Page:Encyclopædia Britannica, Ninth Edition, v. 1.djvu/124

periodic vibrations of the air-particles acting on the ear, and therefore also of the body whence they proceed, each particle passing through the same phase at stated intervals of time. On the other hand, the motion to which noise is due is irregular and flitting, alternately fast and slow, and creating in the mind a bewildering and confusing effect of a more or less unpleasant character. Noise may also be produced by combining in an arbitrary manner several musical notes, as when one leans with the fore-arm against the keys of a piano. In fact, the composition of regular periodic motions, thus effected, is equivalent to an irregular motion.

45.We now proceed to state the laws of musical harmony, and to describe certain instruments by means of which they admit of being experimentally established. The chief of these laws are as follows:—

(1.)The notes employed in music always correspond to certain definite and invariable ratios between the numbers of vibrations performed in a given time by the air when conveying these notes to the ear, and these ratios are of a very simple kind, being restricted to the various permutations of the first four prime numbers $1,2,3,5$, and their powers.

(2.)Two notes are in unison whose corresponding vibrations are executed exactly at the same rate, or for which (denoting by $n,n_{1}$ the numbers per second) ${\tfrac {n_{1}}{n}}=1$. This ratio or interval (as it is termed) is the simplest possible.

(3.)The next interval is that in which ${\tfrac {n_{1}}{n}}=2$, and is termed the octave.

(4.)The interval ${\tfrac {n_{1}}{n}}=3$ is termed the twelfth, and if we reduce the higher note of the pair by an 8^ve, i.e., divide its number of vibrations by $2$, we obtain the interval ${\tfrac {n_{2}}{n}}={\tfrac {3}{2}}$, designated as the interval of the fifth.

(5.)The interval ${\tfrac {n_{1}}{n}}=5$ has no particular name attached to it, but if we lower the higher note by two 8^ves or divide $n_{1}$ by $4$, we get the interval ${\tfrac {n_{3}}{n}}={\tfrac {5}{4}}$, or the interval of the major third.

(6).The interval ${\tfrac {n_{1}}{n}}={\tfrac {5}{3}}$ is termed the major sixth.

(7.)The interval ${\tfrac {n_{1}}{n}}={\tfrac {2\times 3}{5}}={\tfrac {6}{5}}$ is termed the minor third.

(8.)The interval ${\tfrac {n_{1}}{n}}={\tfrac {2\times 2}{3}}={\tfrac {4}{3}}$ is termed the fourth.

(9.)The interval ${\tfrac {9}{8}}$ which, being $={\tfrac {{\tfrac {3}{2}}\times {\tfrac {3}{2}}}{2}}$, may be regarded as formed by taking in the first place a note one-fifth higher than the key-note or fundamental, i.e., higher than the latter by the interval ${\tfrac {3}{2}}$, thence ascending by another fifth, which gives us ${\tfrac {3}{2}}\times {\tfrac {3}{2}}$ and lowering this by an octave, which results in ${\tfrac {9}{8}}$, which is called the second.

(10.)The interval ${\tfrac {15}{8}}$ or ${\tfrac {3}{2}}\times {\tfrac {5}{4}}$ may be regarded as the major third $({\tfrac {5}{4}})$ of the fifth $({\tfrac {3}{2}})$, and is called the interval of the seventh.

46.If the key-note or fundamental be denoted by ${\text{C}}$, and the notes, whose intervals above ${\text{C}}$ are those just enumerated, by ${\text{D}}$, ${\text{E}}$, ${\text{F}}$, ${\text{G}}$, ${\text{A}}$, ${\text{B}}$, ${\text{C}}$, we form what is known in music as the natural or diatonic scale, in which therefore the intervals reckoned from ${\text{C}}$ are successively

$${\tfrac {9}{8}},{\tfrac {5}{4}},{\tfrac {4}{3}},{\tfrac {3}{2}},{\tfrac {5}{3}},{\tfrac {15}{8}},2,$$

and therefore the intervals between each note and the one following are

$${\tfrac {9}{8}},{\tfrac {10}{9}},{\tfrac {16}{15}},{\tfrac {9}{8}},{\tfrac {10}{9}},{\tfrac {9}{8}},{\tfrac {16}{15}}.$$

Of these last intervals the first, fourth, and sixth are each $={\tfrac {9}{8}}$, which is termed a major tone. The second and fifth are each $={\tfrac {10}{9}}$, which is a ratio slightly less than the former, and hence is called a minor tone. The third and seventh are each $={\tfrac {16}{15}}$, to which is given the name of semi-tone.

By interposing an additional note between each pair of notes whose interval is a major or a minor tone, the resulting series of notes may be made to exhibit a nearer approach to equality in the intervals successively separating them, which will be very nearly semi-tones. This sequence of twelve notes forms the chromatic scale. The note interposed between ${\text{C}}$ and ${\text{D}}$ is either ${\text{C}}$ sharp (${\text{C}}$♯) or ${\text{D}}$ flat (${\text{D}}$♭), according as it is formed by raising ${\text{C}}$ a semi-tone or lowering ${\text{D}}$ by the same amount.

47.Various kinds of apparatus have been contrived with a view of confirming experimentally the truth of the laws of musical harmony as above stated.

Savart's toothed wheel apparatus consists of a brass wheel, whose edge is divided into a number of equal projecting teeth distributed uniformly over the circumference, and which is capable of rapid rotation about an axis perpendicular to its plane and passing through its centre, by means of a series of multiplying wheels, the last of which is turned round by the hand. The toothed wheel being set in motion, the edge of a card or of a funnel-shaped piece of common note paper is held against the teeth, when a note will be heard arising from the rapidly succeeding displacements of the air in its vicinity. The pitch of this note will, agreeably to the theory, rise as the rate of rotation increases, and becomes steady when that rotation increases, and becomes steady when that rotation is maintained uniform. It may thus be brought into unison with any sound of which it may be required to determine the corresponding number of vibrations per second, as for instance the note ${\text{A}}_{3}$, three 8^ves higher than the ${\text{A}}$ which is indicated musically by a small circle placed between the second and third lines of the ${\text{G}}$ clef, which ${\text{A}}$ is the note of the tuning-fork usually employed for regulating concert-pitch. ${\text{A}}_{3}$ may be given by a piano. Now, suppose that the note produced with Savart's apparatus is in unison with ${\text{A}}_{3}$, when the experimenter turns round the first wheel the rate of $60$ turns per minute or one per second, and that the circumferences of the various multiplying wheels are such that the rate of revolution of the toothed wheel is thereby increased $44$ times, then the latter wheel will perform $44$ revolutions in a second, and hence, if the number of its teeth be $80$, the number of taps imparted to the card every second will amount to $44\times 80$ or $3520$. This, therefore, is the number of vibrations corresponding to the note ${\text{A}}_{3}$. If we divide this by $2^{3}$ or $8$, we obtain $440$ as the number of vibrations answering to the note ${\text{A}}$. This, however, tacitly assumes that the bands by which motion is transmitted from wheel to wheel do not slip during the experiment. If, as is always more or less the case, slipping occurs, a different mode for determining the rate at which the toothed wheel revolves, such as is employed in the syren of De la Tour (vide below), must be adopted.