Harmonic series

“ By a proper adjustment of the initial amplitude and phase of each of these modes of vibration, so that their resultant shall represent the initial state of the string, we obtain a new representation of the whole motion of the string, in which it is seen to be the resultant of a series of simple harmonic vibrations whose periods are the fundamental period and its submultiples. The determination of the amplitudes and phases of the several simple harmonic vibrations so as to satisfy the initial conditions is an example of harmonic analysis. ”

“ Furthermore, it is easy to prove that the fundamental theorem holds for harmonic systems, in the sense that, if A, B, C are three points on a line l, and A′, B′, C′ are three points on a line l′, and if by any two distinct series of projections A, B, C are projected into A′, B′, C′, then any point of the harmonic system ABC corresponds to the same point of the harmonic system A′B′C′ according to both the projective relations which are thus established between l and l′. ”

“ In parts of the fourth and sixth Nocturnes we can find harmonic schemes which it is probable no other musician would have ever dared to devise, schemes which set at naught our established distinctions of concord and discord, which display in unbroken series artifices that are usually kept for single isolated points of excitement, and which, nevertheless, are as undoubtedly intentional as they are undeniably successful in their aim. ”

“ The values of corresponding to several values of are given in Art. 698.
We are now able to write down the value of for any point , whether on the axis or not, by substituting for , and multiplying each term by the zonal harmonic of of the same order. For must be capable of expansion in a series of zonal harmonics of with proper coefficients. When each of the zonal harmonics becomes equal to unity, and the point lies on the axis. Hence the coefficients are the terms of the expansion of for a point on the axis. ”

“ The vibrations of the steel tongue produce a compound sound composed of a fundamental and its harmonics. By using the cavity of the mouth as a resonator, each harmonic in succession can be isolated and reinforced, giving the instrument the compass shown.
The lower harmonics of the series cannot be obtained, owing to the limited capacity of the resonating cavity. The black notes on the stave show the scale which may be produced by using two harps, one tuned a fourth above the other. ”

“ Har′monist, one skilled in harmony: a musical composer.—Harmonic engine, an invention of Edison's, in which the energy of an electric current is used, by means of two small electro-magnets, to keep up the vibrations of a large and heavily-weighted tuning-fork whose arms are connected with two pistons working a miniature pump; Harmonic progression, a series of numbers the reciprocals of which are in arithmetical progression; Harmonic proportion, the relation of three quantities in harmonic progression—the 2d a harmonic mean between the 1st and 3d, as in the three numbers 2, 3, and 6 ”

“ Now, the process of adding up such a series of simple harmonic curves (for the interference fringes are represented by simple harmonic curves) is very laborious. Hence the instrument shown in Fig. 56, called a harmonic analyzer, was devised to perform this work mechanically. It looks very complex; in reality it is very simple, the apparent complexity arising from the considerable number of elements required. A single element is shown in Fig. ”

“ In other words, no part of the energy depends on the product of the amplitudes of two different harmonics. When two modes of motion of the same system are conjugate to each other, the existence of one of them does not affect the other.
The simplest case of harmonic analysis, that of which the treatment of the vibrating string is an example, is completely investigated in what is known as Fourier’s theorem. ”

“ To enter on the consideration of harmonics of fractional, irrational, or impossible degrees is beyond my purpose, which is to give as clear an idea as I can of what these harmonics are. I have done so by referring the harmonic, not to a system of polar coordinates of latitude and longitude, or to Cartesian coordinates, but to a number of points on the sphere, which I have called the Poles of the harmonic. Whatever be the type of a harmonic of the degree , it is always mathematically possible to find points on the sphere which are its poles. ”

“ The conception of the general harmonic, with its poles placed in any manner on the sphere, is useful rather in fixing our ideas than in making calculations. For the latter purpose it is more convenient to consider the harmonic as the sum of conjugate harmonics of selected types, and the ordinary symmetrical system, in which polar coordinates are used, is the most convenient. In this system the first type is the zonal harmonic , in which all the axes coincide with the axis of polar coordinates. ”

“ It will be found that only at certain places are harmonic notes possible. These places are called nodes or nodal points; they are to be found at the mathematical divisions of the strings into halves, thirds, quarters, etc. The class of harmonics produced in this manner are termed "Natural Harmonics." As each string gives out the same notes relative to the pitch of the open string, one description will suffice.
If a string is lightly touched at its half length during vibration, the octave to the open string will be produced ”

“ It is general to give an all round rule that in playing harmonics the bow must always be drawn close to the bridge, the student will comprehend that the smaller the division of the string used to produce the harmonic, the greater need will there be to attend to this rule. The progression of harmonic notes should be clearly defined; in passages composed wholly of harmonics it may sometimes be found advisable to use a down bow-stroke for each note, slightly striking the commencement of each harmonic ”

“ Closing the holes again and blowing harder, we get the scale an octave higher. By blowing still harder we get an octave higher still. In other words, we are now producing harmonics.
It is possible to produce from a plain tube without finger-holes or valves, such as the French Horn, by tightening the lips and increasing the pressure of the player's breath, the following series of harmonics:
Series of harmonics The harmonics of a pianoforte string can be easily demonstrated by the following experiment: Depress the "loud" pedal and strike any note in the bass a sharp blow. ”

“ When the fundamental note and lowest harmonics predominate in the tone, the quality is soft and flute-like; when the combination is well balanced by the addition of the intermediate harmonics up to the sixth, the quality is rich and sonorous; when the highest harmonics, above the sixth and seventh, predominate, the quality is harsh and screaming. When the high dissonant harmonics are produced in a tolerably even and continuous stream of sound, the quality is said to be 'metallic.' If an instrument is ill-strung or out of order the harmonic scale is disturbed ”

“ The reason why the harmonics of the next fifth are not used, is because that note itself is not a note of the diatonic scale, being a little too sharp, as the fifth of the supertonic, and can only be used as part of a chromatic chord.' The advantages of this system of taking a number of chromatic chords under the head of one key will be obvious to any one who wishes for a complete theory to analyse the progressions of keys in modern music as well as their harmonic structure. ”

“ The various modes of vibration may also be exhibited. If D is removed and the string is bowed in the middle, the fundamental is brought out. If it is touched in the middle with a feather, the edge of a card, or the finger nail, and bowed a quarter of the way along the octave, the first overtone comes out. Each of the first few harmonics may be easily obtained by touching the string at the first node of the harmonic required, and bowing at the first loop, and the presence of the nodes and loops may be verified by putting light paper riders of shape A on the string at the nodes and loops. ”

“ A harmonic form, like a metrical form, was always a cluster of tones that could be both associated and distinguished.
It was a long time before these two means of organizing sound were used in combination. Until the seventeenth century metrical form was chiefly used, quite naturally, to define the gesticulatory part of musical material, the product of active emotion, while harmonic form gave coherence to the vocal part, the product of contemplative or religious emotion. ”

“ The tuning of a harmonograph with independent pendulums is a simple matter. It is merely necessary to move weights up or down until the respective numbers of swings per minute bear to one another the ratio required. This type of harmonograph, if made of convenient size, has its limitations, as it is difficult to get as high a harmonic as 1:2, or the octave with it, owing to the fact that one pendulum must in this case be very much shorter than the other, and therefore is very sensitive to the effects of friction. ”

“ We may now also extend the notion of harmonic elements, viz.
Definition.—Four rays in a flat pencil and four planes in an axial pencil are said to be harmonic if their cross-ratio equals -1, that is, if they are cut by a line in four harmonic points.
If we understand by a “median line” of a triangle a line which joins a vertex to the middle point of the opposite side, and by a “median line” of a parallelogram a line joining middle points of opposite sides, we get as special cases of the last theorem:
The diagonals and median lines of a parallelogram form an harmonic pencil ”

“ The Germans call them overtones. They are always of a frequency which is some multiple of the fundamental Page 22 frequency. That is, the rate of vibration of a harmonic is 2, 3, 4, 5, or some other integral number, times as great as the fundamental itself. A tone having no harmonics is rare in nature and is not an attractive one. The tones of the human voice are rich in harmonics. ”

“ In fig. 8, AA′, BB′ are the points, AA′, BB′, A′B′, B′A, AB, BA′ are the sides, L, M, N are the diagonal points, and LMN is the harmonic triangle of the quadrangle.
The harmonic property of the complete quadrilateral is: Any diagonal or harmonic line is harmonically divided by the other two; and of a complete quadrangle: The angle at any harmonic point is divided harmonically by the joins to the other harmonic points. To prove the first theorem, we have to prove (AA′, βγ) , (BB′, γα) , (CC′, βα) are harmonic. Consider the cross-ratio (CC′, αβ) . ”

“ Of the many possible modes only two are used in our present system of music, the major and the minor. There is only one form of the major mode, and it is the one most people recall when they think of the scale. There are three forms of the minor mode, but of these the harmonic is the most frequently used. ”

“ But the combination of two or three groups cannot be so considered unless they each play an independent part and are not united together at full strength. Incessant four-part harmony is to be deprecated. Satisfactory results will be obtained when the number of harmonic parts is gradually decreased, with some of them sustaining pedal notes, and when the harmony, interspersed with necessary pauses is confined to the limits of one octave, distributed over several octaves, or duplicated in the higher register. ”

“ Looked at in the light of modern knowledge, however, we see in the drone an unconscious groping after the truth of the Harmonic Scale, on which all modern harmony rests. We now perceive that either the Tonic or Dominant, or even both together, may with perfect propriety be sounded through any Tonic, Dominant, or Supertonic harmonies, since these must always consist of harmonics generated by the Tonic or its harmonics, and the generator is therefore always a true bass. ”

“ Now, the modern minor mode is too firmly rooted in its minor tonic chord for the 16th-century feeling of an oblique harmonic scheme to be of more than secondary importance, though that feeling survives, as the discussion of key-relationships will show us. ”

“ The relation of the final note of a scale with its upper 5th or lower 4th thus becomes a fundamental fact of complex harmonic significance—that is to say, of harmony modified by melody in so far as it concerns the succession of sounds as well as their simultaneous combination. In our modern key-system the final note of the scale is called the tonic, and the 5th above or 4th below it is the dominant. ”

“ It is thus seen that the tesseral harmonics of degree n and order m are those which have axes coincident with the z axis, and the other m axis distributed in the equatorial plane, at angular intervals . The sectorial harmonics have all their axes in the equatorial plane.
8. Determination of the Poles of a given Harmonic.—It has been shown that a spherical harmonic can be generated by means of an operator
acting upon ,
the function being so chosen that
this relation shows that if an expression of the form
is added to , the harmonic is unaltered ”

“ For example, if the compound sound is formed of six harmonics, the removal of the source of the sonorous vibrations, from the membrane to a distance equal to 14 of a wave-length of the 1st harmonic, will remove the 2d, 3d, 4th, 5th, and 6th harmonics to distances from the membrane equal respectively to 12, 34, 1, 114, and 112 wave-lengths. ”