Background Of European Music English Language Essay

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Jeans has mentioned that the physics of music is one of the interesting branches of musicology. Books on the physics of music written with the background of European music are in existence. Some scholarly musicians having written with the background of Indian music too. First one thing it is sure that music is also a scientific subject which has some basic qualities of physics. The basis of all fine arts is an attempt to express beauty in from or colour or sounds. A cultivated mind found in a canalized human being can discover this beauty as it is able to discover truth, goodness and godliness. Although the appeal of all arts is primarily to induce our emotions we cannot minimized if one wants to appreciate every work of art and its beauty that particular art should be considered from two points of view, the one is objective and the other is subjective. Similarly Olson (1976) has mentioned that except in Music the objective side can be studied leisurely in all other arts are static, Music is the only art which is only dynamic. The volatile character of the sound which is produced through music makes it difficult to the perceived leisurely. One has to hear the same piece many times in order to comprehend the objective side of it happen music many a times. It is only by giving the closest attention and concentration regarding the piece as a whole one can derive the maximum aesthetic enjoyment of the same. There are three levels of appreciation for any art, and they are the physical, the physiological and the psychical. Carnatic music will easily anyone that rhythm involves considerable thought and alertness in execution.

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According to Thomas, Rossing and Wheeler (2002) European music shows very striking and characteric difference from carnatic music. But there is one fundmental principle under lying in both the music each limits itself to a definite scale or series of notes and its music intervals. There were originally seven notes to the Octave and they have been increased in course of time European music has stopped with twelve notes to the octave with major and minor Peales. That has left their composers free to modulate into a different key for the sake of variety and return again to the original key. In their desire for extend the possibilities of modulation they have abandoned correct intonation. On the other hand, carnatic music has not only kept correct intonation but has included wide range microtonal variations in pitch. It has developed the melody model or ragas to an astounding degree. The development of sales and the devices adopted for modulation have all been well understood on the objective side. But the same tactics cannot be saved of the various ragas of carnatic music there is ample scope for understanding the object side of these ragas. It this study is not done the essence of carnatic music will be missed. In European music an understanding of the phenomena of consonance and dissonance on the objective side has made the choice of consonant that the present development in European music would not have been possible without this knowledge. Hopkin (1996) has described that the objective side of rhythm has been understood in both the rhythm of music. Complicated time-measures are used frequently in Carnatic music. All these have been systematically and methodically classified. The well known treatise named Sangeetha Ratnakara, on Carnatic music explains 120 types of different time measures. It is said many of them have become obsolete now. The prevailing classification has seven talas each of which has fine jatis or classes. The five jatis are further classified according to the number of aksharas. To play all these time measures the technique of percussion instruments has been developed to an enormous degree in Carnatic music. Thus rhythm is more intricate in Carnatic music than in European music.

A comparative study of the objective side of the three elements melody harmony and rhythm in both the systems of music will certainly contribute to the improvement of both. The sound producing parts of a musical instrument will help us in make in general perform two distinct functions. Some parts are designed for the musical vibrations and amplify them by resonance. The resulting sounds depend largely upon the Bind of sympathy existing between the variation parts of the instruments (Kahrs and Brandenburg, 1998).

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According to Ballintijn and Cate (1998) Western scientists particularly D.C. Miller and helonholtize have instruments. But in india that line of research in yet to the pursued. Almost all our instrument in use now has not been altered in the smallest detailed from their ancient forms. The manufacture of the instruments is still in the hands of ordinary cabinet makers who are ignorant of elementary acoastoes considerable improvement is possible both in the shape and the material of them instruments which in bound to react favourable on the quality of the sounds elicited.

1.2 The role of Ear

Benzon (2001) has mentioned that the mechanism of hearing is a subject which touches various branches of science. The ear as a physical instrument possesses remarkable characteristics; It stands foremost among all the receivers of sounds. Its powers of analysis its sensitiveness over wide ranges of intensity and frequency, the perception of direction by means of both the ears have been very helpful in receiving sound. Of all the special feature of the human ear, its power of analyzing a complese note stands supreme. Ear is one of the delicate organs just like the body mouth, eye and nose. The ear has three parts, namely

Outer ear

Middle ear and inner ear

In the outer ear we have first the pinna, the Auditory canal and the Arum skin. Pinna is found only in the ears of men and animals. It is not found in birds. On the sides of Auditing canal is found a kind of wax which prevents ants and other insects from entering the ear. At the end of Auditing canal is found the Drum skin. In the middle ear we find three connected bones called oscicles. One end of the three bones is connected with Drum skin and another end with the inner ear. The range of pitch to which the human ear is sensitive depends upon the individual. It also various with time for the same person. The highest pitch andible is also somewhat uncertain, the lowest frequency sensed as a note is given variously, but may be taken as about 16 vibrations per second. For musical purposes frequencies ranging from 40 to 5000 per second alone are used. Making a comparison with the other sense organ namely the eye it can be observed that while the eye sees only octave the ear hears about eleven octaves of which seven are used in music Thompson, Schellenberg and Husain, 2004).

According to Fee et al. (1998) the ear can manufacture certain tones quite apart from those present outside the ear had been found out as early as the eighteenth century. It was jactini, a famous friend violinist, who first noticed that when two notes a fifth apart in the middle of the scale were played on the organs, together with these notes a new low note was heard whose frequency was the difference of the two higher pitched notes. This is now knoare as the differential tone. Helmholtz further discovered another tone whose frequency is the sum of the original frequencies. These subjective combinational tones are attributed to the non-linear characteristic of the ear. In wind instruments flutes with thicker walls those with longer circumference give lower note and flute with thinner walls and smaller circumference give higher pitch. In percussion instruments, those with thinner skins will give lower notes. Thus is the pitch of sound produced? The same note may be produced with varying loudness.

Juslin and Laukka (2003) has mentioned that these lows can be mathematically expressed as follows.

1f= the frequency of the fundamental note

L= the length of vibrating segment of string

T= the tension of the string (T=mg)

{m = mass of the land, g = acceleration due to gravity}

M = linear density or mass per unit length of the string

Tension on 'T' and linear density (m) being constant, the fundamental frequency of the note 'n' is inversely proportional to the length of the sting.

N is X 1/1 (0s)

nL = K1/L where k is a constant (0s)

nL = K = constant.

'L' and m being kept constant 'n' is directly proportional to T.

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N is x √T (or)

n2 xT (or)

n2/T = constant.

'L' and 'T' being constant,

'n' is inversely proportional to M

I,e n1/m or n2m = constant

1.3 The role of larynx and the Ear in the sphere of Physics and Music

Vocal Chords or Larynx is the organ of speech for man. This can be compared to a musical instrument in fact the vocal chords is considered to be the best of all the instruments in the world. The organ by which voice is produced is known as lazynse. It is popularly known as the voice box. This is situated between the back of the mouth and the top of the wind pipe thus forming the upper part of the tube of communication between external air and the lungs. In general the great is called larynx. The muscles of the throute are formed in such a way that the air breathe in goes through the sound box into the wind pipe, while the food we take reaches the food canal. The air we breathe out comes through the sound box and let off through the threat mouth and nose. The sound box is made of four tender boxes. Inside this box there are two soft skins known as vocal chords. These are pulled by some muscles in such a way that the vocal chords serve like a door to the wind pipe. The vocal chords are longer presides for the teenagers the voice cards will be more tender so that they can sing in shrill voice (Roads and Straws, 1985).

According to Markel and Gray (1976) the voice of a person will depend not only on the nature of his larynx, but also on the shape of the mouth and nose. We will be able to produce sound been without the movement of the larynx. When we speak very softly, we produce the sound first by the movement of the lift. When a person pings, the vibrations of his voice coming from the larynx get strengthened as they pass through the throat, mouth and nose. Man's voice is able to produce the sound which no other musical instrument can produce. The German scientist Hemholtz has made an experiment and has found out how man in able to distinguish between one sound and another.

Robb and Saxman (1988) have described that in the vocal organ the lungs act as a kind of bellows increasing the pressure of the air below the cords in vibration. The vibrations are then communicated in turn to the resonant air charities formed by the larynx, the front and back parts of the moth separated by the frequently is made possible by the muscles which control the width of the glottis and the tension on the cords. Alternations in the intensity of the glottis. The vocal mechanism has been described by some physicists as resembling a string instrument and by others it is likened to a wind instrument. In fact it is a combination of both in that it resembles a string instrument in its vibrator and a wind instrument in its generator. More recently it has been found out that the function of vocal cord is to induce vortex formation in the stream of air as it passes through the gloltis and thus generate the sound. The shrill notes which we hear when the wind blows over the telegraph wire or through stalks of corn or blades of grass are cited as examples. Anyway the vibration of the vocal cords plays an important role in sound production.

The familiar phenomenon of the braking of a boy's voice in his teens is due to the rapid growth of t he larynx and the corresponding increase in the length of the cords. The best human voice has a range of three and a half octaves although in practice few people are able to sing in more than two octaves. The chest Register "and" the Head Register are the recognized voice of men Between these two kinds there is a break in the voice which is disguised by practice. At lower frequency the chest voice is used. It is found then that the slit between the cords is very narrow and long and the cords vibrate as a whole. For the higher notes the head voice is employed and in this case it is found that the vocal cords are wide apart with only their innermost margins vibrating (McKinney, 1982).

Mithen (2005) has mentioned that voice training is an art by itself though its significance has not been fully recognized by our musicians. It is well noticed that among distinguished vocalists only a few have been endowed with good voices. A good voice in the first requisite for a localist. It is quite true that not everybody is lucky enough to possers a rich voice. But even ordinary voices can be improved considerably through proper training. Knowledge of the physics of the human voice will be very much useful regarding this training of voice. In training the voice, proper control over the muscles regulating the air stream and those concerned in the mechanism of the larynx should be obtained first. One has a certain amount of direct control over the muscles which regulate breathing and also over those which cause movements of the tongue, lip and soft palate etc. The muscles of the larynx cannot be exercised independently. The control over them that makes singing possible depends entirely on the ear and the brain centers connected with it. It is because of this the deaf children can be trained to speak but not to sing. This is also obvious from this fact that one is able to sing range perfectly by constant hearing alone without learning the theory of the combinations of notes. It the breathing muscles are not properly controlled, extreme unevenness of the will result. This research intends to investigate in detail the physics associated with musical instruments.

1.4 Problem Identified:

The physics behind the musical instruments is simply astonishing. The sounds coming from musical instruments are available because of standing waves which arise from constructive interference between waves migrating in both directions along a tube or a string. Music and physics are related closely because music is a sound and the sound is a branch of physics but they are also linked in another way. Both are innovative endeavours highly. Heisenberg and Schroedinger gave us quantum theory and Einstein gave us quantum mechanics whereas Chopin gave us an array of wonderful pieces of piano and Beethoven gave us numerous beautiful symphonies. Thus music and physics are both mind products. Physics may form an image of complicated and critical maths for some people but for many people it is an enjoyable and delightful endeavour. This research tries to further explore the relationship by identifying the physics in musical instruments. This study will particularly place its focus on string, wind and percussion instruments.

1.4 Significance of the Study:

This study describes about the stringed instruments, wind instruments and percussion instruments. The string instruments must be played without or with a bow. These instruments generate sounds when plucked, slapped, strummed or struck. The easiest way a string can sound in a standing wave condition is with the two needed nodes at the string ends and an antinode in the mid of the string. All the instruments of string are possible in different sizes that are proportional to child sized bodies. Bowing permits very sustained and big notes with interesting dynamics for people to hear the music. Whereas wind instruments generates sound by a vibrating air column either using the lips of a musician or using a reed. It is categorized into two types brasswinds and woodwinds. The wind instruments are comprised of animal horns in ancient civilization which generates a warning signal. Lastly percussion instruments are struck, shaken or scraped to make a music. Percussion instruments can add intensity and fun to a performance. The percussion instruments have varied pitches that plays as mallets strike the keys. Percussion instruments always generate a better rhythm sense. Thus this study describes about these 3 major instruments which produce various sounds.

1.5 Aims of the study:

The aim of the study is to understand how the concept of physics could be related to musical instruments.

Objectives of the study:

To understand the basic concepts of waves and sounds.

To study about the types of musical scales.

To study the physics of stringed instruments.

To study the physics of wind instruments.

To study the physics of percussion instruments

1.6 Chapterisation Plan:

This study is comprised of the following seven chapters. The following is a summarization of each chapter's contents present in this study

Chapter 1: Introduction: This chapter explains about the background, justification, problem statement, objectives and significance of the study.

Chapter 2: Review of literature: This chapter reviews the literature related to physics in music as well as musical instruments.

Chapter 3: Methodology: This chapter gives a summary of research strategy, research design, sampling plan, and sampling design, types of data and techniques of data analysis and interpretation adopted by researcher to organize this study.

Chapter 4: Physics of string instruments: This chapter explains the physics involved behind the operation of musical string instruments.

Chapter 5: Physics of percussion instruments: This chapter explains the physics involved behind the operation of musical percussion instruments.

Chapter 6: Physics of wind instruments: This chapter explains the physics involved behind the operation of musical wind instruments.

Chapter 7: Conclusion: This chapter describes an overview of findings acquired in the section of data analysis along with the study's conclusion followed by strategies for development and suggestions for future research.

Besides, this research study has a section for bibliography consisting of sources that were used in conducting the research followed by the section of an appendix that has details like tools if any used in the research.

Chapter 2

Review of literature

2.1 Free and forced vibrations of simple systems

Mechanical, acoustical, or electrical vibrations are the sources of sound in musical instruments. Some familiar examples are the vibrations of strings (violin, guitar, piano, etc), bars or rods (xylophone, glockenspiel, chimes, and clarionet reed), membranes (drums, banjo), plates or shells (cymbal, gong, bell), air in a tube (organ pipe, brass and woodwind instruments, marimba resonator), and air in an enclosed container (drum, violin, or guitar body). In most instruments, sound production depends upon the collective behavior of several vibrators, which may be weakly or strongly coupled together. This coupling, along with nonlinear feedback, may cause the instrument as a whole to behave as a complex vibrating system, even though the individual elements are relatively simple vibrators (Hake and Rodwan, 1966).

In the first seven chapters, we will discuss the physics of mechanical and acoustical oscillators, the way in which they may be coupled together, and the way in which they radiate sound. Since we are not discussing electronic musical instruments, we will not deal with electrical oscillators except as they help us, by analogy, to understand mechanical and acoustical oscillators.

According to Iwamiya, Kosygi and Kitamura (1983) many objects are capable of vibrating or oscillating. Mechanical vibrations require that the object possess two basic properties: a stiffness or spring like quality to provide a restoring force when displaced and inertia, which causes the resulting motion to overshoot the equilibrium position. From an energy standpoint, oscillators have a means for storing potential energy (spring), a means for storing kinetic energy (mass), and a means by which energy is gradually lost (damper). vibratory motion involves the alternating transfer of energy between its kinetic and potential forms. The inertial mass may be either concentrated in one location or distributed throughout the vibrating object. If it is distributed, it is usually the mass per unit length, area, or volume that is important. Vibrations in distributed mass systems may be viewed as standing waves. The restoring forces depend upon the elasticity or the compressibility of some material. Most vibrating bodies obey Hooke's law; that is, the restoring force is proportional to the displacement from equilibrium, at least for small displacement.

Simple harmonic motion in one dimension:

Moore (1989) has mentioned that the simplest kind of periodic motion is that experienced by a point mass moving along a straight line with an acceleration directed toward a fixed point and proportional to the distance from that point. This is called simple harmonic motion, and it can be described by a sinusoidal function of time, where the amplitude A describes the maximum extent of the motion, and the frequency f tells us how often it repeats.

The period of the motion is given by

That is, each T seconds the motion repeats itself.

Sundberg (1978) has mentioned that a simple example of a system that vibrates with simple harmonic motion is the mass-spring system shown in Fig.1.1. We assume that the amount of stretch x is proportional to the restoring force F (which is true in most springs if they are not stretched too far), and that the mass slides freely without loss of energy. The equation of motion is easily obtained by combining Hooke's law, F = -Kx, with Newton's second law, F = ma =. Thus,

and

Where

=

The constant K is called the spring constant or stiffness of the spring (expressed in Newton's per meter). We define a constant so that the equation of motion becomes

This well-known equation has these solutions:

)

Figure 2.1: Simple mass-spring vibrating system

Source: Cremer, L., Heckl, M., Ungar, E (1988), "Structure-Borne Sound," 2nd edition, Springer Verlag

Figure 2.2: Relative phase of displacement x, velocity v, and acceleration a of a simple vibrator

Source: Campbell, D. M., and Greated, C (1987), The Musician's Guide to Acoustics, Dent, London

or

From which we recognize as the natural angular frequency of the system.

The natural frequency fo of our simple oscillator is given by and the amplitude by or by A; is the initial phase of the motion. Differentiation of the displacement x with respect to time gives corresponding expressions for the velocity v and acceleration a (Cardle et al, 2003):

,

And

.

Ochmann (1995) has mentioned that the displacement, velocity, and acceleration are shown in Fig. 1.2. Note that the velocity v leads the displacement by radians (90), and the acceleration leads (or lags) by radians (180). Solutions to second-order differential equations have two arbitrary constants. In Eq. (1.3) they are A and; in Eq. (1.4) they are B and C. Another alternative is to describe the motion in terms of constants x0 and v0, the displacement and velocity when t =0. Setting t =0 in Eq. (1.3) gives and setting t = 0 in Eq. (1.5) gives From these we can obtain expressions for A and in terms of xo and vo:

,

and

Alternatively, we could have set t= 0 in Eq. (1.4) and its derivative to obtain B= x0 and C= v0/ from which

.

2.3 Complex amplitudes

According to Cremer, Heckl and Ungar (1990) another approach to solving linear differential equations is to use exponential functions and complex variables. In this description of the motion, the amplitude and the phase of an oscillating quantity, such as displacement or velocity, are expressed by a complex number; the differential equation of motion is transformed into a linear algebraic equation. The advantages of this formulation will become more apparent when we consider driven oscillators.

This alternate approach is based on the mathematical identity where j =. In these terms,

Where Re stands for the "real part of". Equation (1.3) can be written as,

Skrodzka and Sek (2000) has mentioned that the quantity is called the complex amplitude of the motion and represents the complex displacement at t=0. The complex displacement is written

The complex velocity and acceleration become

Desmet (2002) has mentioned that each of these complex quantities can be thought of as a rotating vector or phase rotating in the complex plane with angular velocity, as shown in Fig. 1.3. The real time dependence of each quantity can be obtained from the projection on the real axis of the corresponding complex quantities as they rotate with angular velocity

Figure 2.3: Phase representation of the complex displacement, velocity, and acceleration of a linear oscillator

Source: Bangtsson E, Noreland D and Berggren M (2003), Shape optimization of an acoustic horn, Computer Methods in Applied Mechanics and Engineering, 192:1533-1571

2.4 Continuous systems in one dimension

Strings and bars

This section focuses on systems in which these elements are distributed continuously throughout the system rather than appearing as discrete elements. We begin with a system composed of several discrete elements, and then allow the number of elements to grow larger, eventually leading to a continuum (Karjalainen and Valamaki, 1993).

Linear array of oscillators

According to Mickens (1998) the oscillating system with two masses in Fig. 1.20 was shown to have two transverse vibrational modes and two longitudinal modes. In both the longitudinal and transverse pairs, there is a mode of low frequency in which the masses move in the same direction and a mode of higher frequency in which they move in opposite directions. The normal modes of a three-mass oscillator are shown in Fig. 2.1. The masses are constrained to move in a plane, and so there are six normal modes of vibration, three longitudinal and three transverse. Each longitudinal mode will be higher in frequency than the corresponding transverse mode. If the masses were free to move in three dimensions, there would be 3*3 =9 normal modes, three longitudinal and six transverse.

Increasing the number of masses and springs in our linear array increases the number of normal modes. Each new mass adds one longitudinal mode and (provided the masses move in a plane) one transverse mode. The modes of transverse vibration for mass/spring systems with N=1 to 24 masses are shown in Fig. 2.2; note that as the number of masses increases, the system takes on a wavelike appearance. A similar diagram could be drawn for the longitudinal modes.

Figure 2.4: Normal modes of a three-mass oscillator. Transverse mode (a) has the lowest frequency and longitudinal mode (f) the highest

Source: Jaffe, D and Smith, J (1983), "Extension of the Karplus-Strong

plucked string algorithm," CMJ 7:2, 43-45

Figure 2.5: Modes of transverse vibration for mass/spring systems with different numbers of masses. A system with N masses has N modes

Source: Beranek L (1954), Acoustics. McGraw-Hill, New York

As the number of masses in our linear system increases, we take less and less notice of the individual elements, and our system begins to resemble a vibrating string with mass distributed uniformly along its length. Presumably, we could describe the vibrations of a vibrating string by writing N equations of motion for N equality spaced masses and letting N go to infinity, but it is much simpler to consider the shape of the string as a whole (Bogoliubov, and Mitropolsky, 1961).

Standing waves

Consider a string of length L fixed at x=0 and x= L. The first condition y (0,t) = 0 requires that A = -C and B = -D in Eq. (2.9), so

Using the sum and difference formulas, sin(xy) = sin x cos y cos x sin y and cos(x

Y = 2A sin kx cos

= 2[A cos

The second condition y (L, t) =0 requires that sin kL =0 or . This restricts to values Thus, the string has normal modes of vibration (O'brien, Cook and Essl, 2001):

These modes are harmonic, because each fn is n times f1= c/2L.

The general solution of a vibrating string with fixed ends can be written as a sum of the normal modes:

and the amplitude of the nth mode is. At any point

Alternatively, the general solution could be written as

Where Cn is the amplitude of the nth mode and is its phase (Keefe and Benade, 1982).

2.5 Energy of a vibrating string

McIntyre et al (1981) has mentioned that when a string vibrates in one of its normal modes, the kinetic and potential energies alternately take on their maximum value, which is equal to the total energy. Thus, the energy of a mode can be calculated by considering either the kinetic or the potential energy. The maximum kinetic energy of a segment vibrating in its nth mode is:

Integrating over the entire length gives

The potential and kinetic energies of each mode have a time average value that is En/2. The total energy of the string can be found by summing up the energy in each normal mode:

Plucked string: time and frequency analyses

According to Laroche and Jot (1992) when a string is excited by bowing, plucking, or striking, the resulting vibration can be considered to be a combination of several modes of vibration. For example, if the string is plucked at its center, the resulting vibration will consist of the fundamental plus the odd-numbered harmonics. Fig. 2.5 illustrates how the modes associated with the odd-numbered harmonics, when each is present in the right proportion; add up at one instant in time to give the initial shape of the center-plucked string. Modes 3,7,11, etc., must be opposite in phase from modes, 1, 5, and 9 in order to give maximum displacement at the center, as shown at the top. Finding the normal mode spectrum of a string given its initial displacement calls for frequency analysis or fourier analysis.

Figure 2.6: Time analysis of the motion of a string plucked at its midpoint through one half cycle. Motion can be thought of as due to two pulses travelling in opposite directions

Source: Gokhshtein, A. Y (1981), ''Role of airflow modulator in the excitation of sound in wind instruments,'' Sov. Phys. Dokl. 25, 954-956

Since all the modes shown in Fig.2.6 have different frequencies of vibration, they quickly get out of phase, and the shape of the string changes rapidly after plucking. The shape of the string at each moment can be obtained by adding the normal modes at that particular time, but it is more difficult to do so because each of the modes will be at a different point in its cycle. The resolution of the string motion into two pulses that propagate in opposite directions on the string, which we might call time analysis, is illustrated in Fig.2.6 if the constituent modes are different, of course. For example, if the string is plucked 1/5 of the distance from one end, the spectrum of mode amplitudes shown in Fig. 2.7 is obtained. Note that the 5th harmonic is missing. Plucking the string ¼ of the distance from the end suppresses the 4th harmonic, etc. (Pavic, 2006).

Roads (1989) have mentioned that a time analysis of the string plucked at 1/5 of its length. A bend racing back and forth within a parallelogram boundary can be viewed as the resultant of two pulses (dashed lines) travelling in opposite directions. Time analysis through one half cycle of the motion of a string plucked one-fifth of the distance from one end. The motion can be thought of as due to two pulses moving in opposite directions (dashed curves). The resultant motion consists of two bends, one moving clockwise and the other counter-clockwise around a parallelogram. The normal force on the end support, as a function of time, is shown at the bottom. Each of these pulses can be described by one term in d'Alembert's solution [Eq. (2.5)].

Each of the normal modes described in Eq. (2.13) has two coefficients and Bn whose values depend upon the initial excitation of the string. These coefficients can be determined by Fourier analysis. Multiplying each side of Eq. (2.14) and its time derivative by sin mx/L and integrating from 0 to L gives the following formulae for the Fourier coefficients:

By using these formulae, we can calculate the Fourier coefficients for the string of length L is plucked with amplitude h at one fifth of its length as shown in figure.2.8 time analysis above. The initial conditions are:

y (x,0) = 0

y (x,0) = 5h/L .x, 0 x L/5,

= 5h/4 (1-x/L), L/5 x L.

Using the first condition in first equation gives An=0. Using the second condition in second equation gives

=

=

The individual Bn's become: B1 =0.7444h, B2 =0.3011h, B3 =0.1338h, B4 =0.0465h, B5 =0, B6= -0.0207h, etc. Figure 2.7 shows 20 log for n=0 to 15. Note that Bn=0 for n=5, 10, 15, etc., which is the signature of a string plucked at 1/5 of its length (Shabana, 1990).

Bowed string

Woodhouse (1992) has mentioned that the motion of a bowed string has interested physicists for many years, and much has been written on the subject. As the bow is drawn across the string of a violin, the string appears to vibrate back and forth smoothly between two curved boundaries, much like a string vibrating in its fundamental mode. However, this appearance of simplicity is deceiving. Over a hundred years ago, Helmholtz (1877) showed that the string more nearly forms two straight lines with a sharp bend at the point of intersection. This bend races around the curved path that we see, making one round trip each period of the vibration.

According to Chaigne and Doutaut (1997) to observe the string motion, Helmholtz constructed a vibration microscope, consisting of an eyepiece attached to a tuning fork. This was driven in sinusoidal motion parallel to the string, and the eyepiece was focused on a bright-colored spot on the string. When Helmholtz bowed the string, he saw a Lissajous figure. The figure was stationary when the tuning fork frequency was an integral function of the string frequency. Helmholtz noted that the displacement of the string followed a triangular pattern at whatever point he observed it, as shown in Fig.2.7:

Figure 2.7: Displacement and Velocity of a bowed string at three positions along the length: a) at x = L/4; b) at the center, and c) at x = 3L/4

Source: Smith, J (1986), "Efficient Simulation of the Reed-Bore and Bow-String Mechanisms," Proc. ICMC, 275-280

The velocity waveform at each point alternates between two values. Other early work on the subject was published by Krigar-Menzel and Raps (1891) and by Nobel laureate C. V. Raman (1918). More recent experiments by Schelleng (1973), McIntyre, et al. (1981). Lawergren (1980), Kondo and Kubata (1983), and by others have verified these early findings and have greatly added to our understanding of bowed strings. An excellent discussion of the bowed string is given by Cremer (1981). The motion of a bowed string is shown in Fig.2.8:

Figure 2.8: Motion of a bowed string. A) Time analysis of the motion showing the shape of the string at eight successive times during the cycle. B) Displacement of the bow (dashed line) and the string at the point of contact (solid line) at successive times. The letters correspond to the letters in (A)

Source: McIntyre, M., Woodhouse, J (1979), "On the Fundamentals of Bowed-String Dynamics," Acustica 43:2, 93-108

Dobashi, Yamamoto and Nishita (2003) have described that a time analysis in the above figure 2.8 (A) shows the Helmholtz-type motion of the string; as the bow moves ahead at a constant speed, the bend races around a curved path. Fig. 2.8 (B) shows the position of the point of contact at successive times; the letters correspond to the frames in Figure 2.8(A). Note that there is a single bend in the bowed string. Whereas in the plucked string (fig. 2.8), we had a double bend. The action of the bow on the string is often described as a stick and slip action. The bow drags the string along until the bend arrives [from (a) in figure 2.8 (A)] and triggers the slipping action of the string until it is picked up by the bow once again [frame (c)]. From (c) to (i), the string moves at the speed of the bow. The velocity of the bend up and down the string is the usual . The envelope around which the bend races [the dashed curve in Figure 2.8 (A)] is composed of two parabolas with maximum amplitude that is proportional, within limits, to the bow velocity. It also increases as the string is bowed nearer to one end.

2.6 Vibration of air columns:

According to Moore and Glasberg (1990) the familiar phenomenon of the sound obtained by blowing across the open and of a key shows that vibrations can be set up in an air column. An air column of definite length has a definite natural period of vibrations. When a vibrating tuning fork is held over a tall glass is pured gradually, so as to vary the length of the air column, a length can be obtained which will resound loudly to the note of the tuning fork. Hence it is the air column is the same as that of the tuning fork.

A vibration has three important characteristics namely

Frequency

Amplitude

Phase

2.6.1 Frequency:-

Frequency is defined as the number of vibration in one second. The unit is Hertz. It is normally denoted as HZ. Thus a sound of 1000 HZ means 1000 vibrations in one second. A frequency of 1000 HZ can also be denoted as 1 KHZ. If the frequency range of audio equipment is mentioned as 50 HZ to 3 HZ it means that audio equipment will function within the frequency range between 50HZ and 3000 HZ.

2.6.2 Amplitude:-

Amplitude is defined as the maximum displacement experienced by a particle in figure will help to understand amplitude. Let us consider two vibrating bodies having the same frequency but different amplitudes. The body vibrating with more amplitude will be louder than the body vibrating with less amplitude. The following figure represents two vibrating bodies having the same frequency but different amplitudes (Takala and Hahn, 1992).

2.6.3 Phase:-

Phase is defined as the stage to which a particle has reached in its vibration. Initial phase means the initial stage from which the vibration starts. The following will help to understand the concept of phase. From the source travels in the form of waves before reaching the ear sound cannot travel in vacuum. Sound needs medium for its travel. The medium may be a solid or liquid or gas (Brown and Vaughn, 1993).

Support a glass tube open at both ends in a vertical position, with its lower and dipping into water contained in a wider cylinder. Hold over the upper end of the tube a vibrating tuning form. Adjust the reinforcement of the sound is obtained. Adjust the distance of the air column till we get actually the resonance or sympathetic note. Repeat the adjustments and take the average of the results from the observation. It will be found from the repeated experiments, that the longer the air column is produced when the tuning fork becomes identical.

Vibration of air column in a tube open at both ends:-

O'brien, Shen and Gatchalian (2002) have described that if they think of an air column in a tube open both ends, and try to imagine the ways in which it can vibrate; we shall readily appreciate that the ends will always be antinodes, since here the air is free to move. Between the antinodes there must be at least one node, and the ends, the moving air is either moving towards the center from both ends or away from the centre at both ends. Thus the simplest kind of vibration has a node at the centre and antinodes at the two ends. This can be mathematically expressed as follows:

Wave length of the simplest kind of vibration is four times the distance from node to antinode - 2L where L is the length of the pipe.

Vibration of air column in a tube closed at one end:

The distance from node to antinode in this case is L, the whole length of the pipe, the wavelength is therefore = 4L.

2.7 Resonance-sympathetic vibration

Sloan, Kautz and Synder (2002) have described that everybody which is capable of vibration has natural frequency of its own. When a body is made to vibrate at its neutral frequency, it will vibrate with maximum amplitude. Resonance is a phenomenon in which a body at rest is made to vibrate by the vibrations of another body whose frequency is equal to that of the natural frequency of the first. Resonance can also be called sympathetic vibrations. The following experiment will help to understand resonance:

Consider two stretched stings A and B on a sonometer. With the help of a standard tuning form we can adjust their vibrating lengths [length between the bridges] to have the same frequency. Thus we can place a few paper riders on string B and pluck string A to make it vibrate. The string B will start vibrate and paper riders on it will flutter vigorously and sometimes A can be stopped simply by touching it. Still the string B will continue to vibrate. The vibration in the string B is due to resonance and it can be called as sympathetic vibration. If instead of the fundamental frequency one of the harmonics of string B is equal to the vibrating frequency of string A then the string B will start vibrating at that harmonics frequency. But in the case of harmonics the amplitude of vibration will be less. In Tambura when the sarani is sounded the anusarani also, vibrates thus helping to produce a louder volume of sound. The sarani here makes the anusarani to vibrate. In all musical instruments the material, the shape of the body and enclosed volume of air make use of resonance to bring out increased volume and desired upper partials of harmonics.

2.8 Intonations

Spiegel and Watson (1984) have described that during the course of the history of music, several of music intervals were proposed aiming at a high degree of maturing consonance and dissonance played important role in the evolution of musical scales. Just intonation is the result of standardizing perfect intervals. Just Intonation is limited to one single-key and aims at making the intervals as accordant as possible with both one another and with the harmonics of the keynote and with the closely related tones. The frequency ratio of the musical notes in just Intonation is given below.

Indian note Western note Frequency ratio

r C 1

K2 D 9/8

f2 E 5/4

M1 F 4/3

P G 3/2

D2 A 5/3

N2 B 15/8

S C 2

Ward (1970) has mentioned that most of the frequency ratios are expressed is terms of comparatively small numbers. Constant harmonics are present when frequency ratios are expressed in terms of small numbers. The interval in frequency ratio are:

Between Madhya sthyai C[Sa] and Tara sthayi c[sa] is 2 [1*2=2].

Between Madhya sthyai C[Sa] and Madhya sthayi G[pa] is 3/2 [1*3/2=3/2].

Between Madhya sthayi D[Ri] and Madhya sthayi E[Ga] is 10/9 [9/8*10/9=5/4]

Between Madhya sthyai E[Ga] and Madhya sthayi F[Ma] is 16/15-[5/4*16/15=4/3].

Between Madhya sthyai F[Ma] and Madhya sthayi G[Pa] is 9/8-[4/3*9/8=3/2].

Between Madhya sthyai G[Pa] and Madhya sthayi A[Dha] is 10/9[3/2*10/9=5/3].

Between Madhya sthyai A[Dha] and Madhya sthayi B[Ni] is 9/8-[5/3*9/8=5/8].

Between Madhya sthyai Sa[C] and Ri2[D] there is a svarasthanam [CH]. Hence the interval between Sa[C] and Ri2[D] and Ga2[E] is known as a tone. But there is no svarasthanam [semitone] between Ga2[E] and Ma1[F]. Hence the interval between Ga[E] and Ma1[F] is known as a semitone. Between Pa[G] and Dha[A] we have a tone. Between mathya styayi Ni2[B] and Tara sthyai C[Sa] we have a semitone.

In just Intonation we find that tones are not all equal. But the semitones are equal. In just Intonation the modulation of key of musical notes will be difficult for example, if the keynote is changed from Sa[C] to Pa[G] then the frequency of etatusruthi Dhairatam [A] will change from 1.687, time the frequency of Sa[c]. A musical instrument tuned in just intonation to play sankarabaranam ragam cannot be used to play kalyani ragam. Hence the modulation of key of musical notes will be difficult in just Intonation (Doutaut , Matignon, and Chaigne, 1998).

Equal temperature

Lehr (1997) has described that the above mentioned problem in just Intonation can be solved in the Equal Temperament scale. In Equal temperament all the 12 music intervals in a sthayi [octave] are equal. The frequency ratios of semitones in Equal temperament scale was first calculated by the French Mathematician Mersenne and was published in 'Harmonic Universelle' in the year 1636. But it was not put into use till the latter half of seventeenth century. All keyboard instruments are tuned of Equal Temperature scale. Abraham pandithar strongly advocated Equal Temperament scale and in his famous music treatise 'karunamitha sagaram' he tried to prove that the Equal Temperament scale was in practice in ancient Tamil music.

A simple mathematical exercise will help to under the basis of Equal Temperament scale.

Equal Temperament

Madhya sthayi Sa[c] frequency ratio=1=2 ÌŠ.

Tara sthayi Sa[i] frequency ratio = 2=212/12=2.

Frequency ratios of 12 svarasthanams are given below.

S R1 R2 G1 G2 M1 M2 P D1 D2 N1 N2

↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓

20 21/12 22/12 23/12 24/12 25/12 26/12 27/12 28/12 29/12 210/12 2n/12

S

↓

212/12

All semitones are equal is Equal Temperament scale. Each represents the same frequency ratio 1.05877. The great advantage in Equal Temperament scale is that music can be played equal well in all keys. This means that any of the 12 semitones can be used as 'Sa' in a music instrument tuned to Equal Temperament scale. There is no need to change tuning every time the Raga is changed. Since keyboard instruments are pre-tuned instruments they follow Equal Temperament.

2.9 Production and transmission of sound:-

According to Boulanger (2000) the term sound is related to quite definite and specific sensation caused by the stimulation of the mechanism of the ear. The external cause of the sensation is also related to sound. Anybody in vibration is an external cause of the sensation. A veena [after plucking] or violin [after blowing] in a state of vibration is an external cause of the sensation. A body in a state of vibration becomes a source of sound. A vibration is a periodic to and fro motion about a fixed point

Iwamiya and Fujiwara (1985) have mentioned that the pitch of a musical sound produced on a wind instrument depends on the rate or frequency of the vibrations which cause the sound. In obedience to Nature's law, the column of air in a tube can be made to vibrate only at certain rates, therefore, a tube of any particular length can be made to produce only certain sounds and no others as long as the length of the tube is un-altered. Whatever the length of the tube, these various sounds always bear the same relationship one to the other, but the actual pitch of die series will depend on the length of the tube. The player on a wind instrument, by varying the intensity of the air-stream which he injects into the mouthpiece, can produce at will all or some of the various sounds which that particular length of tube is capable of sounding; thus, by compressing the air-stream with his lips he increases the rate of vibration and produces higher sounds, and by decompressing or slackening the intensity of the air-stream he lowers the rate of vibration and produces lower pitched sounds. In this way the fundamental, or lowest note which a tube is capable of sounding, can be raised becoming higher and higher by intervals which become smaller and smaller as they ascend. These sounds are usually called harmonics or upper partials, and it is convenient to refer to them by number, counting the fundamental as No. t, the octave harmonic as No. 2, and so on. The series of sounds available on a tube approximately 8 feet in length is as follows:

Tsingos et al (2001) has mentioned that a longer tube would produce a corresponding series of sounds proportionately lower in pitch according to its length, and on a shorter tube the same series would be proportionately higher. The entire series available on any tube is an octave lower than that of a tube half its length, or an octave higher than that of a tube double its length ; thus, the approximate lengths of tube required to sound the various notes C are as follows :

Fundamental Length of tube

C, 16 feet

C 8 ,.

c 4,,

c' 2,,

c'' I foot

c''' 1/2,,

Shonle and Horen (1980) has mentioned that the addition of about 6 inches to a 4-foot tube, of a foot to an 8-foot tube, or of 2 feet to a i6-foot tube, will give the series a tone lower (in B flat), and a proportionate shortening of the C tubes will raise the series a tone (D) ; on the same basis, tubes which give any F as the fundamental of a series must be about midway in length between those which give the C above and the C below as fundamental. Examples:

Trumpet (modern) in C-length about 4 feet

,, in F ,, ,, 6 ,,

,, (old) in C ,, ,, 8 ,,

Horn in F ,, ,, 12 ,,

,, ,, C ,, ,, 16 ,,

It will be noticed that the two lower octaves of the harmonic series are very sparsely provided with sounds; the third octave has little more than an arpeggio, and only in the fourth octave do the sounds run consecutively or scale-wise, while semitones only appear at the upper end of the fourth octave. The series, however, does not end there, and is continued in the fifth octave in semitone and smaller intervals, but however favorably proportioned a tube may be, the production of sounds above the 16th note becomes more and more difficult and uncertain, therefore it is only rarely that any wind instrument is required to produce these extremely high harmonics. The sounds of Nature's harmonic series do not all coincide exactly with the notes of the musical or tempered scale ; Nos. 7, II and I3, for example, are noticeably out of tune, but the remainder are either perfectly true or near enough in tune for practical purposes (Von Estorff, 2000).

For the present purpose it is not necessary to enquire into the number of vibrations per second which are required to sound any fundamental or its harmonics, nor need the exact lengths of tubes be taken into consideration. The player on a wind instrument does not count his vibrations nor does he measure his tube-lengths, but in order to understand wind instruments at all, to know their capabilities, their limitations, and why they are fingered and manipulated as they are, it is necessary to be familiar with the harmonic series and to be able to transpose it to suit any fundamental sound (Wand and Strasser, 2004).

Wu (2000) has described although the entire series of harmonics is nominally available on any tube, in actual practice the human lip can hardly vary the pressure to such an extent that all of them can be sounded on the same tube with the same mouthpiece. How many of them can be produced on one instrument depends mainly on the width of the tube in proportion to its length. A tube can be so wide or so narrow that no musical sound can be extracted from it; there must be some sort of reasonable proportion between width and length, and a tube which is not considerably longer than it is wide would be of no practical use as a musical instrument. A wide-bored tube will yield its fundamental more easily than a narrow one, and if not too wide can be made to sound its fundamental and a few of the lower harmonics, whereas a narrower tube can be made to sound up to the 16th note of the series but will then probably fail to sound the first two. For example, a primitive instrument made from an ox-horn or an elephant's tusk may be so short and so wide in proportion that it will only give one note (the fundamental), whereas an orchestral horn is long and proportionately so narrow that it will sound from the second note of the series up to even the I6th note.