The composers to whom I have alluded assume in their hearers a long experience of music in general and also something of that habit of mind previously mentioned which tends to regard music less as a series of actual sounds than as a series of relations between sounds. It may be called a mathematical conception of music, and, like mathematics, it soon comes to deal with irrational quantities. It is an interesting question how far the human mind can advance in this direction. To certain temperaments music of this type is definitely repulsive; but they often feel no less repulsion towards mathematics and philosophy, studies which have been closely associated with music from very early times. We must however beware of being misled by superficial criticism into supposing that the understanding of such musical complexities requires a practical knowledge of mathematical or philosophical technicalities. In the scientific study of musical æsthetics there ultimately arise problems which bring all three branches of learning into contact; but in common practice they do not affect either the composer or the listener. There are writers on music who make use of a philosophical jargon to conceal their incapacity for clear thinking; but the truly philosophical habit of mind aims, if but with rare success, at lucidity. The practical value of this “mathematical” system of composition lies not so much in its employment of technical devices which were practised some five hundred years ago, as in its new method of handling them. It was a great moment in the history of music when someone first discovered that two different tunes could be sung simultaneously and thereby produce harmony. The artistic result of this proceeding depended on two factors which had to be brought into relation――the interest of each tune considered by itself, that is, the driving force which made it perceptible as a continuous tune, and, secondly, the satisfaction derived from the consonance of the two voices where it happened to occur. At one period the interest of the tune predominated, at another it was sacrificed to the interest of consonance. Both interests are however subject to changes of value in the course of time. It is clear enough that such composers as Purcell, Bach and Mozart were deeply interested in the problem of exploiting these two interests, and of finding out how far the driving force of a tune could induce the listener to put up with dissonant harmony. We can see now, at this distance of time, that they positively increased the value of the harmonic interest by the way in which they deliberately tortured the ear of the sensitive listener of their own time. Our ears have become not less but more sensitive to dissonance, more able at any rate to discriminate between varieties of it. But, as I have already indicated, this preoccupation with harmony and with relations between sounds has led to an indifference towards the actual sounds themselves,