different but correct views of music we may arrive at a conception of the possibility of a philosophy of number, such as that of Pythagoras and of the Chinese in Y-King, and then interpret in this sense the saying of the Pythagoreans which Sextus Empiricus quotes (adv. Math., L. vii.): τῳ αριθμῳ δε τα παντ᾽ επεοικεν (_numero cuncta assimilantur_). And if, finally, we apply this view to the interpretation of harmony and melody given above, we shall find that a mere moral philosophy without an explanation of Nature, such as Socrates wanted to introduce, is precisely analogous to a mere melody without harmony, which Rousseau exclusively desired; and, in opposition to this mere physics and metaphysics without ethics, will correspond to mere harmony without melody. Allow me to add to these cursory observations a few more remarks concerning the analogy of music with the phenomenal world. We found in the second book that the highest grade of the objectification of will, man, could not appear alone and isolated, but presupposed the grades below him, as these again presupposed the grades lower still. In the same way music, which directly objectifies the will, just as the world does, is complete only in full harmony. In order to achieve its full effect, the high leading voice of the melody requires the accompaniment of all the other voices, even to the lowest bass, which is to be regarded as the origin of all. The melody itself enters as an integral part into the harmony, as the harmony enters into it, and only thus, in the full harmonious whole, music expresses what it aims at expressing. Thus also the one will outside of time finds its full objectification only in the complete union of all the steps which reveal its nature in the innumerable ascending grades of distinctness. The following analogy is also very remarkable. We have seen in the preceding book that notwithstanding the self-adaptation of all the phenomena of will to each other as regards their species, which constitutes their teleological aspect, there yet remains an unceasing conflict between those phenomena as individuals, which is visible at every grade, and makes the world a constant battle-field of all those manifestations of one and the same will, whose inner contradiction with itself becomes visible through it. In music also there is something corresponding to this. A complete, pure, harmonious system of tones is not only physically but arithmetically impossible. The numbers themselves by which the tones are expressed have inextricable irrationality. There is no scale in which, when it is counted, every fifth will be related to the keynote as 2 to 3, every major third as 4 to 5, every minor third as 5 to 6, and so on. For if they are correctly related to the keynote, they can no longer be so to each other; because, for example, the fifth must be the minor third to the third, &c. For the notes of the scale may be compared to actors who must play now one part, now another.