Logarithmic phi: Difference between revisions

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| Name = logarithmic phi

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'''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''', or [[phi|<math>\varphi</math>]] [[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{{c}}.

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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.

== Approximation ==

== See also ==

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

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-'''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''', or [[phi|<math>\varphi</math>]] [[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{{c}}.
@@ Line 13: / Line 13: @@
 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}}
+{{Interval edo approximation|interval = 353/230}}
 == See also ==
 * [[Generating a scale through successive divisions of the octave by the Golden Ratio]]