at every third, or at every fourth vibration, according to which, then, they are the octave, the fifth, or the fourth of each other, and so on. So long as the vibrations of two notes have a rational relation to each other, which can be expressed in small numbers, they can be connected together in our apprehension through their constantly recurring coincidence: the notes become blended, and are thereby in consonance. If, on the other hand, that relation is an irrational one, or one which can only be expressed in larger numbers, then no coincidence of the vibrations which can be apprehended occurs, but obstrepunt sibi perpetuo, whereby they resist being joined together in our apprehension, and accordingly are called a dissonance. Now, according to this theory, music is a means of making rational and irrational relations of numbers comprehensible, not like arithmetic by the help of the concept, but by bringing them to a knowledge which is perfectly directly and simultaneously sensible. Now the connection of the metaphysical significance of music with this its physical and arithmetical basis depends upon the fact that what resists our apprehension, the irrational relation, or the dissonance, becomes the natural type of what resists our will; and, conversely, the consonance, or the rational relation, which easily adapts itself to our apprehension, becomes the type of the satisfaction of the will. And further, since that rational and irrational element in the numerical relations of the vibrations admits of innumerable degrees, shades of difference, sequences, and variations, by means of it music becomes the material in which all the movements of the human heart, i.e., of the will, movements whose essential nature is always satisfaction and dissatisfaction, although in innumerable degrees, can be faithfully portrayed and rendered in all their finest shades and modifications, which takes place by means of the, invention of the melody. Thus we see here the movements of the will transferred to the province of the mere idea, which is the exclusive scene of the achievements of
