# Edϕ

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Various equal divisions of the octave have close approximations of acoustic phi, or $φ$, ≈833.090296357¢.

If the $m^{th}$ step of $n$ed2 is a close approximation of $φ$, the $n^{th}$ step of $m$ed$φ$ will be a close approximation of 2.

For example, the 7th step of 10ed2 is 840¢, and the 10th step of 7ed$φ$ is ≈1190.128995¢. As another example, the 9th step of 13ed2 is ≈830.7692308¢, and the 13th step of 9ed$φ$ is ≈1203.35265¢.

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

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