From: Dave Rusin <rusin@math.niu.edu>
Date: Wed, 2 Sep 1998 22:32:31 -0500 (CDT)
To: thekumba@aol.com
Subject: series in music

[deletia -- djr]

>  Could you tell me if algebra like 4(A+E)=the sum of n(n+1), or something
>to that effect has any place in music. i am very interested in series and
>sums and would like to know anything you could tell me relating them to
>music.  please write me back thekumba@aol.com.  I greatly appreciate it.

I'm not quite sure to what sum you are referring, but certain sums play
a pivotal role in the analysis of sounds (in particular, musical sounds).
If you watch the motion of an air molecule near a sound source, you'll see
it move back and forth rapidly -- that's how we hear things (the motion is
transfered by molecular collision through the air to our eardrums).
What makes sound pleasant instead of noisy is the regularity of the motion
of neighboring molecules: when they all move together in a completely
periodic pattern, we hear a pure tone -- the frequency of the motion
(e.g. 440 complete paths per second) is called the pitch of the tone.

Now, pure tones are nice but not all that exciting -- they sound like
tuning forks. The  A  on a violin string or piano doesn't jiggle the air
quite so systematically - perhaps a couple of small even motions and then a
rather more forceful one, then repeat. The position of a single molecule
along a line is now not a simple sinusoidal function of time; rather,
it looks like a sine wave with every third bump a little taller.

Here's where the sums come in: assuming the motion is exactly periodic,
that function can be written as a sum of multiples of sin(t) and sin(3t).
The molecule may think it's moving in a complicated dance; we instead
hear it as if it were two molecules (of different sizes perhaps), one moving
with the simple periodicity of a tuning fork set at 440 Hz, the other
behaving like a molecule moved by a 1320Hz fork. This is called the
superposition principle. It's a general phenomenon, enuncuated by
Fourier, allowing the decomposition of "any" periodic function as a
sum of multiples of sin(n t), with n = 1, 2, 3, ... The multiples are
the Fourier coefficients of the given function. This leads to
Fourier Analysis, and more broadly to a branch of mathematics known as,
well, Harmonic Analysis -- not music at all, but as you can see, having
its origins in sums arising from musical topics.

Each instrument has its own characteristic "tone"; this reflects the
Fourier coefficients of the motions associated with a particular note.
Hit an  A  on the piano and you get a sound wave whose dominant summand
is indeed  sin(440 t); but there are the "harmonics" -- smaller multiples
of sin(880 t), sin(1320 t), etc. Different pianos give different mixes.

Oscilloscopes are used to look at the graphs of the particular
periodic functions arising from various sounds; spectral analyzers
compute the weights assigned to the component sine functions.

So indeed, music is all about sums and series from mathematics!

dave