Edϕ: Difference between revisions

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-Various equal divisions of the octave have close approximations of acoustic phi.
+Various equal divisions of the octave have close approximations of acoustic phi, or <span><math>φ</math></span>, ≈833.090296357¢.
-If the mth step of n-edo is a close approximation of φ, the nth step of m-edφ will be a close approximation of an octave.
+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.
-Such m-edφ are interesting as variants of their respective n-edo, introducing some combination tone powers.
+For example, the 7th step of 10ed2 is 840¢, and the 10th step of 7ed<span><math>φ</math></span> 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¢.
+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.
 {| class="wikitable"
@@ Line 9: / Line 12: @@
 |
 | colspan="4" |'''10ed2'''
-| colspan="4" |'''7edφ or 10ed(<math>2^{\frac{10log_2{φ}}{7}}</math>)'''
+| colspan="4" |'''7edφ or 10ed(<math>2^{\frac{10log_2{φ}}{7}} ≈ 1.988629015</math>)'''
 |-
 |'''scale step'''
@@ Line 26: / Line 29: @@
 |120
 |120
-|<math>φ^{\frac{1}{7}}</math> or <math>2^{\frac{1log_2{φ}}{7}}</math>
+|<math>φ^{\frac{1}{7}}</math> or <math>≈1.988629015^{\frac{1}{10}}</math>
 |1.071162542
 |119.0128995
@@ Line 36: / Line 39: @@
 |240
 |120
-|<math>φ^{\frac{2}{7}}</math> or <math>2^{\frac{2log_2{φ}}{7}}</math>
+|<math>φ^{\frac{2}{7}}</math> or <math>≈1.988629015^{\frac{2}{10}}</math>
 |1.147389191
 |238.025799
@@ Line 46: / Line 49: @@
 |360
 |120
-|<math>φ^{\frac{3}{7}}</math> or <math>2^{\frac{3log_2{φ}}{7}}</math>
+|<math>φ^{\frac{3}{7}}</math> or <math>≈1.988629015^{\frac{3}{10}}</math>
 |1.229040323
 |357.0386984
@@ Line 56: / Line 59: @@
 |480
 |120
-|<math>φ^{\frac{4}{7}}</math> or <math>2^{\frac{4log_2{φ}}{7}}</math>
+|<math>φ^{\frac{4}{7}}</math> or <math>≈1.988629015^{\frac{4}{10}}</math>
 |1.316501956
 |476.0515979
@@ Line 66: / Line 69: @@
 |600
 |120
-|<math>φ^{\frac{5}{7}}</math> or <math>2^{\frac{5log_2{φ}}{7}}</math>
+|<math>φ^{\frac{5}{7}}</math> or <math>≈1.988629015^{\frac{5}{10}}</math>
 |1.410187582
 |595.0644974
@@ Line 76: / Line 79: @@
 |720
 |120
-|<math>φ^{\frac{6}{7}}</math> or <math>2^{\frac{6log_2{φ}}{7}}</math>
+|<math>φ^{\frac{6}{7}}</math> or <math>≈1.988629015^{\frac{6}{10}}</math>
 |1.510540115
 |714.0773969
@@ Line 86: / Line 89: @@
 |840
 |120
-|<math>φ^{\frac{7}{7}}</math> or <math>2^{\frac{7log_2{φ}}{7}}</math>
+|<math>φ^{\frac{7}{7}}</math> or <math>≈1.988629015^{\frac{7}{10}}</math>
 |1.618033989
 |833.0902964
@@ Line 96: / Line 99: @@
 |960
 |120
-|<math>φ^{\frac{8}{7}}</math> or <math>2^{\frac{8log_2{φ}}{7}}</math>
+|<math>φ^{\frac{8}{7}}</math> or <math>≈1.988629015^{\frac{8}{10}}</math>
 |1.7331774
 |952.1031958
@@ Line 106: / Line 109: @@
 |1080
 |120
-|<math>φ^{\frac{9}{7}}</math> or <math>2^{\frac{9log_2{φ}}{7}}</math>
+|<math>φ^{\frac{9}{7}}</math> or <math>≈1.988629015^{\frac{9}{10}}</math>
 |1.85651471
 |1071.116095
@@ Line 116: / Line 119: @@
 |1200
 |120
-|<math>φ^{\frac{10}{7}}</math> or <math>2^{\frac{10log_2{φ}}{7}}</math>
+|<math>φ^{\frac{10}{7}}</math> or <math>≈1.988629015^{\frac{10}{10}}</math>
 |1.988629015
 |1190.128995
 |119.0128995
 |}