Acoustic phi - Revision history

Overthink: /* See also */ Use consistent caps

2026-02-27T05:23:02Z

Pailiaq at 22:07, 26 November 2025

2025-11-26T22:07:47Z

← Older revision		Revision as of 22:07, 26 November 2025
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	Acoustic phi is not to be confused with [[logarithmic phi]], which is 1941.6¢ (741.6¢ octave-reduced).		Acoustic phi is not to be confused with [[logarithmic phi]], which is 1941.6¢ (741.6¢ octave-reduced).

			The [[phith root of phi]] is another interval with interesting properties, that divides acoustic phi logarithmically by phi, which creates self similar, fractal-like scales.


	== Approximation ==		== Approximation ==

Pailiaq: /* Approximation */

2025-11-26T09:01:23Z

Approximation

← Older revision		Revision as of 09:01, 26 November 2025
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	== Approximation ==		== Approximation ==
	{{Interval edo approximation\|interval = 1618/1000 \| ~~name~~ = ϕ}}		{{Interval edo approximation\|interval = 1618/1000 \| interval_name = ϕ}}

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

Pailiaq at 08:59, 26 November 2025

2025-11-26T08:59:55Z

← Older revision		Revision as of 08:59, 26 November 2025
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	== Approximation ==		== Approximation ==
	{{Interval edo approximation\|interval = 1618/1000}}		{{Interval edo approximation\|interval = 1618/1000 \| name = ϕ}}

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

Pailiaq at 04:46, 26 November 2025

2025-11-26T04:46:02Z

← Older revision		Revision as of 04:46, 26 November 2025
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	Acoustic phi is not to be confused with [[logarithmic phi]], which is 1941.6¢ (741.6¢ octave-reduced).		Acoustic phi is not to be confused with [[logarithmic phi]], which is 1941.6¢ (741.6¢ octave-reduced).

			== Approximation ==
	{{Interval edo approximation\|interval = 1618/1000}}		{{Interval edo approximation\|interval = 1618/1000}}

Pailiaq at 04:41, 26 November 2025

2025-11-26T04:41:24Z

← Older revision		Revision as of 04:41, 26 November 2025
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	Acoustic phi is not to be confused with [[logarithmic phi]], which is 1941.6¢ (741.6¢ octave-reduced).		Acoustic phi is not to be confused with [[logarithmic phi]], which is 1941.6¢ (741.6¢ octave-reduced).


			{{Interval edo approximation\|interval = 1618/1000}}

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

BudjarnLambeth: /* See also */

2024-10-20T07:15:12Z

BudjarnLambeth: /* See also */ add link to orphan page

2024-10-20T07:15:00Z

FloraC: Fix (i see why it needs to be escaped). Style improvements +1

2024-09-12T17:58:00Z

Fix (i see why it needs to be escaped). Style improvements +1

← Older revision		Revision as of 17:58, 12 September 2024
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	\| Name = acoustic phi		\| Name = acoustic phi
	}}		}}
	ϕ taken as a [[frequency ratio]] (ϕ*''f'' where {{nowrap\|''f'' = 1/1}}) is about 833.1 [[cent]]s. This [[metastable]] interval is sometimes called '''acoustic phi''', or the ''phi neutral sixth''. It is wider than a [[12edo]] minor sixth (800 cents) by about a sixth-tone (33.~~3...~~ cents).		ϕ taken as a [[frequency ratio]] (ϕ⋅''f'' where {{nowrap\|''f'' {{=}} 1/1}}) is about 833.1 [[cent]]s. This [[metastable]] interval is sometimes called '''acoustic phi''', or the ''phi neutral sixth''. It is wider than a [[12edo]] minor sixth (800 cents) by about a sixth-tone (33.3… cents).

	ϕ is the most difficult interval to approximate by rational numbers, as {{w\|Golden ratio #Continued fraction and square root\|its continued fraction}} consists entirely of 1's. The ~~[[wikipedia:~~Convergent (continued fraction)\|convergents]] (rational number approximations, obtained from the continued fractions) are the ratios of successive terms of the Fibonacci sequence converge on ϕ, the just intonation intervals 3/2, [[5/3]] (~884.4¢), [[8/5]] (~814.7¢), [[13/8]] (~840.5¢), [[21/13]] (~830.3¢), … converge on ~833.1 cents.		ϕ is the most difficult interval to approximate by rational numbers, as {{w\|Golden ratio #Continued fraction and square root\|its continued fraction}} consists entirely of 1's. The {{w\|Convergent (continued fraction)\|convergents}} (rational number approximations, obtained from the continued fractions) are the ratios of successive terms of the Fibonacci sequence converge on ϕ, the just intonation intervals 3/2, [[5/3]] (~884.4¢), [[8/5]] (~814.7¢), [[13/8]] (~840.5¢), [[21/13]] (~830.3¢), … converge on ~833.1 cents.

	[[Erv Wilson]] accordingly described ϕ as "the worstest of the worst — and yet somehow with divinity imbued, Lord have mercy!", inspiring the term [[merciful intonation]].		[[Erv Wilson]] accordingly described ϕ as "the worstest of the worst — and yet somehow with divinity imbued, Lord have mercy!", inspiring the term [[merciful intonation]].

FloraC: Partial undo

2024-09-12T17:49:27Z

Partial undo

← Older revision		Revision as of 17:49, 12 September 2024
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	{{Infobox Interval		{{Infobox Interval
	\| Ratio = \varphi = \frac{ 1 + \sqrt{5} }{2}		\| Ratio = \varphi = \frac{ 1 + \sqrt{5} }{2}
	\| Cents = ~~{{#expr: 1200 * ln((1 + sqrt(5)) / 2) / ln(2)}}~~		\| Cents = 833.0902963567409
	\| Name = acoustic phi		\| Name = acoustic phi
	}}		}}
	ϕ taken as a [[frequency ratio]] (ϕ*''f'' where {{nowrap\|''f'' {{=}} 1/1}}) is about 833.1 [[cent]]s. This [[metastable]] interval is sometimes called '''acoustic phi''', or the ''phi neutral sixth''. It is wider than a [[12edo]] minor sixth (800 cents) by about a sixth-tone (33.3... cents).		ϕ taken as a [[frequency ratio]] (ϕ*''f'' where {{nowrap\|''f'' = 1/1}}) is about 833.1 [[cent]]s. This [[metastable]] interval is sometimes called '''acoustic phi''', or the ''phi neutral sixth''. It is wider than a [[12edo]] minor sixth (800 cents) by about a sixth-tone (33.3... cents).

	ϕ is the most difficult interval to approximate by rational numbers, as {{w\|Golden ratio #Continued fraction and square root\|its continued fraction}} consists entirely of 1's. The [[wikipedia:Convergent (continued fraction)\|convergents]] (rational number approximations, obtained from the continued fractions) are the ratios of successive terms of the Fibonacci sequence converge on ϕ, the just intonation intervals 3/2, [[5/3]] (~884.4¢), [[8/5]] (~814.7¢), [[13/8]] (~840.5¢), [[21/13]] (~830.3¢), … converge on ~833.1 cents.		ϕ is the most difficult interval to approximate by rational numbers, as {{w\|Golden ratio #Continued fraction and square root\|its continued fraction}} consists entirely of 1's. The [[wikipedia:Convergent (continued fraction)\|convergents]] (rational number approximations, obtained from the continued fractions) are the ratios of successive terms of the Fibonacci sequence converge on ϕ, the just intonation intervals 3/2, [[5/3]] (~884.4¢), [[8/5]] (~814.7¢), [[13/8]] (~840.5¢), [[21/13]] (~830.3¢), … converge on ~833.1 cents.

← Older revision		Revision as of 05:23, 27 February 2026
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	* [[Edφ]], tunings created by dividing acoustic phi into equally sized smaller steps		* [[Edφ]], tunings created by dividing acoustic phi into equally sized smaller steps
	* [[Phi as a generator]]		* [[Phi as a generator]]
	* [[~~sqrtphi~~]], a temperament based on the square root of phi (~416.5 cents) as a generator.		* [[Sqrtphi]], a temperament based on the square root of phi (~416.5 cents) as a generator.
	* [[~~photosynthesis~~]], a temperament using phi as a “prime” in its subgroup		* [[Photosynthesis]], a temperament using phi as a “prime” in its subgroup

	[[Category:Golden ratio]]		[[Category:Golden ratio]]
	[[Category:Supraminor sixth]]		[[Category:Supraminor sixth]]

← Older revision		Revision as of 07:15, 20 October 2024
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	* [[Phi as a generator]]		* [[Phi as a generator]]
	* [[sqrtphi]], a temperament based on the square root of phi (~416.5 cents) as a generator.		* [[sqrtphi]], a temperament based on the square root of phi (~416.5 cents) as a generator.
	* [[~~Photosynthesis~~]], a temperament using phi as a “prime” in its subgroup		* [[photosynthesis]], a temperament using phi as a “prime” in its subgroup

	[[Category:Golden ratio]]		[[Category:Golden ratio]]
	[[Category:Supraminor sixth]]		[[Category:Supraminor sixth]]

← Older revision		Revision as of 07:15, 20 October 2024
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	== See also ==		== See also ==
	* [[833 Cent Golden Scale (Bohlen)]]		* [[833 Cent Golden Scale (Bohlen)]]
			* [[Edφ]], tunings created by dividing acoustic phi into equally sized smaller steps
	* [[Phi as a generator]]		* [[Phi as a generator]]
	* [[sqrtphi]], a temperament based on the square root of phi (~416.5 cents) as a generator.		* [[sqrtphi]], a temperament based on the square root of phi (~416.5 cents) as a generator.
	* [[~~Edφ~~]], ~~tunings created by dividing acoustic~~ phi ~~into equally sized smaller steps~~		* [[Photosynthesis]], a temperament using phi as a “prime” in its subgroup

	[[Category:Golden ratio]]		[[Category:Golden ratio]]
	[[Category:Supraminor sixth]]		[[Category:Supraminor sixth]]