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Klarinet Archive - Posting 000274.txt from 2001/02

From: George Kidder <>
Subj: Re: [kl] Combination tones......
Date: Thu, 8 Feb 2001 16:37:53 -0500


Good - a useful statement of what I wanted to say. Sorry about the
octave/harmonic confusion, although as you say, it doesn't matter for this

I am not sure about your last statement, however. You are saying that if
you start with two tones (say 1000 and 1001 Hz) you will hear a pulsation
(a variation in amplitude) at a rate of 1 Hz, and that this 1Hz is a new
tone? That is, it is really physically there? So that if I gradually
increase the separation until the tones are 1000 and 1200, I will hear a
200 Hz difference tone, EVEN IN A COMPLETELY LINEAR SYSTEM? I don't think
so. Returning to the radio world that Tom started from, this logic would
imply that when my two local broadcast stations with carrier frequencies at
550 and 1250 MHz are both on the air at once, there is a new carrier at 700
MHz present all the time, even in the absence of non-linear elements. I
certainly hope that this is not true, or the FCC is going to have one awful
problem on its hands. (In fact, of course, there is no such problem, and
the FCC is free to assign 700 MHz to a third station.)

When we discuss audio tones, the whole thing is made more complex by the
fact that our ears are certainly non-linear. (Maybe I better define
non-linear: a system is non-linear when the output is not directly
proportional to the input. Usually, as the input level rises, the output
level rises less than proportionally.) This is a good thing for ears, but
it makes our senses (as is so frequently true) a rather imperfect measuring
device, although very effective detecting devices.

At 11:49 2/8/01 -0700, you wrote:
>>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 =
>Grant Green
>Professional Fool ->
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George Kidder
Bar Harbor, ME

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