Physics and Astronomy Labs/The most beautiful equation in mathematics

In the study of musical harmony we discovered an inability to create a natural scale that could be used on a keyboard instrument. For example, the ratio of do to re is either 9/8 or 10/9, depending on which pair of notes you compare.

If we define f0 as the "first" pitch of a scale, and use integers to define other pitches, we require


 * $$\frac{f_{n+1}}{f_n} = \text{constant}$$

to be the same constant for any pair of notes. From our attempt to fit a natural scale to the keyboard, two possible values for this constant come in mind:

$$\frac{16}{15}\approx1.06667\quad\frac{24}{55}\approx1.04167$$

A good compromise might be to take the geometric or arithmetic means.


 * Calculate both of these means and verify that they fall comfortably between the two numbers.


 * Show that $$2^{1/N}$$ would work for a wide range of N.

An exhaustive effort to create scales for N any number other than 12 generally either fails, or creates a scale with much more than 12 tones in an octave.

Fractional Powers
Powers are usually introduced through multiple products of the "base" number. For example, here the base is 2: