Science Weblecture for Unit 19
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The nature of sound and the relationship between musical tones has long fascinated humans. It was one of the earliest areas where philosophers tried to descirbe natural phenomena in mathematical terms.
We have mentioned the Pythagoreans in our study of the pre-Socratic philosophers. They were among the earliest to propose that mathematics could be used to describe relationships between physical entities. In particular, the Pythagoreans studied the arithmetic and geometrical progressions that are found in musical harmonies.
By the time of the early Roman empire, the study of music, along with arithmetic, geometry, and astronomy, was considered fundamental preparation for further study in the natural sciences. Together, these four "sciences" formed the quadrivium, and with the trivium of language arts, grammar, dialectic, and rhetoric, made up the original seven liberal arts. Praise and description of these seven subjects appears in a long work by the Roman poet Martianus Capella and in Macrobius' commentary Cicero's Dream of Scipio. Boethius too wrote on mathematics and music. Hence we take this opportunity to look at both music and the more general phenomenon of sound.
Sound of any kind involves waves of energy. We think of waves as something made of matter—like waves of water in the ocean. But if you watch the ocean carefully (away from shore), you'll notice that while the water moves up and down, and the pattern of high points in the water goes along in one direction, things floating on the surface move back and forth. The water itself is not actually flowing in the direction of the motion of the high points. What is moving is not matter, but an impulse of energy.
We can describe the characteristics of this energy flow as a "wave". A wave has a crest (high point) and a trough (low point) and repeats its shape over and over. The distance between two high points is the wave length, and the "height" of the crests or depth of the trough is the amplitude of the wave. If the energy flow is the compression and expansion of molecules in the air (or some other compressible medium), the wave is a sound wave, and the length of the wave gives the sound its pitch while the amplitude gives the sound its loudness. If the energy flow is electromagnetic energy, the wave is a light wave involving changes in electrical and magnetic fields, the length of the wave gives the light its color, while the amplitude gives the light wave its brightness.
The concepts of periodicity, something that repeats over and over in the same amount of time and the same way, is an important concept in science. Scientists constantly look for these repeating patterns, because they often indicate that some fundament law is working, and that some particular set of causes —usually forces—governs the generation of the pattern.
The representation of periodic patterns as waves is especially useful. We can take any repeating pattern, whether the actual phenomenon involves physical waves or not, and use the same mathematical treatment to talk about the pattern. Pythagoras would have loved what we do with trigonometry to describe everything from the shock wave spreading out from a supernova to the description of the edge of a leaf as a Mandelbrot pattern. As a scientist, you learn pay attention to anything that happens three times the same way. Twice may be coincidence, but three times is worth investigating to see if something interesting is going on. [This is also a fundamental principle of Celtic mythology!]
[many pages; skim for answers and major concepts]
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