Edϕ: Difference between revisions

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7 steps of 10ed2 closely approximates the 13th harmonic. The 13th harmonic is close to acoustic phi (1.618033989). If we divide acoustic phi into 7 steps, then 10 of those steps gets us close to an octave.
Various equal divisions of the octave have close approximations of [[acoustic phi]], or <math>φ</math>, ≈833.090296357¢.
 
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<math>φ</math> is ≈1190.128995¢.
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><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"
{| class="wikitable"
|+
|+
|
| rowspan="2" |'''scale step'''
| colspan="4" |'''10ed2'''
| colspan="4" |'''10ed2'''
| colspan="4" |'''7edφ'''
| colspan="4" |'''7edφ or 10ed(<math>2^{\frac{10log_2{φ}}{7}} ≈ 1.988629015</math>)'''
|-
|-
|'''scale step'''
|'''frequency multiplier (definition)'''
|'''frequency multiplier (definition)'''
|'''10ed2 frequency multiplier (decimal)'''
|'''10ed2 frequency multiplier (decimal)'''
Line 22: Line 29:
|120
|120
|120
|120
|<math>φ^{\frac{1}{7}}</math>
|<math>φ^{\frac{1}{7}}</math> or <math>≈1.988629015^{\frac{1}{10}}</math>
|1.071162542
|1.071162542
|119.0128995
|119.0128995
Line 32: Line 39:
|240
|240
|120
|120
|<math>φ^{\frac{2}{7}}</math>
|<math>φ^{\frac{2}{7}}</math> or <math>≈1.988629015^{\frac{2}{10}}</math>
|1.147389191
|1.147389191
|238.025799
|238.025799
Line 42: Line 49:
|360
|360
|120
|120
|<math>φ^{\frac{3}{7}}</math>
|<math>φ^{\frac{3}{7}}</math> or <math>≈1.988629015^{\frac{3}{10}}</math>
|1.229040323
|1.229040323
|357.0386984
|357.0386984
Line 52: Line 59:
|480
|480
|120
|120
|<math>φ^{\frac{4}{7}}</math>
|<math>φ^{\frac{4}{7}}</math> or <math>≈1.988629015^{\frac{4}{10}}</math>
|1.316501956
|1.316501956
|476.0515979
|476.0515979
Line 62: Line 69:
|600
|600
|120
|120
|<math>φ^{\frac{5}{7}}</math>
|<math>φ^{\frac{5}{7}}</math> or <math>≈1.988629015^{\frac{5}{10}}</math>
|1.410187582
|1.410187582
|595.0644974
|595.0644974
Line 72: Line 79:
|720
|720
|120
|120
|<math>φ^{\frac{6}{7}}</math>
|<math>φ^{\frac{6}{7}}</math> or <math>≈1.988629015^{\frac{6}{10}}</math>
|1.510540115
|1.510540115
|714.0773969
|714.0773969
Line 82: Line 89:
|840
|840
|120
|120
|<math>φ^{\frac{7}{7}}</math>
|<math>φ^{\frac{7}{7}}</math> or <math>≈1.988629015^{\frac{7}{10}}</math>
|1.618033989
|1.618033989
|833.0902964
|833.0902964
Line 92: Line 99:
|960
|960
|120
|120
|<math>φ^{\frac{8}{7}}</math>
|<math>φ^{\frac{8}{7}}</math> or <math>≈1.988629015^{\frac{8}{10}}</math>
|1.7331774
|1.7331774
|952.1031958
|952.1031958
Line 102: Line 109:
|1080
|1080
|120
|120
|<math>φ^{\frac{9}{7}}</math>
|<math>φ^{\frac{9}{7}}</math> or <math>≈1.988629015^{\frac{9}{10}}</math>
|1.85651471
|1.85651471
|1071.116095
|1071.116095
Line 112: Line 119:
|1200
|1200
|120
|120
|<math>φ^{\frac{10}{7}}</math>
|<math>φ^{\frac{10}{7}}</math> or <math>≈1.988629015^{\frac{10}{10}}</math>
|1.988629015
|1.988629015
|1190.128995
|1190.128995
|119.0128995
|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}}

Latest revision as of 15:32, 1 August 2025

Various equal divisions of the octave have close approximations of acoustic phi, or [math]\displaystyle{ φ }[/math], ≈833.090296357¢.

If the [math]\displaystyle{ m^{th} }[/math] step of [math]\displaystyle{ n }[/math]ed2 is a close approximation of [math]\displaystyle{ φ }[/math], the [math]\displaystyle{ n^{th} }[/math] step of [math]\displaystyle{ m }[/math]ed[math]\displaystyle{ 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]\displaystyle{ φ }[/math] is ≈1190.128995¢. As another example, the 9th step of 13ed2 is ≈830.7692308¢, and the 13th step of 9ed[math]\displaystyle{ φ }[/math] is ≈1203.35265¢.

Such [math]\displaystyle{ m }[/math]ed[math]\displaystyle{ φ }[/math] are interesting as variants of their respective [math]\displaystyle{ n }[/math]ed[math]\displaystyle{ 2 }[/math], introducing some combination tone powers.

scale step 10ed2 7edφ or 10ed([math]\displaystyle{ 2^{\frac{10log_2{φ}}{7}} ≈ 1.988629015 }[/math])
frequency multiplier (definition) 10ed2 frequency multiplier (decimal) pitch (¢) Δ (¢) frequency multiplier (definition) frequency multiplier (decimal) pitch (¢) Δ (¢)
1 [math]\displaystyle{ 2^{\frac{1}{10}} }[/math] 1.071773463 120 120 [math]\displaystyle{ φ^{\frac{1}{7}} }[/math] or [math]\displaystyle{ ≈1.988629015^{\frac{1}{10}} }[/math] 1.071162542 119.0128995 119.0128995
2 [math]\displaystyle{ 2^{\frac{2}{10}} }[/math] 1.148698355 240 120 [math]\displaystyle{ φ^{\frac{2}{7}} }[/math] or [math]\displaystyle{ ≈1.988629015^{\frac{2}{10}} }[/math] 1.147389191 238.025799 119.0128995
3 [math]\displaystyle{ 2^{\frac{3}{10}} }[/math] 1.231144413 360 120 [math]\displaystyle{ φ^{\frac{3}{7}} }[/math] or [math]\displaystyle{ ≈1.988629015^{\frac{3}{10}} }[/math] 1.229040323 357.0386984 119.0128995
4 [math]\displaystyle{ 2^{\frac{4}{10}} }[/math] 1.319507911 480 120 [math]\displaystyle{ φ^{\frac{4}{7}} }[/math] or [math]\displaystyle{ ≈1.988629015^{\frac{4}{10}} }[/math] 1.316501956 476.0515979 119.0128995
5 [math]\displaystyle{ 2^{\frac{5}{10}} }[/math] 1.414213562 600 120 [math]\displaystyle{ φ^{\frac{5}{7}} }[/math] or [math]\displaystyle{ ≈1.988629015^{\frac{5}{10}} }[/math] 1.410187582 595.0644974 119.0128995
6 [math]\displaystyle{ 2^{\frac{6}{10}} }[/math] 1.515716567 720 120 [math]\displaystyle{ φ^{\frac{6}{7}} }[/math] or [math]\displaystyle{ ≈1.988629015^{\frac{6}{10}} }[/math] 1.510540115 714.0773969 119.0128995
7 [math]\displaystyle{ 2^{\frac{7}{10}} }[/math] 1.624504793 840 120 [math]\displaystyle{ φ^{\frac{7}{7}} }[/math] or [math]\displaystyle{ ≈1.988629015^{\frac{7}{10}} }[/math] 1.618033989 833.0902964 119.0128995
8 [math]\displaystyle{ 2^{\frac{8}{10}} }[/math] 1.741101127 960 120 [math]\displaystyle{ φ^{\frac{8}{7}} }[/math] or [math]\displaystyle{ ≈1.988629015^{\frac{8}{10}} }[/math] 1.7331774 952.1031958 119.0128995
9 [math]\displaystyle{ 2^{\frac{9}{10}} }[/math] 1.866065983 1080 120 [math]\displaystyle{ φ^{\frac{9}{7}} }[/math] or [math]\displaystyle{ ≈1.988629015^{\frac{9}{10}} }[/math] 1.85651471 1071.116095 119.0128995
10 [math]\displaystyle{ 2^{\frac{10}{10}} }[/math] 2 1200 120 [math]\displaystyle{ φ^{\frac{10}{7}} }[/math] or [math]\displaystyle{ ≈1.988629015^{\frac{10}{10}} }[/math] 1.988629015 1190.128995 119.0128995
scale step 13ed2 9edφ or 13ed([math]\displaystyle{ 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]\displaystyle{ 2^{\frac{1}{13}} }[/math] 1.054766076 92.30769231 92.30769231 [math]\displaystyle{ φ^{\frac{1}{9}} }[/math] or [math]\displaystyle{ ≈2.003876886^{\frac{1}{13}} }[/math] 1.054923213 92.56558848 92.56558848
2 [math]\displaystyle{ 2^{\frac{2}{13}} }[/math] 1.112531476 184.6153846 92.30769231 [math]\displaystyle{ φ^{\frac{2}{9}} }[/math] or [math]\displaystyle{ ≈2.003876886^{\frac{2}{13}} }[/math] 1.112862986 185.131177 92.56558848
3 [math]\displaystyle{ 2^{\frac{3}{13}} }[/math] 1.17346046 276.9230769 92.30769231 [math]\displaystyle{ φ^{\frac{3}{9}} }[/math] or [math]\displaystyle{ ≈2.003876886^{\frac{3}{13}} }[/math] 1.173984997 277.6967655 92.56558848
4 [math]\displaystyle{ 2^{\frac{4}{13}} }[/math] 1.237726285 369.2307692 92.30769231 [math]\displaystyle{ φ^{\frac{4}{9}} }[/math] or [math]\displaystyle{ ≈2.003876886^{\frac{4}{13}} }[/math] 1.238464025 370.2623539 92.56558848
5 [math]\displaystyle{ 2^{\frac{5}{13}} }[/math] 1.305511698 461.5384615 92.30769231 [math]\displaystyle{ φ^{\frac{5}{9}} }[/math] or [math]\displaystyle{ ≈2.003876886^{\frac{5}{13}} }[/math] 1.306484449 462.8279424 92.56558848
6 [math]\displaystyle{ 2^{\frac{6}{13}} }[/math] 1.377009451 553.8461538 92.30769231 [math]\displaystyle{ φ^{\frac{6}{9}} }[/math] or [math]\displaystyle{ ≈2.003876886^{\frac{6}{13}} }[/math] 1.378240772 555.3935309 92.56558848
7 [math]\displaystyle{ 2^{\frac{7}{13}} }[/math] 1.452422856 646.1538462 92.30769231 [math]\displaystyle{ φ^{\frac{7}{9}} }[/math] or [math]\displaystyle{ ≈2.003876886^{\frac{7}{13}} }[/math] 1.453938184 647.9591194 92.56558848
8 [math]\displaystyle{ 2^{\frac{8}{13}} }[/math] 1.531966357 738.4615385 92.30769231 [math]\displaystyle{ φ^{\frac{8}{9}} }[/math] or [math]\displaystyle{ ≈2.003876886^{\frac{8}{13}} }[/math] 1.533793141 740.5247079 92.56558848
9 [math]\displaystyle{ 2^{\frac{9}{13}} }[/math] 1.615866144 830.7692308 92.30769231 [math]\displaystyle{ φ^{\frac{9}{9}} }[/math] or [math]\displaystyle{ ≈2.003876886^{\frac{9}{13}} }[/math] 1.618033989 833.0902964 92.56558848
10 [math]\displaystyle{ 2^{\frac{10}{13}} }[/math] 1.704360793 923.0769231 92.30769231 [math]\displaystyle{ φ^{\frac{10}{9}} }[/math] or [math]\displaystyle{ ≈2.003876886^{\frac{10}{13}} }[/math] 1.706901614 925.6558848 92.56558848
11 [math]\displaystyle{ 2^{\frac{11}{13}} }[/math] 1.797701946 1015.384615 92.30769231 [math]\displaystyle{ φ^{\frac{11}{9}} }[/math] or [math]\displaystyle{ ≈2.003876886^{\frac{11}{13}} }[/math] 1.800650136 1018.221473 92.56558848
12 [math]\displaystyle{ 2^{\frac{12}{13}} }[/math] 1.896155029 1107.692308 92.30769231 [math]\displaystyle{ φ^{\frac{12}{9}} }[/math] or [math]\displaystyle{ ≈2.003876886^{\frac{12}{13}} }[/math] 1.899547627 1110.787062 92.56558848
13 [math]\displaystyle{ 2^{\frac{13}{13}} }[/math] 2 1200 92.30769231 [math]\displaystyle{ φ^{\frac{13}{9}} }[/math] or [math]\displaystyle{ ≈2.003876886^{\frac{13}{13}} }[/math] 2.003876886 1203.35265 92.56558848

A couple such scales can be found in the Huygens-Fokker Foundation's Scala scale archive. They were described by Walter O'Connell in his 1993 paper 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


Todo: improve synopsis , improve readability
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