Chapter 18

His life was written by his son, Johann Christian Friedrich Harless (1818).

HARLESS, GOTTLIEB CHRISTOPH ADOLF VON(1806-1879), German divine, was born at Nuremberg on the 21st of November 1806, and was educated at the universities of Erlangen and Halle. He was appointed professor of theology at Erlangen in 1836 and at Leipzig in 1845. He was a strong Lutheran and exercised a powerful influence in that direction as court preacher in Dresden and as president of the Protestant consistory at Munich. His chief works wereTheologische EncyklopÃ¤die und Methodologie(1837) andDie christliche Ethik(1842, Eng. trans. 1868). He died on the 5th of September 1879, having, a few years earlier, written an autobiography under the titleBruchstÃ¼cke aus dem Leben eines sÃ¼ddeutschen Theologen.

HARLINGEN,a seaport in the province of Friesland, Holland, on the Zuider Zee, and the terminus of the railway and canal from Leeuwarden (15Â½ m. E.). It is connected by steam tramway by way of Bolswaard with Sneek. Pop. (1900) 10,448. Harlingen has become the most considerable seaport of Friesland since the construction of the large outer harbour in 1870-1877, and in addition to railway and steamship connexion with Bremen, Amsterdam, and the southern provinces there are regular sailings to Hull and London. Powerful sluices protect the inner harbour from the high tides. The only noteworthy buildings are the town hall (1730-1733), the West church, which consists of a part of the former castle of Harlingen, the Roman Catholic church, the Jewish synagogue and the schools of navigation and of design. The chief trade of Harlingen is the exportation of Frisian produce, namely, butter and cheese, cattle, sheep, fish, potatoes, flax, &c. There is also a considerable import trade in timber, coal, raw cotton, hemp and jute for the Twente factories. The local industries are unimportant, consisting of saw-mills, rope-yards, salt refineries, and sail-cloth and margarine factories.

HARMATTAN,the name of a hot dry parching wind that blows during December, January and February on the coast of Upper Guinea, bringing a high dense haze of red dust which darkens the air. The natives smear their bodies with oil or fat while this parching wind is blowing.

HARMODIUS,a handsome Athenian youth, and the intimate friend of Aristogeiton. Hipparchus, the younger brother of the tyrant Hippias, endeavoured to supplant Aristogeiton in the good graces of Harmodius, but, failing in the attempt, revenged himself by putting a public affront on Harmodiusâ€™s sister at a solemn festival. Thereupon the two friends conspired with a few others to murder both the tyrants during the armed procession at the Panathenaic festival (514B.C.), when the people were allowed to carry arms (this licence is denied by Aristotle inAth. Pol.). Seeing one of their accomplices speaking to Hippias, and imagining that they were being betrayed, they prematurely attacked and slew Hipparchus alone. Harmodius was cut down on the spot by the guards, and Aristogeiton was soon captured and tortured to death. When Hippias was expelled (510), Harmodius and Aristogeiton became the most popular of Athenian heroes; their descendants were exempted from public burdens, and had the right of public entertainment in the Prytaneum, and their names were celebrated in popular songs and scolia (after-dinner songs) as the deliverers of Athens. One of these songs, attributed to a certain Callistratus, is preserved in Athenaeus (p. 695). Their statues by Antenor in the agora were carried off by Xerxes and replaced by new ones by Critius and Nesiotes. Alexander the Great afterwards sent back the originals to Athens. It is not agreed which of these was the original of the marble tyrannicide group in the museum at Naples, for which see articleGreek Art, Pl. I. fig. 50.

See KÃ¶pp inNeue Jahrb. f. klass. Altert.(1902), p. 609.

HARMONIA,in Greek mythology, according to one account the daughter of Ares and Aphrodite, and wife of Cadmus. When the government of Thebes was bestowed upon Cadmus by Athena, Zeus gave him Harmonia to wife. All the gods honoured the wedding with their presence. Cadmus (or one of the gods) presented the bride with a robe and necklace, the work of Hephaestus. This necklace brought misfortune to all who possessed it. With it Polyneices bribed Eriphyle to persuade her husband Amphiaraus to undertake the expedition against Thebes. It led to the death of Eriphyle, of Alcmaeon, of Phegeus and his sons. Even after it had been deposited in the temple of Athena Pronoia at Delphi, its baleful influence continued. Phayllus, one of the Phocian leaders in the Sacred War (352B.C.) carried it off and gave it to his mistress. After she had worn it for a time, her son was seized with madness and set fire to the house, and she perished in the flames. According to another account, Harmonia belonged to Samothrace and was the daughter of Zeus and Electra, her brother Iasion being the founder of the mystic rites celebrated on the island (Diod. Sic. v. 48). Finally, Harmonia is rationalized as closely allied to Aphrodite Pandemos, the love that unites all people, the personification of order and civic unity, corresponding to the Roman Concordia.

Apollodorus iii. 4-7; Diod. Sic. iv. 65, 66; Parthenius,Erotica, 25; L. Preller,Griech. Mythol.; Crusius in Roscherâ€™sLexikon.

HARMONIC.In acoustics, a harmonic is a secondary tone which accompanies the fundamental or primary tone of a vibrating string, reed, &c.; the more important are the 3rd, 5th, 7th, and octave (seeSound;Harmony). A harmonic proportion in arithmetic and algebra is such that the reciprocals of the proportionals are in arithmetical proportion; thus, ifa,b,cbe in harmonic proportion then 1/a, 1/b, 1/care in arithmetical proportion; this leads to the relation 2/b=ac/(a+c). A harmonic progression or series consists of terms whose reciprocals form an arithmetical progression; the simplest example is:1 + Â½ +1â„3+ Â¼ + ... (seeAlgebraandArithmetic). The occurrence of a similar proportion between segments of lines is the foundation of such phrases as harmonic section, harmonic ratio, harmonic conjugates, &c. (seeGeometry: II.Projective). The connexion between acoustical and mathematical harmonicals is most probably to be found in the Pythagorean discovery that a vibrating string when stopped at Â½ and2â„3of its length yielded the octave and 5th of the original tone, the numbers, 12â„3, Â½ being said to be, probably first by Archytas, in harmonic proportion. The mathematical investigation of the form of a vibrating string led to such phrases as harmonic curve, harmonic motion, harmonic function, harmonic analysis, &c. (seeMechanicsandSpherical Harmonics).

HARMONICA,a generic term applied to musical instruments in which sound is produced by friction upon glass bells. The word is also used to designate instruments of percussion of the Glockenspiel type, made of steel and struck by hammers (Ger.Stahlharmonika).

The origin of the glass-harmonica tribe is to be found in the fashionable 18th century instrument known as musical glasses (Fr.verrillon), the principle of which was known already in the 17th century.1The invention of musical glasses is generally ascribed to an Irishman, Richard Pockrich, who first played the instrument in public in Dublin in 1743 and the next year in England, but Eisel2described theverrillonand gave an illustration of it in 1738. TheverrillonorGlassspielconsisted of 18 beer glasses arranged on a board covered with cloth, water being poured in when necessary to alter the pitch. The glasses were struck on both sides gently with two long wooden sticks in the shape of a spoon, the bowl being covered with silk or cloth. Eisel states that the instrument was used for church and other solemn music. Gluck gave a concert at the â€œlittle theatre in the Haymarketâ€ (London) in April 1746, at which he performed on musical glasses a concerto of his composition with full orchestral accompaniment. E. H. Delaval is also credited with the invention. When Benjamin Franklin visited London in 1757, he was so much struck by the beauty of tone elicited by Delaval and Pockrich, and with the possibilities of the glasses as musical instruments, that he set to work on a mechanical application of the principle involved, the eminently successful result being the glass harmonica finished in 1762. In this the glass bowls were mounted on a rotating spindle, the largest to the left, and their under-edges passed during each revolution through a water-trough. By applying the fingers to the moistened edges, sound was produced varying in intensity with the pressure, so that a certain amount of expression was at the command of a good player. It is said that the timbre was extremely enervating, and, together with the vibration caused by the friction on the finger-tips, exercised a highly deleterious effect on the nervous system. The instrument was for many years in great vogue, not only in England but on the Continent of Europe, and more especially in Saxony, where it was accorded a place in the court orchestra. Mozart, Beethoven, Naumann and Hasse composed music for it. Marianne Davies and Marianna Kirchgessner were celebrated virtuosi on it. The curious vogue of the instrument, as sudden as it was ephemeral, produced emulation in a generation unsurpassed for zeal in the invention of musical instruments. The most notable of its offspring were Carl Leopold RÃ¶lligâ€™s improved harmonica with a keyboard in 1786, Chladniâ€™s euphon in 1791 and clavicylinder in 1799, Ruffelsenâ€™s melodicon in 1800 and 1803, Franz Leppichâ€™s panmelodiconin1810, Buschmannâ€™s uranion in the same year, &c. Of most of these nothing now remains but the name and a description in theAllgemeine musikalische Zeitung, but there are numerous specimens of the Franklin type in the museums for musical instruments of Europe. One specimen by Emanuel Pohl, a Bohemian maker, is preserved in the Victoria and Albert Museum, London.

For the steel harmonica seeGlockenspiel.

(K. S.)

1See G. P. HarsdÃ¶rfer,Math. und philos. Erquickstunden(Nuremberg, 1677), ii. 147.2MusicusÎ±á½Ï„Î¿Î´Î¯Î´Î±ÎºÏ„Î¿Ï‚(Erfurt, 1738), p. 70.

1See G. P. HarsdÃ¶rfer,Math. und philos. Erquickstunden(Nuremberg, 1677), ii. 147.

2MusicusÎ±á½Ï„Î¿Î´Î¯Î´Î±ÎºÏ„Î¿Ï‚(Erfurt, 1738), p. 70.

HARMONIC ANALYSIS,in mathematics, the name given by Sir William Thomson (Lord Kelvin) and P. G. Tait in their treatise onNatural Philosophyto a general method of investigating physical questions, the earliest applications of which seem to have been suggested by the study of the vibrations of strings and the analysis of these vibrations into their fundamental tone and its harmonics or overtones.

The motion of a uniform stretched string fixed at both ends is a periodic motion; that is to say, after a certain interval of time, called the fundamental period of the motion, the form of the string and the velocity of every part of it are the same as before, provided that the energy of the motion has not been sensibly dissipated during the period.

There are two distinct methods of investigating the motion of a uniform stretched string. One of these may be called the wave method, and the other the harmonic method. The wave method is founded on the theorem that in a stretched string of infinite length a wave of any form may be propagated in either direction with a certain velocity, V, which we may define as the â€œvelocity of propagation.â€ If a wave of any form travelling in the positive direction meets another travelling in the opposite direction, the form of which is such that the lines joining corresponding points of the two waves are all bisected in a fixed point in the line of the string, then the point of the string corresponding to this point will remain fixed, while the two waves pass it in opposite directions. If we now suppose that the form of the waves travelling in the positive direction is periodic, that is to say, that after the wave has travelled forward a distance l, the position of every particle of the string is the same as it was at first, then l is called the wave-length, and the time of travelling a wave-length is called the periodic time, which we shall denote by T, so that l = VT.If we now suppose a set of waves similar to these, but reversed in position, to be travelling in the opposite direction, there will be a series of points, distant Â½l from each other, at which there will be no motion of the string; it will therefore make no difference to the motion of the string if we suppose the string fastened to fixed supports at any two of these points, and we may then suppose the parts of the string beyond these points to be removed, as it cannot affect the motion of the part which is between them. We have thus arrived at the case of a uniform string stretched between two fixed supports, and we conclude that the motion of the string may be completely represented as the resultant of two sets of periodic waves travelling in opposite directions, their wave-lengths being either twice the distance between the fixed points or a submultiple of this wave-length, and the form of these waves, subject to this condition, being perfectly arbitrary.To make the problem a definite one, we may suppose the initial displacement and velocity of every particle of the string given in terms of its distance from one end of the string, and from these data it is easy to calculate the form which is common to all the travelling waves. The form of the string at any subsequent time may then be deduced by calculating the positions of the two sets of waves at that time, and compounding their displacements.Thus in the wave method the actual motion of the string is considered as the resultant of two wave motions, neither of which is of itself, and without the other, consistent with the condition that the ends of the string are fixed. Each of the wave motions is periodic with a wave-length equal to twice the distance between the fixed points, and the one set of waves is the reverse of the other in respect of displacement and velocity and direction of propagation; but, subject to these conditions, the form of the wave is perfectly arbitrary. The motion of a particle of the string, being determined by the two waves which pass over it in opposite directions, is of an equally arbitrary type.In the harmonic method, on the other hand, the motion of the string is regarded as compounded of a series of vibratory motions (normal modesof vibration), which may be infinite in number, but each of which is perfectly definite in type, and is in fact a particular solution of the problem of the motion of a string with its ends fixed.A simple harmonic motion is thus defined by Thomson and Tait (Â§ 53):â€”When a point Q moves uniformly in a circle, the perpendicular QP, drawn from its position at any instant to a fixed diameter AAâ€² of the circle, intersects the diameter in a point P whose position changes by asimple harmonic motion.The amplitude of a simple harmonic motion is the range on one side or the other of the middle point of the course.The period of a simple harmonic motion is the time which elapses from any instant until the moving-point again moves in the same direction through the same position.The phase of a simple harmonic motion at any instant is the fraction of the whole period which has elapsed since the moving-point last passed through its middle position in the positive direction.In the case of the stretched string, it is only in certain particular cases that the motion of a particle of the string is a simple harmonic motion. In these particular cases the form of the string at any instant is that of a curve of sines having the line joining the fixedpoints for its axis, and passing through these two points, and therefore having for its wave-length either twice the length of the string or some submultiple of this wave-length. The amplitude of the curve of sines is a simple harmonic function of the time, the period being either the fundamental period or some submultiple of the fundamental period. Every one of these modes of vibration is dynamically possible by itself, and any number of them may coexist independently of each other.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.We have thus two methods of solving the partial differential equation of the motion of a string. The first, which we have called the wave method, exhibits the solution in the form containing an arbitrary function, the nature of which must be determined from the initial conditions. The second, or harmonic method, leads to a series of terms involving sines and cosines, the coefficients of which have to be determined. The harmonic method may be defined in a more general manner as a method by which the solution of any actual problem may be obtained as the sum or resultant of a number of terms, each of which is a solution of a particular case of the problem. The nature of these particular cases is defined by the condition that any one of them must be conjugate to any other.The mathematical test of conjugacy is that the energy of the system arising from two of the harmonics existing together is equal to the sum of the energy arising from the two harmonics taken separately. 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 Fourierâ€™s theorem.Fourierâ€™s theorem asserts that any periodic function of a single variable period p, which does not become infinite at any phase, can be expanded in the form of a series consisting of a constant term, together with a double series of terms, one set involving cosines and the other sines of multiples of the phase.Thus if Ï†(Î¾) is a periodic function of the variable Î¾ having a period p, then it may be expanded as follows:Ï†(Î¾) = A0+ Î£âˆž1iAicos2iÏ€Î¾+ Î£âˆž1iBisin2iÏ€Î¾.pp(1)The part of the theorem which is most frequently required, and which also is the easiest to investigate, is the determination of the values of the coefficients A0, Ai, Bi. These areA0=1âˆ«p0Ï†(Î¾)dÎ¾; â€ƒ Ai=2âˆ«p0Ï†(Î¾) cos2iÏ€Î¾dÎ¾; â€ƒ Bi=2âˆ«p0Ï†(Î¾) sin2iÏ€Î¾dÎ¾.pppppThis part of the theorem may be verified at once by multiplying both sides of (1) by dÎ¾, by cos (2iÏ€Î¾/p)/dÎ¾ or by sin (2iÏ€Î¾/p)/dÎ¾, and in each case integrating from 0 to p.The series is evidently single-valued for any given value of Î¾. It cannot therefore represent a function of Î¾ which has more than one value, or which becomes imaginary for any value of Î¾. It is convergent, approaching to the true value of Ï†(Î¾) for all values of Î¾ such that if Î¾ varies infinitesimally the function also varies infinitesimally.Lord Kelvin, availing himself of the disk, globe and cylinder integrating machine invented by his brother, Professor James Thomson, constructed a machine by which eight of the integrals required for the expression of Fourierâ€™s series can be obtained simultaneously from the recorded trace of any periodically variable quantity, such as the height of the tide, the temperature or pressure of the atmosphere, or the intensity of the different components of terrestrial magnetism. If it were not on account of the waste of time, instead of having a curve drawn by the action of the tide, and the curve afterwards acted on by the machine, the time axis of the machine itself might be driven by a clock, and the tide itself might work the second variable of the machine, but this would involve the constant presence of an expensive machine at every tidal station.(J. C. M.)For a discussion of the restrictions under which the expansion of a periodic function of Î¾ in the form (1) is valid, seeFourierâ€™s Series. An account of the contrivances for mechanical calculation of the coefficients Ai, Bi... is given underCalculating Machines.A more general form of the problem of harmonic analysis presents itself in astronomy, in the theory of the tides, and in various magnetic and meteorological investigations. It may happen, for instance, that a variable quantity Æ’(t) is known theoretically to be of the formÆ’(t) = A0+ A1cos n1t + B1sin n1t + A2cos n2t + B2sin n2t + ...(2)where the periods 2Ï€/n1, 2Ï€/n2, ... of the various simple-harmonic constituents are already known with sufficient accuracy, although they may have no very simple relations to one another. The problem of determining the most probable values of the constants A0, A1, B1, A2, B2, ... by means of a series of recorded values of the function Æ’(t) is then in principle a fairly simple one, although the actual numerical work may be laborious (seeTide). A much more difficult and delicate question arises when, as in various questions of meteorology and terrestrial magnetism, the periods 2Ï€/n1, 2Ï€/n2, ... are themselves unknown to begin with, or are at most conjectural. Thus, it may be desired to ascertain whether the magnetic declination contains a periodic element synchronous with the sunâ€™s rotation on its axis, whether any periodicities can be detected in the records of the prevalence of sun-spots, and so on. From a strictly mathematical standpoint the problem is, indeed, indeterminate, for when all the symbols are at our disposal, the representation of the observed values of a function, over a finite range of time, by means of a series of the type (2), can be effected in an infinite variety of ways. Plausible inferences can, however, be drawn, provided the proper precautions are observed. This question has been treated most systematically by Professor A. Schuster, who has devised a remarkable mathematical method, in which the action of a diffraction-grating in sorting out the various periodic constituents of a heterogeneous beam of light is closely imitated. He has further applied the method to the study of the variations of the magnetic declination, and of sun-spot records.The question so far chiefly considered has been that of the representation of an arbitrary function of thetimein terms of functions of a special type, viz. the circular functions cos nt, sin nt. This is important on dynamical grounds; but when we proceed to consider the problem of expressing an arbitrary function ofspace-co-ordinatesin terms of functions of specified types, it appears that the preceding is only one out of an infinite variety of modes of representation which are equally entitled to consideration. Every problem of mathematical physics which leads to a linear differential equation supplies an instance. For purposes of illustration we will here take the simplest of all, viz. that of the transversal vibrations of a tense string. The equation of motion is of the formÏâˆ‚Â²y= Tâˆ‚Â²y,âˆ‚tÂ²âˆ‚xÂ²(3)where T is the tension, and Ï the line-density. In a â€œnormal modeâ€ of vibration y will vary as eint, so thatâˆ‚Â²y+ kÂ²y = 0,âˆ‚xÂ²(4)wherekÂ² = nÂ²Ï/T.(5)If Ï, and therefore k, is constant, the solution of (4) subject to the condition that y = 0 for x = 0 and x = l isy = B sin kx(6)providedkl = sÏ€, [s = 1, 2, 3, ...].(7)This determines the variousnormal modesof free vibration, the corresponding periods (2Ï€/n) being given by (5) and (7). By analogy with the theory of the free vibrations of a system offinitefreedom it is inferred that the most general free motions of the string can be obtained by superposition of the various normal modes, with suitable amplitudes and phases; and in particular that any arbitrary initial form of the string, say y = Æ’(x), can be reproduced by a series of the typeÆ’(x) = B1sinÏ€x+ B2sin2Ï€x+ B3sin3Ï€x+ ...lll(8)So far, this is merely a restatement, in mathematical language, of an argument given in the first part of this article. The series (8) may, moreover, be arrived at otherwise, as a particular case of Fourierâ€™s theorem. But if we no longer assume the density Ï of the string to be uniform, we obtain an endless variety of new expansions, corresponding to the various laws of density which may be prescribed. The normal modes are in any case of the typey = Cu(x)eint(9)where u is a solution of the equationdÂ²u+nÂ²Ïu = 0.dxÂ²T(10)The condition that u(x) is to vanish for x = 0 and x = l leads to a transcendental equation in n (corresponding to sin kl = 0 in the previous case). If the forms of u(x) which correspond to the various roots of this be distinguished by suffixes, we infer, on physical grounds alone, the possibility of the expansion of an arbitrary initial form of the string in a seriesÆ’(x) = C1u1(x) + C2u2(x) + C3u3(x) + ...(11)It may be shown further that if r and s are different we have theconjugateororthogonalrelationâˆ«l0Ïur(x) us(x) dx = 0.(12)This enables us to determine the coefficients, thusCr=âˆ«l0ÏÆ’(x) ur(x)dx Ã·âˆ«10Ï {ur(x)}Â² dx.(13)The extension to spaces of two or three dimensions, or to cases where there is more than one dependent variable, must be passed over. The mathematical theories of acoustics, heat-conduction, elasticity, induction of electric currents, and so on, furnish an indefinite supply of examples, and have suggested in some cases methods which have a very wide application. Thus the transverse vibrations of a circular membrane lead to the theory of Besselâ€™s Functions; the oscillations of a spherical sheet of air suggest the theory of expansions in spherical harmonics, and so forth. The physical, or intuitional, theory of such methods has naturally always been in advance of the mathematical. From the latter point of view only a few isolated questions of the kind had, until quite recently, been treated in a rigorous and satisfactory manner. A more general and comprehensive method, which seems to derive some of its inspiration from physical considerations, has, however, at length been inaugurated, and has been vigorously cultivated in recent years by D. Hilbert, H. PoincarÃ©, I. Fredholm, E. Picard and others.References.â€”Schusterâ€™s method for detecting hidden periodicities is explained inTerrestrial Magnetism(Chicago, 1898), 3, p. 13;Camb. Trans.(1900), 18, p. 107;Proc. Roy. Soc.(1906), 77, p. 136. The general question of expanding an arbitrary function in a series of functions of special types is treated most fully from the physical point of view in Lord Rayleighâ€™sTheory of Sound(2nd ed., London, 1894-1896). An excellent detailed historical account of the matter from the mathematical side is given by H. Burkhardt,Entwicklungen nach oscillierenden Funktionen(Leipzig, 1901). A sketch of the more recent mathematical developments is given by H. Bateman,Proc. Lond. Math. Soc.(2), 4, p. 90, with copious references.

If we now suppose a set of waves similar to these, but reversed in position, to be travelling in the opposite direction, there will be a series of points, distant Â½l from each other, at which there will be no motion of the string; it will therefore make no difference to the motion of the string if we suppose the string fastened to fixed supports at any two of these points, and we may then suppose the parts of the string beyond these points to be removed, as it cannot affect the motion of the part which is between them. We have thus arrived at the case of a uniform string stretched between two fixed supports, and we conclude that the motion of the string may be completely represented as the resultant of two sets of periodic waves travelling in opposite directions, their wave-lengths being either twice the distance between the fixed points or a submultiple of this wave-length, and the form of these waves, subject to this condition, being perfectly arbitrary.

To make the problem a definite one, we may suppose the initial displacement and velocity of every particle of the string given in terms of its distance from one end of the string, and from these data it is easy to calculate the form which is common to all the travelling waves. The form of the string at any subsequent time may then be deduced by calculating the positions of the two sets of waves at that time, and compounding their displacements.

Thus in the wave method the actual motion of the string is considered as the resultant of two wave motions, neither of which is of itself, and without the other, consistent with the condition that the ends of the string are fixed. Each of the wave motions is periodic with a wave-length equal to twice the distance between the fixed points, and the one set of waves is the reverse of the other in respect of displacement and velocity and direction of propagation; but, subject to these conditions, the form of the wave is perfectly arbitrary. The motion of a particle of the string, being determined by the two waves which pass over it in opposite directions, is of an equally arbitrary type.

In the harmonic method, on the other hand, the motion of the string is regarded as compounded of a series of vibratory motions (normal modesof vibration), which may be infinite in number, but each of which is perfectly definite in type, and is in fact a particular solution of the problem of the motion of a string with its ends fixed.

A simple harmonic motion is thus defined by Thomson and Tait (Â§ 53):â€”When a point Q moves uniformly in a circle, the perpendicular QP, drawn from its position at any instant to a fixed diameter AAâ€² of the circle, intersects the diameter in a point P whose position changes by asimple harmonic motion.

The amplitude of a simple harmonic motion is the range on one side or the other of the middle point of the course.

The period of a simple harmonic motion is the time which elapses from any instant until the moving-point again moves in the same direction through the same position.

The phase of a simple harmonic motion at any instant is the fraction of the whole period which has elapsed since the moving-point last passed through its middle position in the positive direction.

In the case of the stretched string, it is only in certain particular cases that the motion of a particle of the string is a simple harmonic motion. In these particular cases the form of the string at any instant is that of a curve of sines having the line joining the fixedpoints for its axis, and passing through these two points, and therefore having for its wave-length either twice the length of the string or some submultiple of this wave-length. The amplitude of the curve of sines is a simple harmonic function of the time, the period being either the fundamental period or some submultiple of the fundamental period. Every one of these modes of vibration is dynamically possible by itself, and any number of them may coexist independently of each other.

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.

We have thus two methods of solving the partial differential equation of the motion of a string. The first, which we have called the wave method, exhibits the solution in the form containing an arbitrary function, the nature of which must be determined from the initial conditions. The second, or harmonic method, leads to a series of terms involving sines and cosines, the coefficients of which have to be determined. The harmonic method may be defined in a more general manner as a method by which the solution of any actual problem may be obtained as the sum or resultant of a number of terms, each of which is a solution of a particular case of the problem. The nature of these particular cases is defined by the condition that any one of them must be conjugate to any other.

The mathematical test of conjugacy is that the energy of the system arising from two of the harmonics existing together is equal to the sum of the energy arising from the two harmonics taken separately. 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 Fourierâ€™s theorem.

Fourierâ€™s theorem asserts that any periodic function of a single variable period p, which does not become infinite at any phase, can be expanded in the form of a series consisting of a constant term, together with a double series of terms, one set involving cosines and the other sines of multiples of the phase.

Thus if Ï†(Î¾) is a periodic function of the variable Î¾ having a period p, then it may be expanded as follows:

(1)

The part of the theorem which is most frequently required, and which also is the easiest to investigate, is the determination of the values of the coefficients A0, Ai, Bi. These are

This part of the theorem may be verified at once by multiplying both sides of (1) by dÎ¾, by cos (2iÏ€Î¾/p)/dÎ¾ or by sin (2iÏ€Î¾/p)/dÎ¾, and in each case integrating from 0 to p.

The series is evidently single-valued for any given value of Î¾. It cannot therefore represent a function of Î¾ which has more than one value, or which becomes imaginary for any value of Î¾. It is convergent, approaching to the true value of Ï†(Î¾) for all values of Î¾ such that if Î¾ varies infinitesimally the function also varies infinitesimally.

Lord Kelvin, availing himself of the disk, globe and cylinder integrating machine invented by his brother, Professor James Thomson, constructed a machine by which eight of the integrals required for the expression of Fourierâ€™s series can be obtained simultaneously from the recorded trace of any periodically variable quantity, such as the height of the tide, the temperature or pressure of the atmosphere, or the intensity of the different components of terrestrial magnetism. If it were not on account of the waste of time, instead of having a curve drawn by the action of the tide, and the curve afterwards acted on by the machine, the time axis of the machine itself might be driven by a clock, and the tide itself might work the second variable of the machine, but this would involve the constant presence of an expensive machine at every tidal station.

(J. C. M.)

For a discussion of the restrictions under which the expansion of a periodic function of Î¾ in the form (1) is valid, seeFourierâ€™s Series. An account of the contrivances for mechanical calculation of the coefficients Ai, Bi... is given underCalculating Machines.

A more general form of the problem of harmonic analysis presents itself in astronomy, in the theory of the tides, and in various magnetic and meteorological investigations. It may happen, for instance, that a variable quantity Æ’(t) is known theoretically to be of the form

Æ’(t) = A0+ A1cos n1t + B1sin n1t + A2cos n2t + B2sin n2t + ...

(2)

where the periods 2Ï€/n1, 2Ï€/n2, ... of the various simple-harmonic constituents are already known with sufficient accuracy, although they may have no very simple relations to one another. The problem of determining the most probable values of the constants A0, A1, B1, A2, B2, ... by means of a series of recorded values of the function Æ’(t) is then in principle a fairly simple one, although the actual numerical work may be laborious (seeTide). A much more difficult and delicate question arises when, as in various questions of meteorology and terrestrial magnetism, the periods 2Ï€/n1, 2Ï€/n2, ... are themselves unknown to begin with, or are at most conjectural. Thus, it may be desired to ascertain whether the magnetic declination contains a periodic element synchronous with the sunâ€™s rotation on its axis, whether any periodicities can be detected in the records of the prevalence of sun-spots, and so on. From a strictly mathematical standpoint the problem is, indeed, indeterminate, for when all the symbols are at our disposal, the representation of the observed values of a function, over a finite range of time, by means of a series of the type (2), can be effected in an infinite variety of ways. Plausible inferences can, however, be drawn, provided the proper precautions are observed. This question has been treated most systematically by Professor A. Schuster, who has devised a remarkable mathematical method, in which the action of a diffraction-grating in sorting out the various periodic constituents of a heterogeneous beam of light is closely imitated. He has further applied the method to the study of the variations of the magnetic declination, and of sun-spot records.

The question so far chiefly considered has been that of the representation of an arbitrary function of thetimein terms of functions of a special type, viz. the circular functions cos nt, sin nt. This is important on dynamical grounds; but when we proceed to consider the problem of expressing an arbitrary function ofspace-co-ordinatesin terms of functions of specified types, it appears that the preceding is only one out of an infinite variety of modes of representation which are equally entitled to consideration. Every problem of mathematical physics which leads to a linear differential equation supplies an instance. For purposes of illustration we will here take the simplest of all, viz. that of the transversal vibrations of a tense string. The equation of motion is of the form

(3)

where T is the tension, and Ï the line-density. In a â€œnormal modeâ€ of vibration y will vary as eint, so that

(4)

where

kÂ² = nÂ²Ï/T.

(5)

If Ï, and therefore k, is constant, the solution of (4) subject to the condition that y = 0 for x = 0 and x = l is

y = B sin kx

(6)

provided

kl = sÏ€, [s = 1, 2, 3, ...].

(7)

This determines the variousnormal modesof free vibration, the corresponding periods (2Ï€/n) being given by (5) and (7). By analogy with the theory of the free vibrations of a system offinitefreedom it is inferred that the most general free motions of the string can be obtained by superposition of the various normal modes, with suitable amplitudes and phases; and in particular that any arbitrary initial form of the string, say y = Æ’(x), can be reproduced by a series of the type

(8)

So far, this is merely a restatement, in mathematical language, of an argument given in the first part of this article. The series (8) may, moreover, be arrived at otherwise, as a particular case of Fourierâ€™s theorem. But if we no longer assume the density Ï of the string to be uniform, we obtain an endless variety of new expansions, corresponding to the various laws of density which may be prescribed. The normal modes are in any case of the type

y = Cu(x)eint

(9)

where u is a solution of the equation

(10)

The condition that u(x) is to vanish for x = 0 and x = l leads to a transcendental equation in n (corresponding to sin kl = 0 in the previous case). If the forms of u(x) which correspond to the various roots of this be distinguished by suffixes, we infer, on physical grounds alone, the possibility of the expansion of an arbitrary initial form of the string in a series

Æ’(x) = C1u1(x) + C2u2(x) + C3u3(x) + ...

(11)

It may be shown further that if r and s are different we have theconjugateororthogonalrelation

âˆ«l0Ïur(x) us(x) dx = 0.

(12)

This enables us to determine the coefficients, thus

Cr=âˆ«l0ÏÆ’(x) ur(x)dx Ã·âˆ«10Ï {ur(x)}Â² dx.

(13)

The extension to spaces of two or three dimensions, or to cases where there is more than one dependent variable, must be passed over. The mathematical theories of acoustics, heat-conduction, elasticity, induction of electric currents, and so on, furnish an indefinite supply of examples, and have suggested in some cases methods which have a very wide application. Thus the transverse vibrations of a circular membrane lead to the theory of Besselâ€™s Functions; the oscillations of a spherical sheet of air suggest the theory of expansions in spherical harmonics, and so forth. The physical, or intuitional, theory of such methods has naturally always been in advance of the mathematical. From the latter point of view only a few isolated questions of the kind had, until quite recently, been treated in a rigorous and satisfactory manner. A more general and comprehensive method, which seems to derive some of its inspiration from physical considerations, has, however, at length been inaugurated, and has been vigorously cultivated in recent years by D. Hilbert, H. PoincarÃ©, I. Fredholm, E. Picard and others.

References.â€”Schusterâ€™s method for detecting hidden periodicities is explained inTerrestrial Magnetism(Chicago, 1898), 3, p. 13;Camb. Trans.(1900), 18, p. 107;Proc. Roy. Soc.(1906), 77, p. 136. The general question of expanding an arbitrary function in a series of functions of special types is treated most fully from the physical point of view in Lord Rayleighâ€™sTheory of Sound(2nd ed., London, 1894-1896). An excellent detailed historical account of the matter from the mathematical side is given by H. Burkhardt,Entwicklungen nach oscillierenden Funktionen(Leipzig, 1901). A sketch of the more recent mathematical developments is given by H. Bateman,Proc. Lond. Math. Soc.(2), 4, p. 90, with copious references.

(H. Lb.)

HARMONICHORD,an ingenious kind of upright piano, in which the strings were set in vibration not by the blow of the hammer but by indirectly transmitted friction. The harmonichord, one of the many attempts to fuse piano and violin, was invented by Johann Gottfried and Johann Friedrich Kaufmann (father and son) in Saxony at the beginning of the 19th century, when the craze for new and ingenious musical instruments was at its height. The case was of the variety known asgiraffe. The space under the keyboard was enclosed, a knee-hold being left in which were two pedals used to set in rotation a large wooden cylinder fixed just behind the keyboard over the levers, and covered with a roll-top similar to those of modern office desks. The cylinder (in some specimens covered with chamois leather) tapered towards the treble-end. When a key was depressed, a little tongue of wood, one end of which stopped the string, was pressed against the revolving cylinder, and the vibrations produced by friction were transmitted to the string and reinforced as in piano and violin by the soundboard. The adjustment of the parts and the velocity of the cylinder required delicacy and great nicety, for if the little wooden tongues rested too lightly upon the cylinder or the strings, harmonics were produced, and the note jumped to the octave or twelfth. Sometimes when chords were played the touch became so heavy that two performers were required, as in the early medieval organistrum, the prototype of the harmonichord. Carl Maria von Weber must have had some opinion of the possibilities of the harmonichord, which in tone resembled the glass harmonica, since he composed for it a concerto with orchestral accompaniment.

(K. S.)

HARMONIUM(Fr.harmonium,orgue expressif; Ger.Physharmonika,Harmonium), a wind keyboard instrument, a small organ without pipes, furnished with free reeds. Both the harmonium and its later development, the American organ, are known as free-reed instruments, the musical tones being produced by tongues of brass, technically termed â€œvibratorsâ€ (Fr.anche libre; Ger.durchschlagende Zunge; Ital.anciaorlingua libera). The vibrator is fixed over an oblong, rectangular frame, through which it swings freely backwards and forwards like a pendulum while vibrating, whereas the beating reeds (similar to those of the clarinet family), used in church organs, cover the entire orifice, beating against the sides at each vibration. A reed or vibrator, set in periodic motion by impact of a current of air, produces a corresponding succession of air puffs, the rapidity of which determines the pitch of the musical note. There is an essential difference between the harmonium and the American organ in the direction of this current; in the former the wind apparatus forces the current upwards, and in the latter sucks it downwards, whence it becomes desirable to separate in description these varieties of free-reed instruments.

Theharmoniumhas a keyboard of five octaves compass when complete,and a simple action controlling the valves, &c. The necessary pressure of wind is generated by bellows worked by the feet of the performer upon foot-boards or treadles. The air is thus forced up the wind-trunks into an air-chamber called the wind-chest, the pressure of it being equalized by a reservoir, which receives the excess of wind through an aperture, and permits escape, when above a certain pressure, by a discharge valve or pallet. The aperture admitting air to the reservoir may be closed by a drawstop named â€œexpression.â€ The air being thus cut off, the performer depends for his supply entirely upon the management of the bellows worked by the treadles, whereby he regulates the compression of the wind. The character of the instrument is then entirely changed from a mechanical response to the playerâ€™s touch to an expressive one, rendering what emotion may be communicated from the player by increase or diminution of sound through the greater or less pressure of wind to which the reeds may be submitted. The drawstops bearing the names of the different registers in imitation of the organ, admit, when drawn, the wind from the wind-chest to the corresponding reed compartments, shutting them off when closed. These compartments are of about two octaves and a half each, there being a division in the middle of the keyboard scale dividing the stops into bass and treble. A stop being drawn and a key pressed down, wind is admitted by a corresponding valve to a reed or vibrator (fig. 1). Above each reed in the so-called sound-board or pan is a channel, a small air-chamber or cavity, the shape and capacity of which have greatly to do with the colour of tone of the note it reinforces. The air in this resonator is highly compressed at an even or a varying pressure as the expression-stop may not be or may be drawn. The wind finally escapes by a small pallet-hole opened by pressing down the corresponding key. In Mustel and other good harmoniums, the reed compartments that form the scheme of the instrument are eight in number, four bass and four treble, of three different pitches of octave and double octave distance. The front bass and treble rows are the â€œdiapasonâ€ of the pitch known as 8 ft., and the bourdon (double diapason), 16 ft. These may be regarded as the foundation stops, and are technically the front organ. The back organ has solo and combination stops, the principal of 4 ft. (octave higher than diapason), and bassoon (bass) and oboe (treble), 8 ft. These may be mechanically combined by a stop called full organ. The French maker, Mustel, added other registers for much-admired effects of tone, viz. â€œharpe Ã©olienne,â€ two bass rows of 2 ft. pitch, the one tuned a beat too sharp, the other a beat too flat, to produce a waving tremulous tone that has a certain charm; â€œmusetteâ€ and â€œvoix celeste,â€ 16 ft.; and â€œbaryton,â€ a treble stop 32 ft., or two octaves lower than the normal note of the key. The â€œback organâ€ is usually covered by a swell box, containing louvres or shutters similar to a Venetian blind, and divided into fortes corresponding with the bass and treble division of the registers. The fortes are governed by knee pedals which act by pneumatic pressure. Tuning the reeds is effected by scraping them at the point to sharpen them, or near the shoulder or heel to flatten them in pitch. Air pressure affects the pitch but slightly, being noticeable only in the larger reeds, and harmoniums long retain their tuning, a decided advantage over the organ and the pianoforte. Mechanical contrivances in the harmonium, of frequent or occasional employment, besides those already referred to, are the â€œpercussion,â€ a small pianoforte action of hammer and escapement which, acting upon the reeds of the diapason rows at the moment air is admitted to them, gives prompter response to the depression of the key, or quicker speech; the â€œdouble expression,â€ a pneumatic balance of great delicacy in the wind reservoir, exactly maintaining by gradation equal pressure of the wind; and the â€œdouble touch,â€ by which the back organ registers speak sooner than those of the front that are called upon by deeper pressure of the key, thus allowing prominence or accentuation of certain parts by an expert performer. â€œProlongementâ€ permits selected notes to be sustained after the fingers have quittedtheir keys. Dawesâ€™s â€œmelody attachmentâ€ is to give prominence to an air or treble part by shutting off in certain registers all notes below it. This notion has been adapted by inversion to a â€œpedal substituteâ€ to strengthen the lowest bass notes. The â€œtremoloâ€ affects the wind in the vicinity of the reeds by means of small bellows which increase the velocity of the pulsation according to pressure; and the â€œsourdineâ€ diminishes the supply of wind by controlling its admission to the reeds.By courtesy of Metzler & Co.Fig. 2.â€”Free Reed Vibrator, Mason & Hamlin American Organ.The American Organacts by wind exhaustion. A vacuum is practically created in the air-chamber by the exhausting power of the footboards, and a current of air thus drawn downwards passes through any reeds that are left open, setting them in vibration. This instrument has therefore exhaust instead of force bellows. Valves in the board above the air-chamber give communication to reeds (fig. 2) made more slender than those of the harmonium and more or less bent, while the frames in which they are fixed are also differently shaped, being hollowed rather in spoon fashion. The channels, the resonators above the reeds, are not varied in size or shape as in the harmonium; they exactly correspond with the reeds, and are collectively known as the â€œtube-board.â€ The swell â€œfortesâ€ are in front of the openings of these tubes, rails that open or close by the action of the knees upon what may be called knee pedals. The American organ has a softer tone than the harmonium; this is sometimes aided by the use of extra resonators, termed pipes or qualifying tubes, as, for instance, in Clough & Warrenâ€™s (of Detroit, Michigan, U.S.). The blowing being also easier, ladies find it much less fatiguing. The expression stop can have little power in the American organ, and is generally absent; the â€œautomatic swellâ€ in the instruments of Mason & Hamlin (of Boston, U.S.) is a contrivance that comes the nearest to it, though far inferior. By it a swell shutter or rail is kept in constant movement, proportioned to the force of the air-current. Another very clever improvement introduced by these makers, who were the originators of the instrument itself, is the â€œvox humana,â€ a smaller rail or fan, made to revolve rapidly by wind pressure; its rotation, disturbing the air near the reeds, causes interferences of vibration that produce a tremulous effect, not unlike the beatings heard from combined voices, whence the name. The arrangement of reed compartments in American organs does not essentially differ from that of harmoniums; but there are often two keyboards, and then the solo and combination stops are found on the upper manual. The diapason treble register is known as â€œmelodiaâ€; different makers occasionally vary the use of fancy names for other stops. The â€œsub-bass,â€ however, an octave of 16 ft. pitch and always apart from the other reeds, is used with great advantage for pedal effects on the manual, the compass of American organs being usually down to F (FF, 5 octaves). In large instruments there are sometimes foot pedals as in an organ, with their own reed boxes of 8 and 16 ft. the lowest note being then CC. Blowing for pedal instruments has to be done by hand, a lever being attached for that purpose. The â€œcelesteâ€ stop is managed as in the harmonium, by rows of reeds tuned not quite in unison, or by a shade valve that alters the air-current and flattens one row of reeds thereby.Harmoniums and American organs are the result of many experiments in the application of free reeds to keyboard instruments. The principle of the free reed became widely known in Europe through the introduction of the Chinese cheng1during the second half of the 18th century, and culminated in the invention of the harmonium and kindred instruments. The first step in the invention of the harmonium is due to Professor Christian Gottlieb Kratzenstein of Copenhagen, who had had the opportunity of examining a cheng sent to his native city and of testing its merits.2In 1779 the Academy of Science of St Petersburg had offered a prize for an essay on the formation of the vowel sounds on an instrument similar to the â€œvox humanaâ€ in the organ, which should be capable of reproducing these sounds faithfully. Kratzenstein made as a demonstration of his invention a small pneumatic organ fitted with free reeds, and presented it to the Academy of St Petersburg.3His essay was crowned and was republished with diagrams in Paris4in 1782. Meanwhile, in 1780, a countryman of Kratzensteinâ€™s, an organ-builder named Kirsnick, established in St Petersburg, adapted these reed pipes to some of his organs and to an instrument of his invention called organochordium, an organ combined with piano. When Abt Vogler visited St Petersburg in 1788, he was so delighted with these reeds that in 1790 he induced Rackwitz, an assistant of Kirsnickâ€™s, to come to him and adapt some to an organ he was having built in Rotterdam. Three years later Abt Voglerâ€™s orchestrion, a chamber organ containing some 900 pipes, was completed, and, according to Rackwitz,5was fitted with free-reed pipes. Vogler himself, however, does not mention the free reed when describing this wonderful instrument and his system of â€œsimplificationâ€ for church organs.6To Abt Vogler, who travelled all over Germany, Scandinavia and the Netherlands, exhibiting his skill on his orchestrion and reconstructing many organs, is due the credit of making Kratzensteinâ€™s invention known and inducing the musical world to appreciate the capabilities of the free reed. The introduction of free-reed stops into the organ, however, took a secondary place in his scheme for reform.7Friedrich Kaufmann8of Dresden states that Vogler told him he had imparted to J. N. MÃ¤lzel of Vienna particulars as to the construction of free-reed pipes, and that the latter used them in his panharmonicon,9which he exhibited during his stay in Paris from 1805 to 1807. Kaufmann suggests that it was through him that G. J. GreniÃ© obtained the knowledge which led to his experiments with free reeds in organs. It is more likely that GreniÃ© had read Kratzensteinâ€™s essay and had experimented independently with free reeds. In 1812 his firstorgue expressifwas finished. It was a small organ with one register of free reedsâ€”the expression stop, in fact, added to the pipe organ and having a separate wind-chest and bellows. It would seem from his description of the orchestrion inData zur Akustikthat Vogler knew of no such device. He used the swell shutter borrowed from England and a threefold screen of canvas covered with a blanket arrangedoutside the instrument, neither of which is capable of increasing the volume of sound from the organ, or at least only after having first damped the sound to a pianissimo. Vogler explains minutely the apparatus used to conceal the working of the screen from the eyes of the public.10The credit of discovering in the free reed the capability of dynamic expression was undoubtedly due to GreniÃ©, although Abt Vogler claims to have used compression in 1796,11and Kaufmann in his choraulodion in 1816. A largerorgue expressifwas begun by GreniÃ© for the Conservatoire of Paris in 1812, the construction of which was interrupted and then continued in 1816. Descriptions of GreniÃ©â€™s instrument have been published in French and German.12The organ of the Conservatoire had a pedal free-reed stop of 16 ft., with vibrators 0.240 m. long, 0.035 m. wide, and 0.003 m. thick.13Two compressors, one for the treble and the other for the bass, worked by treadles, enabled the performer to regulate the pressure of wind on the reeds and therefore to obtain the gradations of forte and piano which gained for his instrument the name oforgue expressif. GreniÃ©â€™s instrument was a pipe organ, the pipes terminating in a cone with a hemispherical cap in the top of which was a small hole. There were eight registers including the pedal, and the positive on the first keyboard had reed stops furnished withbeating reeds. Biot insists on the Importance of the regulating wires (Fr.rasettes; Ger.KrÃ¼cken) for determining the vibrating length of the reed tongue and maintaining it invariable. These are clearly shown in his diagram (see articleFree Reed Vibrator, fig. 1); they do not essentially differ from those used with the beating-reed stops in his organ (fig. 76, pl. II.), or indeed from those figured by Praetorius.Isolated specimens of the cheng must have found their way to Europe during the 15th and 16th centuries, for Mersenne14depicts part of one showing the free reed. It would seem that still earlier in the 17th century there was an organ in a monastery in Hesse with free reeds for thePosaunestop, for Praetorius gives a description of the â€œextraordinaryâ€ reed (p. 169); there is no record of the inventor in this case.During the first half of the 19th century various tentative efforts in France and Germany, and subsequently in England, were made to produce new keyboard instruments with free reeds, the most notable of these being the physharmonica15of Anton HÃ¤ckel, invented in Vienna in 1818, which, improved and enlarged, has retained its hold on the German people. The modern physharmonica is a harmonium without stops or percussion action; it does not therefore speak readily or clearly. It has a range of five to six octaves. Other instruments of similar type are the French melophone and the English seraphine, a keyboard harmonica with bellows but no channels for the tongues, for which a patent was granted to Myers and Storer in 1839; the aeoline or aelodicon16of Eschenbach; the melodicon17of Dietz; the melodica18of Rieffelson; the apollonicon;19the new cheng20of Reichstein; the terpodion21of Buschmann, &c. None of these has survived to the present day.The inventor of the harmonium was indubitably Alexandre Debain, who took out a patent for it in Paris in 1840. He produced varied timbre registers by modifying reed channels, and brought these registers on to one keyboard. Unfortunately he patented too much, for he secured even the nameharmonium, obliging contemporary and future experimenters to shelter their improvements under other names, and the venerable name of organ becoming impressed into connexion with an inferior instrument, we have now to distinguish between reed and pipe organs. The compromise of reed organ for the harmonium class of instruments must therefore be accepted. Debainâ€™s harmonium was at first quite mechanical; it gained expression by the expression-stop already described. The Alexandres, well-known French makers, by the ingenuity of one of their workmen, P. A. Martin, added the percussion and the prolongement. The melody attachment was the invention of an English engineer; the introduction of the double touch, now used in the harmoniums of Mustel, Bauer and othersâ€”also in American organsâ€”was due to Tamplin, an English professor.The principle of the American organ originated with the Alexandres, whose earliest experiments are said to have been made with the view of constructing an instrument to exhaust air. The realization of the idea proving to be more in consonance with the genius of the American people, to whom what we may call the devotional tone of the instrument appealed, the introduction of it by Messrs Mason and Hamlin in 1861 was followed by remarkable success. They made it generally known in Europe by exhibiting it at Paris in 1867, and from that time instruments have been exported in large numbers by different makers.

The American Organacts by wind exhaustion. A vacuum is practically created in the air-chamber by the exhausting power of the footboards, and a current of air thus drawn downwards passes through any reeds that are left open, setting them in vibration. This instrument has therefore exhaust instead of force bellows. Valves in the board above the air-chamber give communication to reeds (fig. 2) made more slender than those of the harmonium and more or less bent, while the frames in which they are fixed are also differently shaped, being hollowed rather in spoon fashion. The channels, the resonators above the reeds, are not varied in size or shape as in the harmonium; they exactly correspond with the reeds, and are collectively known as the â€œtube-board.â€ The swell â€œfortesâ€ are in front of the openings of these tubes, rails that open or close by the action of the knees upon what may be called knee pedals. The American organ has a softer tone than the harmonium; this is sometimes aided by the use of extra resonators, termed pipes or qualifying tubes, as, for instance, in Clough & Warrenâ€™s (of Detroit, Michigan, U.S.). The blowing being also easier, ladies find it much less fatiguing. The expression stop can have little power in the American organ, and is generally absent; the â€œautomatic swellâ€ in the instruments of Mason & Hamlin (of Boston, U.S.) is a contrivance that comes the nearest to it, though far inferior. By it a swell shutter or rail is kept in constant movement, proportioned to the force of the air-current. Another very clever improvement introduced by these makers, who were the originators of the instrument itself, is the â€œvox humana,â€ a smaller rail or fan, made to revolve rapidly by wind pressure; its rotation, disturbing the air near the reeds, causes interferences of vibration that produce a tremulous effect, not unlike the beatings heard from combined voices, whence the name. The arrangement of reed compartments in American organs does not essentially differ from that of harmoniums; but there are often two keyboards, and then the solo and combination stops are found on the upper manual. The diapason treble register is known as â€œmelodiaâ€; different makers occasionally vary the use of fancy names for other stops. The â€œsub-bass,â€ however, an octave of 16 ft. pitch and always apart from the other reeds, is used with great advantage for pedal effects on the manual, the compass of American organs being usually down to F (FF, 5 octaves). In large instruments there are sometimes foot pedals as in an organ, with their own reed boxes of 8 and 16 ft. the lowest note being then CC. Blowing for pedal instruments has to be done by hand, a lever being attached for that purpose. The â€œcelesteâ€ stop is managed as in the harmonium, by rows of reeds tuned not quite in unison, or by a shade valve that alters the air-current and flattens one row of reeds thereby.

Harmoniums and American organs are the result of many experiments in the application of free reeds to keyboard instruments. The principle of the free reed became widely known in Europe through the introduction of the Chinese cheng1during the second half of the 18th century, and culminated in the invention of the harmonium and kindred instruments. The first step in the invention of the harmonium is due to Professor Christian Gottlieb Kratzenstein of Copenhagen, who had had the opportunity of examining a cheng sent to his native city and of testing its merits.2In 1779 the Academy of Science of St Petersburg had offered a prize for an essay on the formation of the vowel sounds on an instrument similar to the â€œvox humanaâ€ in the organ, which should be capable of reproducing these sounds faithfully. Kratzenstein made as a demonstration of his invention a small pneumatic organ fitted with free reeds, and presented it to the Academy of St Petersburg.3His essay was crowned and was republished with diagrams in Paris4in 1782. Meanwhile, in 1780, a countryman of Kratzensteinâ€™s, an organ-builder named Kirsnick, established in St Petersburg, adapted these reed pipes to some of his organs and to an instrument of his invention called organochordium, an organ combined with piano. When Abt Vogler visited St Petersburg in 1788, he was so delighted with these reeds that in 1790 he induced Rackwitz, an assistant of Kirsnickâ€™s, to come to him and adapt some to an organ he was having built in Rotterdam. Three years later Abt Voglerâ€™s orchestrion, a chamber organ containing some 900 pipes, was completed, and, according to Rackwitz,5was fitted with free-reed pipes. Vogler himself, however, does not mention the free reed when describing this wonderful instrument and his system of â€œsimplificationâ€ for church organs.6To Abt Vogler, who travelled all over Germany, Scandinavia and the Netherlands, exhibiting his skill on his orchestrion and reconstructing many organs, is due the credit of making Kratzensteinâ€™s invention known and inducing the musical world to appreciate the capabilities of the free reed. The introduction of free-reed stops into the organ, however, took a secondary place in his scheme for reform.7Friedrich Kaufmann8of Dresden states that Vogler told him he had imparted to J. N. MÃ¤lzel of Vienna particulars as to the construction of free-reed pipes, and that the latter used them in his panharmonicon,9which he exhibited during his stay in Paris from 1805 to 1807. Kaufmann suggests that it was through him that G. J. GreniÃ© obtained the knowledge which led to his experiments with free reeds in organs. It is more likely that GreniÃ© had read Kratzensteinâ€™s essay and had experimented independently with free reeds. In 1812 his firstorgue expressifwas finished. It was a small organ with one register of free reedsâ€”the expression stop, in fact, added to the pipe organ and having a separate wind-chest and bellows. It would seem from his description of the orchestrion inData zur Akustikthat Vogler knew of no such device. He used the swell shutter borrowed from England and a threefold screen of canvas covered with a blanket arrangedoutside the instrument, neither of which is capable of increasing the volume of sound from the organ, or at least only after having first damped the sound to a pianissimo. Vogler explains minutely the apparatus used to conceal the working of the screen from the eyes of the public.10The credit of discovering in the free reed the capability of dynamic expression was undoubtedly due to GreniÃ©, although Abt Vogler claims to have used compression in 1796,11and Kaufmann in his choraulodion in 1816. A largerorgue expressifwas begun by GreniÃ© for the Conservatoire of Paris in 1812, the construction of which was interrupted and then continued in 1816. Descriptions of GreniÃ©â€™s instrument have been published in French and German.12The organ of the Conservatoire had a pedal free-reed stop of 16 ft., with vibrators 0.240 m. long, 0.035 m. wide, and 0.003 m. thick.13Two compressors, one for the treble and the other for the bass, worked by treadles, enabled the performer to regulate the pressure of wind on the reeds and therefore to obtain the gradations of forte and piano which gained for his instrument the name oforgue expressif. GreniÃ©â€™s instrument was a pipe organ, the pipes terminating in a cone with a hemispherical cap in the top of which was a small hole. There were eight registers including the pedal, and the positive on the first keyboard had reed stops furnished withbeating reeds. Biot insists on the Importance of the regulating wires (Fr.rasettes; Ger.KrÃ¼cken) for determining the vibrating length of the reed tongue and maintaining it invariable. These are clearly shown in his diagram (see articleFree Reed Vibrator, fig. 1); they do not essentially differ from those used with the beating-reed stops in his organ (fig. 76, pl. II.), or indeed from those figured by Praetorius.

Isolated specimens of the cheng must have found their way to Europe during the 15th and 16th centuries, for Mersenne14depicts part of one showing the free reed. It would seem that still earlier in the 17th century there was an organ in a monastery in Hesse with free reeds for thePosaunestop, for Praetorius gives a description of the â€œextraordinaryâ€ reed (p. 169); there is no record of the inventor in this case.

During the first half of the 19th century various tentative efforts in France and Germany, and subsequently in England, were made to produce new keyboard instruments with free reeds, the most notable of these being the physharmonica15of Anton HÃ¤ckel, invented in Vienna in 1818, which, improved and enlarged, has retained its hold on the German people. The modern physharmonica is a harmonium without stops or percussion action; it does not therefore speak readily or clearly. It has a range of five to six octaves. Other instruments of similar type are the French melophone and the English seraphine, a keyboard harmonica with bellows but no channels for the tongues, for which a patent was granted to Myers and Storer in 1839; the aeoline or aelodicon16of Eschenbach; the melodicon17of Dietz; the melodica18of Rieffelson; the apollonicon;19the new cheng20of Reichstein; the terpodion21of Buschmann, &c. None of these has survived to the present day.

The inventor of the harmonium was indubitably Alexandre Debain, who took out a patent for it in Paris in 1840. He produced varied timbre registers by modifying reed channels, and brought these registers on to one keyboard. Unfortunately he patented too much, for he secured even the nameharmonium, obliging contemporary and future experimenters to shelter their improvements under other names, and the venerable name of organ becoming impressed into connexion with an inferior instrument, we have now to distinguish between reed and pipe organs. The compromise of reed organ for the harmonium class of instruments must therefore be accepted. Debainâ€™s harmonium was at first quite mechanical; it gained expression by the expression-stop already described. The Alexandres, well-known French makers, by the ingenuity of one of their workmen, P. A. Martin, added the percussion and the prolongement. The melody attachment was the invention of an English engineer; the introduction of the double touch, now used in the harmoniums of Mustel, Bauer and othersâ€”also in American organsâ€”was due to Tamplin, an English professor.

The principle of the American organ originated with the Alexandres, whose earliest experiments are said to have been made with the view of constructing an instrument to exhaust air. The realization of the idea proving to be more in consonance with the genius of the American people, to whom what we may call the devotional tone of the instrument appealed, the introduction of it by Messrs Mason and Hamlin in 1861 was followed by remarkable success. They made it generally known in Europe by exhibiting it at Paris in 1867, and from that time instruments have been exported in large numbers by different makers.

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