Sound possesses a musical quality only when the waves or pulses follow each other at absolutely regular intervals.

262. The Effect of the Rapidity of Motion on the Musical Tone Produced. If the disk is rotated so slowly that less than about 16 puffs are produced in one second, only separate puffs are heard, and a musical tone is lacking; if, on the other hand, the disk is rotated in such a way that 16 puffs or more are produced in one second, the separate puffs will blend together to produce a continuous musical note of very low pitch. If the speed of the disk is increased so that the puffs become more frequent, the pitch of the resulting note rises; and at very high speeds the notes produced become so shrill and piercing as to be disagreeable to the ear. If the speed of the disk is lessened, the pitch falls correspondingly; and if the speed again becomes so low that less than 16 puffs are formed per second, the sustained sound disappears and a series of intermittent noises is produced.

263. The Pitch of a Note. By means of an apparatus called the siren, it is possible to calculate the number of vibrations producing any given musical note, such, for example, as middle C on the piano. If air is forced continuously against the disk as it rotates, a series of puffs will be heard (Fig. 177).

If the disk turns fast enough, the puffs blend into a musical sound, whose pitch rises higher and higher as the disk moves faster and faster, and produces more and more puffs each second.

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

[Ill.u.s.tration: FIG. 177.--A siren.]

The pitch of pianos, from the lowest ba.s.s 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 ba.s.s 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:--

Key of C

C

D

E

F

G

A

B

C'

No. of vibrations

per sec.

256

288

320

341

384

427

480

512

Interval

9/8

5/4

4/3

3/2

5/3

15/8

2

The intervals of F and A are not strictly 4/3 and 5/3, but are nearly so; if F made 341.3 vibrations per second instead of 341; and if A made 426.6 instead of 427, then the intervals would be exactly 4/3 and 5/3. Since the real difference is so slight, we can a.s.sume the simpler ratios without appreciable error.

Any eight notes whose frequencies are in the ratio of 9/8, 5/4, etc., will when played in succession give the familiar musical scale; for example, the deepest ba.s.s voice starts a musical scale whose notes have the frequencies 80, 90, 100, 107, 120, 133, 150, 160, but the intervals here are identical with those of a higher scale; the interval between C and D, 80 and 90, is 9/8, just as it was before when the frequencies were much greater; that is, 256 and 288. In singing "Home, Sweet Home," for example, a ba.s.s voice may start with a note vibrating only 132 times a second; while a tenor may start at a higher pitch, with a note vibrating 198 times per second, and a soprano would probably take a much higher range still, with an initial frequency of 528 vibrations per second. But no matter where the voices start, the intervals are always identical. The air as sung by the ba.s.s voice would be represented by _A_. The air as sung by the tenor voice would be represented by _B_. The air as sung by the soprano voice would be represented by _C_.

[Ill.u.s.tration: FIG. 178.--A song as sung by three voices of different pitch.]

CHAPTER XXVIII

MUSICAL INSTRUMENTS

265. Musical instruments maybe divided into three groups according to the different ways in which their tones are produced:--

_First._ The stringed instruments in which sound is produced by the vibration of stretched strings, as in the piano, violin, guitar, mandolin.

_Second._ The wind instruments in which sound is produced by the vibrations of definite columns of air, as in the organ, flute, cornet, trombone.

_Third._ The percussion instruments, in which sound is produced by the motion of stretched membranes, as in the drum, or by the motion of metal disks, as in the tambourines and cymbals.

266. Stringed Instruments. If the lid of a piano is opened, numerous wires are seen within; some long, some short, some coa.r.s.e, some fine.

Beneath each wire is a small felt hammer connected with the keys in such a way that when a key is pressed, a string is struck by a hammer and is thrown into vibration, thereby producing a tone.

If we press the lowest key, that is, the key giving forth the lowest pitch, we see that the longest wire is struck and set into vibration; if we press the highest key, that is, the key giving the highest pitch, we see that the shortest wire is struck. In addition, it is seen that the short wires which produce the high tones are fine, while the long wires which produce the low tones are coa.r.s.e. The shorter and finer the wire, the higher the pitch of the tone produced.

The longer and coa.r.s.er the wire, the lower the pitch of the tone produced.

[Ill.u.s.tration: FIG. 179.--Piano wires seen from the back.]

The constant striking of the hammers against the strings stretches and loosens them and alters their pitch; for this reason each string is fastened to a screw which can be turned so as to tighten the string or to loosen it if necessary. The tuning of the piano is the adjustment of the strings so that each shall produce a tone of the right pitch.

When the strings are tightened, the pitch rises; when the strings are loosened, the pitch falls.

What has been said of the piano applies as well to the violin, guitar, and mandolin. In the latter instruments the strings are few in number, generally four, as against eighty-eight in the piano; the hammer of the piano is replaced in the violin by the bow, and in the guitar by the fingers; varying pitches on any one string are obtained by sliding a finger of the left hand along the wire, and thus altering its length.

Frequent tuning is necessary, because the fine adjustments are easily disturbed. The piano is the best protected of all the stringed instruments, being inclosed by a heavy framework, even when in use.

[Ill.u.s.tration: FIG. 180.--Front view of an open piano.]

267. Strings and their Tones. Fasten a violin string to a wooden frame or box, as shown in Figure 181, stretching it by means of some convenient weight; then lay a yardstick along the box in order that the lengths may be determined accurately. If the stretched string is plucked with the fingers or bowed with the violin bow, a clear musical sound of definite pitch will be produced. Now divide the string into two equal parts by inserting the bridge midway between the two ends; and pluck either half as before. The note given forth is of a decidedly higher pitch, and if by means of the siren we compare the pitches in the two cases, we find that the note sounded by the half wire is the octave of the note sounded by the entire wire; the frequency has been doubled by halving the length. If now the bridge is placed so that the string is divided into two unequal portions such as 1:3 and 2:3, and the shorter portion is plucked, the pitch will be still higher; the shorter the length plucked, the higher the pitch produced. This movable bridge corresponds to the finger of the violinist; the finger slides back and forth along the string, thus changing the length of the bowed portion and producing variations in pitch.

[Ill.u.s.tration: FIG. 181.--The length of a string influences the pitch.]

[Ill.u.s.tration: FIG. 182.--Only one half of the string is bowed, but both halves vibrate.]

If there were but one string, only one pitch could be sounded at any one time; the additional strings of the violin allow of the simultaneous production of several tones.

268. The Freedom of a String. Some stringed instruments give forth tones which are clear and sweet, but withal thin and lacking in richness and fullness. The tones sounded by two different strings may agree in pitch and loudness and yet produce quite different effects on the ear, because in one case the tone may be much more pleasing than in the other. The explanation of this is, that a string may vibrate in a number of different ways.

Touch the middle of a wire with the finger or a pencil (Fig. 182), thus separating it into two portions and draw a violin bow across the center of either half. Only one half of the entire string is struck, but the motion of this half is imparted to the other half and throws it into similar motion, and if a tiny A-shaped piece of paper or rider is placed upon the unbowed half, it is hurled off.

[Ill.u.s.tration: FIG. 183.--The string vibrates in three portions.]

If the wire is touched at a distance of one third its length and a bow is drawn across the middle of the smaller portion, the string will vibrate in three parts; we cannot always see these various motions in different parts of the string, but we know of their existence through the action of the riders.

Similarly, touching the wire one fourth of its length from an end makes it vibrate in four segments; touching it one fifth of its length makes it vibrate in five segments.

In the first case, the string vibrated as a whole string and also as two strings of half the length; hence, three tones must have been given out, one tone due to the entire string and two tones due to the segments. But we saw in Section 267 that halving the length of a string doubles the pitch of the resulting tone, and produces the octave of the original tone; hence a string vibrating as in Figure 183 gives forth three tones, one of which is the fundamental tone of the string, and two of which are the octave of the fundamental tone.

Hence, the vibrating string produces two sensations, that of the fundamental note and of its octave.

[Ill.u.s.tration: FIG. 184.--When a string vibrates as a whole, it gives out the fundamental note.]

When a string is plucked in the middle without being held, it vibrates simply as a whole (Fig. 184), and gives forth but one note; this is called the fundamental. If the string is made to vibrate in two parts, it gives forth two notes, the fundamental, and a note one octave higher than the fundamental; this is called the first overtone. When the string is made to move as in Figure 183, three distinct motions are called forth, the motion of the entire string, the motion of the portion plucked, and the motion of the remaining unplucked portion of the string. Here, naturally, different tones arise, corresponding to the different modes of vibration. The note produced by the vibration of one third of the original string is called the second overtone.

The above experiments show that a string is able to vibrate in a number of different ways at the same time, and to emit simultaneously a number of different tones; also that the resulting complex sound consists of the fundamental and one or more overtones, and that the number of overtones present depends upon how and where the string is plucked.

[Ill.u.s.tration: FIG. 185.--A string can vibrate in a number of different ways simultaneously, and can produce different notes simultaneously.]

269. The Value of Overtones. The presence of overtones determines the quality of the sound produced. If the string vibrates as a whole merely, the tone given out is simple, and seems dull and characterless. If, on the other hand, it vibrates in such a way that overtones are present, the tone given forth is full and rich and the sensation is pleasing. A tuning fork cannot vibrate in more than one way, and hence has no overtones, and its tone, while clear and sweet, is far less pleasing than the same note produced by a violin or piano.

The untrained ear is not conscious of overtones and recognizes only the strong dominant fundamental. The overtones blend in with the fundamental and are so inconspicuously present that we do not realize their existence; it is only when they are absent that we become aware of the beauty which they add to the music. A song played on tuning forks instead of on strings would be lifeless and unsatisfying because of the absence of overtones.

It is not necessary to hold finger or pencil at the points 1:3, 1:4, etc., in order to cause the string to vibrate in various ways; if a string is merely plucked or bowed at those places, the result will be the same. It is important to remember that no matter where a string of definite length is bowed, the note most distinctly heard will be the fundamental; but the quality of the emitted tone will vary with the bowing. For example, if a string is bowed in the middle, the effect will be far less pleasing than though it were bowed near the end. In the piano, the hammers are arranged so as to strike near one end of the string, at a distance of about 1:7 to 1:9; and hence a large number of overtones combine to reenforce and enrich the fundamental tone.