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"
@@ Line 124: / Line 124: @@
 |119.0128995
 |}
 {| class="wikitable"
 |+
 | rowspan="2" |'''scale step'''
 | colspan="4" |'''13ed2'''
-| colspan="4" |'''9edφ or 13ed(<math>2^{\frac{10log_2{φ}}{7}} ≈ 1.988629015</math>)<math>2^{\frac{10log_2{φ}}{7}} ≈ 1.988629015</math>'''
+| colspan="4" |'''9edφ or 13ed(<math>2^{\frac{13log_2{φ}}{9}} ≈ 2.003876886</math>)'''
 |-
 |'''frequency multiplier (definition)'''
@@ Line 140: / Line 141: @@
 |-
 |'''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
@@ Line 150: / Line 151: @@
 |-
 |'''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
@@ Line 160: / Line 161: @@
 |-
 |'''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
@@ Line 170: / Line 171: @@
 |-
 |'''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
@@ Line 180: / Line 181: @@
 |-
 |'''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
@@ Line 190: / Line 191: @@
 |-
 |'''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
@@ Line 200: / Line 201: @@
 |-
 |'''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
@@ Line 210: / Line 211: @@
 |-
 |'''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
@@ Line 220: / Line 221: @@
 |-
 |'''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
@@ Line 230: / Line 231: @@
 |-
 |'''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
@@ Line 240: / Line 241: @@
 |-
 |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
@@ Line 250: / Line 251: @@
 |-
 |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
@@ Line 260: / Line 261: @@
 |-
 |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}}