According to legend, Pythagoras discovered the foundations of music by listening to the sounds of four blacksmith's hammers, which produced consonance and dissonance when they were struck simultaneously. Specifically, he noticed that hammer A produced consonance with hammer B when they were struck together, and hammer C produced consonance with hammer A, but hammers B and C produced dissonance with each other. Hammer D produced such perfect consonance with hammer A that they seemed to be "singing" the same note! Pythagoras rushed into the blacksmith to discover why, and he found that the explanation was in the weight ratios. The hammers weighed 12, 9, 8, and 6 pounds respectively. Hammers A and D were in a ratio of 2:1, which is the ratio of the octave. Hammers B and C weighed 9 and 8 pounds. Their ratios with hammer A were (12:9 = 4:3 = musical fourth) and (12:8 = 3:2 = musical fifth). Interestingly, if you invert the ratio of hammer B (making it 3:4 = 12:8) it becomes the ratio of hammer C, and vise-verse, thus, the musical fourth can be described as an inverted fifth, and vise verse. The space between B and C is a ratio of 9:8, which is equal to musical whole tone, or whole step interval.

More sources:

James, Jamie The Music of the Spheres: Music, Science, and the Natural Order of the Universe. New York, Grove Press, 1993

"Pythagorean hammers" Harvard Dictionary of Music. Second Edition, Massachusets: Harvard University Press

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