Edϕ
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 phi scale doubles the resolution of the 9th root scale featured above, and notably introduces a good 3/2 and a good 7/4.
cet33.scl 25 25th root of phi, Walter O´Connell (1993) cet46.scl 18 18th root of phi, Walter O´Connell (1993)