View source for Physics and Astronomy Labs/The most beautiful equation in mathematics

[[File:Natural scale ratios.svg|thumb|350px|A systematic ordering of rational fractions between 1 and 2 are shown.  The boxed ratios correspond to the  [[w:major third|major third]].,  [[w:perfect fourth|perfect fourth]], [[w:perfect fifth|perfect fifth]] and [[w:major sixth|major sixth]] of a major scale.  The rounded boxes correspond to accidentals that are minor intervals from the [[w:Tonic (music)|tonic]].]][[File:Natural_piano_scale.svg|thumb|The white notes can play [[w:Twinkle, Twinkle, Little Star|"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 f<sub>0</sub> as the "first" pitch of a scale, and use integers to define other pitches, we require

:<math>\frac{f_{n+1}}{f_n} = \text{constant}</math>

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:

<math>\frac{16}{15}\approx1.06667\quad\frac{24}{55}\approx1.04167</math>

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 <math>2^{1/N}</math> 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: