The figures below the lines denote how many times greater the
number of vibrations is than that of the first tone. In the
first octave we find only one tone; in the second, two; in the
third, all the tones of the major chord with the minor seventh.
In the fourth octave we find sixteen tones (which, however, we
divide in our system of music into twelve). Likewise, we find
in the fifth octave thirty-two tones, which number is doubled in
the sixth. Hence, the Greeks had quarter and eighth tones, which
we in our equal-tempered tuning have done away with.[9]
The production of a higher pitch in a tone rests in all sounding
bodies upon the uniform law which we may observe in the strings
of musical instruments, whose tones ascend either by greater
tension, by shortening, or through a diminution of the density
of the strings.