repeated reflections from the sides of the tube (see Expt. 64, p. 100), or as a result of keeping the disturbance constantly within the walls of the tube.

EXAMPLES ON CHAPTERS I-IV.

1. State exactly what you mean by the 'density' of a medium, and explain what you understand by its ' 'elasticity.' Is the elasticity of water greater or less than that of air? How would you prove to any one

that your answer is correct?

2. Describe and explain an experiment illustrating the use of soundboards in making the vibrations of a wire audible.

3. Explain any method by means of which the ticking of a watch may be made audible to a person at the other end of a large room.

4. What is the velocity of sound in air? On what grounds can it be asserted that musical tones of high and low pitch travel at the same speed? 5. What is meant by a wave of sound and by the length of a wave. Explain how sound is transmitted through air.

6. A gun is fired on a cold winter's day at a certain distance from an observer who hears the report five seconds after seeing the flash. Would the interval between seeing the flash and hearing the sound have been the same on a hot day in summer? Give reasons for your answer.

7. A street with houses on both sides runs north and south, and a church is situated at a little distance to the east of it. As I walk down the eastern side of the street the sound from a peal of bells in the church-tower seems to come from the west. Explain this, drawing a diagram to illustrate your answer.

CHAPTER V

PITCH AND MUSICAL INTERVALS

38. Musical sounds may differ in respect of— (1) Intensity or Loudness.

(2) Pitch.

(3) Quality.

At this
What

We have already stated that the intensity or loudness of a sound depends upon amplitude of vibration (Art. 10). Pitch we shall consider more fully in the present chapter. stage it is not possible to treat of quality of sounds. is meant by difference in quality may be readily appreciated on producing the same note by means of different instruments, say a piano, a violin a flute and the human voice. Even though the sounds be of exactly the same pitch and, as nearly as possible, of the same loudness, you can easily tell which is which. As we go on to consider the different ways in which musical sounds are produced we may be able to point out some of the causes which produce these differences in quality.

39. Pitch is that which distinguishes a high or shrill note from a deep or low one. Most people can tell when two notes are of the same pitch or height; and can decide which is the higher of two notes. People who are able to judge correctly of differences in pitch are said to have 'a good ear for music.' If you are blessed with such, you will already know what is meant by pitch; and if you are not, it is scarcely possible to help you to such knowledge.

40. We proceed to show by experiment (1) that any series

of regularly recurring impulses (within a certain range) produces a musical note; and (2) that the more rapidly these impulses follow each other, the higher is the pitch of the note.

EXPT. 16. Savart's Wheel.-On a centrifugal machine or whirling-table (see Fig. 15) fix a toothed wheel, such as a large clock-wheel. Make the wheel rotate, at first very slowly, then

Fig. 14.

more and more rapidly, at the same time holding a thin card lightly against the teeth. At first you hear only a series of separate taps. As these succeed each other more rapidly they begin to blend together and produce a low note the pitch of which increases as the rapidity of rotation increases. The quality of the sound produced is poor and thin: it is scarcely correct to call it a musical note.

EXPT. 17. The Disc- siren.— Perform a similar experiment by blowing a jet of air against a rotating disc having a circular row of holes pierced in it (Fig. 14). Here again the quality of the sound is poor: it is mixed up with the noise made by the air in rushing against the disc.

The disc may be made of stiff smooth cardboard or sheet-metal and of about the following dimensions. Radius of disc, 15 cm. Radius of row of holes, 13 cm. Holes about 0.5 cm. in diameter and 2 cm, apart. Jet made of glass-tubing drawn out at the point to same diameter as the holes or somewhat narrower.

What properly known as a 'siren' is an instrument which does not differ greatly in principle from the above. It is, however, more elaborate in construction and contains a 'counter,' which registers the number of revolutions made in a given time. When this is known we can easily calculate the number of puffs of air per second that are required to produce a given note.

In neither of the above experiments is the sound produced directly by a vibratory motion. In the case of the disc-siren, each time a hole comes in front of the jet the air rushes through and produces a pulse of compression in the space beyond. In virtue of the elasticity of the air, this is succeeded by a pulse of rarefaction during the interval that elapses before the next hole comes into position. Thus the air is set into vibration much

as it would be by the motion of a tuning-fork or vibrating string. The pitch of the note produced increases as the vibration-number or frequency increases (Art. 3).

41. How Pitch is expressed.-We may express the pitch of a note—

(1) Relatively, as when we say that one note is an octave or a fifth higher than some other note which is chosen as a convenient standard of reference. This is the method generally adopted in Music.

(2) Absolutely, as when we say that a certain number of vibrations per second is required to produce a note of a given pitch. This is the method adopted in Physics; the pitch of any note being expressed by stating its vibration-number or frequency (Art. 3). Thus the pitch of the note produced by an open organ-pipe 2 feet long is 280; for such a pipe, at the ordinary temperature of the air, produces vibrations at the rate of 280 per second.

42. Musical Intervals.-The interval between any two notes is measured physically by the ratio between the vibrationnumber of the higher note and that of the lower one. If the vibration-number of the lower note be n, and that of the higher note n', the interval is measured by the ratio

I

n'

n

Unison.-Two notes are said to be in unison when they have exactly the same pitch. The ratio between the vibrationnumbers of two such notes is clearly Thus although there is no difference of pitch between two notes in unison, the interval is expressed by 1 and not by o.

Octave. If the vibration-number of one note is double that of another, the first note is said to be an octave above the

2

second. This interval is represented by the ratio. Like all other musical intervals, its value depends, not on the absolute vibration-numbers, but on their ratios. The notes represented by the numbers 100, 200, 400, 800 . . . are each an octave

above the one below.

43. The Major Diatonic Scale.-Musicians divide the interval between a note and its octave into seven smaller intervals of unequal value known as tones and semi-tones.

S

The notes which occur in this 'scale' may be typified (although they are not exactly represented) by the white keys of a pianoforte. In the old or 'staff' system of notation these notes are known as

In the new or 'Tonic Solfa' system of notation (for which we shall always use thick letters), the notes are represented as m f s 1 t

d

r

d'

which are read as
Doh Ray Me Fah Soh

Lah Te

doh.

These seven notes (or eight, with the addition of the octave1) form the musical scale in common use. To distinguish it from a somewhat different scale, used chiefly in solemn and mournful music (the minor scale, called by Solfaists the 'lah mode'), it is known as the major diatonic scale.

At this stage no better advice can be given to a non-musical student than that he should play these notes on the piano, or, better still, get some one to sing them to him, until he knows the different notes and the intervals between them. Without such knowledge any discussion of the nature of musical intervals must be as unintelligible as Chinese. He should also listen carefully to the notes when sounded in pairs, and notice what combinations of the notes are harmonious (or produce a pleasing effect upon the ear) and which are dissonant (or produce a harsh or disagreeable effect).

44. Intervals which occur in the Common Chord.We now proceed to illustrate how some of the more important intervals that occur in music can be measured and expressed by numbers. For this purpose we shall choose the most harmonious intervals, viz. those which occur in the Common or Major Chord. The notes which form this chord are

We shall show in two ways that in order to produce this series of notes the vibration-numbers must be in the proportion of

4,

5, 6, 8.

EXPT. 18.-Four toothed

1. By Savart's Wheels

wheels will be required, having respectively 80, 100, 120, and

1 Latin octavus = eighth.