Circles of Dissonance
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Ivars Peterson (MathTrek)  
Much of today's music rests upon ancient tradition going back thousands of years to the time of the Greek mathematician and mystic Pythagoras. The Pythagoreans observed that tones an octave apart are pleasing to the ear. In modern terms, one can say that consonance arises when the frequency of one tone is precisely twice the frequency of the other. Hence, a ratio of 2:1 produces harmonious tones. Similarly, musical intervals involving tones in the ratios of 3:2 (a fifth) and 4:3 (a fourth) are also pleasing...  


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