source file: m1482.txt
Date: Wed, 22 Jul 1998 18:56:20 -0600 (MDT)

Subject: Re: TUNING digest 1481

From: John Starrett <jstarret@tiger.cudenver.edu>

On Wed, 22 Jul 1998 tuning@eartha.mills.edu wrote:

> Can somebody in this forum help me understand the concepts of overtones and
> tone color?  For example, A440 produced by a tuning fork consists of
> vibrations having frequencies of mostly A440. But A440 played on a piano or a
> tuba will have more overtones, frequencies that are multiples of 440  - 880,
> 1320 etc.  This is were I lose it. Are overtones different frequencies
> produced by different aspects of whatever materials are producing the sound?
> So if the sound is being created by simpler materials such as a tuning fork,
> the will be fewer other frequencies involved, whereas if the sound is created
> by a piano, the hammer and string are more complex materials and therefore
> have more frequencies or overtones? 

It is not so much that rich tones are composed of a fundamental 
frequency and its overtones as that they can be decomposed this way. 
Sound is frequently visualized as a graph of air pressure vs time:
                                _____       _____       _____
    /|    /|    /|    /|       |     |     |     |     |     |
  /  |  /  |  /  |  /  |       |     |     |     |     |     |
/    |/    |/    |/    |  or   |     |_____|     |_____|     |

for instance. Joseph Fourier figured out how to add up cosine and sine 
waves, which are mathematical representations of the the simplest kinds of 
sounds, to obtain any type of periodic function whatsoever, such as the 
ramp and square waves pictured above. 
	Describing a complex waveform in terms of its decomposition into a 
sum of sine and cosine waves whose frequencies are simple multiples of 
the fundamental (such as A 440) is useful, but not necessary, for 
describing a tone and understanding why a waveform looks and sounds as it 
does. For instance, the first waveform pictured is similar to 
that of a bowed instrument. The bow pulls the string until the rosin can 
no longer hold it, and then the string quickly snaps free, travelling a 
distance until it is recaptured by the rosined horsehair, to be pulled 
again until it escapes, etc. The ramp up represents the string being 
pulled by the bow, and the quick dropoff is where the string escapes and 
quickly springs back. The motion of the string is transferred to the top 
of the instrument, which moves in and out with the change in tension on 
the string, transfering its motion to the air. The changes in air 
pressure are graphed by the helpful scientist and pulled apart into 
separate simpler (read smoother) graphs, which when added together give 
back the original. The separate frequencies people talk about are not 
really separate properties of the instrument (except in the case of 
synthesizers that make complex sounds by adding together simple ones) but 
rather a convenient mathematical way to analyze the complex vibrations of 
musical instruments. 
    This is a simplification for the sake of a short(?) answer to the 
complex question you asked. Certainly something that vibrates in a 
complex way produces a complex sound, but simple waveforms such as 
triangle waves and square waves have complex representations in terms of 
the number of higher harmonics that have to be added together to build 
them from scratch. I am sure a number of others will respond to your 
question, so I am signing off.

John Starrett
http://www-math.cudenver.edu/~jstarret