Phi to the phi

Interval information

Expression	[math]\varphi^{\varphi}[/math]
Size in cents	1347.968¢
Name	phi to the phi
Harmonic entropy (Shannon, [math]\sqrt{nd}[/math])	~4.21818 bits

Phi to the phi is the interval, which if used as an interval of equivalence, equates acoustic phi with logarithmic phi - in other words, when this interval is divided by logarithmic phi, the result is acoustic phi. The interval measures 1347.9684152 cents, making it a neutral ninth.

Theory

Golden ratio raised to the power of itself is equal to about 2.1784.

Concoctic scales made of two Fibonacci numbers (8&13, 13&21, 21&34, etc.) have both the amount of notes to the period approaching phi. and a generator that increasingly approaches logarithimic phi. In this case, phi to the phi is used as an interval of equivalence, and the generator also approaches the acoustic phi.

Useful divisions

21edφ^φ

Not only it has an interval 13\21 approaching acoustic phi, it also corresponds to 18.6948edo, which makes it sound quite close to the Rectified Hebrew's 19-tone scale (18.579-edo). It has a very precise major third (as opposed to conventional 19edo's precise minor third of 6/5) and a superpythagorean fifth of 706 cents.

Approximation of harmonics in 21EdPhi^Phi

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+19.6	+23.7	-25.0	-26.2	-20.9	-31.0	-5.4	-16.8	-6.6	+21.0	-1.3
	Relative (%)	+30.5	+36.9	-39.0	-40.8	-32.5	-48.3	-8.4	-26.1	-10.3	+32.7	-2.0
Steps (reduced)		19 (19)	30 (9)	37 (16)	43 (1)	48 (6)	52 (10)	56 (14)	59 (17)	62 (20)	65 (2)	67 (4)

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	64.2	23/22
2	128.4	14/13, 15/14
3	192.6	9/8, 19/17
4	256.8	15/13, 23/20
5	320.9
6	385.1
7	449.3	13/10, 17/13
8	513.5
9	577.7	7/5
10	641.9	19/13
11	706.1	3/2
12	770.3
13	834.5	21/13
14	898.6
15	962.8
16	1027	20/11
17	1091.2
18	1155.4
19	1219.6
20	1283.8	21/10, 23/11
21	1348