Edϕ

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)