This means that 3/2 ≈ 2^(7/12) [accurate to about 0.1%]
And also 4/3 ≈ 2^(5/12) [also accurate to about 0.1%]
And also 9/8 = (3/2) / (4/3) ≈ 2^(2/12) [accurate to about 0.2%; putting two factors of 3 in makes the approximation only half as good]
You also get another nice coincidence, that 5^3 = 125 ≈ 128 = 2^7
This means that 5/4 ≈ 2^(4/12) [accurate to about 1%]
There are some people who have written music in a 41-note equal tempered scale, because then you get an even better approximation to the 3/2 ratio:
3^41 = 36472996377170786403 ≈ 36893488147419103232 = 2^65
(3/2) ≈ 2^(24/41) [accurate to about 0.03%]
This means that 3/2 ≈ 2^(7/12) [accurate to about 0.1%]
And also 4/3 ≈ 2^(5/12) [also accurate to about 0.1%]
And also 9/8 = (3/2) / (4/3) ≈ 2^(2/12) [accurate to about 0.2%; putting two factors of 3 in makes the approximation only half as good]
You also get another nice coincidence, that 5^3 = 125 ≈ 128 = 2^7
This means that 5/4 ≈ 2^(4/12) [accurate to about 1%]
There are some people who have written music in a 41-note equal tempered scale, because then you get an even better approximation to the 3/2 ratio:
3^41 = 36472996377170786403 ≈ 36893488147419103232 = 2^65
(3/2) ≈ 2^(24/41) [accurate to about 0.03%]