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THE CONNECTION between common musical intervals and simple numbers has been an intriguing mystery since ancient times. There has indeed been little difficulty over the facts showing this connection, which are not mysterious, and which are hardly subject to dispute at this time. But there remains considerable uncertainty as to the meaning of these facts, whether for the and practice of the art of music or more urgently for certain fundamental problems in aesthetics and in general philosophy. The interpretation of this connection entails numerous ramifications and a whole series of paradoses. Thus, what would appear at first sight to involve merely some technical details about the elementary relations of musical tones has acquired a very large significance for the whole range of art, science and human thought. These ramifications, rather than the immediate technical concern with musical harmony, have been responsible for the very great interest which musicians and others have shown in the problem. True, practitioners of the art of music of late have not notably been drawn to questions of deep theoretical import. The trend of music theory tests has been increasingly in the direction of rule-of-thumb manuals, more or less ingenious schemes for the painless teaching of standard routines, and new systems of classification or procedure said to reveal the craft secrets of the art or the unconscious processes of composers no longer here to defend themselves. Current musicological research seems bent upon the obtaining of adequate documentation of no matter what, and is thus correspondingly chary of possibly speculative generalizations or inter-connections. Thereby we obtain new presentations and embroideries of past theoretical thought, and but little illumination or new thought. Yet every lively student of music is apt at some time to have been confronted with the stated connection between music and mathematics. Most have been excited to wonder about it, but have given up the quest under cover of one or another hasty but saving dismissal. Many have filed the matter away in their minds either as something to be pursued when and if the opportunity arises, or as a subject far too technical for attention and solution by mere musicians, little aware that it was not always so considered. Similarly, students of physics, who have likewise been informed regularly about the bearing of acoustics on certain musical relations, most often accept the facts as though they were adequately explained and unquestiona.bly applied, thereafter continuing to argue neither the scientific nor the human approaches to music which ought properly to concern them, but rather the frequency response curves of amplifiers and the relative merits of Brahms quartets and musique conGrete. It has usually been the more philosophical among scientists, and
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