Forced Vibrations Of Simple Systems English Language Essay

Mechanical, acoustical, or electrical quivers are the beginnings of sound in musical instruments. Some familiar illustrations are the quivers of strings fiddle, guitar, piano, etc, bars or rods xylophone, orchestral bells, bells, and clarionet reed, membranes ( membranophones, banjo ) , plates or shells ( cymbal, gong, bell ) , air in a tubing ( organ pipe, brass and woodwind instruments, marimba resonating chamber ) , and air in an enclosed container ( membranophone, fiddle, or guitar organic structure ) . In most instruments, sound production depends upon the corporate behaviour of several vibrators, which may be decrepit or strongly coupled together. This yoke, along with nonlinear feedback, may do the instrument as a whole to act as a complex vibrating system, even though the single elements are comparatively simple vibrators ( Hake and Rodwan, 1966 ) .

In the first seven chapters, we will discourse the natural philosophies of mechanical and acoustical oscillators, the manner in which they may be coupled together, and the manner in which they radiate sound. Since we are non discoursing electronic musical instruments, we will non cover with electrical oscillators except as they help us, by analogy, to understand mechanical and acoustical oscillators.

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Harmonizing to Iwamiya, Kosygi and Kitamura ( 1983 ) many objects are capable of vibrating or hovering. Mechanical quivers require that the object possess two basic belongingss: a stiffness or spring like quality to supply a restoring force when displaced and inactiveness, which causes the ensuing gesture to overshoot the equilibrium place. From an energy point of view, oscillators have a agency for hive awaying possible energy ( spring ) , a agency for hive awaying kinetic energy ( mass ) , and a agency by which energy is bit by bit lost ( damper ) . vibratory gesture involves the jumping transportation of energy between its kinetic and possible signifiers. The inertial mass may be either concentrated in one location or distributed throughout the vibrating object. If it is distributed, it is normally the mass per unit length, country, or volume that is of import. Vibrations in distributed mass systems may be viewed as standing moving ridges. The restoring forces depend upon the snap or the squeezability of some stuff. Most vibrating organic structures obey Hooke ‘s jurisprudence ; that is, the reconstructing force is relative to the supplanting from equilibrium, at least for little supplanting.

Simple harmonic gesture in one dimension:

Moore ( 1989 ) has mentioned that the simplest sort of periodic gesture is that experienced by a point mass traveling along a consecutive line with an acceleration directed toward a fixed point and relative to the distance from that point. This is called simple harmonic gesture, and it can be described by a sinusoidal map of clip, where the amplitude A describes the maximal extent of the gesture, and the frequence degree Fahrenheit tells us how frequently it repeats.

The period of the gesture is given by

That is, each T seconds the gesture repeats itself.

Sundberg ( 1978 ) has mentioned that a simple illustration of a system that vibrates with simple harmonic gesture is the mass-spring system shown in Fig.1.1. We assume that the sum of stretch ten is relative to the reconstructing force F ( which is true in most springs if they are non stretched excessively far ) , and that the mass slides freely without loss of energy. The equation of gesture is easy obtained by uniting Hooke ‘s jurisprudence, F = -Kx, with Newton ‘s 2nd jurisprudence, F = mom = . Thus,

and

Where

=

The changeless K is called the spring invariable or stiffness of the spring ( expressed in Newton ‘s per metre ) . We define a invariable so that the equation of gesture becomes

This well-known equation has these solutions:

)

Figure 2.1: Simple mass-spring vibrating system

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

Figure 2.2: Relative stage of supplanting ten, speed V, and acceleration a of a simple vibrator

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

or

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

The natural frequence field-grade officer of our simple oscillator is given by and the amplitude by or by A ; is the initial stage of the gesture. Differentiation of the supplanting ten with regard to clip gives corresponding looks for the speed V and acceleration a ( Cardle et al, 2003 ) :

,

And

.

Ochmann ( 1995 ) has mentioned that the supplanting, speed, and acceleration are shown in Fig. 1.2. Note that the speed V leads the supplanting by radians ( 90 ) , and the acceleration leads ( or lags ) by radians ( 180 ) . Solutions to second-order differential equations have two arbitrary invariables. In Eq. ( 1.3 ) they are A and ; in Eq. ( 1.4 ) they are B and C. Another option is to depict the gesture in footings of invariables x0 and v0, the supplanting and speed when T =0. Puting t =0 in Eq. ( 1.3 ) gives and puting t = 0 in Eq. ( 1.5 ) gives From these we can obtain looks for A and in footings of xo and vo:

,

and

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

.

2.3 Complex amplitudes

Harmonizing to Cremer, Heckl and Ungar ( 1990 ) another attack to work outing additive differential equations is to utilize exponential maps and complex variables. In this description of the gesture, the amplitude and the stage of an oscillatory measure, such as supplanting or speed, are expressed by a complex figure ; the differential equation of gesture is transformed into a additive algebraic equation. The advantages of this preparation will go more evident when we consider goaded oscillators.

This alternate attack is based on the mathematical individuality where J = . In these footings,

Where Re stands for the “ existent portion of ” . Equation ( 1.3 ) can be written as,

Skrodzka and Sek ( 2000 ) has mentioned that the measure is called the complex amplitude of the gesture and represents the complex supplanting at t=0. The complex supplanting is written

The complex speed and acceleration become

Desmet ( 2002 ) has mentioned that each of these complex measures can be thought of as a revolving vector or stage rotating in the complex plane with angular speed, as shown in Fig. 1.3. The existent clip dependance of each measure can be obtained from the projection on the existent axis of the corresponding complex measures as they rotate with angular speed

Figure 2.3: Phase representation of the complex supplanting, speed, and acceleration of a additive oscillator

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

2.4 Continuous systems in one dimension

String sections and bars

This subdivision focuses on systems in which these elements are distributed continuously throughout the system instead than looking as distinct elements. We begin with a system composed of several distinct elements, and so let the figure of elements to turn larger, finally taking to a continuum ( Karjalainen and Valamaki, 1993 ) .

Linear array of oscillators

Harmonizing to Mickens ( 1998 ) the hovering system with two multitudes in Fig. 1.20 was shown to hold two cross vibrational manners and two longitudinal manners. In both the longitudinal and cross braces, there is a manner of low frequence in which the multitudes move in the same way and a manner of higher frequence in which they move in opposite waies. The normal manners of a three-mass oscillator are shown in Fig. 2.1. The multitudes are constrained to travel in a plane, and so there are six normal manners of quiver, three longitudinal and three transverse. Each longitudinal manner will be higher in frequence than the corresponding transverse manner. If the multitudes were free to travel in three dimensions, there would be 3*3 =9 normal manners, three longitudinal and six transverse.

Increasing the figure of multitudes and springs in our additive array increases the figure of normal manners. Each new mass adds one longitudinal manner and ( provided the multitudes move in a plane ) one transverse manner. The manners of cross quiver for mass/spring systems with N=1 to 24 multitudes are shown in Fig. 2.2 ; note that as the figure of multitudes additions, the system takes on a crinkled visual aspect. A similar diagram could be drawn for the longitudinal manners.

Figure 2.4: Normal manners of a three-mass oscillator. Transverse manner ( a ) has the lowest frequence and longitudinal manner ( degree Fahrenheit ) the highest

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

plucked twine algorithm, ” CMJ 7:2, 43-45

Figure 2.5: Manners of cross quiver for mass/spring systems with different Numberss of multitudes. A system with N multitudes has N manners

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

As the figure of multitudes in our additive system additions, we take less and less notice of the single elements, and our system begins to resemble a vibrating twine with mass distributed uniformly along its length. Presumably, we could depict the quivers of a vibrating twine by composing N equations of gesture for N equality spaced multitudes and allowing N travel to eternity, but it is much simpler to see the form of the twine as a whole ( Bogoliubov, and Mitropolsky, 1961 ) .

Standing moving ridges

See a twine of length L fixed at x=0 and x= L. The first status Y ( 0, T ) = 0 requires that A = -C and B = -D in Eq. ( 2.9 ) , so

Using the amount and difference expressions, wickedness ( xy ) = sin x cos Y cos x wickedness Y and cos ( ten

Y = 2A wickedness kx cos

= 2 [ A cos

The 2nd status Y ( L, T ) =0 requires that wickedness kL =0 or. This restricts to values Thus, the twine has normal manners of quiver ( O’brien, Cook and Essl, 2001 ) :

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

The general solution of a vibrating twine with fixed terminals can be written as a amount of the normal manners:

and the amplitude of the n-th manner is. At any point

Alternatively, the general solution could be written as

Where Cn is the amplitude of the n-th manner and is its stage ( Keefe and Benade, 1982 ) .

2.5 Energy of a vibrating twine

McIntyre et Al ( 1981 ) has mentioned that when a twine vibrates in one of its normal manners, the kinetic and possible energies alternately take on their maximal value, which is equal to the entire energy. Therefore, the energy of a manner can be calculated by sing either the kinetic or the possible energy. The maximal kinetic energy of a section vibrating in its n-th manner is:

Integrating over the full length gives

The possible and kinetic energies of each manner have a clip mean value that is En/2. The entire energy of the twine can be found by summing up the energy in each normal manner:

Plucked twine: clip and frequence analyses

Harmonizing to Laroche and Jot ( 1992 ) when a twine is excited by bowing, tweaking, or contact, the ensuing quiver can be considered to be a combination of several manners of quiver. For illustration, if the twine is plucked at its centre, the ensuing quiver will dwell of the cardinal plus the odd-numbered harmonics. Fig. 2.5 illustrates how the manners associated with the odd-numbered harmonics, when each is present in the right proportion ; add up at one blink of an eye in clip to give the initial form of the center-plucked twine. Modes 3,7,11, etc. , must be opposite in stage from manners, 1, 5, and 9 in order to give maximal supplanting at the centre, as shown at the top. Finding the normal manner spectrum of a twine given its initial supplanting calls for frequence analysis or fourier analysis.

Figure 2.6: Time analysis of the gesture of a twine plucked at its center through one half rhythm. Gesture can be thought of as due to two pulsations going in opposite waies

Beginning: Gokhshtein, A. Y ( 1981 ) , ”Role of airi¬‚ow modulator in the excitement of sound in air current instruments, ” Sov. Phys. Dokl. 25, 954-956

Since all the manners shown in Fig.2.6 have different frequences of quiver, they rapidly get out of stage, and the form of the twine alterations quickly after tweaking. The form of the twine at each minute can be obtained by adding the normal manners at that peculiar clip, but it is more hard to make so because each of the manners will be at a different point in its rhythm. The declaration of the twine gesture into two pulsations that propagate in opposite waies on the twine, which we might name clip analysis, is illustrated in Fig.2.6 if the component manners are different, of class. For illustration, if the twine is plucked 1/5 of the distance from one terminal, the spectrum of manner amplitudes shown in Fig. 2.7 is obtained. Note that the 5th harmonic is losing. Plucking the twine A? of the distance from the terminal suppresses the 4th harmonic, etc. ( Pavic, 2006 ) .

Roads ( 1989 ) have mentioned that a clip analysis of the twine plucked at 1/5 of its length. A crook rushing back and Forth within a parallelogram boundary can be viewed as the end point of two pulsations ( dotted lines ) going in opposite waies. Time analysis through one half rhythm of the gesture of a twine plucked fifth part of the distance from one terminal. The gesture can be thought of as due to two pulsations traveling in opposite waies ( dotted curves ) . The attendant gesture consists of two decompression sicknesss, one traveling clockwise and the other counter-clockwise around a parallelogram. The normal force on the terminal support, as a map of clip, is shown at the underside. Each of these pulsations can be described by one term in d’Alembert ‘s solution [ Eq. ( 2.5 ) ] .

Each of the normal manners described in Eq. ( 2.13 ) has two coefficients and Bn whose values depend upon the initial excitement of the twine. These coefficients can be determined by Fourier analysis. Multiplying each side of Eq. ( 2.14 ) and its clip derivative by wickedness mx/L and incorporating from 0 to L gives the undermentioned expression for the Fourier coefficients:

By utilizing these expressions, we can cipher the Fourier coefficients for the twine of length L is plucked with amplitude H at one fifth of its length as shown in figure.2.8 clip analysis above. The initial conditions are:

Y ( x,0 ) = 0

Y ( x,0 ) = 5h/L.x, 0 ten L/5,

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

Using the first status in first equation gives An=0. Using the 2nd status in 2nd equation gives

=

=

The single 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 twine plucked at 1/5 of its length ( Shabana, 1990 ) .

Bowed twine

Woodhouse ( 1992 ) has mentioned that the gesture of a bowed twine has interested physicists for many old ages, and much has been written on the topic. As the bow is drawn across the twine of a fiddle, the twine appears to vibrate back and forth swimmingly between two curving boundaries, much like a twine vibrating in its cardinal manner. However, this visual aspect of simpleness is lead oning. Over a hundred old ages ago, Helmholtz ( 1877 ) showed that the twine more about forms two consecutive lines with a crisp crook at the point of intersection. This crook races around the curving way that we see, doing one unit of ammunition trip each period of the quiver.

Harmonizing to Chaigne and Doutaut ( 1997 ) to detect the twine gesture, Helmholtz constructed a quiver microscope, dwelling of an ocular attached to a tuning fork. This was driven in sinusoidal gesture analogue to the twine, and the ocular was focused on a bright-colored topographic point on the twine. When Helmholtz bowed the twine, he saw a Lissajous figure. The figure was stationary when the tuning fork frequence was an built-in map of the twine frequence. Helmholtz noted that the supplanting of the twine followed a triangular form at whatever point he observed it, as shown in Fig.2.7:

Figure 2.7: Supplanting and Velocity of a bowed twine at three places along the length: a ) at x = L/4 ; B ) at the centre, and degree Celsius ) at x = 3L/4

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

The speed wave form at each point alternates between two values. Other early work on the topic 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 apprehension of bowed strings. An first-class treatment of the bowed twine is given by Cremer ( 1981 ) . The gesture of a bowed twine is shown in Fig.2.8:

Figure 2.8: Gesture of a bowed twine. A ) Time analysis of the gesture demoing the form of the twine at eight consecutive times during the rhythm. B ) Supplanting of the bow ( dotted line ) and the twine at the point of contact ( solid line ) at consecutive times. The letters correspond to the letters in ( A )

Beginning: McIntyre, M. , Woodhouse, J ( 1979 ) , “ On the Fundamentalss of Bowed-String Dynamics, ” Acustica 43:2, 93-108

Dobashi, Yamamoto and Nishita ( 2003 ) have described that a clip analysis in the above figure 2.8 ( A ) shows the Helmholtz-type gesture of the twine ; as the bow moves in front at a changeless velocity, the crook races around a curving way. Fig. 2.8 ( B ) shows the place of the point of contact at consecutive times ; the letters correspond to the frames in Figure 2.8 ( A ) . Note that there is a individual crook in the bowed twine. Whereas in the plucked twine ( fig. 2.8 ) , we had a dual crook. The action of the bow on the twine is frequently described as a stick and slip action. The bow drags the twine along until the crook arrives [ from ( a ) in figure 2.8 ( A ) ] and triggers the stealing action of the twine until it is picked up by the bow one time once more [ frame ( degree Celsius ) ] . From ( degree Celsius ) to ( I ) , the twine moves at the velocity of the bow. The speed of the crook up and down the twine is the usual. The envelope around which the crook races [ the dotted curve in Figure 2.8 ( A ) ] is composed of two parabolas with maximal amplitude that is relative, within bounds, to the bow speed. It besides increases as the twine is bowed nearer to one terminal.

2.6 Vibration of air columns:

Harmonizing to Moore and Glasberg ( 1990 ) the familiar phenomenon of the sound obtained by blowing across the unfastened and of a key shows that quivers can be set up in an air column. An air column of definite length has a definite natural period of quivers. When a vibrating tuning fork is held over a tall glass is pured bit by bit, so as to change the length of the air column, a length can be obtained which will echo aloud to the note of the tuning fork. Hence it is the air column is the same as that of the tuning fork.

A quiver has three of import features viz.

Frequency

Amplitude

Phase

2.6.1 Frequency: –

Frequency is defined as the figure of quiver in one second. The unit is Hertz. It is usually denoted as HZ. Thus a sound of 1000 HZ means 1000 quivers in one second. A frequence of 1000 HZ can besides be denoted as 1 KHZ. If the frequence scope of audio equipment is mentioned as 50 HZ to 3 HZ it means that audio equipment will work within the frequence scope between 50HZ and 3000 HZ.

2.6.2 Amplitude: –

Amplitude is defined as the maximal supplanting experienced by a atom in figure will assist to understand amplitude. Let us see two vibrating organic structures holding the same frequence but different amplitudes. The organic structure vibrating with more amplitude will be louder than the organic structure vibrating with less amplitude. The undermentioned figure represents two vibrating organic structures holding the same frequence but different amplitudes ( Takala and Hahn, 1992 ) .

2.6.3 Phase: –

Phase is defined as the phase to which a atom has reached in its quiver. Initial stage means the initial phase from which the quiver starts. The followers will assist to understand the construct of stage. From the beginning travels in the signifier of moving ridges before making the ear sound can non go in vacuity. Sound needs medium for its travel. The medium may be a solid or liquid or gas ( Brown and Vaughn, 1993 ) .

Support a glass tubing unfastened at both terminals in a perpendicular place, with its lower and dunking into H2O contained in a wider cylinder. Keep over the upper terminal of the tubing a vibrating tuning signifier. Adjust the support of the sound is obtained. Adjust the distance of the air column boulder clay we get really the resonance or sympathetic note. Repeat the accommodations and take the norm of the consequences from the observation. It will be found from the perennial experiments, that the longer the air column is produced when the tuning fork becomes indistinguishable.

Vibration of air column in a tubing unfastened at both terminals: –

O’brien, Shen and Gatchalian ( 2002 ) have described that if they think of an air column in a tubing unfastened both terminals, and seek to conceive of the ways in which it can vibrate ; we shall readily appreciate that the terminals will ever be antinodes, since here the air is free to travel. Between the antinodes there must be at least one node, and the terminals, the traveling air is either traveling towards the centre from both terminals or off from the Centre at both terminals. Thus the simplest sort of quiver has a node at the Centre and antinodes at the two terminals. This can be mathematically expressed as follows:

Wave length of the simplest sort of quiver is four times the distance from node to antinode – 2L where L is the length of the pipe.

Vibration of air column in a tubing closed at one terminal:

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

2.7 Resonance-sympathetic quiver

Sloan, Kautz and Synder ( 2002 ) have described that everybody which is capable of quiver has natural frequence of its ain. When a organic structure is made to vibrate at its impersonal frequence, it will vibrate with maximal amplitude. Resonance is a phenomenon in which a organic structure at remainder is made to vibrate by the quivers of another organic structure whose frequence is equal to that of the natural frequence of the first. Resonance can besides be called sympathetic quivers. The undermentioned experiment will assist to understand resonance:

See two stretched stings A and B on a audiometer. With the aid of a standard tuning signifier we can set their vibrating lengths [ length between the Bridgess ] to hold the same frequence. Thus we can put a few paper riders on twine B and pluck threading A to do it vibrate. The twine B will get down vibrate and paper riders on it will flit smartly and sometimes A can be stopped merely by touching it. Still the twine B will go on to vibrate. The quiver in the twine B is due to resonance and it can be called as sympathetic quiver. If alternatively of the cardinal frequence one of the harmonics of twine B is equal to the vibrating frequence of threading A so the twine B will get down vibrating at that harmonics frequence. But in the instance of harmonics the amplitude of quiver will be less. In Tambura when the sarani is sounded the anusarani besides, vibrates therefore assisting to bring forth a louder volume of sound. The sarani here makes the anusarani to vibrate. In all musical instruments the stuff, the form of the organic structure and enclosed volume of air make usage of resonance to convey out increased volume and desired upper partials of harmonics.

2.8 Intonations

Spiegel and Watson ( 1984 ) have described that during the class of the history of music, several of music intervals were proposed taking at a high grade of maturating consonant rhyme and disagreement played of import function in the development of musical graduated tables. Just modulation is the consequence of standardising perfect intervals. Just Intonation is limited to one single-key and aims at doing the intervals every bit accordant as possible with both one another and with the harmonics of the keynote and with the closely related tones. The frequence ratio of the musical notes in merely 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 frequence ratios are expressed is footings of relatively little Numberss. Changeless harmonics are present when frequence ratios are expressed in footings of little Numberss. The interval in frequence ratio are:

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

Between Madhya sthyai C [ Sa ] and Madhya sthayi G [ dad ] 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 [ half step ] between Ga2 [ E ] and Ma1 [ F ] . Hence the interval between Ga [ E ] and Ma1 [ F ] is known as a half step. Between Pa [ G ] and Dha [ A ] we have a tone. Between mathya styayi Ni2 [ B ] and Tara sthyai C [ Sa ] we have a half step.

In merely Intonation we find that tones are non all equal. But the half steps are equal. In merely Intonation the transition of key of musical notes will be hard for illustration, if the keynote is changed from Sa [ C ] to Pa [ G ] so the frequence of etatusruthi Dhairatam [ A ] will alter from 1.687, clip the frequence of Sa [ c ] . A musical instrument tuned in merely modulation to play sankarabaranam ragam can non be used to play kalyani ragam. Hence the transition of key of musical notes will be hard in merely Intonation ( Doutaut, Matignon, and Chaigne, 1998 ) .

Equal temperature

Lehr ( 1997 ) has described that the above mentioned job in merely Intonation can be solved in the Equal Temperament graduated table. In Equal disposition all the 12 music intervals in a sthayi [ octave ] are equal. The frequence ratios of half steps in Equal disposition graduated table was foremost calculated by the Gallic Mathematician Mersenne and was published in ‘Harmonic Universelle ‘ in the twelvemonth 1636. But it was non put into usage till the latter half of 17th century. All keyboard instruments are tuned of Equal Temperature graduated table. Abraham pandithar strongly advocated Equal Temperament graduated table and in his celebrated music treatise ‘karunamitha sagaram ‘ he tried to turn out that the Equal Temperament graduated table was in pattern in ancient Tamil music.

A simple mathematical exercising will assist to under the footing of Equal Temperament graduated table.

Equal Disposition

Madhya sthayi Sa [ c ] frequence ratio=1=2 IS .

Tara sthayi Sa [ I ] frequence 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

a†“ a†“ a†“ a†“ a†“ a†“ a†“ a†“ a†“ a†“ a†“ a†“

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

Second

a†“

212/12

All half steps are equal is Equal Temperament graduated table. Each represents the same frequence ratio 1.05877. The great advantage in Equal Temperament graduated table is that music can be played equal good in all keys. This means that any of the 12 half steps can be used as ‘Sa ‘ in a music instrument tuned to Equal Temperament graduated table. There is no demand to alter tuning every clip the Raga is changed. Since keyboard instruments are pre-tuned instruments they follow Equal Disposition.

2.9 Production and transmittal of sound: –

Harmonizing to Boulanger ( 2000 ) the term sound is related to quite definite and specific esthesis caused by the stimulation of the mechanism of the ear. The external cause of the esthesis is besides related to sound. Anybody in quiver is an external cause of the esthesis. A veena [ after tweaking ] or fiddle [ after blowing ] in a province of quiver is an external cause of the esthesis. A organic structure in a province of quiver becomes a beginning of sound. A quiver is a periodic to and fro gesture about a fixed point

Iwamiya and Fujiwara ( 1985 ) have mentioned that the pitch of a musical sound produced on a air current instrument depends on the rate or frequence of the quivers which cause the sound. In obeisance to Nature ‘s jurisprudence, the column of air in a tubing can be made to vibrate merely at certain rates, hence, a tubing of any peculiar length can be made to bring forth merely certain sounds and no others every bit long as the length of the tubing is un-altered. Whatever the length of the tubing, these assorted sounds ever bear the same relationship one to the other, but the existent pitch of die series will depend on the length of the tubing. The participant on a air current instrument, by changing the strength of the air-stream which he injects into the mouthpiece, can bring forth at will all or some of the assorted sounds which that peculiar length of tubing is capable of sounding ; therefore, by compacting the air-stream with his lips he increases the rate of quiver and produces higher sounds, and by uncompressing or slowing the strength of the air-stream he lowers the rate of quiver and produces lower pitched sounds. In this manner the cardinal, or lowest note which a tubing is capable of sounding, can be raised going higher and higher by intervals which become smaller and smaller as they ascend. These sounds are normally called harmonics or upper partials, and it is convenient to mention to them by figure, numbering the cardinal as No. T, the octave harmonic as No. 2, and so on. The series of sounds available on a tubing about 8 pess in length is as follows:

Tsingos et Al ( 2001 ) has mentioned that a longer tubing would bring forth a corresponding series of sounds proportionally lower in pitch harmonizing to its length, and on a shorter tubing the same series would be proportionally higher. The full series available on any tubing is an octave lower than that of a tubing half its length, or an octave higher than that of a tubing dual its length ; therefore, the approximative lengths of tubing required to sound the assorted notes C are as follows:

Cardinal Length of tubing

C, 16 pess

C 8, .

degree Celsius 4, ,

c’ 2, ,

degree Celsius ” I foot

degree Celsius ” ‘ 1/2, ,

Shonle and Horen ( 1980 ) has mentioned that the add-on of about 6 inches to a 4-foot tubing, of a pes to an 8-foot tubing, or of 2 pess to a i6-foot tubing, will give the series a tone lower ( in B flat ) , and a proportionate shortening of the C tubings will raise the series a tone ( D ) ; on the same footing, tubings 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 cardinal. Examples:

Trumpet ( modern ) in C-length about 4 pess

, , 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 really sparsely provided with sounds ; the 3rd octave has little more than an arpeggio, and merely in the 4th octave do the sounds run consecutively or scale-wise, while half steps merely appear at the upper terminal of the 4th octave. The series, nevertheless, does non stop at that place, and is continued in the fifth octave in half step and smaller intervals, but nevertheless favourably proportioned a tubing may be, the production of sounds above the 16th note becomes more and more hard and unsure, therefore it is merely seldom that any air current instrument is required to bring forth these highly high harmonics. The sounds of Nature ‘s harmonic series do non all coincide precisely with the notes of the musical or treated graduated table ; Nos. 7, II and I3, for illustration, are perceptibly out of melody, but the balance are either absolutely true or close plenty in melody for practical intents ( Von Estorff, 2000 ) .

For the present intent it is non necessary to ask into the figure of quivers per second which are required to sound any cardinal or its harmonics, nor need the exact lengths of tubings be taken into consideration. The participant on a air current instrument does non number his quivers nor does he mensurate his tube-lengths, but in order to understand air current instruments at all, to cognize their capablenesss, their restrictions, 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 permute it to accommodate any cardinal sound ( Wand and Strasser, 2004 ) .

Wu ( 2000 ) has described although the full series of harmonics is nominally available on any tubing, in existent pattern the human lip can barely change the force per unit area to such an extent that all of them can be sounded on the same tubing with the same mouthpiece. How many of them can be produced on one instrument depends chiefly on the breadth of the tubing in proportion to its length. A tubing can be so broad or so narrow that no musical sound can be extracted from it ; there must be some kind of sensible proportion between breadth and length, and a tubing which is non well longer than it is broad would be of no practical usage as a musical instrument. A wide-bored tubing will give its cardinal more easy than a narrow one, and if non excessively broad can be made to sound its cardinal and a few of the lower harmonics, whereas a narrower tubing can be made to sound up to the 16th note of the series but will so likely fail to sound the first two. For illustration, a crude instrument made from an ox-horn or an elephant ‘s ivory may be so short and so broad in proportion that it will merely give one note ( the fundamental ) , whereas an orchestral horn is long and proportionally so narrow that it will sound from the 2nd note of the series up to even the I6th note.

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