Acoustics (Gr. to hear), that branch of physical science which explains the phenomena and laws of sound. For the production of these phenomena three conditions are required: 1, a sonorous body; 2, a medium to propagate, and 3, an organism to perceive the sound. From these conditions the science of acoustics is naturally divided into three branches, of which the last belongs entirely to the field of physiology, or rather biology, while in the first two the most intricate and at the same time most successful application of mathematics to mechanical science is to be found. A superficial examination into the cause of sound shows that it originates in vibrations of the sounding body, and is thus a result of its elasticity. The air, being very elastic, is ordinarily the medium by which sound is transmitted to our ears; but most other bodies, solid as well as liquid, transmit sound as well and even better than air, while in a vacuum transmission ceases, as is proved by the well-known experiment of exhausting by means of an air pump the air from around a continuously ringing bell.

The phases of the sonorous vibrations are appropriately called undulations or waves; they are communicated to the body transmitting the sound by one or more impulses from the sonorous body, and are transmitted by alternate compressions and expansions of the parts. The velocity of this transmission for air at the freezing point of Fahrenheit is 1,090 feet per second, and about one foot more for every degree above. Very violent sounds, however, travel faster, as proved by Boyden in Boston and Earnshaw in Sheffield, England; the cause of this is the heat developed by strong compression of the air by a powerful wave of sound. Heavy gases transmit sound slower and light gases faster than air: carbonic acid 858 feet, hydrogen 4,164 feet per second. Water transmits sound with about the same velocity as the latter, while alcohol, ether, and turpentine transmit it slower (3,800 feet), and saline solutions in water faster (from 5,000 to 6,500 feet per second). Through metals the transmission is in round numbers as follows: lead, 4,000 feet per second; copper, 11,000; iron and steel, 16,000. If a wave is violent enough to produce a shock against the drum of the ear, a sound is always heard even if there be but a single wave; such is the case with a clap of thunder, the explosion of a gun, or the crack of a whip.

But if the waves are weak, such as those produced by the vibration of a string, there must be a succession of them at a certain rate of rapidity, in order to make the sound audible. If these waves succeed one another at regular intervals and thus have equal lengths, we have a musical tone; if irregular, they produce merely a noise. - The lowest tone used in music is produced by an organ pipe nearly 32 feet long, in which the tone is produced on the same principle as in the flute, by blowing a current of air against a sharp edge; the friction causing a vibration of the air column in the pipe, on the same principle as the friction of a violin bow causes the vibration of a string. The length of the wave produced in an organ pipe is equal to the length of the pipe; and as sound travels through air with a velocity of about 1,090 feet per second, it must pass through a pipe 32 feet long in nearly the 32d part of a second, and thus produce 32 waves per second. If the pipe is 10 feet long, we must have, 64 waves per second; for an 8-feet pipe, 128 waves; 4 feet, 250; 2 feet, 512; 1 foot, 1,024; 6 inches, 2,048; 3 inches, 4,096; and 1 1/2 inch, 8,192 waves.

These are the correct velocities of vibrations of the tones represented by the note called C, Ut, or Do, from octave to octave, according to the so-called theoretical pitch. In Handel's time the lower C corresponded to 31 vibrations per second, and the Italian opera in London had it in 1859 at 34 vibrations; while the pitch recently established by the French conservatory of music and by a congress of musicians in London agreed to nearly 33 vibrations, corresponding to the Stuttgart pitch. Only the eight octaves mentioned above are used in music. The capacity of the ear, however, extends an octave below the lowest and more than two above the highest of these figures, being between 16 and 38,000 vibrations per second; but there is a difference in this regard between individuals, some persons being perfectly deaf for very low or very high tones distinctly heard by others. The seven different tones of the so-called diatonic scale are interpolated between the octaves given above, and expressed by the customary notes and staff of five lines with clef, or by the letters C, D, E, F, G, A, B, C. They correspond for the lower octave with the velocity of vibrations 32, 36, 40, 42f, 48, 54, 60, and 64 vibrations per second respectively; by multiplying either of these numbers by 2, 4, 8, 16, etc, we obtain the velocities of any other octave.

It is seen that some of these numbers bear simple ratios to one another, as C: C = 1:2, C:G = 2:3, C: F = 3: 4, 0:E = 4:5, E: G = 5: 6; these tones harmonize, the others are discordant. The further comparison of the numbers shows that the differences between the 3d and 4th and between the 7th and 8th of the scale are less than those preceding or following. This has given reason for the interpolation of five other tones between those of which the differences are greater, so as approximately to equalize these differences; in this way 12 tones in each octave have been obtained, forming a scale called chromatic. These interpolated tones are inappropriately called semitones, and designated with the same sign as the next note, but preceded by a ft (sharp) or b (fiat). This scale is represented in the velocity of vibrations and in name as follows:

The keyed instruments give a material representation of this scale. The relation of progression between its tones, when tuned according to the proportions given here, is so irregular, that when transposing the diatonic scale, that is, when commencing it at another tone than C, very impure harmonies are obtained. This is corrected, or rather compromised, by making the mutual proportions of the 12 numbers representing the chromatic scale such as to obtain a regular geometrical series; this is the so-called equal temperament. In order to accomplish this with strict mathematical accuracy, we have only to interpolate 11 terms of such a series between the numbers 1 and 2, which express the relations between a tone and its octave; this is mathematically expressed by the series 2°, 2 1/12, 2 2/12, 2 3/12 2 4/12, etc, to 212/12; or by logarithms: log. 12/12, log. 13/12, log. 14/12, etc, to log. 24/12, which by calculation gives the series 1.000, 1.0594, 1.1225, 1.1892, 1.2599, 1.3348, 1.4142, 1.4983, 1.5874, 1.6818, 1.7818, 1.8877, 2.000. Multiplying each of these numbers by 32, we obtain the velocity of vibration for the lower octave, for the absolute equal temperament: