TUNING digest 1481

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

Greetings All,

This goes back to the microtonal flute thread from May.
Please forgive my ignorance about the details of the workings
of wind instruments. I've just begun to seriously think about
microtonal wind instruments and have a few questions.

>From what I understand, many players achieve different tunings
by using various fingerings. From this list, I know this has been
done on flute and bassoon. To which other wind instruments has this
approach been applied ? And, what types of tunings are possible ?

The other approach is of course instruments that are physically
altered from their std. forms, to achieve non-std. pitches. Logic
would argue that there probably aren't many users of these instruments
around. Any thoughts/ comments ?

Is there a good source of info out there for someone wanting
to write parts for these instruments ? Though ultimately it would
require personally working with the people who are to actually
perform the parts, to fine tune the writing to match their actual
physical abilities. Thoughts ?

Regards,
Drew

"Isn't it funny
How a bear likes honey ?
Buzz! Buzz! Buzz!
I wonder why he does ?"
- Winnie-the-Pooh