160 teeth (or numbers in the proportion of 4, 5, 6, and 8); and the wheels are to be fixed (at a little distance apart) on the axis of a whirling-table, as shown in Fig. 15.

Rotate the wheels and touch each in succession with a thin card as in Expt. 16. As the speed of rotation cannot be kept constant for any length of time it is best to touch the wheels very lightly and rapidly one after the other; the notes are easily recognised as being those which form the Common Chord.

Change the rate of rotation. The notes all alter; but your ear perceives that they always bear to each other the same ' relative pitch.' The intervals do not depend upon absolute pitch, but only upon the ratios of the vibration-numbers.

II. By the Disc-siren-EXPT. 19.-For this purpose we require a disc of the same size as that used in Expt. 17, but pierced with four circular rows of holes (Fig. 15). The innermost row should have 24 holes, the next 30, the next 36, and the outer row 48 (the numbers being in the proportion of 4, 5, 6, and 8).

The jet may be made of glass tubing bent twice at right angles and mounted as shown in Fig. 14: this arrangement is convenient for swinging the jet quickly over the four rows of holes without fear of breaking it.

45. How Intervals are Compounded. -Musical intervals are compounded by multiplication and not by addition.

The experiments just performed show us that the interval between d and m, or C and E, is represented by any of the equal ratios

The interval between m and s, or E and G, is represented by any of the equal ratios

for

Again the interval between d and s, or C and G, is represented by

Now this last is also the interval obtained by compounding the first two,

46. Intervals which occur in the Scale.-By experiments similar to those above described, or by careful experiments made with more elaborate apparatus, the ratios of the vibration-numbers of the notes forming the scale have been determined. The names and values of the various intervals, counting from C (the tonic or key-note) are given in the following table. The first column indicates the notes numbered 1, 2, 3 . . . in order, and the second and third columns their names. The fourth column gives the names of the intervals, and the fifth their numerical values.

Of these intervals the Second (21) and the Seventh (15)

are dissonant.

The others are consonant or harmonious, the most perfect consonance

being given by the Fifth (2) and the Octave (2). Thus we see that the

most harmonious intervals are those in which small numbers (from I to 5) occur.

47. Standards of Pitch.-Once we settle upon the absolute pitch of

the tonic or key-note of our scale, the absolute pitch of every other note in the scale is thereby fixed. Thus if we agree to take C=256, the vibration-number of G (a fifth above it) is 256 × 2=384, and so on. This is the pitch usually adopted by writers on acoustics and by makers of acoustical apparatus. The number 256 has the advantage of being a power of 2 (viz. 28), and the choice is mainly a matter of convenience.

The most convenient standard of pitch is a tuning-fork made so as to execute a known number of vibrations per second. In our country there is no legal standard of pitch. What is vaguely referred to as ' concert pitch' may be taken to represent C=264, or thereabouts. (The C here referred to is what is known as the 'middle C' of a piano.)

48. Intervals between successive Notes in the Scale.-The various intervals given in Art. 46 have as their least common denominator the number 24. If we take this as representing the vibration-number of the key-note, the other notes in the scale will be represented by the following whole numbers :--

30

ΙΟ

The numbers in the last row represent the intervals between each successive pair of notes. They are obtained by dividing each number in the top row by the one before it. Thus the interval rm is equal to or This is not quite the same as the preceding interval (2), but both are

16

27

9

known as tones. The interval is called a semi-tone. Thus the major

15

diatonic scale consists of a series of intervals in the following order: two tones and a semi-tone, three tones and a semi-tone.

49. A Harmonic Series is a series of notes whose vibration-numbers are in the following proportion

All the notes in such a series (at any rate up to the sixth) harmonise well with the first (or fundamental) note and with each other. It will be a useful exercise to find out the relations between these notes.

Let us call the lowest or fundamental note d. The interval between this and the next note is an octave : hence the second note

is d'. The third note is a fifth (2) above the second, `or a twelfth above the first; it is therefore the note s'. The fourth note is an

octave

(4) above the second, or two octaves (4) above the first; it is therefore d''. The fifth is a major third (5) above the fourth, and is

therefore m". The sixth is a fifth (2) above number 4, and is there

4

fore s'. Thus the first six notes of the harmonic series are

CHAPTER VI

TRANSVERSE VIBRATIONS OF STRINGS

50. By a string is here meant any elastic and flexible cord (such as the twisted cat-gut used in violins) or metallic wire stretched between fixed supports. When such a stretched string is pulled to one side it tends to return to its position of rest. When let go it flies back, but, like a spring or pendulum, it overshoots the mark, and goes on swinging from side to side. A string can be set into vibration by striking, plucking, or bowing it. The vibrations thus produced are transverse vibrations, and these are the only ones that we shall consider. The vibrations are further said to be stationary: certain points (e.g. the fixed ends) remain permanently at rest. These points are called nodes.

We shall see presently that a string may vibrate transversely in many different ways, dividing up into a number of smaller vibrating parts or segments. The lowest or fundamental note of the string is produced when it vibrates as a whole (or in one segment). In this case the only nodes are at the two fixed ends. All other points are in motion, and the amplitude of the motion is greatest at the centre, which is called an antinode.

51. Laws. The number of vibrations executed per second by a string when sounding its fundamental note is found to depend upon

(1) Its length.

(2) Its diameter.

(3) Its density.

(4) The stretching force.