Physics and Astronomy Labs/The most beautiful equation in mathematics

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"

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:

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

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: