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When you think trigonometry you should think triangles -- not just geometric
triangles but musical triangles -- because trigonometry is the mathematics of
sound and music.
Sound is vibrating air. When air vibrates at a high frequency we hear a high pitched tone. When air vibrates at a low frequency we hear a low pitched tone. In general we can hear vibrations whose frequency is between 20 and 20,000 hertz (cycles per second). If your workstation has a CBL and a microphone, click the icon at the right. |
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A microphone measures air pressure. Sound is just the variation of air pressure. Sound can be represented mathematically by a function f(t) giving the air pressure at time t . More precisely, f(t) represents the difference between the air pressure at time t and the average air pressure.
The simplest sounds, called pure tones are represented by functions of the form
f(t) = A sin(2 pi w t)
| The movie at the left looks different values of the amplitude. We hear the amplitude as the loudness of the sound. |
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| The movie at the left looks different values of the frequency. We hear the frequency as pitch or tone. |
f(t) = sin t g(t) = 2 sin t h(t) = 0.5 sin t p(t) = -1.5 sin t
Describe the visual effects of the amplitude.
f(t) = sin t g(t) = sin 2t h(t) = sin 0.5t p(t) = sin (-1.5t)
Describe the visual effects of the frequency.
Mathematicians often add another parameter to the pure tone
to get tones described by
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| The new parameter -- d -- is called the phase parameter and shifts the graph as shown in the movie at the left. |
f(t) = sin (t)
g(t) = sin (t + Pi/10)
h(t) = sin (t + Pi/2)
p(t) = sin (t + Pi)
Describe the visual effects of the phase.
Listening to a single pure tone one cannot hear the phase. It does, however, have a major impact when two pure tones are combined or when a single pure tone is played through two speakers. The computer algebra system files for this module demonstrate these effects both visually and (if you are using Mathematica) audibly.
f(t) = sin t + sin (t)
g(t) = sin t + sin (t + Pi/10)
h(t) = sin t + sin (t + Pi/2)
p(t) = sin t + sin (t + Pi)
Describe the visual effects of combining two sounds of different phase.
We have been looking at or, more precisely, listening to the way that functions of the form
describe sound but the same functions describe many other phenomena, for example, light and radio waves. In the CAS files you observed the way in which sound waves of different phase can interact -- sometimes constructively and sometimes destructively. The same mathematics describes and explains some very puzzling optical happenings -- for example, diffraction and interference patterns. Click the picture below to go to a physics module on interference.
We have been looking at what happens when two sounds of the same frequency but different phase are combined. Now we want to see what happens when two sounds of different frequencies are combined.
f(t) = sin t + sin (2t)
g(t) = sin t + sin (1.05t)
h(t) = sin t + sin (1.1t
p(t) = sin t + sin (0.95t)
Warning: Be sure to try different ranges for t. You may get misleading results if you use a range for t that is too small.
Describe the visual effects of combining two sounds of different frequency.
We can get some insight into what happens when we combine two sounds with different frequencies by using a little algebra together with the formula for the sin of the sum of two angles.
Suppose that we combine two sounds of different frequency.
g(t) = sin(2 pi v t)
We begin by writing
w = (u + v)/2 d = (u - v)/2 u = w + d v = w - d
so that
f(t) + g(t) = sin(2 pi w t + 2 pi d t) + sin(2 pi w t - 2 pi d t)
= sin(2 pi w t) cos( 2 pi d t) + cos(2 pi w t) sin( 2 pi d t) +
sin(2 pi w t) cos(-2 pi d t) + cos(2 pi w t) sin(-2 pi d t)
= sin(2 pi w t) cos( 2 pi d t) + cos(2 pi w t) sin( 2 pi d t) +
sin(2 pi w t) cos( 2 pi d t) - cos(2 pi w t) sin( 2 pi d t)
= 2 sin(2 pi w t) cos( 2 pi d t)
Now looking at the bottom line
f(t) + g(t) = 2 sin(2 pi w t) cos( 2 pi d t)
we can see algebraically the result of combining two sounds of different frequency.
The frequency of the first factor is the average of the frequencies of the two separate sounds. Without the second factor this factor would have an amplitude of 2 and would oscillate back-and-forth between +2 and -2. The frequency of the second factor is the difference between the two individual frequencies. When the two individual sounds are close in frequency then the second factor oscillates slowly. This produces the low frequency "warbling" we heard earlier. We hear the high frequency first factor as the tone or pitch of the sound and we hear the lower frequency second factor as changing volume.
This phenomenon is called beats. It is the basis by which piano tuners tune pianos. They play a note on the piano and strike a tuning fork at the same time. If the piano is off just a little bit they hear a very noticeable beat.