466 HARMONISTS HARMONY entitled to any higher rank than that of a mu- sical curiosity or toy. HARMONISTS. See RAPP, GEOKG. HARMONY (Gr. dp/zovta, agreement or con- cord), in music, the agreeable sensation pro- duced on the ear by the simultaneous sounding of various accordant notes. The discussion of this subject in its more general bearings would include the consideration of the whole theory of music ; but we shall confine ourselves to an account of the conditions necessary to produce harmonious effects, and to an explanation of the reason of those conditions. From the days of Pythagoras to the year 1862 no true expla- nation had been given of the facts that the sounding together of notes forming certain mu- sical intervals gives rise to agreeable sensations, while the simultaneous sounding of the notes of other intervals causes disagreeable or disso- nant effects. It is true that Pythagoras, 2,400 years ago, had shown the relations existing between harmonious chords and the lengths of the vibrating strings producing their con- stituent notes. About the same time Tso-kin- ming, a friend of Confucius, taught that the five sounds of the ancient Chinese gamut cor- responded to the five elements of their natural philosophy, water, fire, wood, metal, and earth, and that the numbers 1, 2, 3, and 4 are the source of all perfection. In the middle ages "the music of the spheres" of Pythagoras played an important part in the discussions on harmony ; and according to Athanasius Kir- cher, music is the product of both the " macro- cosm" and the "microcosm." Even a mind so profoundly scientific as that of Kepler was entangled in such mysticism ; and such occult relations even in these days charm many mu- sicians, more disposed to the pleasures of the imagination than to the toil of scientific rea- soning. Euler, in his Tentamen Novce Theories MusicoB (1739), attempts to explain the facts of musical harmony by the hypothesis that the mind takes a delight in the sentiment of sim- C ratios of vibration. After Euler, D'Alem- t, in his Elements de musique (1762), adopt- ed and developed the hypothesis of Rameau, who thought that he saw in the harmonics which exist in nearly all sounds suitable for music a rational explanation of the main prin- ciples of harmony. Another system of har- mony was brought out in 1754 by Tartini, the celebrated violinist, who rediscovered the re- sultant tones of Sorge, and fancied that he had found in them a clue to the long sought ex- planation of consonance and dissonance. The honor attending the solution of this problem was reserved for H. Helmholtz, professor of physi- ology in the university of Heidelberg. In 1862 he published a work entitled Die Lehre von den Tonempfindungen ah physiologische Grundlage fur die Theorie der Musilc, in which is laid the true physical basis of musical harmony, founded on a minute study of the auditory sensations. The main distinction between his views and the hypotheses of those who preceded him is, that he refers the causes of consonance and dissonance to the sensations produced by continuous and discontinuous sounds, while all before him referred the facts of harmony to a psychological cause. In order fully to appre- ciate Helmholtz's discovery, it will be neces- sary to preface an account of it with a few considerations on the causes and nature of sound ; on the distinction between a simple and a composite sound; on the phenomena of interference and beats ; and on the power of the ear to analyze a composite sound into its sonorous elements. Sound is the sensation caused by tremors sent from rapidly vibrating bodies through the air or other elastic medium to the ear. The vibrating body at the source of the sound, and the elastic medium between that body and the ear, may be of either solid, liquid, or gaseous matter ; but generally the vibrating body is either a solid, as a string or tuning fork, or a mass of air, as in the case of organ pipes and nearly all wind instruments. But only vibrations the number of which in a second is comprised within a definite range can produce on the ear the sensation of sound. This range is between about 40 and 40,000 vi- brations per second, the pitch of sounds rising with the number of vibrations producing them. As the velocity of sound in air having a tem- perature of 32 F. is 1,090 ft. per second, it follows that if we divide 1,090 by the number of vibrations the sounding body makes in one second, we shall have the distance from the sounding body through which the air is affect- ed, or vibrated, after the body has made its first vibration ; and here we take a vibration in the German and English sense, as a motion to and fro, and not to or fro as it is understood by the French. Thus, suppose a body to make one vibration in ^ of a second, and then in- stantly to come to rest ; the air in front of this vibrating body will be moved to a depth of iffS or 27i ft. ; and this depth of air affect- ed by one vibration is called a wave length of sound. The half of this wave length nearest the body was formed by the body receding from the air in front of it, and therefore this half of the wave length is composed of rarefied air, or air the molecules of which are separated by more than their natural distances, while the other half of the wave is formed of condensed air, or air the molecules of which are forced near together. But this wave progresses forward with a velocity of 1,090 ft. per second, and as it passes through the air it causes those mole- cules over which it passes to oscillate once for- ward and once backward; and it follows tlmt the air touching the drum of the ear will force this membrane inward and then outward, and thus a tremor is given to the fibrilbe of the auditory nerve. But if, instead of making only one vibration, the body continuously vibrates. then the waves succeed each other with perfect regularity, and, producing continuous oscilla- tions in the air and ear, cause the continuous sensation necessary for the perception of a mu-
