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Klarinet Archive - Posting 000270.txt from 2001/02

From: Grant Green <gdgreen@-----.com>
Subj: Re: [kl] Combination tones......
Date: Thu, 8 Feb 2001 14:18:18 -0500

>Tom and all,
>
>Yes, indeed, and this seems to be the crux of the matter. A single tone
>(the fundamental) passed through a perfect (linear) system will result in
>only the fundamental; there will be no additional tones produced. A single
>tone through a non-linear system will contain harmonics (2, 3, 4 ,,, times
>the fundamental) and the sums and differences between these harmonics.
>However, this harmonic series is all octaves, and the differences between
>members of this series are also octaves, so there is no tone produced which

Yes, except that the harmonic series is not all octaves: only the
powers of 2 are octaves (2, 4, 8, 16, etc), while the others are
other intervals. For example, the third harmonic is a fifth (plus an
octave). Still, none of the heterodyne components can be less than
the fundamental.

>***
>For example, suppose two tones of 1000 and 1200 Hz. The first tone
>produces harmonics at 1000, 2000, 3000 .... The second tone produces
>harmonics at 1200, 2400, 3600 .... 1200 - 1000 = 200, 2400 - 2000 = 400,
>2000 - 1200 @-----. Also, these resultant tones can interact; 200 +
>400 @-----.

And if you put all the combination/difference tones in order, you get
a series that goes: 200, 400, 600, 800, 1000, 1200, 1400, ... which
you immediately recognize is a harmonic series based on 200 Hz. You
perceive this as a single note at 200 Hz, just as you perceive the
1000 and 1200 Hz tones as single notes, with the upper harmonics
absorbed into the timbre (or "character") of the tone.

The higher one goes in the series, the smaller the amplitude: the
higher harmonics have such low amplitude that they are hard to hear
at all.

>Bottom line - a whole cacophony of tones can be produced by interactions of
>two pure sine waves in a non-linear system The ear is such a non-linear
>system. A vibrating panel (wallboard, sheet metal) is a non-linear system.
> And (to go back to the initial observation) it is likely that the tuner is
>also non-linear, at least for loud input tones. And to further complicate
>the matter, the fundamental tone from a musical instrument is not a pure
>sine wave, but already contains harmonics (at least) and other frequencies.

For a well-made musical instrument, nearly all of the audible sound
will be at harmonic frequencies, fortunately. This makes the
frequency calculation identical to the example above: only the
amplitudes (the amount of each sound) are changed.

>I guess what surprises me with the organ example is not that difference
>tones are produced, but that there aren't so many of them that the result
>is a mess!

Because the difference tones all line up in a harmonic series, you
perceive the whole mess as a single note.

>BTW, the "beats" phenomenon referred to below is, as stated, amplitude
>changes only, does not generate any additional tones, and does not require
>a non-linear system.

Actually, the beats are really the difference tone taken down to
infrasonic frequencies. You could take two trombones (or two
continuously-variable oscillators) tuned to an interval, and
gradually bring the pitches to unison: you would hear the difference
tone get lower and lower, until it became an infrasonic flutter =
beats.

Grant

++++++++++++++++++++++++++++++++++++++++++++++++
Grant Green gdgreen@-----.com
ecode:contrabass http://www.contrabass.com
Professional Fool -> http://www.mp3.com/ProFools
++++++++++++++++++++++++++++++++++++++++++++++++

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