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

== 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 meantone]]

* [[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]]

* [[8L 5s]]

*[[8L 5s]]

* [[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]]

; 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 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 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 is not to be confused with [[acoustic phi]], which is 833.1¢.
-== 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 meantone]]
-* [[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]]
-* [[8L 5s]]
+*[[8L 5s]]
-* [[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]]
-; 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]]