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)