When Numbers Became Sound — From String Ratios to Digital Music
Why is dividing an octave exactly into a usable scale unexpectedly difficult? Travel from string ratios in Croton and listening in Alexandria through instruments in Baghdad, polyphony in Venice, independent twelve-tone calculations in Qinyang and Leiden, measured vibration in Paris, twenty-four keys in Köthen, mathematical taste in Saint Petersburg, Fourier and Helmholtz, transmission limits in New York, computer music in Urbana and Murray Hill, and MP3 in Erlangen. This is not a story of mathematics dictating harmony, but of instruments, ears, institutions, and machines negotiating ratios that cannot all fit at once.
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c. 500 BCE
Crotone
Hearing Octaves and Fifths in String-Length Ratios — the Pythagorean Tradition
For otherwise identical strings, halving length doubles frequency and produces an octave; a 2:3 length ratio produces the 3:2 frequency ratio of a perfect fifth. These small integers made a powerful bridge between number and sound. The story that Pythagoras discovered them from blacksmiths’ hammers appears centuries later, and hammer weight does not determine pitch in the simple proportions of the legend.
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