Edϕ: Difference between revisions

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-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"
 |+
-|
+| rowspan="2" |'''scale step'''
 | colspan="4" |'''10ed2'''
 | colspan="4" |'''7edφ or 10ed(<math>2^{\frac{10log_2{φ}}{7}} ≈ 1.988629015</math>)'''
 |-
-|'''scale step'''
 |'''frequency multiplier (definition)'''
 |'''10ed2 frequency multiplier (decimal)'''
@@ Line 124: / Line 124: @@
 |119.0128995
 |}
+{| class="wikitable"
+|+
+| rowspan="2" |'''scale step'''
+| colspan="4" |'''13ed2'''
+| colspan="4" |'''9edφ or 13ed(<math>2^{\frac{13log_2{φ}}{9}} ≈ 2.003876886</math>)'''
+|-
+|'''frequency multiplier (definition)'''
+|'''10ed2 frequency multiplier (decimal)'''
+|'''pitch (¢)'''
+|'''Δ (¢)'''
+|'''frequency multiplier (definition)'''
+|'''frequency multiplier (decimal)'''
+|'''pitch (¢)'''
+|'''Δ (¢)'''
+|-
+|'''1'''
+|<math>2^{\frac{1}{13}}</math>
+|1.054766076
+|92.30769231
+|92.30769231
+|<math>φ^{\frac{1}{9}}</math> or <math>≈2.003876886^{\frac{1}{13}}</math>
+|1.054923213
+|92.56558848
+|92.56558848
+|-
+|'''2'''
+|<math>2^{\frac{2}{13}}</math>
+|1.112531476
+|184.6153846
+|92.30769231
+|<math>φ^{\frac{2}{9}}</math> or <math>≈2.003876886^{\frac{2}{13}}</math>
+|1.112862986
+|185.131177
+|92.56558848
+|-
+|'''3'''
+|<math>2^{\frac{3}{13}}</math>
+|1.17346046
+|276.9230769
+|92.30769231
+|<math>φ^{\frac{3}{9}}</math> or <math>≈2.003876886^{\frac{3}{13}}</math>
+|1.173984997
+|277.6967655
+|92.56558848
+|-
+|'''4'''
+|<math>2^{\frac{4}{13}}</math>
+|1.237726285
+|369.2307692
+|92.30769231
+|<math>φ^{\frac{4}{9}}</math> or <math>≈2.003876886^{\frac{4}{13}}</math>
+|1.238464025
+|370.2623539
+|92.56558848
+|-
+|'''5'''
+|<math>2^{\frac{5}{13}}</math>
+|1.305511698
+|461.5384615
+|92.30769231
+|<math>φ^{\frac{5}{9}}</math> or <math>≈2.003876886^{\frac{5}{13}}</math>
+|1.306484449
+|462.8279424
+|92.56558848
+|-
+|'''6'''
+|<math>2^{\frac{6}{13}}</math>
+|1.377009451
+|553.8461538
+|92.30769231
+|<math>φ^{\frac{6}{9}}</math> or <math>≈2.003876886^{\frac{6}{13}}</math>
+|1.378240772
+|555.3935309
+|92.56558848
+|-
+|'''7'''
+|<math>2^{\frac{7}{13}}</math>
+|1.452422856
+|646.1538462
+|92.30769231
+|<math>φ^{\frac{7}{9}}</math> or <math>≈2.003876886^{\frac{7}{13}}</math>
+|1.453938184
+|647.9591194
+|92.56558848
+|-
+|'''8'''
+|<math>2^{\frac{8}{13}}</math>
+|1.531966357
+|738.4615385
+|92.30769231
+|<math>φ^{\frac{8}{9}}</math> or <math>≈2.003876886^{\frac{8}{13}}</math>
+|1.533793141
+|740.5247079
+|92.56558848
+|-
+|'''9'''
+|<math>2^{\frac{9}{13}}</math>
+|1.615866144
+|830.7692308
+|92.30769231
+|<math>φ^{\frac{9}{9}}</math> or <math>≈2.003876886^{\frac{9}{13}}</math>
+|1.618033989
+|833.0902964
+|92.56558848
+|-
+|'''10'''
+|<math>2^{\frac{10}{13}}</math>
+|1.704360793
+|923.0769231
+|92.30769231
+|<math>φ^{\frac{10}{9}}</math> or <math>≈2.003876886^{\frac{10}{13}}</math>
+|1.706901614
+|925.6558848
+|92.56558848
+|-
+|11
+|<math>2^{\frac{11}{13}}</math>
+|1.797701946
+|1015.384615
+|92.30769231
+|<math>φ^{\frac{11}{9}}</math> or <math>≈2.003876886^{\frac{11}{13}}</math>
+|1.800650136
+|1018.221473
+|92.56558848
+|-
+|12
+|<math>2^{\frac{12}{13}}</math>
+|1.896155029
+|1107.692308
+|92.30769231
+|<math>φ^{\frac{12}{9}}</math> or <math>≈2.003876886^{\frac{12}{13}}</math>
+|1.899547627
+|1110.787062
+|92.56558848
+|-
+|13
+|<math>2^{\frac{13}{13}}</math>
+|2
+|1200
+|92.30769231
+|<math>φ^{\frac{13}{9}}</math> or <math>≈2.003876886^{\frac{13}{13}}</math>
+|2.003876886
+|1203.35265
+|92.56558848
+|}
+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}}