In the present work an attempt will be made to form a connection between the boundaries of two sciences, which, although drawn together by many natural relations, have hitherto remained sufficiently distinct-the boundaries of physical and physiological acoustics on the one side, and of musical science and esthetics on the other. The class of readers addressed will, consequently, have had very different cultivation, and will be affected by very different interests. It will therefore be best for the author at the outset distinctly to state his intention in undertaking the work, and the aim he has sought to attain. The horizons of physics, philosophy, and art have of late been too widely separated, and, as a consequence, the language, the methods, and the aims of any one of these studies present a certain amount of difficulty for the student of any other of them; and possibly this is the principal cause why the problem here undertaken has not been long ago more thoroughly considered and advanced towards its solution.
It is true that acoustics constantly employs conceptions and names borrowed from the theory of harmony, and speaks of the 'scale,' intervals,' consonances,' and so forth; and similarly, manuals of Thorough Bass generally begin with a physical chapter which speaks of the numbers of vibrations,' and fixes their 'ratios' for the different intervals; but, up to the present time, this apparent connection of acoustics and music has been wholly external, and may be regarded rather as an expression given to the feeling that such a connection must exist, than its actual formulation. Physical knowledge may indeed have been useful for musical instrument makers, but for the development and foundation of the theory of harmony it has hitherto been totally barren. And yet the essential facts within the field here to be explained and turned to profit, have been known from the earliest times. Even Pytha