Logarithmic phi - Revision history

Overthink: /* Approximation */ simpler but more accurate rational approximation

2025-11-26T22:09:31Z

Approximation: simpler but more accurate rational approximation

← Older revision		Revision as of 22:09, 26 November 2025
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	Logarithmic phi is well-approximated in equal divisions of the octave corresponding to the Fibonacci sequence: [[8edo]], [[13edo]], [[21edo]], [[34edo]], [[55edo]], etc.		Logarithmic phi is well-approximated in equal divisions of the octave corresponding to the Fibonacci sequence: [[8edo]], [[13edo]], [[21edo]], [[34edo]], [[55edo]], etc.
	== Approximation ==		== Approximation ==
	{{Interval edo approximation\|interval = ~~30695~~/~~20000~~}}		{{Interval edo approximation\|interval = 353/230}}

	== See also ==		== See also ==
	* [[Generating a scale through successive divisions of the octave by the Golden Ratio]]		* [[Generating a scale through successive divisions of the octave by the Golden Ratio]]

Pailiaq at 22:08, 26 November 2025

2025-11-26T22:08:40Z

← Older revision		Revision as of 22:08, 26 November 2025
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	Logarithmic phi is not to be confused with [[acoustic phi]], which is 833.1{{c}}.		Logarithmic phi is not to be confused with [[acoustic phi]], which is 833.1{{c}}.

			The [[phith root of phi]] is another interval with interesting properties, that divides acoustic phi logarithmically by phi (in the same way that logarithmic phi divides the octave by logarithmically by phi), which creates self similar, fractal-like scales.


	Logarithmic phi is well-approximated in equal divisions of the octave corresponding to the Fibonacci sequence: [[8edo]], [[13edo]], [[21edo]], [[34edo]], [[55edo]], etc.		Logarithmic phi is well-approximated in equal divisions of the octave corresponding to the Fibonacci sequence: [[8edo]], [[13edo]], [[21edo]], [[34edo]], [[55edo]], etc.

Pailiaq at 04:52, 26 November 2025

2025-11-26T04:52:20Z

← Older revision		Revision as of 04:52, 26 November 2025
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	Logarithmic phi is well-approximated in equal divisions of the octave corresponding to the Fibonacci sequence: [[8edo]], [[13edo]], [[21edo]], [[34edo]], [[55edo]], etc.		Logarithmic phi is well-approximated in equal divisions of the octave corresponding to the Fibonacci sequence: [[8edo]], [[13edo]], [[21edo]], [[34edo]], [[55edo]], etc.
			== Approximation ==
			{{Interval edo approximation\|interval = 30695/20000}}
	== See also ==		== See also ==
	* [[Generating a scale through successive divisions of the octave by the Golden Ratio]]		* [[Generating a scale through successive divisions of the octave by the Golden Ratio]]

FloraC: Don't invoke cents when there's no need

2025-07-31T13:47:55Z

Don't invoke cents when there's no need

← Older revision		Revision as of 13:47, 31 July 2025
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	\| Name = logarithmic phi		\| Name = logarithmic phi
	}}		}}
	'''Logarithmic phi''', or 1200*[[~~Phi~~\|<math>\varphi</math>]] ~~cents~~ = 1941.6 ~~cents~~ (or, octave-reduced, 741.6 cents) is useful as a generator, for example in [[Erv Wilson]]'s "Golden Horagrams". As a frequency relation it is <math>2^{\varphi}</math>, or <math>2^{\varphi - 1} = 2^{1/\varphi}</math> when octave-reduced. Logarithmic phi is notable for being the most difficult interval to approximate by [[~~EDO~~]]s, and as such a "small equal division of logarithmic phi" [[nonoctave]] tuning would minimize pseudo-octaves.		'''Logarithmic phi''', or [[phi\|<math>\varphi</math>]] [[2/1\|octave]]s = 1941.6 [[cent]]s (or, octave-reduced, 741.6 cents) is useful as a generator, for example in [[Erv Wilson]]'s "Golden Horagrams". As a frequency relation it is <math>2^{\varphi}</math>, or <math>2^{\varphi - 1} = 2^{1/\varphi}</math> when octave-reduced. Logarithmic phi is notable for being the most difficult interval to approximate by [[edo]]s, and as such a "small equal division of logarithmic phi" [[nonoctave]] tuning would minimize pseudo-octaves.

	Logarithmic phi is not to be confused with [[acoustic phi]], which is 833.1¢.		Logarithmic phi is not to be confused with [[acoustic phi]], which is 833.1{{c}}.

	Logarithmic phi is well-approximated in equal divisions of the octave corresponding to the Fibonacci sequence: [[8edo]], [[13edo]], [[21edo]], [[34edo]], [[55edo]], etc.		Logarithmic phi is well-approximated in equal divisions of the octave corresponding to the Fibonacci sequence: [[8edo]], [[13edo]], [[21edo]], [[34edo]], [[55edo]], etc.

	==See also==		== See also ==
	* [[Generating a scale through successive divisions of the octave by the Golden Ratio]]		* [[Generating a scale through successive divisions of the octave by the Golden Ratio]]
	* [[Golden sequences and tuning]]		* [[Golden sequences and tuning]]
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	* [[Metallic MOS]]		* [[Metallic MOS]]

	;The MOS patterns generated by logarithmic phi		; The MOS patterns generated by logarithmic phi
	* [[3L 2s]]		* [[3L 2s]]
	* [[5L 3s]]		* [[5L 3s]]
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	* [[13L 8s]]		* [[13L 8s]]
	* [[21L 13s]]		* [[21L 13s]]
	* ~~...~~		* …

	;Related regular temperaments		; Related regular temperaments
	*[[Father family\|Father temperament]]		* [[Father family\|Father temperament]]
	*[[Keegic temperaments #Aurora\|Aurora temperament]]		* [[Keegic temperaments #Aurora\|Aurora temperament]]
	*[[Triforce]] divides an 1/3 octave period into logarithmic-phi-sized fractions.		* [[Triforce]] divides an 1/3 octave period into logarithmic-phi-sized fractions.

	;Music		; Music
	*[http://www.elvenminstrel.com/music/tuning/horagrams/horagram_intro.htm 5- to 9-tone, octave-repeating scales from Wilson's Golden Horagrams of the Scale Tree], by [[David Finnamore]]		* [http://www.elvenminstrel.com/music/tuning/horagrams/horagram_intro.htm 5- to 9-tone, octave-repeating scales from Wilson's Golden Horagrams of the Scale Tree], by [[David Finnamore]]

	[[Category:Golden ratio]]		[[Category:Golden ratio]]

Sintel: quick note about edo approximations

2025-03-28T23:50:51Z

quick note about edo approximations

← Older revision		Revision as of 23:50, 28 March 2025
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	Logarithmic phi is not to be confused with [[acoustic phi]], which is 833.1¢.		Logarithmic phi is not to be confused with [[acoustic phi]], which is 833.1¢.

			Logarithmic phi is well-approximated in equal divisions of the octave corresponding to the Fibonacci sequence: [[8edo]], [[13edo]], [[21edo]], [[34edo]], [[55edo]], etc.

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

Lériendil: added golden sequences & tuning link

2025-03-28T15:35:33Z

added golden sequences & tuning link

← Older revision		Revision as of 15:35, 28 March 2025
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	==See also==		==See also==
	*[[Generating a scale through successive divisions of the octave by the Golden Ratio]]		* [[Generating a scale through successive divisions of the octave by the Golden Ratio]]
	*[[Golden meantone]]		* [[Golden sequences and tuning]]
	*[[Metallic MOS]]		* [[Golden meantone]]
			* [[Metallic MOS]]

	;The MOS patterns generated by logarithmic phi		;The MOS patterns generated by logarithmic phi
	*[[3L 2s]]		* [[3L 2s]]
	*[[5L 3s]]		* [[5L 3s]]
	*[[8L 5s]]		* [[8L 5s]]
	*[[13L 8s]]		* [[13L 8s]]
	*[[21L 13s]]		* [[21L 13s]]
	*...		* ...

	;Related regular temperaments		;Related regular temperaments

Lériendil: deleted the mmtm theory link

2025-03-28T15:34:13Z

deleted the mmtm theory link

← Older revision		Revision as of 15:34, 28 March 2025
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	\| Name = logarithmic phi		\| Name = logarithmic phi
	}}		}}
	'''Logarithmic phi''', or 1200*[[Phi\|<math>\varphi</math>]] cents = 1941.6 cents (or, octave-reduced, 741.6 cents) is useful as a generator, for example in [[Erv Wilson]]'s "Golden Horagrams". As a frequency relation it is <math>2^{\varphi}</math>, or <math>2^{\varphi - 1} = 2^{1/\varphi}</math> when octave-reduced. Logarithmic phi is notable for being the most difficult interval to approximate by [[EDO]]s, and as such a "small equal division of logarithmic phi" [[nonoctave]] tuning would minimize pseudo-octaves~~. With or without pseudo-octaves, an "equal division of logarithmic phi" [[nonoctave]] tuning forms an [[Modal systematization of soid-family scales\| Intense Phrygian-Subpental Aeolian]] mode~~.		'''Logarithmic phi''', or 1200*[[Phi\|<math>\varphi</math>]] cents = 1941.6 cents (or, octave-reduced, 741.6 cents) is useful as a generator, for example in [[Erv Wilson]]'s "Golden Horagrams". As a frequency relation it is <math>2^{\varphi}</math>, or <math>2^{\varphi - 1} = 2^{1/\varphi}</math> when octave-reduced. Logarithmic phi is notable for being the most difficult interval to approximate by [[EDO]]s, and as such a "small equal division of logarithmic phi" [[nonoctave]] tuning would minimize pseudo-octaves.

	Logarithmic phi is not to be confused with [[acoustic phi]], which is 833.1¢.		Logarithmic phi is not to be confused with [[acoustic phi]], which is 833.1¢.

Cmloegcmluin: extracting info from main phi page which was only relevant to logarithmic phi

2024-07-01T08:51:25Z

extracting info from main phi page which was only relevant to logarithmic phi

← Older revision		Revision as of 08:51, 1 July 2024
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	*[[Father family\|Father temperament]]		*[[Father family\|Father temperament]]
	*[[Keegic temperaments #Aurora\|Aurora temperament]]		*[[Keegic temperaments #Aurora\|Aurora temperament]]
			*[[Triforce]] divides an 1/3 octave period into logarithmic-phi-sized fractions.

	;Music		;Music

Cmloegcmluin: fix math code in infobox (avoid double brace, which ends the infobox)

2024-03-06T05:14:29Z

fix math code in infobox (avoid double brace, which ends the infobox)

← Older revision		Revision as of 05:14, 6 March 2024
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	{{Infobox Interval		{{Infobox Interval
	\| Ratio = 2^{\varphi} = 2^{\frac{1+\sqrt{5)} {2} }		\| Ratio = 2^{\varphi} = 2^{\frac{1+\sqrt{5} } {2} }
	\| Cents = 1941.640786499874		\| Cents = 1941.640786499874
	\| Name = logarithmic phi		\| Name = logarithmic phi
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	[[Category:Golden ratio]]		[[Category:Golden ratio]]

	~~{{todo\|review\|comment=broken math code in infobox}}~~

BudjarnLambeth at 10:27, 3 March 2024

2024-03-03T10:27:49Z

← Older revision		Revision as of 10:27, 3 March 2024
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	[[Category:Golden ratio]]		[[Category:Golden ratio]]

	{{todo\|review\|broken math code in infobox}}		{{todo\|review\|comment=broken math code in infobox}}