Harmony , (Gr. agreement or concord), in music, the agreeable sensation produced 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 explanation had been given of the facts that the sounding together of notes forming certain musical intervals gives rise to agreeable sensations, while the simultaneous sounding of the notes of other intervals causes disagreeable or dissonant 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 constituent notes. About the same time Tso-kin-ming, a friend of Confucius, taught that the five sounds of the ancient Chinese gamut corresponded 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 "macrocosm" 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 musicians, more disposed to the pleasures of the imagination than to the toil of scientific reasoning. Euler, in his Tentamen Norm Theorioe Musicoe (1739), attempts to explain the facts of musical harmony by the hypothesis that the mind takes a delight in the sentiment of simple ratios of vibration. After Euler, D'Alem-bert, in his Elements de musique (1762), adopted 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 principles of harmony. Another system of harmony was brought out in 1754 by Tartini, the celebrated violinist, who rediscovered the resultant tones of Sorge, and fancied that he had found in them a clue to the long sought explanation of consonance and dissonance.
The honor attending the solution of this problem was reserved for II. Helmholtz, professor of physiology in the university of Heidelberg. In 1862 he published a work entitled Die Lehre von den Tonempfindungen als physiologische Grundlage far die Theorie der Musik, 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 appreciate Helmholtz's discovery, it will be necessary 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 vibrations per second, the pitch of sounds rising with the number of vibrations producing them. As the velocity of sound in air having a temperature 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 affected, 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 1/40 of a second, and then instantly to come to rest; the air in front of this vibrating body will be moved to a depth of 1090/40, or 27 1/2 ft.; and this depth of air affected 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 molecules over which it passes to oscillate once forward and once backward; and it follows that the air touching the drum of the ear will force this membrane inward and then outward, and thus a tremor is given to the fibrillae 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 oscillations in the air and ear, cause the continuous sensation necessary for the perception of a musical sound.
If the body, instead of making 40 vibrations, made 8,000 per second (which corresponds to the highest note used in music), the wave length would amount to only 1 6/10 inch; yet this very short wave and the long one of 27 1/2 ft. travel with the same velocity of 1,090 ft. per second. The sounds produced by these vibrations are either simple or composite. A simple sound is a sound having only one pitch, while a composite sound is one composed of two or more definite and separable sounds having pitches generally in the ratio of 1 : 2 : 3 : 4: 5, etc. This series of sounds is called the harmonic series. Thus, the sound of a tuning fork when mounted on its resonant box, or that of a gently blown closed organ pipe, is simple, for the ear can distinguish but one pitch in these sounds; while the sounds of piano or violin strings, or of reed organ pipes, are highly composite, and the ear can separate them into simple sounds whose numbers of vibration are to each other as 1 : 2 : 3 : 4: 5, etc. For example, if we take a reed pipe giving C below the middle C of the piano (which note we will designate as Co), we can separate the sound of this pipe into the following simple sounds : C2, C3, G3, C4, E4, G4, Bb4, C5, D5, E5,; or, expressed in musical notation :
These simple sounds all coexist in the sound of the reed pipe, but their relative intensities diminish as they ascend in pitch; that is, the lowest in pitch is the loudest, and serves to designate the position of the pipe in the musical scale. Now it has for a long time been known that those musical sounds which were best adapted to render the effects of musical composition, and which we distinguish for their brilliant or plaintive qualities, arc always composite, and contain besides the fundamental sound the harmonic series; and indeed the timbre of a sound depends entirely on the number and relative intensities of its harmonics. On minute examination it has been found that a simple sound is produced only when the air near the ear oscillates forward and backward with the same kind of motion as exists in a freely swinging pendulum. If, however, the ear experiences the sensation of a composite sound, the air near it has a reciprocating motion, which is the resultant of as many pendulum vibrations as there are harmonics in the sound.
Yet the ear is a powerful and subtle instrument for decomposing such complex motions into their simple vibratory components; for the ear, properly aided, can separate the. composite sound of a reed pipe or of a vibrating string into 12 and more distinct simple harmonic vibrations. Those who are interested in this subject of the analysis and synthesis of sound will find a full description of various experimental methods in a paper by Prof. A. M. Mayer "On an Experimental Confirmation of Fourier's Theorem, as applied to the decomposition of the vibrations of a composite sonorous wave into its elementary pendulum vibrations," etc, in the "American Journal of Science" (1874). According to Helmholtz, the ear accomplishes this analysis of sound by means of 3,000 little rods or cords, existing in the ductus cochlearis of the inner ear, and known as the rods of Corti. These rods are of graduated lengths and thicknesses like the strings in a piano, and appear to be tuned to 3,000 simple notes, equally distributed throughout the range of the seven octaves of musical sounds. Each rod is connected with a filament of the auditory nerve.
The mode of action of this highly organized part of the auditory apparatus is as follows : the vibrations of a composite sound reaching the rods of Corti, each rod, being in tune with a simple sound or harmonic existing in the composite sound, enters into vibration and shakes its attached nerve filament, and thus the ear receives a sensation formed of as many simple sounds as really existed in the composite vibration. Indeed, it appears that the rods of Corti are set in vibration exactly as the strings of a piano vibrate to the elements of a note when we sing over the strings of the instrument. - We may now consider the manner of production of beats, and the effects they produce on the ear; and then we shall be in possession of the main facts necessary to explain the fundamental prin-ples of musical harmony. When two sounds nearly in unison fall upon the ear, they produce alternate risings and fallings in the inten-sity of their resultant effect on the ear. These alternations of intensity are called beats, and are caused in the following manner: Suppose two sounds, produced by two bodies, one giving 2,000 vibrations in a second, the other 2,001. It is evident that if both bodies vibrate together at the beginning of a second, they will again vibrate together at the end of the second; therefore at these two instants the action of one of them on the air conspires with the action of the other, and thus we have an impression given to the air which is the sum of the two vibrations; but at the half seconds the motions of the two bodies are opposed, and therefore at those instants they will neutralize each other's action if their intensities of vibration are equal, and at the instant of the half seconds we shall have entire silence.
Hence it follows that the number of beats per second given by any two vibrations will equal the difference in the number of vibrations these bodies separately give in one second. Their beats produce on the ear an intermittent action similar to that experienced by the eye when successive flashes of light fall upon it. These intermittent actions on the sensorium are always unpleasant, and even irritating. The degree of unpleasantness, however, depends on the number of the beats or 'flashes per second, and also varies with the pitch of the sound or the color of the light. But when the beats have reached a certain number in a second, they no longer produce intermittent effects on the nerves; for the action produced by one beat lasts, without perceptible diminution, until the arrival of the following one, and the sensation becomes continuous; in other words, when the beats follow with sufficient rapidity, they blend together and form a smooth, sonorous effect, like a simple musical sound. This relation between discontinuous and continuous impressions on the nerves, and unpleasant and pleasant sensations, is at the foundation of nelmholtz"s theory of musical harmony. - We must now consider the effects resulting when, instead of producing only simple sounds together, as above, we simultaneously produce composite sounds differing slightly in pitch.
If we sound two tuning forks, each giving the middle C of the piano, we shall have two simple sounds in unison. Now gradually elevate one of them in pitch and observe the changing sensations. The harshness increases until they are separated about a tone; then the disagreeable sensation diminishes, and entirely vanishes when the notes have been separated by an interval equal to a minor third. But if, instead of sounding the forks, we use two reed pipes giving the same notes, we observe that the slightest departure from unison at once causes a very unpleasant sensation; the reason of this is, that besides the beats of the fundamental simple sounds of the pipes, we have the sensations produced by the beating of some 20 harmonics of their fundamentals. Therefore the tuning of reed pipes is difficult, but their intervals are defined with an extraordinary degree of sharpness. It is here also to be remarked that the number of beats per second given by any pair of harmonics is directly as their height in the harmonic series. Thus if the fundamental or first harmonics give 3 beats per second, the sixth harmonics will give 18 beats per second.
Therefore, in sounding two such pipes, each giving 20 harmonics, we should have produced on the ear 632 beats per second, 3 belonging to the first pair of harmonics, and 60 to the 20th pair. - Helmholtz's discovery consists in the demonstration of the fact that the degree of smoothness or consonance of any given chord depends entirely on the number of elementary harmonics and resultant tones which beat together in the given notes, on the intensities of these beats, and on the number per second of beats produced by each dissonant pair of harmonics. This fact he proved by nearly every means known to modern science, and thus established a real physical cause for the harmonious or dissonant sensations we experience on combining various notes. We can best illustrate the truth of Helmholtz's theory and show his main results by giving in musical notation the principal intervals of fundamental notes, indicated in minims, with their accompanying harmonics written over them in crotchets. Only the first six harmonics are indicated, because those of higher order are generally either absent from a musical sound, or exist with such feeble intensity as not greatly to affect the degree of consonance.
The respective harmonics which beat we have connected together by straight lines, so that at a glance one can approximately determine the degree of consonance of a given interval. The intervals here given are the true intervals of the natural scale, and not the false intervals of the tempered scale. On the latter scale the only consonant interval is the octave. The intervals we have selected are the octave, the fifth, the fourth, the major third, the major sixth, and the minor seventh; the ratios of the vibrations giving the notes of these intervals are respectively as 1 : 2, 2 : 3, 3 : 4, 4 : 5, 3 : 5, and 9 : 16.
No dissonance here occurs because the harmonics of both notes are in unison.
We have here two pairs in unison, 3-2 and 6-4; but a slight departure from perfect smoothness of effect is caused by the third harmonic of the higher note beating with the fourth and fifth of the lower. If the vibrations of the two fundamental notes of this interval are not rigorously as 2 : 3, there will be discord. Hence, on all instruments of fixed equal-tempered scales, as the organ or piano, even the interval of the fifth is slightly discordant, only the octave intervals being in tune.
The dissonance of this interval is greater than in the case of the fifth, because the harmonics 3-2 are both vibrations of intensity, and therefore give louder beats than the pairs 3-4 and 3-5 of the fifth. In the fourth we have also the additional beats of pairs 6-4 and 6-5.
The Major Third and the Major Sixth.
The major third and the major sixth are written together as they are about equally consonant, for the dissonance caused by the beats of pair 3-2, separated by a tone, in the sixth, about equals that of the weaker beating pair 4-3, separated by a semitone, in the major third.
The Minor Seventh.
The minor seventh is the smoothest of that class of chords sometimes denominated discords, and is less dissonant than the minor sixth. Besides the beats of the harmonics existing as described in the above intervals, we have also the influence of the beats of the resultant tones, which are the products of the combined vibrations of the fundamental notes and of their harmonics. These resultant tones can produce beats either with harmonics or with other resultant tones. These resultant tones are of two kinds, viz.: difference tones and summation tones. Difference tones were discovered by Sorge in 1740, and their pitch is equal to the difference of the two vibrations of the sounds producing them. Summation tones were discovered by Helmholtz, and their pitch is equal to the sum of the vibrations of the two sounds producing them. It will be observed that Helmholtz's work is to a great extent merely qualitative; and although he indicates the existence of beats as the cause of discord, yet he does not give laws capable of quantitative expression, by which to determine beforehand the degree of consonance or dissonance existing in any given chord. - The recent research of Prof. Mayer of Hoboken, N. J., "On the Experimental Determination of the Law connecting the Pitch of a Note with the Duration of the Residual Sensation it produces in the Ear" (American Journal of Science, 1874), first gave the duration in absolute time of the sensation of sounds after the exciting vibrations had ceased to exist outside the ear, and thus afforded the means of determining with quantitative exactness the smallest number of beats that two sounds must produce in order that they form a consonant interval.
This latter condition will of course be fulfilled when the beats become just rapid enough in their succession to produce a continuous sensation in the ear. The following is the important law discovered by Prof. Mayer: If N equal the number of vibrations producing any given note, and D equal, in the fraction of a second, the duration of the residual sensation (that is, the time during which the sensation remains after the vibrations outside the ear have ceased), then D =(53248/N+23 +24).0001. The denominator of the (vulgar) fraction thus determined will be the smallest number of beats per second which one simple sound must make with another in order that harshness or dissonance shall entirely disappear from the interval. Thus the simple note giving the middle C of the piano makes 264 vibrations per second, and the residual sensation of its sound remains on the ear 1/48 of a second; therefore the note which will make 48 beats per second with this 0 will form an interval free from all harshness. The number of vibrations of this note will be 264 + 48, or 312, which is D, and forms with C the interval of the minor third. Hence the nearest note to this C which will form with it a harmonious combination is its minor third.
If we in like manner calculate the nearest interval to form a consonance with the C below the middle C, we shall find it to be the major third. This nearest consonant interval contracts as the pitch ascends, so that for the C of the fifth octave above the middle C (the highest octave used in music) the interval has contracted to 6/10 of a semitone. Prof. Mayer has also determined the other limit of the effects of beats by ascertaining in the different octaves the number of beats which produce the greatest harshness or dissonance on the ear. We give above a curve which at a glance shows the connection between the pitch of a note and the duration of the residual sensation. The curve approaches closely to an equilateral hyperbola (which latter curve is also given in a dotted line as a means of comparison); it would indeed coincide with the hyperbola if the duration of the residual sensation were simply inversely as the pitch. The units of division on the horizontal line equal 64 vibrations per second, while the units on the vertical line equal 1/200 of a second. To find by means of this curve the duration of a simple sound, obtain the point on the horizontal line corresponding to its number of vibrations, and then erect from this point a perpendicular reaching to the curve.
The length of this perpendicular in units of the vertical scale will give the duration of the residual sensation of the sound in the fraction of a second; and the denominator of this (vulgar) fraction gives the number of beats which the note will have to make with a neighboring one to form the smallest consonant interval. - Although the science of counterpoint is based upon the principles of harmony, yet the discussion of this subject leads into the higher aesthetic principles of musical composition; we therefore refer to the article Music for information on that subject.
Curve showing the Eolation of Pitch and Duration.