A few words, however, must be said on the subject of equal temperament. Previous to the time of Bach and Jean Philippe Rameau, the scale of the piano was arranged according to the laws of acoustics. It is impossible to enter into this fully, but the result was that a piano could not be tuned to play in all the twenty-four keys. This is difficult to explain, but I shall endeavor to make it clear. The pitch of a tone is determined by the number of vibrations it makes in a second, and it follows that there must be a regular ratio of increase in the number of vibrations of the notes of a scale as we proceed upward. This establishes the scientific basis of the scale. Now, any one who is at all acquainted with the piano knows that the same black key is struck to produce either C sharp or D flat. But this has been true only since Bach's day. Previous to that time instruments were timed according to the scientific laws, and by these we find that the C of the third octave has 256 vibrations, the C sharp 266.66, and the D flat 276.48. Thus D flat is a higher note than C sharp, and scientifically requires a differently tuned string and a separate key. The same trouble confronts us with most other notes, so that "theory requires no less than seventy-two keys to the octave in order that the musician may have complete command over all the keys employed in modern music." In order to reduce the octave to twelve semi-tones with twelve keys and to make the sharps and flats agree, as they do now, the system of equal temperament deliberately puts out of tune every interval except the octave. By slightly lowering some and raising others, the present scale was obtained. Its advantage is that it makes it practicable to play in all twenty-four keys, and because of the identity of the sharps and flats it becomes easy to modulate from one key to another. For instance, C sharp, which is the distinguishing note of the scale of D major, is also D flat, and thus it becomes easy to modulate from D to D flat, which leads to G flat, a very remote key. This gives the modern composer immense freedom of style, and adds greatly to the key complexity of music, whereas, before the adoption of the system of equal temperament, composers had to confine themselves to a few closely related keys. Who invented the system of equal temperament, no one knows. It is mentioned in the "Harmonic Universelle" of Marin Mersenne, the French writer before quoted, but it is quite certain that it was not extensively employed before the time of Bach, who brought the system to practical perfection, and demonstrated it in his "Well-Tempered Clavichord," a set of twenty-four preludes and fugues going through all the keys. The science of equal temperament was first set forth in a satisfactory manner by Jean Philippe Rameau, the French opera composer, in his "Traité de l'Harmonie," Paris, 1722. Chapter VIII The Evolution of Piano Playing