Logarithmic phi: Difference between revisions

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| Name = logarithmic phi

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'''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.

==See also==

== See also ==

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

* [[Golden sequences and tuning]]

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* [[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]]

* [[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]]

@@ Line 4: / Line 4: @@
 | 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.
-==See also==
+== See also ==
 * [[Generating a scale through successive divisions of the octave by the Golden Ratio]]
 * [[Golden sequences and tuning]]
@@ Line 16: / Line 16: @@
 * [[Metallic MOS]]
-;The MOS patterns generated by logarithmic phi
+; The MOS patterns generated by logarithmic phi
 * [[3L 2s]]
 * [[5L 3s]]
@@ Line 22: / Line 22: @@
 * [[13L 8s]]
 * [[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]]