Defense Acquisition University’s Wonder Waves

Seeing Music, Hearing Math

Every sound you hear is made up of “pulses” of pressure. When you hear a noise, your ear is really detecting tiny changes in air pressure  coming from the noise source. When you hear a firecracker explode, you are hearing a single “pulse” where the air pressure goes up and comes back down.dau 1

dau 3When you hear a continuous sound like a musical note, your ears are detecting the constant up and down changes in air pressure being generated by the voice, musical instrument, or loudspeaker producing the sound. These sounds
can be visualized as “waves” of pressure.dau 2


Music is made of tones and rhythms.  Tones are sound waves created by the musical instrument or voice and propagated through the air to your ears.  Have you notices that some musical sounds or chords seem harsh and dissonant while others seem open and harmonic?  There is a reason for that: two tones that are related by simple fractions sound good to our ears. When one pitch is twice as high as another, we call that an “octave” (a 2:1 ratio).

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When we plot these two waves against each, something pretty neat happens: we get a simple, pleasing shape to go with our simple, pleasing harmony. This is the “Lissajous” shape for an “octave”.dau5

At Panoply, you can use our “lissajous generator” to “draw” different sound waves and seethe interaction that produces pleasing tone pairs. The lissajous generator is a device made from speakers, sticks, flexible plastic, a laser pointer and a mirror.dau 6

When a tone is played on one speaker, the vibrations cause the mirror to deflect making the laser dot form a line.  The same for the other speaker except one line is vertical, the other horizontal.  When a tone is played on both speakers, they interact to form patterns called “lissajous figures”.  If the same tone is played on both speakers, the figure will be circle, ellipse or line (it depends on the phase relationship of the tones).  When the tone on one speaker is changed (like walking up the keys on a piano) the generator draws patterns showing the interaction.

The first two tones in “Taps” are a “fourth” apart. (That’s a ratio of 4:3)  If you played them together they sound more open or “harmonious”.  Their frequencies have a least common multiple that is small and their lissajous figure looks like this:dau 7

The first note pair in “Chopsticks” is a “major second” and they sound harsh; that is because the least common multiple of their frequencies is large (A ratio of 9:8).  The pattern made by a major second looks like this:


It is busy!  It looks just as “dissonant” as it sounds!

Animations courtesy of Dr. Dan Russell, Grad. Prog. Acoustics, Penn State.

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