Edϕ

The octave-reduced 13th harmonic, 13/8 (≈840.5276618¢), is closely approximated by 7 steps of 10ed2 (840¢).

13/8 (1.625) is also close to acoustic phi (≈1.618033989), because 8 and 13 are sequential Fibonacci numbers.

So, if we divide acoustic phi into 7 steps, then 10 of those steps will bring us quite close to an octave.

This tuning therefore could also be thought of as 10edo with a stretched octave. Since the ratio between acoustic phi and the 10ed2 approximation of the 13th harmonic is ≈0.9917741623, the resulting stretched octave is 2^0.9917741623 = 1.988629015, or ≈1190.128995¢.

	10ed2				7edφ or 10ed([math]\displaystyle{ 2^{\frac{10log_2{φ}}{7}} }[/math])
scale step	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{ 2^{\frac{1log_2{φ}}{7}} }[/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{ 2^{\frac{2log_2{φ}}{7}} }[/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{ 2^{\frac{3log_2{φ}}{7}} }[/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{ 2^{\frac{4log_2{φ}}{7}} }[/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{ 2^{\frac{5log_2{φ}}{7}} }[/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{ 2^{\frac{6log_2{φ}}{7}} }[/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{ 2^{\frac{7log_2{φ}}{7}} }[/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{ 2^{\frac{8log_2{φ}}{7}} }[/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{ 2^{\frac{9log_2{φ}}{7}} }[/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{ 2^{\frac{10log_2{φ}}{7}} }[/math]	1.988629015	1190.128995	119.0128995