Logarithmic phi: Difference between revisions

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{{Infobox Interval

| Ratio = 2^{\varphi} {{=}} 2^{\frac{1+\sqrt{5} }{2} }

| Ratio = 2^{\varphi} = 2^{\frac{1+\sqrt{5} } {2} }

| Cents = 1941.640786499874

| 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 an "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.

== See also ==

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

* [[Golden sequences and tuning]]

* [[Golden meantone]]

* [[Metallic MOS]]

; The MOS patterns generated by logarithmic phi:

; The MOS patterns generated by logarithmic phi

* [[3L 2s]]

* [[5L 3s]]

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* [[13L 8s]]

* [[21L 13s]]

* ~~...~~

* …

; Related regular temperaments:

; Related regular temperaments

* [[Father family|Father temperament]]

* [[Keegic temperaments #Aurora|Aurora temperament]]

* [[Triforce]] divides an 1/3 octave period into logarithmic-phi-sized fractions.

; Music

@@ Line 1: / Line 1: @@
 {{Infobox Interval
-| Ratio = 2^{\varphi} {{=}} 2^{\frac{1+\sqrt{5} }{2} }
+| Ratio = 2^{\varphi} = 2^{\frac{1+\sqrt{5} } {2} }
 | Cents = 1941.640786499874
 | 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 an "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.
 == See also ==
 * [[Generating a scale through successive divisions of the octave by the Golden Ratio]]
+* [[Golden sequences and tuning]]
 * [[Golden meantone]]
 * [[Metallic MOS]]
-; The MOS patterns generated by logarithmic phi:
+; The MOS patterns generated by logarithmic phi
 * [[3L 2s]]
 * [[5L 3s]]
@@ Line 19: / Line 22: @@
 * [[13L 8s]]
 * [[21L 13s]]
-* ...
+* …
-; Related regular temperaments:
+; Related regular temperaments
 * [[Father family|Father temperament]]
 * [[Keegic temperaments #Aurora|Aurora temperament]]
+* [[Triforce]] divides an 1/3 octave period into logarithmic-phi-sized fractions.
 ; Music