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From: sandell@aristotle.ils.nwu.edu
Newsgroups: comp.music
Subject: Freq'cy vs. pitch shifting & "missing fundamental"
Message-ID: <10133@accuvax.nwu.edu>
Date: 26 Jul 90 16:44:30 GMT
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Lines: 74

From: Greg Sandell <sandell@aristotle.ils.nwu.edu>

Johan Thornton writes:

> Well, yes.  Let's look at a signal that contains a 100Hz, a 200Hz and a 
> 300Hz sine.  The spectrum of this signal is:
> 
> A|
>  |   |   |   | 
>  |   |   |   |
>  ---------------------
>  0   1   2   3   4   5  (x 100Hz) 
 
 	[ some stuff deleted here... ]
 
> If we do a frequency shift on the original signal, say by 50Hz, we do
> 
>        f =  f(old) + 50
> 
> and get 150, 250 and 350 Hz.  Note that these no longer have the same
>                               ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
> harmonic relationship.  The spectrum now looks like:
> ^^^^^^^^^^^^^^^^^^^^^
 
Yes, this is true.
 
> A|
>  |     |   |   |
>  |     |   |   |
>  ---------------------
>  0   1   2   3   4   5
> 
> A pitch change stretches the spectrum while a frequency shift slides it.
> A frequency shift will generally change a  harmonic sound into an 
>                        ^^^^^^^^^
> inharmonic sound.
> 
> Johan Thornton, Esq.
> jthornto@fs1.ee.ubc.ca

The choice of the word "generally" is interesting.  The transformation
created above suggests a harmonic sound with a 50 Hz fundamental,
with components at the first, second, 4th and 6th harmonic missing.  In this
case, most people aren't likely to experience this as a harmonic tone,
but if four or five higher harmonics (preferably consecutive) were
present, the listener would probably 'hear' the 50 Hz fundamental
and the signal would fuse as a harmonic sound.

That's right, you hear harmonics that aren't even there...amazingly,
even harmonics quite a bit below the lowest actually-present harmonic.
This is called the "missing fundamental" effect. (In older writings,
it was called the "residue pitch".)  

So any frequency shift always creates a shift to a new potentially
harmonic sound with missing components, and depending upon the
number of consecutive components present and the range in which they appear,
the sound may actually be perceived as harmonic, with a new pitch.
I believe that Ritsma(1967) found that the "missing fundamental"
effect is most effective when at least some of the components are
below 5000Hz, and even more specifically, when the 3rd through 5th
harmonics are present and happen to fall in the range of 300-2000Hz.
(JASA 42, 191-198).

Although nearly everybody is skeptical when they first hear this, they
are usually won over by the "transistor radio" demonstration.  The
bass guitar has fundamental frequencies as low as 41 Hz:  the tiny
speaker in your walkman headphones is obviously inadequate to
project a tone this low effectively.  But do we have any trouble 
making out the bass part?  No...it would appear that we reconstruct
the signal perceptually from the higher frequency information which
is present.

Greg Sandell
sandell@ils.nwu.edu