Excerpt from The Evolution of the Beautiful in Sound: A Treatise, in Two Sections IN this treatise it has been deemed expedient in alluding to the most ancient series of sounds the Pentatonic), to Show the width and relationship of intervals by giving the lengths of the elastic bodies or pipes which generated the sounds. The more general plan has been to describe the width or space existing between certain intervals by the ratios of the widths or Spaces. Pythagoras is credited with the introduction of the ratio system into Greece, but it does not appear to have been adopted by the best known Grecian writers on musical science. Their methods, as well as those of the earlier Greek periods, consisted in giving the respective fractional parts of the string which produced the intervals. AS the figures in the ratio system are the same as those which Show the fractional parts of a string, no mistake can arise by adopting either the one or the other system when describing ancient methods. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.