Edφ

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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]ed2 is a close approximation of [math]φ[/math], the [math]n^{th}[/math] step of [math]m[/math]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 9ed[math]φ[/math] is ≈1203.35265¢.

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

scale step 10ed2 7edφ or 10ed([math]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]2^{\frac{1}{10}}[/math] 1.071773463 120 120 [math]φ^{\frac{1}{7}}[/math] or [math]≈1.988629015^{\frac{1}{10}}[/math] 1.071162542 119.0128995 119.0128995
2 [math]2^{\frac{2}{10}}[/math] 1.148698355 240 120 [math]φ^{\frac{2}{7}}[/math] or [math]≈1.988629015^{\frac{2}{10}}[/math] 1.147389191 238.025799 119.0128995
3 [math]2^{\frac{3}{10}}[/math] 1.231144413 360 120 [math]φ^{\frac{3}{7}}[/math] or [math]≈1.988629015^{\frac{3}{10}}[/math] 1.229040323 357.0386984 119.0128995
4 [math]2^{\frac{4}{10}}[/math] 1.319507911 480 120 [math]φ^{\frac{4}{7}}[/math] or [math]≈1.988629015^{\frac{4}{10}}[/math] 1.316501956 476.0515979 119.0128995
5 [math]2^{\frac{5}{10}}[/math] 1.414213562 600 120 [math]φ^{\frac{5}{7}}[/math] or [math]≈1.988629015^{\frac{5}{10}}[/math] 1.410187582 595.0644974 119.0128995
6 [math]2^{\frac{6}{10}}[/math] 1.515716567 720 120 [math]φ^{\frac{6}{7}}[/math] or [math]≈1.988629015^{\frac{6}{10}}[/math] 1.510540115 714.0773969 119.0128995
7 [math]2^{\frac{7}{10}}[/math] 1.624504793 840 120 [math]φ^{\frac{7}{7}}[/math] or [math]≈1.988629015^{\frac{7}{10}}[/math] 1.618033989 833.0902964 119.0128995
8 [math]2^{\frac{8}{10}}[/math] 1.741101127 960 120 [math]φ^{\frac{8}{7}}[/math] or [math]≈1.988629015^{\frac{8}{10}}[/math] 1.7331774 952.1031958 119.0128995
9 [math]2^{\frac{9}{10}}[/math] 1.866065983 1080 120 [math]φ^{\frac{9}{7}}[/math] or [math]≈1.988629015^{\frac{9}{10}}[/math] 1.85651471 1071.116095 119.0128995
10 [math]2^{\frac{10}{10}}[/math] 2 1200 120 [math]φ^{\frac{10}{7}}[/math] or [math]≈1.988629015^{\frac{10}{10}}[/math] 1.988629015 1190.128995 119.0128995
scale step 13ed2 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 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)