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Pythagorean, Meantone, and Equal Temperament Musical Scales
Pythagorean, Meantone, and Equal Temperament Musical Scales
The Pythagorean scale is generated from just two integers (2 and 3). Multiplying the frequency of any tone by 2 produces the characteristic sound of an octave, and multiplying by 3 then dividing by 2 produces the characteristic sound of a "fifth" (the fifth tone in the diatonic scale: Do Re Mi Fa Sol). All of the 12 tones of the Pythagorean scale are produced by repeatedly multiplying by 3/2 until you reach a tone close to (but not the same as) an octave of the original. Visualizing the resulting "circle" of fifths in Mathematica reveals the beautiful structure and mathematical nature of the Pythagorean scale. You can also explore the structure and nature of other musical scales invented and used by musicians, tuners, composers, and mathematicians throughout musical history.