~~ of ~~the octave. In this way it is a useful~~ generator ~~if you wish to avoid octaves~~ and ~~maximize the golden properties of the resulting scale~~.		The golden ratio can be used as a frequency multiplier or as a pitch fraction; in the former case it is known as [[acoustic phi]] and in the latter case it is known as [[logarithmic phi]]. These two versions of phi have completely different musical applications which can be read about in detail on their separate pages. [[Lemba]] is a notable [[regular temperament]] for approximating both versions of phi simultaneously, requiring only two of its [[generators]] for logarithmic phi, and only one each of its generator and [[period]] for acoustic phi.

	== External links ==		== External links ==
	* [http://tonalsoft.com/enc/p/phi.aspx Phi Φ / phi φ] on [[Tonalsoft Encyclopedia]]		* [http://tonalsoft.com/enc/p/phi.aspx Phi Φ / phi φ] on [[Tonalsoft Encyclopedia]]

	[[Category:Golden ratio\| ]]		[[Category:Golden ratio]]
	[[Category:Irrational intervals]]		[[Category:Irrational intervals]]

Fredg999: External link

2024-06-30T21:43:36Z

External link

← Older revision		Revision as of 21:43, 30 June 2024
Line 6:		Line 6:

	The phith root of phi (<math>\sqrt[\varphi]{\varphi}</math> or approximately 514.878 cents) is another useful interval for generating golden scales. Similarly to [[logarithmic phi]], <math>\sqrt[\varphi]{\varphi}</math> can be used as a generator interval to produce MOS scales whose sizes are Fibonacci numbers, where the equave is the acoustic phi instead of the octave. In this way it is a useful generator if you wish to avoid octaves and maximize the golden properties of the resulting scale.		The phith root of phi (<math>\sqrt[\varphi]{\varphi}</math> or approximately 514.878 cents) is another useful interval for generating golden scales. Similarly to [[logarithmic phi]], <math>\sqrt[\varphi]{\varphi}</math> can be used as a generator interval to produce MOS scales whose sizes are Fibonacci numbers, where the equave is the acoustic phi instead of the octave. In this way it is a useful generator if you wish to avoid octaves and maximize the golden properties of the resulting scale.

			== External links ==
			* [http://tonalsoft.com/enc/p/phi.aspx Phi Φ / phi φ] on [[Tonalsoft Encyclopedia]]

	[[Category:Golden ratio\| ]]		[[Category:Golden ratio\| ]]
	[[Category:Irrational intervals]]		[[Category:Irrational intervals]]

Fredg999 category edits: Categories (order)

2022-08-03T03:03:31Z

Categories (order)

← Older revision		Revision as of 03:03, 3 August 2022
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	The phith root of phi (<math>\sqrt[\varphi]{\varphi}</math> or approximately 514.878 cents) is another useful interval for generating golden scales. Similarly to [[logarithmic phi]], <math>\sqrt[\varphi]{\varphi}</math> can be used as a generator interval to produce MOS scales whose sizes are Fibonacci numbers, where the equave is the acoustic phi instead of the octave. In this way it is a useful generator if you wish to avoid octaves and maximize the golden properties of the resulting scale.		The phith root of phi (<math>\sqrt[\varphi]{\varphi}</math> or approximately 514.878 cents) is another useful interval for generating golden scales. Similarly to [[logarithmic phi]], <math>\sqrt[\varphi]{\varphi}</math> can be used as a generator interval to produce MOS scales whose sizes are Fibonacci numbers, where the equave is the acoustic phi instead of the octave. In this way it is a useful generator if you wish to avoid octaves and maximize the golden properties of the resulting scale.

			[[Category:Golden ratio\| ]]
	[[Category:Irrational intervals]]		[[Category:Irrational intervals]]
	~~[[Category:Golden ratio\| ]]~~

Fredg999 category edits: Categories

2022-08-03T03:03:17Z

Sintel at 12:28, 24 February 2022

2022-02-24T12:28:29Z

← Older revision		Revision as of 12:28, 24 February 2022
Line 5:		Line 5:
	The golden ratio can be used as a frequency multiplier or as a pitch fraction; in the former case it is known as [[acoustic phi]] and in the latter case it is known as [[logarithmic phi]]. [[Lemba]] is particularly notable for approximating both simply and accurately simultaneously, at a generator + a period for acoustic and 2 generators for logarithmic, making it an excellent choice for experimenting with phi based composition. [[Triforce]] is also essentially based on dividing the 1/3 octave period into logarithmic phi sized fractions.		The golden ratio can be used as a frequency multiplier or as a pitch fraction; in the former case it is known as [[acoustic phi]] and in the latter case it is known as [[logarithmic phi]]. [[Lemba]] is particularly notable for approximating both simply and accurately simultaneously, at a generator + a period for acoustic and 2 generators for logarithmic, making it an excellent choice for experimenting with phi based composition. [[Triforce]] is also essentially based on dividing the 1/3 octave period into logarithmic phi sized fractions.

	The phith root of phi (<math>\sqrt[\varphi]{\varphi}</math> or approximately 514.878 cents) is another useful interval for generating golden scales. Similarly to logarithmic phi ~~(<math>\sqrt[2~~]~~{\varphi}</math>)~~, <math>\sqrt[\varphi]{\varphi}</math> can be used as a generator interval to produce MOS scales whose sizes are Fibonacci numbers, where the equave is the acoustic phi instead of the octave. In this way it is a useful generator if you wish to avoid octaves and maximize the golden properties of the resulting scale.		The phith root of phi (<math>\sqrt[\varphi]{\varphi}</math> or approximately 514.878 cents) is another useful interval for generating golden scales. Similarly to [[logarithmic phi]], <math>\sqrt[\varphi]{\varphi}</math> can be used as a generator interval to produce MOS scales whose sizes are Fibonacci numbers, where the equave is the acoustic phi instead of the octave. In this way it is a useful generator if you wish to avoid octaves and maximize the golden properties of the resulting scale.

	[[Category:Golden ratio\| ]]		[[Category:Golden ratio\| ]]
	[[Category:Theory]]		[[Category:Theory]]

Sevish: Add paragraph about phith root of phi as a generator

2022-02-24T12:06:29Z

Add paragraph about phith root of phi as a generator

← Older revision		Revision as of 12:06, 24 February 2022
Line 4:		Line 4:
	== Musical applications ==		== Musical applications ==
	The golden ratio can be used as a frequency multiplier or as a pitch fraction; in the former case it is known as [[acoustic phi]] and in the latter case it is known as [[logarithmic phi]]. [[Lemba]] is particularly notable for approximating both simply and accurately simultaneously, at a generator + a period for acoustic and 2 generators for logarithmic, making it an excellent choice for experimenting with phi based composition. [[Triforce]] is also essentially based on dividing the 1/3 octave period into logarithmic phi sized fractions.		The golden ratio can be used as a frequency multiplier or as a pitch fraction; in the former case it is known as [[acoustic phi]] and in the latter case it is known as [[logarithmic phi]]. [[Lemba]] is particularly notable for approximating both simply and accurately simultaneously, at a generator + a period for acoustic and 2 generators for logarithmic, making it an excellent choice for experimenting with phi based composition. [[Triforce]] is also essentially based on dividing the 1/3 octave period into logarithmic phi sized fractions.

			The phith root of phi (<math>\sqrt[\varphi]{\varphi}</math> or approximately 514.878 cents) is another useful interval for generating golden scales. Similarly to logarithmic phi (<math>\sqrt[2]{\varphi}</math>), <math>\sqrt[\varphi]{\varphi}</math> can be used as a generator interval to produce MOS scales whose sizes are Fibonacci numbers, where the equave is the acoustic phi instead of the octave. In this way it is a useful generator if you wish to avoid octaves and maximize the golden properties of the resulting scale.

	[[Category:Golden ratio\| ]]		[[Category:Golden ratio\| ]]
	[[Category:Theory]]		[[Category:Theory]]

Xenllium at 03:52, 24 January 2022

2022-01-24T03:52:37Z

← Older revision		Revision as of 03:52, 24 January 2022
Line 1:		Line 1:
	{{Wikipedia}}		{{Wikipedia}}
	The '''golden ratio''' or '''phi''' (Greek letter φ / ϕ) may be defined by a/b such that a/b = (a+b)/a. It follows that ϕ-1 = 1/ϕ, and also that ϕ = (1+sqrt(5))/2, or approximately 1.6180339887... ϕ is an irrational number that appears in many branches of mathematics.		The '''golden ratio''' or '''phi''' (Greek letter <math>\phi</math> / <math>\varphi</math>) may be defined by <math>\frac{a}{b}</math> such that <math>\frac{a}{b} = \frac{a+b}{a}</math>. It follows that <math>\varphi - 1 = 1 / \varphi</math>, and also that <math>\varphi = \frac{1+\sqrt{5}}{2}</math>, or approximately 1.6180339887... <math>\varphi</math> is an irrational number that appears in many branches of mathematics.

	== Musical applications ==		== Musical applications ==
			The golden ratio can be used as a frequency multiplier or as a pitch fraction; in the former case it is known as [[acoustic phi]] and in the latter case it is known as [[logarithmic phi]]. [[Lemba]] is particularly notable for approximating both simply and accurately simultaneously, at a generator + a period for acoustic and 2 generators for logarithmic, making it an excellent choice for experimenting with phi based composition. [[Triforce]] is also essentially based on dividing the 1/3 octave period into logarithmic phi sized fractions.

	~~The golden ratio can be used as a frequency multiplier or as a pitch fraction; in the former case it is known as~~ [[~~acoustic phi]] and in the latter case it is known as [[logarithmic phi]]. [[Lemba~~]] is particularly notable for approximating both simply and accurately simultaneously, at a generator + a period for acoustic and 2 generators for logarithmic, making it an excellent choice for experimenting with phi based composition. [[~~Triforce~~]] ~~is also essentially based on dividing the 1/3 octave period into logarithmic phi sized fractions.~~		[[Category:Golden ratio\| ]]
			[[Category:Theory]]