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"
{| class="wikitable"
Line 9: Line 12:
|
|
| colspan="4" |'''10ed2'''
| 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'''
|'''scale step'''
Line 26: Line 29:
|120
|120
|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
|1.071162542
|119.0128995
|119.0128995
Line 36: Line 39:
|240
|240
|120
|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
|1.147389191
|238.025799
|238.025799
Line 46: Line 49:
|360
|360
|120
|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
|1.229040323
|357.0386984
|357.0386984
Line 56: Line 59:
|480
|480
|120
|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
|1.316501956
|476.0515979
|476.0515979
Line 66: Line 69:
|600
|600
|120
|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
|1.410187582
|595.0644974
|595.0644974
Line 76: Line 79:
|720
|720
|120
|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
|1.510540115
|714.0773969
|714.0773969
Line 86: Line 89:
|840
|840
|120
|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
|1.618033989
|833.0902964
|833.0902964
Line 96: Line 99:
|960
|960
|120
|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
|1.7331774
|952.1031958
|952.1031958
Line 106: Line 109:
|1080
|1080
|120
|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
|1.85651471
|1071.116095
|1071.116095
Line 116: Line 119:
|1200
|1200
|120
|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
|1.988629015
|1190.128995
|1190.128995
|119.0128995
|119.0128995
|}
|}

Revision as of 00:03, 9 February 2020

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.

10ed2 7edφ or 10ed([math]\displaystyle{ 2^{\frac{10log_2{φ}}{7}} ≈ 1.988629015 }[/math])
scale step 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
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