Gamut.

The ear not only dis

242. Musical Scale. tinguishes between given sounds, — which is most grave, and which is most acute, but it also appreciates the relations between the number of vibrations corresponding to each. We cannot recognize whether for one sound the number of vibrations is precisely two, three, or four times as great as for another, but when the number of vibrations corresponding to two successive or simultaneous sounds have to each other a simple ratio, these sounds excite an agreeable impression, which varies with the relation between the two sounds.

From this principle there results a series of sounds characterized by relations which have their origin in the nature of our mental organization, and which constitute what is called a musical scale.

The whole series of musical tones is divided into octaves, or groups of eight tones each. Each group constitutes what is called the gamut, or diatonic scale.

The notes are named do, re, mi, fa, sol, la, si, do; but they are designated by the letters C, D, E, F, G, A, B, C. In the table below is given the relative number of vibrations for each note, 1 denoting the number corresponding to C:

The relative lengths of strings required to produce the eight notes of the scale are expressed by the reciprocal of these quantities, as follows:

If we know the number of vibrations of C, we can find the others by multiplying those of C by the fractions placed over the other notes in the first table. Let 256 represent the vibrations of C, then the following numbers will denote the vibrations for each note:

There are really but seven notes in what is called the diatonic scale, the eighth note, C, being truly the first of seven other notes above, having relations to one another similar to those of the notes below, and constituting another octave.

The results obtained in these tables can be verified by the Siren and Sonometer.

243. Intervals. The interval between any two notes is

called a musical interval.

The numerical value of any interval is found by dividing the number of vibrations in a given tone by the number of vibrations in that preceding it.

The intervals between consecutive notes, called seconds, is given in the following table :

C to D, D to E, E to F, F to G, G to A, A to B,

B to C.

If the interval comprise two, three, four, etc., seven notes, it is called a third, a fourth, a fifth, etc., an eighth or an octave; thus, the interval between C and E is a third, and is equal to ; the interval from C to F is a fourth, and is equal to ; the interval from any note to the next note of the same name is an octave, and is always equal to 2.

In the following table is a summary of the results already given, for one octave of the diatonic scale, arranged on the musical staff:

Absolute number ( 256 288 320 341 384 426 480 512

of vibrations

Scale of intervals

244. Melody. A number of

A number of tones of like quality,

varying more or less in pitch, following one another with regularity, is called a melody.

The air in a piece of music is an example of melody.

245. Chords.

Harmony. - Discord. - When two or more sounds are produced at the same time, having agreeable relations to one another, we have a chord.

A succession of chords in melodious order constitutes harmony.

The air, in music, with accompaniment, is an example of harmony.

When these agreeable relations do not exist, we have discord.

The simplest and most agreeable harmony occurs when the vibrations are equal in number; then comes the octave, in which the number of vibrations corresponding to one sound is double that corresponding to the other; then the fifth, in which the numbers are as 3 to 2; then the fourth, in which the numbers are as 4 to 3; and finally the third, in which the ratio is that of 5 to 4.

The more frequent the coincidences between the vibrations, the greater the harmony.

Summary.

Transverse Vibrations of Coras.

Investigation of the Laws of Vibrations.

Description of the Sonometer.

Laws of Vibrations.

Verification of the Laws.

Formation of Nodes.

Illustrated with the Sonometer.

Position of Nodes on a String.

Vibration of the String as a Whole or in Segments. Longitudinal Vibrations of Wires and Rods.

Experiments.

Vibration of Plates.

Experiment with Plate and Sand.

Chladni's Nodal Forins.

Overtones, or Harmonics.

Quality, or Timbre, of Sounds.

Musical Scale.

Names of Notes.

Letters used in designating Notes.

Relative number of Vibrations of each Note, in
Tabulated Form.

Relative length of Strings to give each Note, in
Tabulated Form.

Absolute number of Vibrations for each Note, in
Tabulated Form.

A Musical Interval.

Tabulated results on the Musical Staff.

Melody. Harmony. Discord.

SECTION III.

OPTICAL STUDY OF SOUNDS.

- THE HUMAN VOICE AND EAR.

MUSICAL INSTRUMENTS.

THE PHONOGRAPH.

246. Optical Study of Sounds. It has been shown in a previous article how the vibrations executed by a sonorous body can be counted. The Siren and SAVART'S Wheel are instruments used for this purpose.

During the last few years physicists have studied carefully the vibratory motions of sounding bodies by means of the eye, and have thus been independent of the aid of the ear in determining the relationship of sounds. A deaf person, by this optical method, can become skilful in judging of the character and pitch of sound-waves.

247. Lissajous' Representation of Vibrations. - One of the best methods of making vibrations apparent has been devised by M. LISSAJOUS, a French physicist. He attaches a small metallic mirror to one prong of a tuning-fork, and to the other a counterpoise to secure regularity of vibrations. A ray of light from a hole in a darkened chimney, a few yards distant, is made to strike this mirror, and from this it is reflected to another mirror, which sends it to an achromatic,

convergent lens; this lens is so placed as to project the images on a screen.

When the fork is at rest, we have on the screen a luminous point, the image of the hole in the chinney; when it vibrates the mirror vibrates with it, and the point moves up and down with such rapidity as to leave a line of light on the screen. If we rotate the fork while it is vibrating, we get instead of the straight line a bright sinuous The position of the parts is shown in Fig. 170, except that the fixed mirror takes the place of the vertical tuning fork.

one.

248. Vibratory Motions at Right Angles. If we use two forks, one horizontal and the other vertical, both provided with mirrors and arranged as in Fig. 170, we shall have thrown on the screen a variety of images.

If the vertical fork vibrates, we perceive a luminous line in a vertical direction; if the horizontal one vibrates, while the vertical fork is at rest, the luminous line is horizontal.

If both forks vibrate at the same time, the two movements at right angles will combine and produce a luminous curve, the form of which will depend upon the number of vibrations of the two forks in a given time. The arrows show the direction of the ray of light in its passage to the screen. Some varieties of curve are represented in Fig. 171.

By the aid of these principles, tuning-forks can be compared with a standard fork with greater precision than would be the case with the most susceptible ear.