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: He identified that simple integer ratios produce the most "consonant" or pleasing sounds: Octave : 2:1 ratio (halving the string length). Perfect Fifth : 3:2 ratio. Perfect Fourth : 4:3 ratio.
As music evolved from "science" to "art" around the 17th century, composers began using more sophisticated mathematical logic to organize sound. Fractal patterns in music - ScienceDirect.com Music and Mathematics: From Pythagoras to Fractals
The mathematical study of music began with Pythagoras , who discovered that musical intervals correspond to precise numerical ratios of vibrating string lengths. : He identified that simple integer ratios produce