The instrument is so constructed that clockwork at the top registers the number of revolutions made by the disk in one second.  The number of holes in the disk multiplied by the number of revolutions a second gives the number of puffs of air produced in one second.  If we wish to find the number of vibrations which correspond to middle C on the piano, we increase the speed of the disk until the note given forth by the siren agrees with middle C as sounded on the piano, as nearly as the ear can judge; we then calculate the number of puffs of air which took place each second at that particular speed of the disk.  In this way we find that middle C is due to about 256 vibrations per second; that is, a piano string must vibrate 256 times per second in order for the resultant note to be of pitch middle C. In a similar manner we determine the following frequencies:—­

|do  |re  |mi  |fa  |sol |la  |si  |do  |
|C   |D   |E   |F   |G   |A   |B   |C’  |
|256 |288 |320 |341 |384 |427 |480 |512 |

[Illustration:  FIG. 177.—­A siren.]

The pitch of pianos, from the lowest bass note to the very highest treble, varies from 27 to about 3500 vibrations per second.  No human voice, however, has so great a range of tone; the highest soprano notes of women correspond to but 1000 vibrations a second, and the deepest bass of men falls but to 80 vibrations a second.

While the human voice is limited in its production of sound,—­rarely falling below 80 vibrations a second and rarely exceeding 1000 vibrations a second,—­the ear is by no means limited to that range in hearing.  The chirrup of a sparrow, the shrill sound of a cricket, and the piercing shrieks of a locomotive are due to far greater frequencies, the number of vibrations at times equaling 38,000 per second or more.

264.  The Musical Scale.  When we talk, the pitch of the voice changes constantly and adds variety and beauty to conversation; a speaker whose tone, or pitch, remains too constant is monotonous and dull, no matter how brilliant his thoughts may be.

While the pitch of the voice changes constantly, the changes are normally gradual and slight, and the different tones merge into each other imperceptibly.  In music, however, there is a well-defined interval between even consecutive notes; for example, in the musical scale, middle C (do) with 256 vibrations is followed by D (re) with 288 vibrations, and the interval between these notes is sharp and well marked, even to an untrained ear.  The interval between two notes is defined as the ratio of the frequencies; hence, the interval between C and D (do and re) is 288/256, or 9/8.  Referring to Section 263, we see that the interval between C and E is 320/256, or 5/4, and the interval between C and C’ is 512/256, or 2; the interval between any note and its octave is 2.

The successive notes in one octave of the musical scale are related as follows:—­