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harmonic

In Wagner's harmonic style we encounter the entire problem of modern musical texture.

82A similar method has frequently been applied to the study of variations of soil-temperatures by harmonic analysis of the annual waves.

30are the amplitudes of the component harmonic waves of periods 24, 12, 8 and 6 hours; al, a2, a 3, a 4, are the corresponding phase angles.

10Let the two tones with their harmonic overtones be 256 512 768 1024 1280 1536 512 1024 1536.

11There is some reason to hope that the day of these misconceptions is passed; although there is also some reason to fear that on other grounds the present era may be known to posterity as an era of instrumentation comparable, in its gorgeous chaos of experiment and its lack of consistent ideas of harmony and form, only to the monodic period at the beginning of the 17th century, in which no one had ears for anything but experiments in harmonic colour.

00Discords must not be taken unprepared, because a singer can only find his note by a mental judgment, and in attacking a discord he has to find a note of which the harmonic meaning is at variance with that of other notes sung at the same time.

00But, when we look at the many passages in which the violas double the basses, we shall do well to consider whether there is room in the harmonic scheme for the violas to do anything else, and whether the effect would not be thin without them.

00A subject so vast and so incapable of classification cannot be discussed here, but its aesthetic principles may be illustrated by the extreme case of the trumpets and horns, which in classical times had no scale except that of the natural harmonic series.

00In another party line system a harmonic principle is employed: the ringing machines deliver alternating currents of four frequencies, while each bell is constructed to operate at a particular frequency only.

00In mathematics, he was the first to draw up a methodical treatment of mechanics with the aid of geometry; he first distinguished harmonic progression from arithmetical and geometrical progressions.

00Apart from the gain in tragic force resulting from Wagner's masterly development of the character of Brangaene, the raw material of the story was already suggestive of that astounding combination of the contrasted themes of love and death, the musical execution of which involves a harmonic range almost as far beyond that of its own day as the ordinary harmonic range of the 19th century is beyond that of the 16th.

00Its harmonic style is, except in the Grail music, even more abstruse than in Tristan; and the intense quiet of the action is far removed from the forces which in that tumultuous tragedy carry the listener through every difficulty.

00But elsewhere there are few passages in which the extremely recondite harmonic style can be with certainty traced to anything but habit.

00The last two examples at the end of the article on Harmony show almost all that is new in Wagner's harmonic principles.

00We have seen (in the articles on Harmony and Music) how harmonic music originated in just this habit of regarding combinations of sound as mere sensations, and how for centuries the habit opposed itself to the intellectual principles of contrapuntal harmony.

00The only illogical point in his system is that the beauty of his dreamlike chords depends not only on his artful choice of a timbre that minimizes their harshness, but also on the fact that they enter the ear with the meaning they have acquired through centuries of harmonic evolution on classical lines.

00Haydn uses a true Straussian discord in The Seasons, in order to imitate the chirping of a cricket; but the harshest realism in Gatterdammerung (the discord produced by the horns of Hagen and his churls in the mustering-scene in the second act) has a harmonic logic which would have convinced Corelli.

00The brilliant success of Humperdinck's Hansel and Gretel, in which Wagnerian technique is applied to the diatonic style of nursery songs with a humorous accuracy undreamed of by Wagner's imitators, points a moral which would have charmed Wagner himself; but until the revival of some rudiments of musical common sense becomes widespread, there is little prospect of the influence of Wagner's harmonic style being productive of anything better than nonsense.

00Under the general heading "Analysis" occur the subheadings "Foundations of Analysis," with the topics theory of functions of real variables, series and other infinite processes, principles and elements of the differential and of the integral calculus, definite integrals, and calculus of variations; "Theory of Functions of Complex Variables," with the topics functions of one variable and of several variables; "Algebraic Functions and their Integrals," with the topics algebraic functions of one and of several variables, elliptic functions and single theta functions, Abelian integrals; "Other Special Functions," with the topics Euler's, Legendre's, Bessel's and automorphic functions; "Differential Equations," with the topics existence theorems, methods of solution, general theory; "Differential Forms and Differential Invariants," with the topics differential forms, including Pfaffians, transformation of differential forms, including tangential (or contact) transformations, differential invariants; "Analytical Methods connected with Physical Subjects," with the topics harmonic analysis, Fourier's series, the differential equations of applied mathematics, Dirichlet's problem; "Difference Equations and Functional Equations," with the topics recurring series, solution of equations of finite differences and functional equations.

00For the subjects of this general heading see the articles Mechanics; Dynamics, Analytical; Gyroscope; Harmonic Analysis; Wave; HYDROMechanics; Elasticity; Motion, Laws Of; Energy; Energetics; Astronomy (Celestial Mechanics); Tide.

00The linear transformation replaces points on lines through the origin by corresponding points on projectively corresponding lines through the origin; it therefore replaces a pencil of lines by another pencil, which corresponds projectively, and harmonic and other properties of pencils which are unaltered by linear transformation we may expect to find indicated in the invariant system.

00There is no linear covariant, since it is impossible to form a symbolic product which will contain x once and at the same time appertain to a quadratic. (v.) is the Jacobian; geometrically it denotes the bisectors of the angles between the lines ax, or, as we may say, the common harmonic conjugates of the lines and the lines x x .

000= {(n +I)Ar" - where P. denotes the zonal harmonic of the nth order; also, in the exceptional case of =Ao cos 0, 4) = Ao/r; 4'= Bor, 49 = - Bo log tan 2B sh - lx/y.

00The simplest form of wave, so far as our sensation goes - that is, the one giving rise to a pure tone - is, we have every reason to suppose, one in which the displacement is represented by a harmonic curve or a curve of sines, y=a sin m(x - e).

00The chief experimental basis for supposing that a train of longitudinal waves with displacement curve of this kind arouses the sensation of a pure tone is that the more nearly a source is made to vibrate with a single simple harmonic motion, and therefore, presumably, the more nearly it sends out such a harmonic train, the more nearly does the note heard approximate to a single pure tone.

00Now we can see that two notes of the same pitch, but of different quality, or different form of displacement curve, will, when thus analysed, break up into a series having the same harmonic wave-lengths; but they may differ as regards the members of the series present and their amplitudes and epochs.

00We may regard quality, then, as determined by the members of the harmonic series present and their amplitudes and epochs.

00For the superposition of these trains will give a stationary wave between A H A (16) Y which is an equation characteristic of simple harmonic motion.

00But where it is appropriate, the disturbance sent out into the air contains the same harmonic series as the source.

00But the harmonics are most readily heard if we fortify the ear by an air cavity with a natural period equal to that of the harmonic to be sought.

00Now suppose that in addition to the internal force represented by, ux, an external harmonic force of period 27r/p is applied.

00Each 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.

00When the harmonic is sounded the riders at the loops are thrown off, while those at the nodes remain seated.

0036, for the fundamental and the first harmonic. When a string is struck or bowed at a point, any harmonic with a node at that point is absent.

00The first overtone has frequency 6.25 that of the fundamental, and is not in the harmonic series.

00If the wire is stretched across a room and stroked in the middle with a damp cloth the fundamental is easily obtained, and the first harmonic can be brought out by stroking it at a quarter the length from one end.

00If it is clamped at one-quarter and threequarters of the length from the ends, and is stroked in the middle, the first harmonic sounds.

00If we measure the time from an instant at which the two are in the same phase the resultant disturbance is y=a sin i t+a sin 27rn2t =2a cos ir(n i - n 2)t sin ir(nl-t-n2)t, which may be regarded as a harmonic disturbance of frequency (ni+n2)/2 but with amplitude 2a cos 7r(n i - n 2)t slowly varying with the time.

00Take as a further example the fifth with harmonic overtones as under The fundamental and overtones of the second either coincide with or fall midway between overtones in the first, and there is no approach to a dissonant frequency of beats, and the concord is perfect.

00Mehler, who proved that a simple relation existed between the function of zero order and the zonal harmonic of order n.

00Also W = (V +IA)w i; or w1=W/(V+/A), w p =W/(V+plA), and wn =W/(Vd-nIA), or the densities of the several liquids vary inversely as the respective volumes of the instrument immersed in them; and, since the divisions of the scale correspond to equal increments of volume immersed, it follows that the densities of the several liquids in which the instrument sinks to the successive divisions form a harmonic series.

00A spectroscope may be compared to a mechanical harmonic analyser which when fed with an irregular function of one variable represented by a curve supplies us with the sine curves into which the original function may be resolved.

00In other spectra such " harmonic " ratios were also discovered, but their search was abandoned when it was found that their number did not exceed that calculated by the laws of probability on the supposition of a chance distribution.

00(2) On the arithmetic, geometric and harmonic means between two straight lines, and the problem of representing all three in one and the same geometrical figure.

00The three commonest means are the arithmetical, geometrical, and harmonic; of less importance are the contraharmonical, arithmetico-geometrical, and quadratic.

00The harmonic mean of n quantities is the arithmetical mean of their reciprocals.

00in the proportion a: b ::b :c or b' = ac; and the harmonic mean places the quantities in harmonic proportion, i.e.

00Among the remaining buildings must be mentioned the town hall (17th century; restored 1823), the court-house, the concert-hall of the "Harmonic" club, the record office (1900), the leeskabuiet, or subscription library and reading-rooms, and the ten-storeyed Witte Huis (1897), which is used for offices and is one of the highest private buildings on the Continent.

00The steam pressure in the heater may be periodically varied by the gauge in such a manner as to produce an approximately simple harmonic oscillation of temperature at the hot end, while the cool end is kept at a steady temperature.

00This may be described as a simple harmonic oscillation whose amplitude diminishes asymptotically to zero according to the law ehlr.

00The path is therefore an ellipse of which a, b are conjugate semi-diameters, and is described in the period 24 Ju; moreover, the velocity at any point P is equal to ~ OD, where OD is the semi-diameter conjugate to OP. ~,This type of motion;,s called elliptic harmonic. If the co-ordinate axes are the principal axes of the ellipse, the angle ft in (I o) is identical with the excentric angle.

00II the coefficient of x be positive the variations of x are simple harmonic, and x can remain permanently small; the circular orbit is then said to be stable.

00In elliptic harmonic motion the velocity of P is parallel and proportional to the semi-diameter CD which is conjugate to the radius CP; the hodograph is therefore an ellipse similar to the actual orbit.

00In the particular case where the motion is of the kind known as simple harmonic the disturbing force on the frame due to the reciprocation of the weight is equal to the component of the centrifugal force in the line of stroke due to a weight equal to the reciprocated weight supposed concentrated at the crank pin.

00In balancing the mechanism of a steam engine it is often sufficiently accurate to consider the motion of the pistons as simple harmonic, and the effect on the framework of the acceleration of the connecting rod may be approximately allowed for by distributing the weight of the rod between the crank pin and the piston inversely as the centre of gravity of the rod divides the distance between the centre of the cross head pin and the centre of the crank pin.

00The moving parts of the engine are then divided into two complete and independent systems, namely, one system of revolving weights consisting of crank pins, crank arms, &c., attached to and revolving with the crank shaft, and a second system of reciprocating weights consisting of the pistons, cross-heads, &c., supposed to be moving each in its line of stroke with simple harmonic motion.

00The assumption that the pistons of an engine move with simple harmonic motion is increasingly erroneous as the ratio of the length of the crank r, to the length of the con oecting rod 1 increases.

00In details of execution and harmonic combinations they illustrate the precision, logic, lucidity and cheerful spirit of the national genius.

00This is a simple harmonic wave-line, whose mean distance from the axis is a, whose wave-length is lira, and whose amplitude is c. The internal pressure corresponding to this unduloid is as before p = T/a.

0013) are the disks, and if x the distance between the disks is equal to irr half the wave-length of the harmonic curve, the disks will be at the points where the curve is at its mean 1 3.

00Whatever the deformation of the originally straight boundary of the axial section may be, it can be resolved by Fourier's theorem into deformations of the harmonic type.

00Disturbances of the former kind lead to vibrations of harmonic type, whose amplitudes always remain small; but disturbances, whose wave-length exceeds the circumference, result in a greater and greater departure from the cylindrical figure.

00The relative importance of two harmonic disturbances depends upon their initial magnitudes, and upon the rate at which they grow.

00If, as before, the frequency be p7211, and a the radius of the sphere, we have p 2 =n(n-1)(n+2)P a3, (6) n denoting the order of the spherical harmonic by which the deviation from a spherical figure is expressed.

00The cartesian equation, referred to the fixed diameter and the tangent at B as axes may be expressed in the forms x= a6, y=a(I -cos 0) and y-a=a sin (x/afir); the latter form shows that the locus is the harmonic curve.

00The periodic variations can be represented algebraically as the resultant of a series of harmonic motions in the following way: Let L be an angle which is increasing uniformly with the time, and let n be its rate of increase.

00The first book deals with the generation of the three conics; the second with the asymptotes, axes and diameters; the third with various metrical relations between transversals, chords, tangents, asymptotes, &c.; the fourth with the theory of the pole and polar, including the harmonic division of a straight line, and with systems of two conics, which he shows to intersect in not more than four points; he also investigates conics having single and double contact.

00Chandler afterwards made an important addition to the subject by showing that the motion was represented by the superposition of two harmonic terms, the first having a period of about 430 days, the other of one year.

00Thomson's tide gauge, tidal harmonic analyser and tide predicter are famous, and among his work in the interest of navigation must be mentioned his tables for the simplification of Sumner's method for determining the position of a ship at sea.

00A ray of light is reflected from this mirror and from another mirror which is rocked by a small motor driven off the same circuit, so that the ray has two vibratory motions imparted to it at right angles, one a simple harmonic motion and the other a motion imitating the variation of the current or electromotive force under test.

00It has wit, economy and intellectual control, in a richly expressive harmonic idiom.

00The polarized signal is recovered from the first harmonic at 40 kHz using lock-in amplifiers.

00The Oxford Harmonic Society is a mixed voice choral society, based in Oxford, England with about 150 members.

00At the harmonic convergence of 1987, this movement reached a flowering and a breakthrough.

00damped harmonic oscillation.

00RMS both channels driven into 8 ohms from 20 to 20,000 Hz with no more than 0.03 total harmonic distortion.

00harmonic flute for the proposed Vox Humana, thus compromising the Choir division too.

00formalism of quantum mechanics for bound systems, in particular, the simple harmonic oscillator and the hydrogen atom.

00You move the saddles until the note at the 12th fret and the 12th fret harmonic are identical.

00harmonic oscillator.

00harmonic voltage distortions at points remote from the active shunt filter itself.

00harmonic wavelets.

00harmonic morphisms with fibers of dimension one, Comm.

00harmonic overtones is then to modify the quality or character 5 of the note, independently of pitch.

00harmonic tremor which follows the build up of repeated hybrid events.

00The polarized signal is recovered from the first harmonic at 40 kHz using lock-in amplifiers.

00In 1876, Heaviside took account of self inductance, obtaining the equation of damped harmonic motion known as Heaviside's Equation of Telegraphy.

00The first movement is in sonata allegro form and uses traditional harmonic progressions.

00iodized oil and complete necrosis on contrast-enhanced wideband harmonic gray-scale sonography.

00Dr. Chris Wood Chris's interest lies in the study of harmonic mappings between Riemannian manifolds.

00In particular, harmonic amplitudes for the various errors show minima at specific loads as expected.

00In the harmonic chart all 3 aspects are within a 12 degree orb.

00A nitrogen molecule is thus a good approximation to a harmonic oscillator.

00The effect of the harmonic overtones is then to modify the quality or character 5 of the note, independently of pitch.

00My mother who is a piano teacher was once taken in by an infant prodigy who had the most incredible grasp of harmonic relationships.

00Magical harmonic progressions in the delicate coda prepare the way for the energy of the finale.

00Her music often combines chiseled rhythmic pulsation with a bittersweet melodic and harmonic language.

00The octave quint which stands on the former harmonic flute slide is the old fifteenth, with five extra pipes at the bottom.

00The restraint energy represents the effect of the harmonic restraints we have imposed.

00On the palate the wine is velvety, soft, rich, harmonic, with the characteristic hint of citrus rind on the finish.

00The finale is another substantial movement, full of surprise harmonic sidesteps and enormous energy.

00With the first stop of the harmonic series it will stand in the front of the instrument on an open soundboard.

00subtletystener deserves time to hear Bach's harmonic subtleties.

00He had developed the " harmonic telegraph " which could send more than one message at a time over a single telegraph wire.

00These lines are seen in harmonic tremor which follows the build up of repeated hybrid events.

00Harmonic: There are two minor triads in the harmonic minor scale.

00Wavelet methods make this possible and good preliminary results have already been achieved using harmonic wavelet methods make this possible and good preliminary results have already been achieved using harmonic wavelets (Newland & Butler, 1998 ).

00are the amplitudes of the component harmonic waves of periods 24, 12, 8 and 6 hours; al, a2, a 3, a 4, are the corresponding phase angles.

00There is some reason to hope that the day of these misconceptions is passed; although there is also some reason to fear that on other grounds the present era may be known to posterity as an era of instrumentation comparable, in its gorgeous chaos of experiment and its lack of consistent ideas of harmony and form, only to the monodic period at the beginning of the 17th century, in which no one had ears for anything but experiments in harmonic colour.

00Discords must not be taken unprepared, because a singer can only find his note by a mental judgment, and in attacking a discord he has to find a note of which the harmonic meaning is at variance with that of other notes sung at the same time.

00Moreover, the opening theme is formed of slow arpeggios; and the more modern harmonic elements, though technically chromatic, consist, from the modern point of view, rather in swift changes between nearly related keys than in chromatic blurring of the main key.

00Those instruments of which the tones and compass are most suitable for polyphonic melody are for the most part high in pitch; a circumstance which, in conjunction with the practice (initiated by the monodists and ratified by science and common sense) of reckoning chords upwards from the bass, leads to the conclusion that the instruments which hold the main threads in the design shall be supported where necessary by a simple harmonic filling-out on some keyed instrument capable of forming an unobtrusive background.

00But, when we look at the many passages in which the violas double the basses, we shall do well to consider whether there is room in the harmonic scheme for the violas to do anything else, and whether the effect would not be thin without them.

00A subject so vast and so incapable of classification cannot be discussed here, but its aesthetic principles may be illustrated by the extreme case of the trumpets and horns, which in classical times had no scale except that of the natural harmonic series.

00In another party line system a harmonic principle is employed: the ringing machines deliver alternating currents of four frequencies, while each bell is constructed to operate at a particular frequency only.

00In mathematics, he was the first to draw up a methodical treatment of mechanics with the aid of geometry; he first distinguished harmonic progression from arithmetical and geometrical progressions.

00Apart from the gain in tragic force resulting from Wagner's masterly development of the character of Brangaene, the raw material of the story was already suggestive of that astounding combination of the contrasted themes of love and death, the musical execution of which involves a harmonic range almost as far beyond that of its own day as the ordinary harmonic range of the 19th century is beyond that of the 16th.

00Its harmonic style is, except in the Grail music, even more abstruse than in Tristan; and the intense quiet of the action is far removed from the forces which in that tumultuous tragedy carry the listener through every difficulty.

00But elsewhere there are few passages in which the extremely recondite harmonic style can be with certainty traced to anything but habit.

00In Wagner's harmonic style we encounter the entire problem of modern musical texture.

00The last two examples at the end of the article on Harmony show almost all that is new in Wagner's harmonic principles.

00We have seen (in the articles on Harmony and Music) how harmonic music originated in just this habit of regarding combinations of sound as mere sensations, and how for centuries the habit opposed itself to the intellectual principles of contrapuntal harmony.

00The only illogical point in his system is that the beauty of his dreamlike chords depends not only on his artful choice of a timbre that minimizes their harshness, but also on the fact that they enter the ear with the meaning they have acquired through centuries of harmonic evolution on classical lines.

00Haydn uses a true Straussian discord in The Seasons, in order to imitate the chirping of a cricket; but the harshest realism in Gatterdammerung (the discord produced by the horns of Hagen and his churls in the mustering-scene in the second act) has a harmonic logic which would have convinced Corelli.

00The brilliant success of Humperdinck's Hansel and Gretel, in which Wagnerian technique is applied to the diatonic style of nursery songs with a humorous accuracy undreamed of by Wagner's imitators, points a moral which would have charmed Wagner himself; but until the revival of some rudiments of musical common sense becomes widespread, there is little prospect of the influence of Wagner's harmonic style being productive of anything better than nonsense.

00Under the general heading "Analysis" occur the subheadings "Foundations of Analysis," with the topics theory of functions of real variables, series and other infinite processes, principles and elements of the differential and of the integral calculus, definite integrals, and calculus of variations; "Theory of Functions of Complex Variables," with the topics functions of one variable and of several variables; "Algebraic Functions and their Integrals," with the topics algebraic functions of one and of several variables, elliptic functions and single theta functions, Abelian integrals; "Other Special Functions," with the topics Euler's, Legendre's, Bessel's and automorphic functions; "Differential Equations," with the topics existence theorems, methods of solution, general theory; "Differential Forms and Differential Invariants," with the topics differential forms, including Pfaffians, transformation of differential forms, including tangential (or contact) transformations, differential invariants; "Analytical Methods connected with Physical Subjects," with the topics harmonic analysis, Fourier's series, the differential equations of applied mathematics, Dirichlet's problem; "Difference Equations and Functional Equations," with the topics recurring series, solution of equations of finite differences and functional equations.

00For the subjects of this general heading see the articles Mechanics; Dynamics, Analytical; Gyroscope; Harmonic Analysis; Wave; HYDROMechanics; Elasticity; Motion, Laws Of; Energy; Energetics; Astronomy (Celestial Mechanics); Tide.

00The linear transformation replaces points on lines through the origin by corresponding points on projectively corresponding lines through the origin; it therefore replaces a pencil of lines by another pencil, which corresponds projectively, and harmonic and other properties of pencils which are unaltered by linear transformation we may expect to find indicated in the invariant system.

00There is no linear covariant, since it is impossible to form a symbolic product which will contain x once and at the same time appertain to a quadratic. (v.) is the Jacobian; geometrically it denotes the bisectors of the angles between the lines ax, or, as we may say, the common harmonic conjugates of the lines and the lines x x .

000= {(n +I)Ar" - where P. denotes the zonal harmonic of the nth order; also, in the exceptional case of =Ao cos 0, 4) = Ao/r; 4'= Bor, 49 = - Bo log tan 2B sh - lx/y.

00This work included the "Logometria," the trigonometrical theorem known as "Cotes' Theorem on the Circle" (see TRIGONOMETRY), his theorem on harmonic means, subsequently developed by Colin Maclaurin, and a discussion of the curves known as "Cotes' Spirals," which occur as the path of a particle described under the influence of a central force varying inversely as the cube of the distance.

00The simplest form of wave, so far as our sensation goes - that is, the one giving rise to a pure tone - is, we have every reason to suppose, one in which the displacement is represented by a harmonic curve or a curve of sines, y=a sin m(x - e).

00The chief experimental basis for supposing that a train of longitudinal waves with displacement curve of this kind arouses the sensation of a pure tone is that the more nearly a source is made to vibrate with a single simple harmonic motion, and therefore, presumably, the more nearly it sends out such a harmonic train, the more nearly does the note heard approximate to a single pure tone.

00Any periodic curve may be resolved into sine or harmonic curves by Fourier's theorem.

00It follows from this that any periodic disturbance in air can be resolved into a definite series of simple harmonic disturbances of wave-lengths equal to the original wave-length and its successive submultiples, and each of these would separately give the sensation of a pure tone.

00Now we can see that two notes of the same pitch, but of different quality, or different form of displacement curve, will, when thus analysed, break up into a series having the same harmonic wave-lengths; but they may differ as regards the members of the series present and their amplitudes and epochs.

00We may regard quality, then, as determined by the members of the harmonic series present and their amplitudes and epochs.

00For the superposition of these trains will give a stationary wave between A H A (16) Y which is an equation characteristic of simple harmonic motion.

00But where it is appropriate, the disturbance sent out into the air contains the same harmonic series as the source.

00But the harmonics are most readily heard if we fortify the ear by an air cavity with a natural period equal to that of the harmonic to be sought.

00If the harmonic corresponding to the resonator is present its tone swells out loudly.

00Suppose that a mass M is controlled by some sort of spring, so that moving freely it executes harmonic vibrations given by -µx, where µx is the restoring force to the centre of vibration.

00Now suppose that in addition to the internal force represented by, ux, an external harmonic force of period 27r/p is applied.

00Each 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.

00When the harmonic is sounded the riders at the loops are thrown off, while those at the nodes remain seated.

0036, for the fundamental and the first harmonic. When a string is struck or bowed at a point, any harmonic with a node at that point is absent.

00By reducing the tension to one quarter of its previous amount, the number of ventral segments will be seen to be increased to two, indicating that the first harmonic of the thread is now in unison with the solid, and consequently that its fundamental is an octave lower than it was with the former tension; thus confirming the law that n varies as S IT.

00The first overtone has frequency 6.25 that of the fundamental, and is not in the harmonic series.

00If the wire is stretched across a room and stroked in the middle with a damp cloth the fundamental is easily obtained, and the first harmonic can be brought out by stroking it at a quarter the length from one end.

00If it is clamped at one-quarter and threequarters of the length from the ends, and is stroked in the middle, the first harmonic sounds.

00If we measure the time from an instant at which the two are in the same phase the resultant disturbance is y=a sin i t+a sin 27rn2t =2a cos ir(n i - n 2)t sin ir(nl-t-n2)t, which may be regarded as a harmonic disturbance of frequency (ni+n2)/2 but with amplitude 2a cos 7r(n i - n 2)t slowly varying with the time.

00Let the two tones with their harmonic overtones be 256 512 768 1024 1280 1536 512 1024 1536.

00Take as a further example the fifth with harmonic overtones as under The fundamental and overtones of the second either coincide with or fall midway between overtones in the first, and there is no approach to a dissonant frequency of beats, and the concord is perfect.

00Mehler, who proved that a simple relation existed between the function of zero order and the zonal harmonic of order n.

00Also W = (V +IA)w i; or w1=W/(V+/A), w p =W/(V+plA), and wn =W/(Vd-nIA), or the densities of the several liquids vary inversely as the respective volumes of the instrument immersed in them; and, since the divisions of the scale correspond to equal increments of volume immersed, it follows that the densities of the several liquids in which the instrument sinks to the successive divisions form a harmonic series.

00A spectroscope may be compared to a mechanical harmonic analyser which when fed with an irregular function of one variable represented by a curve supplies us with the sine curves into which the original function may be resolved.

00In other spectra such " harmonic " ratios were also discovered, but their search was abandoned when it was found that their number did not exceed that calculated by the laws of probability on the supposition of a chance distribution.

00(2) On the arithmetic, geometric and harmonic means between two straight lines, and the problem of representing all three in one and the same geometrical figure.

00The three commonest means are the arithmetical, geometrical, and harmonic; of less importance are the contraharmonical, arithmetico-geometrical, and quadratic.

00The harmonic mean of n quantities is the arithmetical mean of their reciprocals.

00in the proportion a: b ::b :c or b' = ac; and the harmonic mean places the quantities in harmonic proportion, i.e.

00Among the remaining buildings must be mentioned the town hall (17th century; restored 1823), the court-house, the concert-hall of the "Harmonic" club, the record office (1900), the leeskabuiet, or subscription library and reading-rooms, and the ten-storeyed Witte Huis (1897), which is used for offices and is one of the highest private buildings on the Continent.

00A similar method has frequently been applied to the study of variations of soil-temperatures by harmonic analysis of the annual waves.

00The steam pressure in the heater may be periodically varied by the gauge in such a manner as to produce an approximately simple harmonic oscillation of temperature at the hot end, while the cool end is kept at a steady temperature.

00Some sections of the latter, especially those on the connexion between music and architecture, the scale of harmonic proportions, and the Greek use of bronze vases to reverberate and strengthen the actors' voices in the theatre, are now almost wholly unintelligible.

00This may be described as a simple harmonic oscillation whose amplitude diminishes asymptotically to zero according to the law ehlr.

00The path is therefore an ellipse of which a, b are conjugate semi-diameters, and is described in the period 24 Ju; moreover, the velocity at any point P is equal to ~ OD, where OD is the semi-diameter conjugate to OP. ~,This type of motion;,s called elliptic harmonic. If the co-ordinate axes are the principal axes of the ellipse, the angle ft in (I o) is identical with the excentric angle.

00II the coefficient of x be positive the variations of x are simple harmonic, and x can remain permanently small; the circular orbit is then said to be stable.

00In elliptic harmonic motion the velocity of P is parallel and proportional to the semi-diameter CD which is conjugate to the radius CP; the hodograph is therefore an ellipse similar to the actual orbit.

00In the particular case where the motion is of the kind known as simple harmonic the disturbing force on the frame due to the reciprocation of the weight is equal to the component of the centrifugal force in the line of stroke due to a weight equal to the reciprocated weight supposed concentrated at the crank pin.

00In balancing the mechanism of a steam engine it is often sufficiently accurate to consider the motion of the pistons as simple harmonic, and the effect on the framework of the acceleration of the connecting rod may be approximately allowed for by distributing the weight of the rod between the crank pin and the piston inversely as the centre of gravity of the rod divides the distance between the centre of the cross head pin and the centre of the crank pin.

00The moving parts of the engine are then divided into two complete and independent systems, namely, one system of revolving weights consisting of crank pins, crank arms, &c., attached to and revolving with the crank shaft, and a second system of reciprocating weights consisting of the pistons, cross-heads, &c., supposed to be moving each in its line of stroke with simple harmonic motion.

00The assumption that the pistons of an engine move with simple harmonic motion is increasingly erroneous as the ratio of the length of the crank r, to the length of the con oecting rod 1 increases.

00In details of execution and harmonic combinations they illustrate the precision, logic, lucidity and cheerful spirit of the national genius.

00This is a simple harmonic wave-line, whose mean distance from the axis is a, whose wave-length is lira, and whose amplitude is c. The internal pressure corresponding to this unduloid is as before p = T/a.

0013) are the disks, and if x the distance between the disks is equal to irr half the wave-length of the harmonic curve, the disks will be at the points where the curve is at its mean 1 3.

00Whatever the deformation of the originally straight boundary of the axial section may be, it can be resolved by Fourier's theorem into deformations of the harmonic type.

00Disturbances of the former kind lead to vibrations of harmonic type, whose amplitudes always remain small; but disturbances, whose wave-length exceeds the circumference, result in a greater and greater departure from the cylindrical figure.

00The relative importance of two harmonic disturbances depends upon their initial magnitudes, and upon the rate at which they grow.

00If, as before, the frequency be p7211, and a the radius of the sphere, we have p 2 =n(n-1)(n+2)P a3, (6) n denoting the order of the spherical harmonic by which the deviation from a spherical figure is expressed.

00The cartesian equation, referred to the fixed diameter and the tangent at B as axes may be expressed in the forms x= a6, y=a(I -cos 0) and y-a=a sin (x/afir); the latter form shows that the locus is the harmonic curve.

00The periodic variations can be represented algebraically as the resultant of a series of harmonic motions in the following way: Let L be an angle which is increasing uniformly with the time, and let n be its rate of increase.

00The first book deals with the generation of the three conics; the second with the asymptotes, axes and diameters; the third with various metrical relations between transversals, chords, tangents, asymptotes, &c.; the fourth with the theory of the pole and polar, including the harmonic division of a straight line, and with systems of two conics, which he shows to intersect in not more than four points; he also investigates conics having single and double contact.

00Chandler afterwards made an important addition to the subject by showing that the motion was represented by the superposition of two harmonic terms, the first having a period of about 430 days, the other of one year.

00Thomson's tide gauge, tidal harmonic analyser and tide predicter are famous, and among his work in the interest of navigation must be mentioned his tables for the simplification of Sumner's method for determining the position of a ship at sea.

00A ray of light is reflected from this mirror and from another mirror which is rocked by a small motor driven off the same circuit, so that the ray has two vibratory motions imparted to it at right angles, one a simple harmonic motion and the other a motion imitating the variation of the current or electromotive force under test.

00Her music often combines chiseled rhythmic pulsation with a bittersweet melodic and harmonic language.

00The octave quint which stands on the former harmonic flute slide is the old fifteenth, with five extra pipes at the bottom.

00Only £ 15 View details Harmonic Translation System The next generation in radionics technology.

00An harmonic clarion sounds at the clarion pitch (4 '), but its resonator pipes are those of a trumpet 8 '.

00The RESTRAINT energy represents the effect of the harmonic restraints we have imposed.

00On the palate the wine is velvety, soft, rich, harmonic, with the characteristic hint of citrus rind on the finish.

00The slow movement ventured boldly into dark harmonic realms, the finale scampered out into the sunlight.

00The finale is another substantial movement, full of surprise harmonic sidesteps and enormous energy.

00With the first stop of the harmonic series it will stand in the front of the instrument on an open soundboard.

00The listener deserves time to hear Bach 's harmonic subtleties.

00He had developed the " harmonic telegraph " which could send more than one message at a time over a single telegraph wire.

00Harmonic: There are two minor triads in the harmonic minor scale.

00French composer, whose harmonic innovations helped pave the way for the musical upheavals of the 20th century.

00Pressure is building that could be released through earthquakes or through harmonic tremors, to use volcanology terms.

00Wavelet methods make this possible and good preliminary results have already been achieved using harmonic wavelets (Newland & Butler, 1998).

00Allow yourself to be enveloped by the soft melody as it blends perfectly with the harmonic jazz undertones.

00When strings get clogged, their sound is greatly reduced, and the overall harmonic picture produced when they vibrate is limited.

00The combination of that many strings with the many different bending possibilities introduced by the pedals creates a range of harmonic possibilities that can be overwhelming for the novice player.

00If a bass player can develop this kind of harmonic knowledge the instrument will be a lot more interesting for him or her.

00You will know to play a harmonic if the fret number is surrounded by a less-than and greater-than sign.

00For example, a 12th fret harmonic would be written like this: "<12>."

00Even today, the appearance of the 'mothership' and the eeriness of the repeating harmonic theme they played holds up very well.

00If the harmonic corresponding to the resonator is present its tone swells out loudly.

01Suppose that a mass M is controlled by some sort of spring, so that moving freely it executes harmonic vibrations given by -µx, where µx is the restoring force to the centre of vibration.

01

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