the volume of air thrown into pulsations. The loudness of a drum, of a cannonade, or of thunder, is due to the immense mass of air set in motion. For a like reason the sounding-board of a piano, the body of a violin, &c., increase the intensity of the sound by taking up the same vibration as the strings, and by their large surface bringing a greatly increased mass of air into play. The power and quality of stringed instruments generally depend altogether on the facility with which this auxiliary vibrating mass assumes the same rate of pulsation as the string. This reinforcement of sound will be again considered under the head of Resonance.

Lastly, the intensity of any note, as of any sound, depends on the distance of the sounding body: and diminishes at the square of the rate that the distance increases.

This (see Art. 494) is an immediate consequence of the fact that sound in the open air, spreads uniformly in all directions round about its origin.

514. (ii.) The pitch or height of a musical tone-sensibility to which constitutes a musical ear-depends on the rate of the aërial vibration set up by the sounding body; or, which is merely another way of expressing the same fact, on the number of air-pulsations produced by the sounding body in any space of time, such as a second.

A low, grave, or bass, tone is one of comparatively slow and few vibrations; while a high, shrill, or sharp tone has quicker and more numerous vibrations.

This general connection between the height or pitch of a musical tone and the number of vibrations may be readily illustrated in various ways :—

It may be roughly shown by drawing the finger-nail slowly and then rapidly across the teeth of a comb. See also for another kind of illustration, Art. 509.

Or, again, by having in the disc referred to in Art. 512, a series of circles, A, B, C, D (fig. 144), round the common centre, each containing different numbers of punched holes, we may show by blowing into the smaller ring, A, first, and then into the others; B, C, D, in succession while the disc is being rapidly whirled, that the height of the note rises with the number of air-pulses produced during one turn of the disc.

515. By an ingenious instrument called the Syren, shown in

The Syren.

341 fig. 143, which was contrived by the French philosopher Cagniard de la Tour, the passage of the wind through the holes of the disc is made to cause at once the necessary rotation and pulsation. The wind or air enters by the pipe, A, from a bellows or other source, into the close box, B, in the lid of which is a series of holes corresponding with those in the rotating disc, D, seen above. The holes are not bored directly through, but obliquely slanting in one direction through the lid of the box, B, and slanting in the opposite direction through the disc, D. Thus, according to the principle of "oblique action," already explained (Arts. 375, &c.) the disc is impelled

round its axis by the wind as it issues through the lower set of holes into the upper. But, again, the wind cannot pass except when the one set of holes is opposite the other; thus it will escape by a succession of puffs, and during one turn of the disc there will be as many puffs as there are holes in the circle, and they will thus increase in number as the rotation of the disc accelerates. An endless screw on the axle of the syren disc gears into the teeth of a counting-wheel, C, which, with another wheel, E, serves to indicate the rate at which the syren is turning, and consequently the number of pulses made per second.

With this instrument we can both build up a musical tone out of a number of individual pulsations of air, and we can analyze the number of pulses corresponding to any given musical tone, by bringing the pitch of the syren to coincide with it, and then estimating by means of the counter, C, the pulse-rate.

So long as the rate of rotation, as shown by the indicating hands, remains the same, and therefore the number of air-pulses remains the same, the ear recognizes the same pitch or height of musical tone; but with a variation of the velocity, the ear at once recognizes a difference in the pitch of the tone.

342

The Musical Scale.

The syren thus demonstrates the cause of, and defines the exact relation between, those differences of pitch which are the basis of the musical sense, and which are discernible, with more or less nicety, by most individuals.

516. Any succession of tones cf different pitch is more or less pleasing; but there is a natural selection of tones employed for the creation of musical pleasure, which rise step by step one above another, forming what is called the musical scale. These have been very much the same in all ages and among all nations, and they depend on the physiological structure and capabilities of the human ear.

"The Musical scale."

517. The natural steps by which the voice rises, and with which the ear is pleased, form the musical scale; it consists, as is well known, really of eight steps or intervals, by which, starting from any arbitrary note, the voice rises or falls to the satisfaction of the

ear.

The eighth note, or octave,* to the first, has this peculiarity, that it blends indistinguishably with the first ; and if we rise, in the same way, to the octave to this second note, and then to the octave to this third note, and so on, the whole set of octaves when sounded together blend most pleasantly. Thus the whole series of musical tones is naturally divided into groups of eight notes each, or octaves, the notes of each group all bearing the same relation to each other.

The notes are commonly named by the seven letters C, D, E, F, G, A, B, the eighth note, or octave, being named C, or C', and the subsequent notes D', E', F', G', A', B', C", D", E", &c. The musical steps or intervals are named as follows :

Relations between the Notes of the Scale. 343

C to G a fifth.

C to A a sixth.

C to B a seventh.

C to C an octave, or eighth.

The interval between each successive pair of tones is not precisely the same; and it has been long an interesting speculation to what cause the apparently arbitrary selection of notes could be assigned. As long ago as in the age of the Grecian philosopher Pythagoras (B.C. 530), some approximations to the truth had been attained. But with the aid of the modern physical instrument described above as the syren, or even with the simple perforated disc (Art. 512), the following remarkable relations underlying the musical sequence of notes, may be readily demonstrated :

Relations between the Notes of the Scale.

518. (i.) Suppose the number of holes in the third circle, C (fig. 144), to be sixteen, or double of that in the first circle, a (eight), it is found by blowing first into the one set of holes and then into the other, that the second note is the octave above the first. Hence

One note is the octave of another when its pulse-rate is double that of the former.

(ii.) Suppose that, while the first circle, A, contains eight holes,

the second, B, contains twelve, or that B has three holes for every two that A has; then we find, on sounding first A and then B, that the latter is a fifth above the former. Hence

One note is the fifth above another when it pulses three times, while the former pulses twice.

(iii.) Suppose that we sound first the set of holes in B, and then

those in C, the latter is recognized as the fourth above the former. Hence, since the numbers of their holes are as 4: 3,

One note is the fourth above another when it makes four pulses while the latter makes three.

By a number of such experiments it may be demonstrated that there is always some simple relation between the number of vibrations or pulses corresponding to the different notes of the scale. This pulse-ratio may be expressed by a fraction, such as, which means that the number of aërial vibrations per second corresponding to the first note bears to the number corresponding to the second note the ratio or relation of 4 to 3.

344

Pulse-ratios of the Diatonic Scale.

The following are the pulse-ratios for the major scale :

In what is called the minor scale, the interval of a third has the pulse-ratiog instead of; the minor interval of a sixth has the ratio § instead of §; and the minor seventh 16 instead of 15.

48

24 24 24 24 24 24 24

519. Reducing by ordinary arithmetic the major scale fractions to one common denominator (24) we obtain the following set of fractions: 24, 27, 30, 32, 30, 40, 45, 4. Hence we see that, if we had a syren disc with eight circles of holes, containing the numbers 24, 27, 30, 32, 36, 40, 45, 48, respectively, we should be able to play the eight notes of the major scale on the instrument. The exact number of air-pulses corresponding to any one note, say the first, which we call C, would depend of course on the rate at which the disc is whirling. But whatever this rate, the others would all follow in regular musical sequence. Thus, given the “vibrationnumber," or pulse-rate, corresponding to the fundamental, key, or starting note, we can at once determine the pulse-rate or vibrationnumbers corresponding to the whole octave.

If the middle C of the pianoforte make, as it is now generally tuned, 264 vibrations per second, then the vibrations of the whole

In the same way we might find the vibration-numbers for all the notes of a piano. From the first A of the bass-of a common seven-octave piano-to the last A of the treble we have a range of from 27 vibrations or pulses per second to as many as 3520. These are not by any means the limits of the musical scale, or of the perception of a musical tone, for the pulses of the Syren begin to assume the character of a tone when they reach 16 per second, and 20 pulses per second may be regarded as giving a sufficiently decided tone. On the other hand, the shrillest note in the orchestra is estimated to have 4750 vibrations per second, and the limits of