C.V. Roman wrote the following six papers on Indian musical instruments:
  1. "The Ectara". Jour. Indian Math. Club. 170.1909.

  2. "Escalations of the Stretched Strings". J. Indian Math Club, U. 1910.

  3. "Musical Drums with Harmonic Overtones". Nature. 104,500(1920) (with S. Kumar).

  4. "On some Indian Stringed instruments". Proc. Indian Assoc. Cultiv. Sc. 7,29,1921.

  5. The Acoustical knowledge of the Ancient Hindus" (Asutosh.Mookerjee, Silver
    Jubilee Volume), Calcutta University, 2,179.

  6. "The IndianMusicalDrums', Proc.mdran Acad.ot Sciences Al, 179- 188, (1934).

Raman also wrote about thirteen papers on the theory of violin and one on the Pianoforte. In fact his monograph on "Musical Instruments and Their Tones" in the Handbuch der Physrk (8.354,1927) makes only passing references to his own work on the Indian musical instruments.

Still Raman's work on the Indian musical instruments is a pioneering effort at understanding the scientific principles behind the construction of these instruments, especially the way they achieve the desired tonal structure. We therefore reproduce extracts from papers 4, 5,6.As it happened with many other lines of investigation pioneered by Raman, there has been very little work done by way of follow-up. We have not come across any reference to published work based on Raman's ideas on the Indian percussion instruments; we may cite the following references which appear to take Raman's analysis further.

  1. R.N. Ghosh, Phys.Rev. 20,526-7 (1922)

  2. K.N. Rao. Proc. Indian Acad. Sciences. 7A, 75-84 (1938)

  3. B.S. Ramakrishna and Manmohan Sondin, "Vibration of Indian Musical Drums Regarded as composite Membranes"! Jour. Acoust. Soc. America, 26, 523-529 (1954)

  4. B.S. Ramakrishna, "Modes of Vibration of the Indian Drum Dugga or Left-Hand Thabla", Jour. Acoust. Soc. America, 29,234-238 (1957)

  5. B.S. Ramakrishna, M.M. Sondin and Y. Devadas, "Some Recent Studies in Indian Musical Drums", Jour. Hist.Telcom. Eng. 3,285-290(1957)

  6. T. Sarojini and A. Rahman, "Variational method for the Vibrations of Indian Drums", Jour. Acoust. Soc. America, 30,191 -196 (1958)

  7. Thomas D. Rossing and W. Arnold Sykes. "Acoustics of Indian Drums", Percussive
    Notes (Urbana, USA) 19(3) 58-67 (1982).

In the Appendix, we reproduce parts of the article (3) by B.S. Ramakrishna and Manmohan Sondin. As regards the open problems in the subject, we can do no better than quote the following from a personal communication received from Prof. B.S. Ramakrishna:

"As for the open problems in the Indian Musical instruments about all I can say is that the whole field is wide open. In a very real sense the studies initiated by Raman have not been advanced substantially since then. In the case of violin, for instance, one can cite at least two hundred references to work done in the last fifty years, but nothing comparable exists for the Indian musical instruments. Incidentally, Raman himself contributed very significantly to our knowledge of the physics of violin.

Since Raman's time new experimental techniques and theoretical tools have become available if one really wants to explore this still virgin field. For example, there are extraordinarily sensitive pickups now available for studying the vibration patterns of the sound radiating surfaces of the instruments. The laser has created techniques (e.g., time averaged holography) which can map the vibration amplitudes with great precision. There are now instruments for displaying the sound spectra of the various notes in real time, and then, there are computers to synthesize musical notes of different instruments. On the theoretical side too there are completely new analytical techniques like finite element analysis, model analysis, etc. developed elsewhere which were wholly unknown during Raman's time. The only thing that is not available today in the country is, unfortunately, scientists, interested in pursuing such studies.

On some Indian Stringed Instruments

(Extracted from C.V. Raman: Proc. Indian Assoc. Cultiv. Sc. 7, 29, 1921)


Afascinating field for research offers itself in the scientific study of the numerous kinds of musical instruments to be found in India. Some of these instruments of indigenous origin are of undoubted antiquity and disclose a remarkable appreciation of acoustic principles. An investigation of their special features in comparison with those of instruments of other countries may be expected to yield results of great interest. An instance of the fruitfulness of the line of work here suggested is to be found in the present author's research on the Indian Musical Drums, which have been found to embody in a practical form the solution of the problem of loading a circular drum-head in such a manner as to make it give a harmonic succession of overtones in the same way as a stringed instrument. In the present paper it is proposed to offer a preliminary note on the results of the author's study of some Indian stringed instruments.

The Form of the Bridge in the "Tanpura" and the "Veena"

The "Tanpura" and the "Veena" are two of the most highly valued indigenous stringed instruments intended to be excited by plucking. Plate 1; figure 1 illustrates the form of the 'Tanpura". This instrument has no frets and is intended merely to be used as a drone in accompaniment with vocal or other music. It has four metal "strings" which are stretched over a large resonant body and can be accurately tuned up to the right pitch by a simple device for continuous adjustment of tension. The remarkable feature of the 'Tanpura" to which I wish to draw attention is the special form of bridge fixed to the resonant body over which’ the strings pass. The strings do not come clear off the edge of a sharp bridge as in European stringed instruments, but pass over a cured wooden surface fixed to the body which forms the bridge. The exact length of the string which actually touches the upper surface of the bridge is adjusted by slipping in a woolen or silken thread of suitable thickness between each string and the bridge below it and adjusting its position by trial. Generally the thread is moved forwards or backwards to such a position that the metal "string" just grazes the surface of the bridge- The description will be clearer oh a reference to figure 3 above where the bridge and the string passing over it are indicated diagrammatically.

The "Veena" on the other hand is a fretted instrument intended for use in playing melodies (figure 2 in plate 1). The form of the bridge adopted in it differs from that of the 'Tanpura" in two respects. The upper curved surface of the bridge in the "Veena" is of metal, and the special mode of adjustment of contact by means of a thread used in the 'Tanpura" is dispensed with, and the string merely comes off the curved upper surface of the bridge at a tangent; as indicated diagrammatically in figure 4- {No attempt is made in This figure to indicate the form of the lower part of the bridge). The bridge of the "Veena" is also much higher above the body of the instrument than in the 'Tanpura". Even when the strings are pressed down on the frets when the instrument is being played, the curvature of the upper surface of the bridge ensures the string always leaving the bridge at a tangent to it as shown.

The Failure of the Young-Helmholtz Law

The special form of bridge illustrated above has a very remarkable influence on the tone-quality. This can be most readily demonstrated in the 'Tanpura". When the adjustment of contact of string and bridge is made carefully by trial, the instrument is highly sonorous, giving a tone of fine musical quality. If on the other hand the grazing contact of string and bridge is rendered inoperative (as for instance by inserting a small piece of metal between the string and the surface of the bridge) the tone becomes dull and insipid. A similar remark applies also to the case of the "Veena", though the difference is less striking in the latter case. In attempting to find an explanation for the difference in tone-quality produced by the special form of bridge, the author made a surprising observation, namely, that in the tone of the 'Tanpura" or the "Veena", overtones may be heard powerfully which according to known acoustical principles should have been entirely absent. According to the law enunciated by Young and Helmholtz, if the string is plucked at a point of excitation division, the harmonics having a node at the point of excitation should be entirely absent. This law may be readily verified on an ordinary sonometer with the usual form of bridge.

For This purpose, the position of the node should first be found exactly by trial by putting the finger in contact with the string and plucking elsewhere so as to elicit the overtones desired. Having found the position of the node, the string should be plucked exactly at that point and then again touched with the finger at the same point. On an ordinary sonometer, these results in the sound being immediately quenched in as much as the finger damps out all the partials except those having a node at the point touched, and the latter are not excited in the first instance in accordance with the Young-Helmholtz law. On trying the same experiment with the "Veena" or the 'Tanpura", it will be found that the overtone having a node at the plucked point sings out powerfully. In fact the position of the plucked point hardly appears to make a difference in regard to the intensity of the overtones in the 'Tanpura". This remarkable result is not due to any indefiniteness in the position of the node point, as the latter is found to be quite well defined as is shown by the fact that in order to demonstrate the effect successfully, the string must be plucked and then touched exactly at that right point, otherwise the sound is quenched. We are thus forced to the conclusion that the effect of the special form of bridge is completely to set aside the validity of the Young-Helmholtz law and actually to manufacture a powerful sequence of overtones including those which ought not to have been elicited according to that law.

Outline of a Mechanical Theory

Some photographs of the vibration-curve of a 'Tanpura" strings were made at the suggestion of the author by Mr. Ahmed Shah Bukhari at the Government College, Lahore, and last November. They showed that in consequence of the grazing contact at the bridge, the vibration of the string decreased in amplitude and altered its form at a much more rapid rate than when the grazing contact was rendered ineffective. A more complete investigation is obviously desirable. From first principles, however, it is obvious that in the 'Tanpura" the forces exerted by the vibrating string on the bridge must be very different from what they would be for a bridge of ordinary form. It seems probable that by far the greater portion of the communication of energy to the bridge occurs at or near the point of grazing contact. The forces exerted by the string on the bridge near this point are probably the nature of impulses occurring once in each vibration of the string. This would explain the powerful retinue of overtones including even those absent initially in the vibration of the string. At a slightly later stage, the reaction of the bridge on the string would result in a modification of the vibration form of the latter and bring into existence partials absent initially in it. There would in fact be a continual transformation of the energy of vibration of the fundamental vibration into the overtones.

The foregoing explanation of the character of the tones of the 'Tanpura" would not be fully appliable to the 'Veena" as the forces exerted by the string on the bridge in This case would not be purely of an impulsive character. There is however a certain portion of the bridge over which the string comes into intermittent contact during the vibration, and it seems very probable that the theory for this case is intermediate in character between that for the 'Tanpura" and those for stringed instruments with bridges of the ordinary type. Further experimental work is needed in support of this view.

The Acoustical Knowledge of the Ancient Hindus

(Extracted from C.V.Raman: Ashutosh Mookherjee, Silver Jubilee Volume, Calcutta, 1922.)

Music, both vocal and instrumental, undoubtedly played an important part in the cultural life of ancient India. Sanskrit literature, both secular and religious, makes numerous references to instruments of various kinds, and it is, I believe, generally held by archaeologists that some of the earliest mentions of such instruments to be found anywhere are those contained in the ancient Sanskrit works. Certain it is that at a very early period in the history of the country, the Hindus were acquainted with the use of stringed instruments excited by plucking or bowing, with the transverse form of flute, with wind and reed instruments of different types and with percussion instruments. It is by no means improbable that India played an important role in the progressive Evolution and improvement of these instruments and might have served as a source from which their knowledge spread both eastwards and westwards.

It would form a fascinating chapter of history to try and trace the gradual development of musical instruments and musical knowledge, from the rhythmic chanting of the Rig Veda in the ancient home of the Aryan race to the Indian music of the present day. But the materials available for the writing of this history seem to be all too meager. Much of the long period over which the gradual evolution must have spread lies in the dim and remote past of which but the vaguest glimpses can be obtained from such records as exist. Something more definite regarding the acoustical developments in Ancient India might perhaps be gleaned from a study of the musical instruments, the models of which have been handed down as heirlooms for untold generations. Several of the Indian stringed instruments, for example, disclose in their design, even on a superficial examination, a quite remarkable appreciation of the principles of sound-production and of resonance. A fuller study seemed likely to lead to results of considerable interest.

It was this hope that induced me some two years ago to commence a systematic examination by modern scientific methods of the ancient Indian musical instruments. The objects I set before myself were to investigate the traditional designs according to which these Indian instruments are constructed and the variations of these designs that exist in the different parts of the country, to discover the raison d'etre of the methods of construction employed and to find the special tone-characters which were held in esteem by the designers. It seemed that such an examination might also prove useful from the practical stand-point by disclosing the best designs and indicating the directions in which any improvements might be possible.

Various circumstances have delayed the complete carrying out of the projected work, and it is probable that little progress might have been made with it up to date, but for the fact that my attention was recently drawn somewhat forcibly to the musical qualities of the ancient Indian instruments of percussion. Through the kindness of an enthusiastic fellow-worker, Mr. Sivakali Kumar, some good specimens of the Indian percussion instruments were put at my disposal and I have been enabled to carry out a scientific examination of their acoustical properties. The results obtained are very remarkable and significant and are being described in detail in a monograph "On Musical Drums" which will be published by the Indian Asso. Cultiv. Sci. I propose in this short essay to indicate the main results of this investigation and to show how far they throw light on the state of acoustical knowledge in ancient and medieval India.

Acoustics of Percussion Instruments

By way of preface, I shall first refer to a few facts regarding the vibrations of stretched membranes which are familiar to students of physics and which it is useful here to recall. As is well known, the vibrations of a circular stretched membrane or drum-head excited by impact are generally of an extremely complex character. Besides the gravest or fundamental tone of the membrane, we have a large retinue of overtones which stand to each other in no sort of musical relation. These overtones are always excited in greater or less degree and produce a discordant effect.

All the instruments of percussion known to European physicists in which a circular drum-head is employed have therefore to be regarded more as noise producers introduced for making the rhythm than as musical instruments. This is true even of the kettle-drum which is tuned to a definite pitch and occasionally used in European orchestral music. As has been shown by the late Lord Rayleigh in a paper published some time ago, the air enclosed in the shell of the kettle-drum does not produce any advantageous alterations of the pitch relations of the overtones. All the instruments of percussion known to European science are thus essentially non-musical and can only be tolerated in open air music or in large orchestras where a little noise more or less makes no difference. Indian musical instruments of percussion however stand in an entirely different category. Times without number we have heard the best singers or performers on the flute or violin accompanied by the well known indigenous musical drums, and the effect with a good instrument is always excellent. It was this, in fact, that conveyed to me the hint that the Indian instruments of percussion possess interesting acoustic properties, and stimulated the research.

The Indian Musical Drums (Extracted from C.V. Raman: Proc. Ind. Acad. ScAI, 179, 1934.).


Many years ago, I drew attention to the remarkable acoustic properties of the natural drums »-which are used as accompaniments to vocal or instrumental music and are extremely popular in India even at the present time. My investigations showed clearly that this instrument contained the solution in a practical form of the acoustical problem of transforming circular drum-head giving in-harmonic overtones into a harmonic musical instrument. In its classical form, the instrument is known as the Mridangam and is referred to in ancient Sanskrit works and is also pictured in the paintings on the walls of the Ajanta Caves. It is thus clearly a very ancient invention, and is acoustical perfection must be considered a remarkable testimony to the inventiveness and musical taste of its progenitors.

The physical study of the Mridanga, however, possesses more than a merely archaeological interest. As was pointed out in my earliest note on the subject and somewhat more fully in my article on musical instruments in Handb. Phys., the successful conversion of an inharmonic sequence of tones into a harmonic one has been' effected in a very interesting manner. The drum has the special property of vibrating freely in different forms but with identical frequencies which can be superposed on each other. Sorrie of the superposition forms have a striking simplicity, and indicate an analogy between the musical drum and the harmonic vibrations of a uniform stretched string. In view of the extreme brevity of the accounts previously published, it appeared desirable to set out more fully the results obtained. The subject is, however, far from being exhausted by the present report, and it is hoped that the paper is only the precursor of a complete treatise on the musical drums of India.

Description of the Instruments

An immense variety of drums of various forms and shapes are to be found in use in different parts of the country. The musical drum, however, stands apart in a class by itself, and is used exclusively for high class chamber music where the noise of an ordinary percussion instrument would be intolerable. The classical form of the instrument known as the Mridanga is a two-sided drum which is played with both hands. Its shape resembles that of two truncated cones or flower-pots placed together end to end with the narrow ends outwards. The construction usually takes the form of a hollow wooden shell of the shape mentioned with both ends open over which the drum-skins are stretched. To enable the drum-skins to be tightened to any desired tension, and at the same time ensure a uniform tension in all directions, the following device is adopted. The drum-heads are firmly attached to circular hoops by interlaced thongs of leather. The hoops are then put over the ends of the drum. A long band of leather repeatedly passes through both the hoops and to and fro over the full length of the drum, in all exactly sixteen times at equal intervals along its circumference. The ends of the leather band are then tied together.

The tuning of the drum is roughly effected by tightening up the leather cord by adjusting the position of 8 movable cylindrical blocks of wood over which it passes. The final adjustment is made by the strokes of a hammer which force down the hoop over which the drum-head is stretched to the extent desired. The arrangement enables the drum-head to be accurately adjusted to any desired tension, and what is equally important, enables the tensions in different directions to be equalized with meticulous precision. One can travel from one end of India to the other and seek in vain for a Mridanga which has either more or less than sixteen tightening straps. It is clear, therefore, that the inventors of the drum not only realized the importance of equalizing the tensions, but laid stress on having exactly the right number of tension equalizers, namely, sixteen.

Apart from the details mentioned above, the special feature of the Mridanga consists in the construction of the drum-head played with the right hand. As originally put on, the drum-head is not a single piece of leather, but consists of three layers of drum-skin superposed on each other. In the final stages of construction, all the layers except one are taken out leaving only rings round the margin to reinforce one drum-skin which is left intact and is capable of vibration. Externally one such ring of leather is visible over the drum-head. The latter is then loaded symmetrically with a firmly adherent are elaborate processes which take a great deal of time. Actually, the composition is put on layer by layer and pressed down by rubbing with a smooth piece of stone or metal. The thickness of the layer is greatest in the centre and shades down towards the margin. In some cases, it is found that the thickness is stepped down by three, Ave or seven stages towards the margin. Watching the process of putting it on, it is found that the thickness and distribution are determined by testing the tone of the drum continuously as the work proceeds.

The left-hand drum-head of the Mridanga is usually larger in size than the right hand one. It is constructed in a similar manner to that described above, but without the central loading. In playing the instrument, however, the left-hand drum skin is loaded with a piece of dough (kneaded wheat flour) which is moistened and put on in sufficient quantity towards the centre to bring the pitch down to the desired value.

A modern variant of the Mridanga is known as the Thabla. This really consists of two drums played simultaneously with the right-hand and left-hand respectively. Both consist of wooden or metal shells open at one end only and covered with drum-skins. The drum-head of the Thabla played with the right-hand is very similar to that of Mridanga. The drum played with the left-hand has a firmly adherent composition which is, however, unsymmetrical placed on the membrane. The purpose of such unsymmetrical loading is quite different from that of the symmetrical loading used in the right-hand drum, with which alone we are concerned in the present paper. The tension arrangements in the Thabla are similar to those in the Mridanga, with the difference that the tightening cords simply pass round the closed end of the Thabla. The number of tightening straps is exactly 16 as in the Mridanga. In some very recent forms of Thabla, the tightening is effected by 16 iron rods placed at equal intervals round the drums, each having a hook which goes over the circular hoop of the drum-skin and is provided with a tightening nut and bolt at the lower end. With This arrangement again it is possible to adjust the tensions very accurately to equality in all directions.

The description of the Mridanga and of the Thabla given above is sufficiently comprehensive to cover all cases met with in practice. It must not be imagined, however, that all instruments going by these names are exact copies of each other. This is far from being the case. The individual examples differ notably in the size and shape of the wooden shell used, as also in the nature of the wood it self and the thickness of the shell. Notable differences also occur in the thickness of the leather used for the drum-skin, in the exact area and distribution of the central load, and specially also in the width of the marginal ring of the leather which is left superposed on the vibrating drum-head. In some forms of Mridanga or Thabla, the marginal ring is left very wide. In others, it is cut down to the barest minimum.

Its Acoustic Character

The most striking feature which distinguishes the Mridanga and the Thabla from other forms of drum is the sustained character of the tones. This is evidently the result of two features in the construction, namely, the heavy wooden shell on which the drum-head is stretched and the symmetrical loading of the latter by a firmly adherent composition. A drum-head which is stretched on a frame of small mass is obviously incapable of prolonged vibration, owing to the rapid communication of movement to the supporting frame. The heavy rigid shell in the Mridanga or Thabla, on the other hand, is favorable to the sustained vibration of the drum-head. The loading of the drum head greatly increases the energy of vibration and is therefore a factor which favors the emission of a sustained tone. The presence of the enclosed air within the shell is probably also a factor tending in the same direction.

It is empirically observed that the width of the marginal ring of leather superposed on the drumhead has a notable influence on the duration of tone. The ring in fact acts as a kind of damper, and its width is adjusted to obtain the desired kind of tone. With a broad ring, we obtain a muffled tone of short duration in which few overtones are present, winkle with a narrow ring, the tone is prolonged and is also brighter, containing more overtones. From these observations, it is to be inferred that the purpose served by the marginal ring is mainly to suppress high tones which are not desired. The mechanism of such suppression is not difficult to understand. It is well known that in the case of a circular drum-head, the amplitude of vibration is relatively greatest towards the centre in the case of lower tones, but increases relatively towards the margin in the case of higher overtones having several nodal diameters. The leather ring, therefore, acts as a damper for these high overtones without sensibly influencing the lower tones. Too broad again, however, carries the suppression to an undesirable extend, cutting out even the tones of lower pitch.

As will be seen later in the paper, the construction of the drum-head seeks to arrange the first nine normal modes of the membrane into a harmonic sequence of five tones. The existence of normal modes of still higher pitch can only serve to injure the final result, and their elimination may there fore be described as the purpose of the marginal ring. The latter cannot greatly affect the pitch of the graver tones in as much as the amplitude of vibration is rather small for such tones towards the margin, and the effect of the marginal ring considered as a load must therefore be unimportant. The contact between the ring and the drum-head is probably imperfect, and this should tend to make the ring act as a damper rather than as a load influencing the pitch.

It should be mentioned that the Mridanga and Thabla, though they have much in common, are by no means identical in their acoustic properties. In the playing of the Mridanga, the flat of the hand is used more frequently, winkle with the Thabla, the finger types are usually employed

The Five Tones of the Drum

The sustained character of the vibrations of the drum makes it possible to excite and observe them very readily. Indeed, one of the most striking properties of the harmonic drum is that any desired mode of vibration may be excited by simple percussion quite as easily as a stretched string may be caused to vibrate in one or other of its harmonic modes by touching it at a nodal point and plucking it suitably. The analogy is indeed very close as will be presently made clear.

The gravest mode of vibration of the drum-head is, of course, that without any interior nodal lines. This is best excited by bringing down the flat of the palm of the hand smartly on the centre of the drum-head and then quickly removing it. Produced in This way, the deep hum-tone obtained is quite free from overtones, whereas the tone obtained by striking the drum with the finger tips contains overtones.

The second tone of the drum-head is that having one nodal diameter. The professional drummer excites This by smartly striking the membrane with the edge of his palm laying his liitle finger along a diameter so as to bring it to rest, winle the edge strikes the membrane and rapidly recoils from it. A clear sustained tone is obtained in this way. That the membrane thus excited vibrates with one nodal diameter at rest is readily demonstrated by strewing liitle fine sand on it either before or immediately after the stroke. The sand gathers itself into a clear cut straight line along a nodal diameter coinciding with the position of the liitle finger in striking. For the success of the experiment, one has, of course, to cultivate the professional touch in the manner of striking the drum. It is also very important to adjust the tensions of the membrane in different directions to equality with great care. If this is not done, the experiment succeeds only if the drum is struck along with nodal diameter having the greatest or the least tension. When struck along a diameter with an intermediate tension, beats are heard, and the nodal diameters as indicated by the sand rotates to and fro about the centre periodically. If the beats are very fast, the sand is visible only as a liitle pile at the centre of the drum-head.

There is another method of exciting the mode of vibration with one nodal diameter which is very simple and does not need any professional skill. This is merely to touch the membrane gently with one finger of the left-hand laid along a diameter near the margin, and to strike the membrane smartly with a finger of the right-hartd at a suitable point on the perpendicular diameter. The finger touching the drum determines the position of the nodal diameter which is indicated by the sand forming a line across the drum-head.

For exciting the tinrd tone of the drum by itself, the simplest method is to touch the membrane gently with the fingers at two points near each other on the edge of the black central load and then strike the drum smartly with the finger at a point removed 90 degree away; a clear ringing tone is obtained, and if the two points touched are at a suitable distance apart, two parallel nodal lines stretching across the drums are formed by the sand. The significance of this form relatively to the usual modes of vibrations of a circular drum-head will be considered later.

The fourth tone of the drum is similarly excited by touching the edge of the loaded area lightly at three points, and striking the drum near its outer edge smartly with the finger at a point 90 degree away from the middle of the three points touched. If the three points touched are at suitable equal distances apart, the drum-head vibrates with three parallel nodal lines stretching across it, the position of which is indicated by the lines of the sand. A clear ringing tone is heard at the same time.

The fifth tone may similarly be excited by touching the edge of the loaded area at four points, and striking the drum smartly at a point some distance away on the marginal ring of leather. Except in large and specially well-made instruments, the duration of This tone is rather small, and it is not quite so easy to’ obtain its sand figures by percussion as in the case of the graver tones.

The harmonic relationship between the five tones of the drum is readily appreciated when they are excited one after another in the manner described above. It will be noticed that the fundamental corresponds to the drum-head vibrating as a whole. The second harmonic corresponds to the drum-head vibrating in two equal parts separated by a nodal diameter. The tinrd harmonic corresponds to a mode of vibration in which the drum-head divides into three parts separated by two parallel nodal lines. The fourth harmonic corresponds to a mode in which the drum-head divides into four parts separated by three parallel nodal lines. The fifth harmonic similarly corresponds to a case in which the drum-head vibrates in five parts separated by four nodal lines. The analogy with the simple case of a vibrating stretched string is thus remarkably close.

Superposition Figures of the Tinrd Harmonic

We have now to consider the relationship between the normal modes of Vibration of the drum-head and the harmonic tones given by it. As regards the first and second harmonics, no special remarks are necessary as the modes of vibration are unique in each case. The tinrd harmonic, on the other hand, is produced by a combination, in any desired ratio of amplitudes, of the modes of vibration of the drum-head with one nodal circle, and the mode with two nodal diameters. The proof of this statement is very easy and is illustrated in figures 7 to 12. It depends on the fact that by touching the drum-head gently at suitable points and exciting it by percussion, the mode of vibration with one nodal circle, and the mode with two nodal diameters may be excited, either each by itself or together in any desired ratio of amplitude. In either case, the pitch of the tone obtained is identical, but the superposition gives rise to nodal diagrams which are observed as sand figures and assume varying shapes.

The mode of vibration with one nodal circle is most readily obtained by touching the drum-head at some liitle distance from its centre with the tip of a pencil, and tapping the centre with a light hammer. If the point of damping has been suitably chosen, a nodal circle is obtained. If it is too near or too far from the centre, an elliptic sand figure is found. To obtain a very elongated ellipse, we touch two points which are very close to each other on the edge of the loaded area and strike the drum with the finger at a point removed from them by 90 degree. On increasing the distance between the two points touched, the ellipse straightens out and we obtain two parallel nodal lines running across the drum-head, and dividing it into three parts of which the middle has a smaller area than two outer ones. On further increasing the distance between the two points touched, the nodal lines cure outwards and assume the form of hyperbolae. Finally, when two points 90 degree apart on the edge of the loaded area are damped with the fingers and the drum is struck at the mid-point of the adjacent quadrant, we get two nodal diameters passing through the centre.

Superposition Figures of the Fourth Harmonic

The forth harmonic is given by the drum-head vibrating in one or other of two forms: (a) a mode with one nodal diameter and one nodal circle (b) a mode with three nodal diameters, or by superposition of both these forms. This is demonstrated by the sand figures reproduced as figures 13 -18. Most of the figures for the fourth harmonic are obtained by touching three points on the edge of the loaded area and tapping the drum with the finger just on the inner edge of the marginal ring of leather. If the three points touched are exactly 60 apart from each other, we get the mode of vibration with three nodal diameters. If they are closer together but at equal distances the diagram takes the form of three lines, one of which is a diameter running across the drum-head and the other two are hyperbolae curved outwards. As a special case, we have the figure consisting of three parallel straight lines running across the drum. All these figures are evidently obtained by the superposition of the normal two modes mentioned above. To obtain by itself the mode with one nodal diameter and a nodal circle, the device is adopted of touching the loaded area at two points 90 degree apart, one at the edge of the loaded area and the other on the nodal circle itself and of tapping the drum-head near its edge.

A specially interesting case is that shown in figure 18 where we have a nodal figure consisting' of three lines, two of them running perpendicular to the tinrd which is a diameter. This is really the same case as that of three parallel lines running across the drum-head and may be derived from it be reversing the relative phase of the two superposed modes of vibration. It is obtained by touching the edge of the loaded area at three points, one of which is removed by 90 degree from the midpoint of the other two. In general, when the three points touched are unequally spaced, we get curious curved nodal lines of which figure 17 is an example. These are clearly due to superposition of the two normal modes with the nodal diameters not coincident but inclined to each other.

Superposition Figures of the Fifth Harmonic

From the fact that the fifth harmonic is obtained when four points on the drum-head are damped, it may be inferred that it arises from a super-position of at least two modes, namely, (a) one with four nodal diameters only, (b) one with two nodal diameters and a nodal circle. Experiments made at Calcutta in 1919 with a fine large Mridanga showed that in reality we have also a tinrd mode superposed on the above, namely, the mode with two nodal circles only. By touching the drum at suitable points, it was found possible to excite any of the foregoing three modes by itself, and obtain the relative sand figures, the pitch of the modes being the same in all the three cases. If it is possible to excite the drum in sustained vibrations of this frequency, a great variety of superposition figures should evidently be capable” of being obtained.

The paper gives a detailed description of the results obtained by the author in the year 1919 which showed that in: the Indian musical drums we have a circular, drum-head which is loaded and damped in such a manner that all the overtones above the ninth are suppressed and these nine are grouped in such a manner as to give a succession of five tones in harmonic sequence. The vibrations of the drum-head present a striking analogy to the case of a stretched string giving one or the other of its first harmonics; the drum-head divides up into 1,2,3,4 or 5 sections giving the respective harmonies. The tinrd, fourth and fifth harmonics are obtained by super-position of 2,2 and 3 respectively of the normal modes of vibration. The corresponding superposed forms of vibration are readily obtained and demonstrated by means of sand figures. Numerous figures illustrate the paper.

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