Logarithmic phi: Difference between revisions

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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}}.

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.

== Approximation ==

== See also ==

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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}}.
+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.
+== Approximation ==
+{{Interval edo approximation|interval = 353/230}}
 == See also ==