Klarinet Archive - Posting 000502.txt from 2000/08

From: "Keith" <100012.1302@-----.com>
Subj: [kl] RE: klarinet Digest 17 Aug 2000 20:15:00 -0000 Issue 2474
Date: Thu, 17 Aug 2000 21:50:05 -0400

Bill,

No, Tony is right. You are confusing "harmonic" with "simple harmonic" and
specifically the
mechanical description of "simple harmonic motion". Nobody said this was
simple <grin>. Except maybe an undergraduate text!

The Fourier theorem shows that any periodic function, whatever its shape,
can be represented by a series of sine and cosine waves with simple
multiples of frequency, as in Tony's post below. Thus, by definition and
rigorously, harmonic. This has nothing to do with the mechanical forces, if
any, causing the waveform; it applies equally to light, X-rays and any other
wave motion including periodic squiggles on paper. You simply cannot build a
PERIODIC wave any other way. (definition: f(x) is always equal to f(x+t)
where t is the period).

Tony is on slightly thin ice by equating this to "sustained" tone, even
though he qualified this later by saying "at constant dynamic level". He
would usually be right, but one could construct a sustained tone (with
electronics) that had a constant dynamic level but continuously changing
frequency mixture, which would not have to be harmonically related. But this
wouldn't be periodic (have the same wave form repeated over and over again
with the same period).

Keith Bowen

> ----- Original Message -----
> From: "Tony Pay" <Tony@-----.uk>
> To: <klarinet>
> Sent: Thursday, August 17, 2000 12:06 PM
> Subject: Re: [kl] inharmonic (anharmonic) vs. decay vs periodic vs recipe
>
> > On Thu, 17 Aug 2000 09:30:05 -0700 (PDT), Bilwright@-----.net said:
> >
> > > It seems to me (but I am not presenting myself as an expert in this
> > > stuff) that we are confusing different aspects of waveform with each
> > > other because of imprecise definitions:
> >
> > No, I'm not:-)
> >
> > > > The sequence, f, 2f, 3f, 4f, 5f, etc is called the 'harmonic
> > > > series', and sounds that can be analysed in this way -- periodic
> > > > sounds -- are called 'harmonic'.
> > >
> > > No, 'harmonic' means that the _FREQUENCIES_ of the component sine
> > > waves have whole number relationships. Thus sine waves are periodic,
> > > but not all periodic waves are harmonic.
>
> ------------------------------
>
> Date: Thu, 17 Aug 2000 12:56:52 -0700 (PDT)
> To: klarinet@-----.org
> From: Bilwright@-----.net (William Wright)
> Subject: Re: [kl] inharmonic (anharmonic) vs. decay vs periodic vs recipe
> Message-ID: <11865-399C4384-4314@-----.net>
>
> <><> Bill wrote:
> Thus sine waves are periodic, but not all periodic waves are harmonic.
>
> <><> and Tony replied:
> Yes, all periodic waves are harmonic.
>
> Well <friendly smile, I'm not a troll> my college physics text
> (Sears & Zemansky) doesn't agree. It says:
>
> "Any sort of motion which repeats itself in equal intervals of time is
> called periodic, and if the motion is back and forth over the same path,
> it is also called oscillatory. A complete vibration or oscillation
> means one round trip...."
>
> and my text goes on to say that harmonic refers to a special sort of
> oscillatory motion where the "elastic restoring force" [acting on the
> body in motion] is not constant but instead "varies during the motion"
> such that "the acceleration is directly proportional to the object's
> displacement [from the position where no force is active on the body]."
>
> The result of all this is our friend, the sinusoidal wave.
>
> But there are plenty of periodic motions, both discontinuous and
> continuous, that are caused by forces whose strength at any moment is
> *not* proportional to the displacement. Such periodic motions are not
> sinusoidal or harmonic; and without the sine and cosine functions, all
> of the remaining 'harmonic' analysis falls apart and leads to
> misunderstandings.
>
> A simple example would be the pendulum on a clock, where the pendulum
> swings harmonically if unhindered, but the periodic interference of the
> escapement inside the clock converts the second hand's periodic harmonic
> motion into a periodic but not harmonic motion -- repeated periodically
> once every second.
>
> More germane to our discussion, the inertia of air molecules -- which
> has a significant effect in a clarinet -- is not directy linked to the
> reed's displacement at any moment in time. Nor is the reed's own
> inertia, for that matter. Inertia may not relate to the 'equal
> temperament' topic that began this thread, but it is one example of how
> a periodic motion can (is) modified to produce motion that is still
> periodic but no longer harmonic.
>
> ....before I go any further with this, I'll stop talking and give you an
> opprtunity to respond to this most basic assertion, if you disagree with
> it.
>
> Thanks,
> Bill
>

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