As a saxophone player, I have always wondered how exactly sound waves work and why some notes sound good together while others don't. For example, when notes that are a half step apart are played simultaneously, "wobbles" are produced. If two sound waves interfere when they have frequencies that are not identical but very close, there is a resulting modulation in amplitude. When the waves interfere constructively, we say that there is a beat. The number of beats per second is known as the beat frequency, which is simply the absolute value of the difference in the frequencies of the two pitches. From a music theory standpoint, intervals can be referred to as consonances or dissonances. Consonances occur when tones of different frequencies are played simultaneously and sound pleasing together. Dissonances occur when tones of different frequencies are played simultaneously and sound displeasing together. According to a lecture by Professor Steven Errede from the University of Illinois, the Greek scholar Pythagoras studied consonance and dissonance using a device known as a monochord, a one stringed instrument with a movable bridge, which divides, "the string of length L into two segments, x and L–x. Thus, the two string segments can have any desired ratio, R = x/(L–x). When the monochord is played, both string segments vibrate simultaneously. Since the two segments of the string have a common tension, T, and the mass per unit length, mu = M/L is the same on both sides of the string, then the speed of propagation of waves on each of the two segments of the string is the same..." Basically, the ratio of the lengths of the two string segments is also the ratio of the two frequencies. Consonance occurs when the lengths of the string segments are in unique integer ratios. To learn more about the physics of consonances and dissonances, read his lecture here: https://courses.physics.illinois.edu/phys406/lecture_notes/p406pom_lecture_notes/p406pom_lect8.pdf.

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