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(These pages will
explain damping functions and the very cool way they are used
in music!)
(The black and white graphs are screen captures from a
TI-92 graphing calculator. The other graphical images are
screen captures from the digital editing software SAW Plus.)
What are damping functions?
The best way to explain them
is to show you some examples...
Look at the function f(x)
= x*sin(10x).
(The * is being
used to indicate multiplication.)
Ignoring the first factor,
x, for a minute, the
graph of g(x)
= sin(10x)
looks like:
(graphing window: x on [-7,7], y on [-4,4])
So, what does multiplying by
the x
do? Let's find out by graphing the whole thing!
f(x) = x*sin(10x)
(graphing window: x on [-7,7], y on [-4,4])
Man! It really changed! Look back
up at our first graph. Do you see what happened?
The graph of g(x)
= sin(10x)
is getting squished (or damped) between the graphs of
y = x
and y = -x
!!
Don't believe me? Then check
it out! Let's graph f(x)
= x*sin(10x),
y = x
and y =
-x
all on the same graph...
(y=x and y= -x are
the thicker lines.)
We see that our sine graph
is, indeed, bounded between them! Pretty cool, huh?
In the function
this x is called
the damping factor.
Let's try something more complicated.
What would the graph of
f(x) = (log
x)*cos(15x) look
like?
Well, g(x)
= cos(15x)
looks like:
(graphing window: x on [-.235, 7], y on [-1.5, 1.5])
Our damping factor is
log
x. So, g(x)
= cos(15x)
is going to get bounded by
y = log
x and
y = -log
x . (Note
that g(x)=cos(15x) is also going to get restricted to the
domain of y=log(x) which is x>0.)
Let's see what it will look
like!
(graphing window: x on [-.235, 7], y on [-1.5, 1.5])
It sure looks like it's bounded
by those logs... Let's graph them to be sure!
Yep!
(Looks like the messy end of a fish, doesn't it?)
Now, how are damping factors
used? Well, one way they are used is in the only thing as
cool as math: MUSIC!
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the music on this site, you'll need to download
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Music is an
intricate combination of sine waves. This is
what 5 seconds of Autumn Fell looks like: |
If we
zoom in to look at just .07 seconds (7/100 of a second)
of this song, we find something that looks just like a messy
trig function!
It looks
similar to something I can graph on my calculator...
(Graphing window: x on [-15.58, 15.58], y on [-4, 4])
So, where
do damping functions come in? They're used at the end of
the song, where the sound "fades out!"
When the
song's final mix is done, the producer decides if and how
the song should be faded out. The process is all done by
computer! The software used, SAW
Plus in this case, gives the engineer three main choices:
1)
Linear Fade
2) Logarithmic Fade
3) Inverse Logarithmic Fade
(Remember that inverse logs are
just exponentials!) |
This is how the end of
Autumn Fell looks and sounds before it is faded:
|
press to listen |
Inverse Logarithmic
Fades:
Here's an example
of an inverse logarithmic fade.
We can see the
effects of the fade in the next picture.
Here are two examples
of possible damping factors for this type of fade:
and 
But this type
of fade isn't used very often. The software doesn't give the
engineer control over the rate of the fade. As you can see,
it fades out pretty abruptly!
Linear Fades:
Below are two
examples of Autumn Fell being linearly faded at different
rates. Notice that the damping factor with the steeper slope
fades out a lot faster. Engineers can adjust the slope so
that the fade sounds just right. Listen to the difference!
Logarithmic Fades:
This type of fading
is often a favorite of engineers because of the natural sound
it produces. Here's what Autumn Fell sounds like with a logarithmic
fade. Can you tell the difference? See if you can by comparing
it to the slower linear fade. You can hear the music in the
log fade a lot longer.
Can you figure out the mathematics
behind the logarithmic fade? How would we graph an example?
Let's do
some reviewing and exploring.
If we graph
 ,
and 
...
(Graphing window: x on [-2.58,
13.98], y on [-2, 5.1])
Notice that
the smaller the base (3/2,
2 and 10 in our example),
the steeper the graph.
Now, how
can we get our logs to flip over the y-axis like they are
in our fades? By using f(-x) instead of f(x)!
Let's check
by graphing 
(Graphing window: x on [-13.98, 2.38], y on [-2,
5.1])
Getting
back to our logarithmic fades... When the engineer decides
how fast the song should fade, the program adjusts the base
of the logarithm!
Music
is just applied mathematics...
A good musician is an applied mathematician!
Eddie
Van Halen? Math geek? You be the judge!
Questions
to be pondered:
| Since music is
made up of combinations of sine waves, what instrument
do you think can produce a graph that is the closest
to the standard f(x)=sin(x)? (Another way to think about
it: What instrument produces the purest tone?) |
| What instrument
do you think would produce the most complex sine waves? |
| Since music is made
up of combinations of sine waves, what instrument do
you think can produce a graph the closest to the standard
f(x)=sin(x)? (Another way to think about it: What instrument
produces the purest tone? |
The instrument that produces
the purest tone (a wave closest to the standard sine wave)
is the flute!
Here's a zoom-in
so you can see what's really going on!
Note: If you consider
the range of instruments used in Hillbilly music...... then
the bottle really gives the purest tone! Check it out!
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(Please ignore the fact that this picture says "flute"
-
It's really a bottle!)
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If we zoom-in,
we can see what a perfect sine wave blowing into a bottle
can produce!
| What instrument do
you think would produce the most complex sine waves? |
The instrument that produces
the most complex sine wave is the cymbal! (Or the human voice,
if you don't consider a cymbal an instrument.)
Look how messy
this zoom-in is!
Here's a great
related site to check out: How
CD's Work
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