Information on the history of music and mathematics.
In the classical period, and until around 1650, the theory of music and sound was part of the scientific curriculum along with arithmetic, geometry and astronomy. Pythagoras (580-500 BC) considered the basis of music to be mathematics and mathematics to be the governing principle of reality. If mathematics governed the universe then harmony, based on mathematical principles, was divine.
This led to the theory of the music of the spheres: the planets and other extraterrestrial bodes moved around the earth in perfect circular orbits, producing celestial harmonies. Attacks on this theory by Copernicus and others, who claimed not only that the earth moved round the sun but that planetary orbits were elliptical, was therefore an attack on classical as well as Christian principles.
This world-view led Pythagoras and his followers to conduct practical experiments on sound. They discovered that strings of different lengths produced sounds of different pitch, according to a mathematical ratio. Thus a string of two-thirds the length (ratio 2:1) sounded an octave higher.
Other philosophers and scientists laid the basis of modern musical theory. Aristotle (383-322 BC) theorised that the transmission of sound was based on motion and that a sound somehow thrust air forward towards the hearer. Galileo (1564-1642), besides his famous involvement in the argument over planetary orbits, conducted experiments on pendulums which showed the importance of frequency and suggested a means of transferring frequencies through vibrations. He also related frequency to pitch by producing notes using a scraper across a grooved board.