Pythagorean Hammers

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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. Supposedly, 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.

Ostensibly, 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). The space between B and C is a ratio of 9:8, which is equal to the musical whole tone, or whole step interval.

The legend - which can be traced as far back as Nicomachus'*Enchiridion harmonices*, from the 2nd century CE - is demonstrably false, at least with respect to the hammers. These proportions are indeed relevant to string length (i.e. that of a monochord) - using these founding intervals, it is possible to...

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Ostensibly, 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). The space between B and C is a ratio of 9:8, which is equal to the musical whole tone, or whole step interval.

The legend - which can be traced as far back as Nicomachus'

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