Harmonics question
Steve
daedalus at tezcat.com
Wed May 20 00:34:19 CEST 1998
>I understand that a square wave is made up of sine waves, but what are the
>exact frequencies that the harmonics oscillate at? For example, if you
>have a square wave at 130Hz are all the harmonics octaves up and down from it?
>
First, understand that a harmonic (when it is "in tune") is the same thing
as a partial, meaning that it's a standing wave whose wavelength is a
whole-number division of the fundamental frequency (you divide the string
into 2 parts for the first partial, 3 parts for the 2nd, and so on.)
So: let's say that the fundamental is 100 Hz. The first partial (or 2nd
harmonic) is 200Hz (1/2 the wavelength so twice the frequency), the second
is 300Hz (1/3 the wavelength), the third is 400Hz, and so on. Since an
octave up is twice the initial frequency, you can see that not all of the
harmonics fall on "octaves up" from the fundamental.
Simple waveforms are expressible in terms of the strength of each of these
harmonics. A sine wave is simple - it's all fundamental. A sawtooth wave
has all of the harmonics, but the strength of each successive one decreases
exponentially. A triangle wave has only the even harmonics. A square wave
has only the odd harmonics. As a general rule, the greater the strength of
the upper harmonics relative to the lower ones, the more "bright" the
sound. Also, it's possible to have harmonics that *don't* line up at the
partial frequencies, which gives a dissonant, bell-like effect. This is
called "inharmonicity" and can be heard in the lower notes of upright
pianos, or any other stringed instrument where some of the strings are
exceptionally thick (and therefore resistant to being bent).
Most of this and lots more can be found in any basic/introductory text on
acoustics. If you're at a loss there, try "The Master Handbook of
Acoustics" by F. Alton Everest, 3rd edition, ISBN# 0-8306-4438-5 (or
-4437-7 for the paperback). Spend the $22 and feed your head.
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