PREFACE.

It was already understood clearly enough in the earliest times when Music began to be cultivated, that those things by which music is rendered pleasing to the ear and affects the mind with pleasure are neither placed within the arbitrary judgment of men, nor depend upon custom. For Pythagoras A Greek philosopher (c. 570–495 BC) credited with discovering that musical intervals correspond to mathematical ratios., who first laid the foundations of music, already recognized that the underlying principle of the consonances intervals that sound stable and pleasant together which delight the ears lies hidden in perceptible proportions, even if it was not yet clear to him in what manner these ratios are perceived by the sense of hearing. However, because he had perceived the true principles of harmony less distinctly, he had attributed too much to his proportions and did not know how to establish the proper limits for them; for which reason he was deservedly criticized by Aristoxenus A Greek theorist (4th century BC) who argued that the human ear, rather than mathematical ratios alone, should be the judge of musical intervals.: who indeed, just as the doctrine of Pythagoras...