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.

A systematic ordering of rational fractions between 1 and 2 are shown. The boxed ratios correspond to the major third., perfect fourth, perfect fifth and major sixth of a major scale. The rounded boxes correspond to accidentals that are minor intervals from the tonic.

The white notes can play "Twinkle"

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:

${\frac {16}{15}}\approx 1.06667\quad {\frac {24}{55}}\approx 1.04167$

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: