Figure 1

It is a remarkable(!) coincidence that
2^{7/12} is very close to 3/2.
Why?
Harmony occurs in music when two pitches vibrate
at frequencies in small integer ratios.
For instance,
the notes of middle C and high C sound good together
(concordant) because the latter has
TWICE the frequency of the former.
Middle C and the G above it sound good together
because the frequencies of G and C are in a 3:2 ratio.
Well, almost!
In the 16th century the popular method for tuning a
piano was to a justtoned scale. What this means
is that harmonies with the fundamental note (tonic)
of the scale were pure; i.e., the frequency
ratios were pure integer ratios.
But because of this, shifting the melody to other keys
would make the music sound different (and bad) because
the harmonies in other keys were impure!
So, the equaltempered scale (in common use today),
popularized by Bach,
sets out to "even out" the badness by making the
frequency ratios the same between all
12 notes of the chromatic scale
(the white and the black keys on a piano).
Thus, harmonies shifted to other
keys would sound exactly the same, although a really good
ear might be able to tell that the harmonies in the
equaltempered scale are not quite pure.
So to divide the ratio 2:1 from high C to middle C into
12 equal parts,
we need to make the ratios between successive note
frequencies 2^{1/12}:1. The startling
fact that 2^{7/12} is very close to 3/2
ensures that the interval between
C and G, which are 7 notes apart in the chromatic scale,
sounds "almost" pure! Most people cannot tell the
difference!
What a harmonious coincidence!
The Math Behind the Fact:
It is possible that our octave might be divided into
something other than 12 equal parts if the above
coincidence were not true!
It is worth noting that on a stringed instrument, a player
has complete control over the frequency of notes. So
she can produce pure harmonies. Very good string
players will actually play Asharp and Bflat differently.
Sometimes they don't even realize they are doing it
they just do what their ear tells them. Playing pure
harmonies on stringed instruments also means that the
same note will sound different
depending on the key the music is played in;
for instance, notes based on 'C' will produce slightly
different frequencies than those based on 'A'.
There are a lot of interesting connections of
mathematics with music. Take courses in music theory
to learn more!
How to Cite this Page:
Su, Francis E., et al. "Music Math Harmony."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
