The high octave of a tone has in the same time exactly double the number of vibrations of the tone itself. Suppose, therefore, that a tone has 50 vibrations in a second, its octave has 100 in the same time; i.e., twice as many. The octave above this has 200 vibrations, &c. The Pythagoreans knew this acoustic law of the ascending tones, and that the octave of a tone had twice as many vibrations in a second as the tone itself, and that the fifth above the first octave had three times as many; the second octave, four times; the major third above the second octave, five times as many; the fifth of the same octave, six times; the small seventh of the same octave, seven times. In notation it would be thus, if we take as the lowest note C, for example: 1:C 2:c 3:g 4:c¹ 5:e¹ 6:g¹ 7:b♭¹ 8:c² 9:d² 16:c³ 32:c⁴ 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. THE TIMBRE (KLANGFARBE) OF TONES Strength and pitch were the first two distinctions of different tones. The third is the timbre. When we hear one and the same tone sounded successively upon a violin, trumpet, clarionet, oboe, upon a piano, or by a human voice, &c., although it is of the same strength and of the same pitch, yet the tone of all these instruments is different, and we very easily distinguish the instrument from which it comes. The changes of the timbre seem to be infinitely manifold; for, not to mention the fact that we have a multitude of different musical instruments, all which can give the same tone, letting alone also that different instruments of the same kind as well as different voices show certain differences of timbre, the very same tone can be given upon one and the same instrument, or by one and the same voice, with manifold differences of timbre.[10]