Transcription of Mark Petersen - Applied Mathematics
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Mark Petersen - Applied Mathematics
1 Mathematical HarmoniesMark PetersenWhat is music? When you hear a flutist, a signal is sent from her fingers to your ears. Asthe flute is played, it vibrates. The vibrations travel through the air and vibrate youreardrums. These vibrations are fast oscillations in air pressure, which your ear detects BasicsThe simplest model of a musical sound is a sine wave, were the domain (x-axis) is timeand the range (y-axis) is pressure.)2sin(ftAP where: P pressure, in decibels or Pascalst time, in secondsAamplitude (height of the wave) or volume, in decibels or Pascalsf frequency or pitch, in period, in seconds is the duration of one wave. fT1 Figure 1. A sine wave with amplitude A = 60 dB and frequency f = 100 general, a sound has two characteristics: pitch and volume. The pitch, or note played,corresponds to the frequency of the wave. High notes have high frequencies, so thepressure varies quickly.
10 20 30 40 60 80 100 200 300 400 600 800 1000 2000 3000 4000 6000 8000 10000 PIANO VIOLIN TUBA PICCOLO SOPRANO ... frequency is an octave higher. Using these facts, we may construct the following chart. Note Frequency Diagram of vibrating string low low low A f = 55 Hz ... Notice that the number of arithmetic terms between each geometric is 0 ...
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