Edϕ: Difference between revisions

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Various equal divisions of the octave have close approximations of acoustic phi, or ~~~~<math>φ</math~~></span~~>, ≈833.090296357¢.

Various equal divisions of the octave have close approximations of [[acoustic phi]], or <math>φ</math>, ≈833.090296357¢.

If the ~~~~<math>m^{th}</math~~></span~~> step of ~~~~<math>n</math>ed2 is a close approximation of ~~~~<math>φ</math~~></span~~>, the ~~~~<math>n^{th}</math~~></span~~> step of ~~~~<math>m</math>ed~~~~<math>φ</math~~></span~~> will be a close approximation of 2.

If the <math>m^{th}</math> step of <math>n</math>ed2 is a close approximation of <math>φ</math>, the <math>n^{th}</math> step of <math>m</math>ed<math>φ</math> will be a close approximation of 2.

For example, the 7th step of 10ed2 is 840¢, and the 10th step of 7ed~~~~<math>φ</math~~></span~~> is ≈1190.128995¢.

For example, the 7th step of 10ed2 is 840¢, and the 10th step of 7ed<math>φ</math> is ≈1190.128995¢.

As another example, the 9th step of 13ed2 is ≈830.7692308¢, and the 13th step of 9ed~~~~<math>φ</math>~~~~ is ≈1203.35265¢.

As another example, the 9th step of 13ed2 is ≈830.7692308¢, and the 13th step of [[9edϕ|9ed<math>φ</math>]] is ≈1203.35265¢.

Such ~~~~<math>m</math>ed~~~~<math>φ</math~~></span~~> are interesting as variants of their respective ~~~~<math>n</math>ed~~~~<math>2</math>, introducing some combination tone powers.

Such <math>m</math>ed<math>φ</math> are interesting as variants of their respective <math>n</math>ed<math>2</math>, introducing some combination tone powers.

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A couple such scales can be found in the [[Scala Scale Archive|Huygens-Fokker Foundation's Scala scale archive]]. They were described by Walter O'Connell in his 1993 paper [http://anaphoria.com/oconnell.pdf The Tonality of the Golden Section]. The 18th root of ~~phi~~ scale doubles the resolution of the 9th root scale featured above, ~~and~~ notably ~~introduces a good~~ 3/2 and ~~a good~~ 7/4.

A couple such scales can be found in the [[Scala Scale Archive|Huygens-Fokker Foundation's Scala scale archive]]. They were described by Walter O'Connell in his 1993 paper [http://anaphoria.com/oconnell.pdf The Tonality of the Golden Section]. The 18th root of φ scale doubles the resolution of the 9th root scale featured above, as so as the 9th root of φ scale is similar to 13ed2 the 18th root of φ scale is similar to 26edo (which does a notably better job of approximating 3-, 5-, and 7- limit harmonies).

cet33.scl 25 25th root of phi, Walter O´Connell (1993)

cet46.scl 18 18th root of phi, Walter O´Connell (1993)

== See also ==

* [[EDe]]

* [[Acoustic pi]]

* [[User:Eliora/Phi to the phi]]

[[Category:Golden ratio]]

@@ Line 1: / Line 1: @@
-Various equal divisions of the octave have close approximations of acoustic phi, or <span><math>φ</math></span>, ≈833.090296357¢.
+Various equal divisions of the octave have close approximations of [[acoustic phi]], or <math>φ</math>, ≈833.090296357¢.
-If the <span><math>m^{th}</math></span> step of <span><math>n</math><span>ed2 is a close approximation of <span><math>φ</math></span>, the <span><math>n^{th}</math></span> step of <span><math>m</math><span>ed<span><math>φ</math></span> will be a close approximation of 2.
+If the <math>m^{th}</math> step of <math>n</math><span>ed2 is a close approximation of <math>φ</math>, the <math>n^{th}</math> step of <math>m</math><span>ed<math>φ</math> will be a close approximation of 2.
-For example, the 7th step of 10ed2 is 840¢, and the 10th step of 7ed<span><math>φ</math></span> is ≈1190.128995¢.
+For example, the 7th step of 10ed2 is 840¢, and the 10th step of 7ed<math>φ</math> is ≈1190.128995¢.
-As another example, the 9th step of 13ed2 is ≈830.7692308¢, and the 13th step of 9ed<span><math>φ</math></span> is ≈1203.35265¢.
+As another example, the 9th step of 13ed2 is ≈830.7692308¢, and the 13th step of [[9edϕ|9ed<math>φ</math>]] is ≈1203.35265¢.
-Such <span><math>m</math><span>ed<span><math>φ</math></span> are interesting as variants of their respective <span><math>n</math><span>ed<span><math>2</math><span>, introducing some combination tone powers.
+Such <math>m</math><span>ed<math>φ</math> are interesting as variants of their respective <math>n</math><span>ed<math>2</math><span>, introducing some combination tone powers.
 {| class="wikitable"
@@ Line 271: / Line 271: @@
 |}
-A couple such scales can be found in the [[Scala Scale Archive|Huygens-Fokker Foundation's Scala scale archive]]. They were described by Walter O'Connell in his 1993 paper [http://anaphoria.com/oconnell.pdf The Tonality of the Golden Section]. The 18th root of phi scale doubles the resolution of the 9th root scale featured above, and notably introduces a good 3/2 and a good 7/4.
+A couple such scales can be found in the [[Scala Scale Archive|Huygens-Fokker Foundation's Scala scale archive]]. They were described by Walter O'Connell in his 1993 paper [http://anaphoria.com/oconnell.pdf The Tonality of the Golden Section]. The 18th root of φ scale doubles the resolution of the 9th root scale featured above, as so as the 9th root of φ scale is similar to 13ed2 the 18th root of φ scale is similar to 26edo (which does a notably better job of approximating 3-, 5-, and 7- limit harmonies).
   cet33.scl                      25  25th root of phi, Walter O´Connell (1993)
   cet46.scl                      18  18th root of phi, Walter O´Connell (1993)
+== See also ==
+* [[EDe]]
+* [[Acoustic pi]]
+* [[User:Eliora/Phi to the phi]]
+[[Category:Golden ratio]]
+{{todo|inline=1|improve synopsis|improve readability}}