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music and mathematics
Music theorists sometimes use mathematics to understand music, and although music has no axiomatic foundation in modern mathematics, mathematics is "the basis of sound" and sound itself "in its musical aspects... exhibits a remarkable array of number properties", simply because nature itself "is amazingly mathematical".〔Reginald Smith Brindle, ''The New Music'', Oxford University Press, 1987, pp 42-3〕
The attempt to structure and communicate new ways of composing and hearing music has led to musical applications of set theory, abstract algebra and number theory. Some composers have incorporated the golden ratio and Fibonacci numbers into their work.〔Reginald Smith Brindle, ''The New Music'', Oxford University Press, 1987, Chapter 6 ''passim''〕
==History==
Though ancient Chinese, Indians, Egyptians and Mesopotamians are known to have studied the mathematical principles of sound,〔Reginald Smith Brindle, ''The New Music'', Oxford University Press, 1987, p 42〕 the Pythagoreans (in particular Philolaus and Archytas) of ancient Greece were the first researchers known to have investigated the expression of musical scales in terms of numerical ratios,〔Plato, (Trans. Desmond Lee) ''The Republic'', Harmondsworth Penguin 1974, page 340, note.〕 particularly the ratios of small integers. Their central doctrine was that "all nature consists of harmony arising out of numbers".〔Sir James Jeans, ''Science and Music'', Dover 1968, p. 154.〕
From the time of Plato, harmony was considered a fundamental branch of physics, now known as musical acoustics. Early Indian and Chinese theorists show similar approaches: all sought to show that the mathematical laws of harmonics and rhythms were fundamental not only to our understanding of the world but to human well-being.〔Alain Danielou, ''Introduction to the Study of Musical Scales'', Mushiram Manoharlal 1999, Chapter 1 ''passim''.〕 Confucius, like Pythagoras, regarded the small numbers 1,2,3,4 as the source of all perfection.〔Sir James Jeans, ''Science and Music'', Dover 1968, p. 155.〕
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