accounts through the employment of a continued progression upward in pitch by the interval of the fifth.[1] We are here reminded of the Pythagorean derivation of the diatonic interval order of the Greeks in like manner from a progression of fifths; but if Chinese sources are to be trusted, its application in their music antedates the Pythagorean discovery by a score of centuries.[2] The musical theory of the Greeks had its origin in the division of a string; that of China, in the measurement of pipes. The vibration ratio of the fundamental tones of two pipes will, other things being equal, be very nearly if not exactly the inverse of the ratio of their lengths. Thus a pipe giving a tone an octave above another like one (vibration ratio 21) will be of half its length; a pipe giving a tone a fifth above another (vibration ratio 32) will be of two-thirds its length. This connection between the simplest ratios and striking facts of pitch-relation seems from time immemorial to have been employed in giving a philosophical basis to the Chinese musical system. Père Amiot quotes (p. 117) a speculation about the foundations of music written by Hoai-nan-tsee, King of Hoainan (B.C. 105), which begins as follows: "The principle of all science is unity. Unity as single cannot produce anything, but it engenders everything, insomuch as it includes within itself the two principles of which the harmony and the union produce everything. It is in this sense that one can say, unity engenders duality, duality triplicity; and from triplicity all things are engendered."[3]
According to these principles, a pipe whose note is at the interval of the fifth (32) above that of another, will be a natural
- ↑ Amiot, p. 112; Van Aalst, p. 14.
- ↑ Van Aalst, p. 7. The Emperor Huangti (B.C. 2697) and his minister Ling-lun are by one account credited with the invention of the fifth progression.
- ↑ We find similar ideas made the foundation of musical theory in Europe two millenniums later in the Harmonik und Metrik of Moritz Hauptmann (2d ed., Leipzig, 1873, p. 8). "Rightness — that is, reasonableness — of musical structure has for its law of formation unity with the negation of it and the resolution of this contradiction: an original unity which passes through a state of inner contrariety to become a unity restored. This process must continually repeat itself, either upon original unity or upon the result of a previous process. So the unity of a single note becomes in its restoration the triad, the unity of the triad becomes in its restoration the scale" (since the scale may be viewed as a chain of three triads).
