Edϕ

Various equal divisions of the octave have close approximations of acoustic phi, or , ≈833.090296357¢. [math]\displaystyle{ φ }[/math]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{10log_2{φ}}{7}} ≈ 1.988629015 }[/math])[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		1.054766076	92.30769231	92.30769231		1.054923213	92.56558848	92.56558848
2		1.112531476	184.6153846	92.30769231		1.112862986	185.131177	92.56558848
3		1.17346046	276.9230769	92.30769231		1.173984997	277.6967655	92.56558848
4		1.237726285	369.2307692	92.30769231		1.238464025	370.2623539	92.56558848
5		1.305511698	461.5384615	92.30769231		1.306484449	462.8279424	92.56558848
6		1.377009451	553.8461538	92.30769231		1.378240772	555.3935309	92.56558848
7		1.452422856	646.1538462	92.30769231		1.453938184	647.9591194	92.56558848
8		1.531966357	738.4615385	92.30769231		1.533793141	740.5247079	92.56558848
9		1.615866144	830.7692308	92.30769231		1.618033989	833.0902964	92.56558848
10		1.704360793	923.0769231	92.30769231		1.706901614	925.6558848	92.56558848
11		1.797701946	1015.384615	92.30769231		1.800650136	1018.221473	92.56558848
12		1.896155029	1107.692308	92.30769231		1.899547627	1110.787062	92.56558848
13		2	1200	92.30769231		2.003876886	1203.35265	92.56558848