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Teaching
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This teaching guide is
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Saw Me a Tune:
Music from handtools and other stuff
Music and math have been associated throughout history. The Greek
mathematician Pythagoras used numbers to model everything in the physical world, including
music. He noted music's regularities, including tempos and scales. And the great composer,
Johann Sebastian Bach, reportedly sometimes treated the composition of music as exercises
in solving mathematical puzzles.
Sound of Math
Today, the language of mathematics is used to represent most musical ideas. Musicians know
what notes to play and how to perform a certain piece of music thanks to time signatures,
note representation and scale progressions. Just as important, math is used to analyze
music and other sounds.
Music can be thought of as organized sound. Sound begins when some physical body, in
contact with the air, vibrates. The surrounding atmosphere then begins to vibrate, causing
periodic variations in atmospheric pressure, which are called sound waves. These
waves are much like the ripples seen when a rock is dropped into a pond. The bones and
skin of the ear are pushed by these rippling waves, and nerves send a message to the brain
where the sound is interpreted. Because sound can't be seen, scientists make graphical
representations on paper. Thus, sound becomes math.
A simple way to produce a sound wave is with a tuning fork. When struck, a tuning fork
vibrates and produces an audible tone. If these vibrations are graphed, a sine curve
is formed. By studying this curve, the frequency, loudness and quality of a sound can be
determined.
Harmony &
Noise
Each cycle of a sound wave includes one compression and one rarefaction. The
frequency is the number of cycles the sound wave completes per second. Frequency,
measured in Hertz, is an important element in music because sounds that have frequencies
with simple mathematical relationships are usually interpreted by people as being
harmonious.
Not all sounds are pleasant ones. For instance, striking two stones together produces a
vibration, but the vibration is not harmonious. That's because sound vibrates from the
stones in all three dimensions. The frequencies of the sound wave are of great variety and
have nothing in common. Such sound usually is called noise.
Making Waves
Musical instruments produce their own waves, which, in most cases, are set up in one
dimension only. Because the waves travel in one dimension, the frequencies are related
mathematically, and therefore, we hear a harmonious sound. For example, a guitar produces
vibrations along its strings. The sound of a drum is less harmonious than other
instruments because it vibrates in two dimensions.
When a musician places a finger on the string of an instrument, the wavelength is
shortened, increasing the frequency. The frequency of a tone is inversely related to the
length of its vibrating string. Pythagoras noted that musical tones are related to the
length of the string by exact ratios. Certain simple ratios give the most harmonious
intervals, according to Pythagoras, and that is the basis for the musical scales we use
today. By holding down a string at its midpoint and plucking a free half, the tone
produced is exactly one octave above the tone of the entire string.
A sine curve also can illustrate the volume or intensity of a sound. The higher the
volume, the higher will be the amplitude of the sound wave. However, the frequency is not
affected by the volume.
Musical Glasses
The same mathematical ideas can be applied to all musical instruments to help designers
produce the most enjoyable music possible. Furthermore, almost any object can be turned
into a musical instrument with a little help from mathematics. For example, a set of
drinking glasses filled with different amounts of water will create various tones if hit
lightly with a wooden spoon. By using different amounts of water, the glasses’
wavelengths are changed much like when a guitar string is depressed.
Musical Saws
Sometimes ordinary musical instruments are made from not-so-ordinary objects. Who would
have thought that a handsaw could make music? Someone did. In fact, musical saws date back
to the 12th century. Woodcutters from Argentina were among the first to discover that by
bending handsaws and hitting them, the tools made interesting sounds. Today's musical saw
has many devoted players as well as fans.
These days, even the most unusual musical instruments can be simulated in a studio.
With the advance of electronic and computer-generated sound in the music industry,
mathematics is becoming even more useful to musicians.
Sources
Havlena, Dennis. "Musical Saw--How to Find One and Play
It." June 1998. Online. America Online.
"Sound." Compton’s Encyclopedia Online v. 2.0. The Learning
Company, Inc. 1997. Online. America Online. June 1998.
Wagner, Pete. "The Pond and the Philosopher’s Stone." Online.
Netscape. June 1998.
"What Is the Frequency of Sound Waves?" National Museum of Science and
Technology. 1997. Online. America Online. June 1998.
"What Makes Digital Music?" National Museum of Science. 1995. Online.
America Online. June 1998.
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Last Updated: 02/16/03
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