Most modern synthesisers, use a sample and synthesis(S+S) technique to generate the timbres heard. One sample is taken of a particular instrument, the sample is then shifted to a pre-set pitch, depending on which key is pressed. With most instruments the characteristics of the timbre are dependent on the pitch of the note being played. In the S+S method of synthesis such characteristics are lost. Other characteristics, which we will call performance parameters, such as where the string is plucked, or bowed, how hard it done, are all lost, unless extra samples are included with such variations. Filters and dynamics' processors are used to simulate, these ‘performance parameters’. Some top of the range S+S, keyboards use complex 14-pole filters, as opposed to a basic 4-pole filter arrangement. Consider though, filters are modifiers of the frequencies already present in the sample. They cannot add any frequencies. Take something like a trumpet who’s, frequency content is increased, when blown hard. It can be arranged to have the volume of the sample playback to be proportional to keyboard velocity. This would not add the necessary frequencies. The best we could do in this situation, is a have a sample of hard blown trumpet and arrange to have the filter cut off frequency, which was increased with keyboard velocity and volume. This would allow the full frequencies through, when played hard, and have them filter out when played softly. This may not be an accurate sound of a soft blown trumpet. Although capable of producing some very convincing sounds, S+S is a pale imitation of the real thing. The next step, from the S+S technique, is that of physical modelling. This is not a new technique, models of traffic flow, model for stress analysis of mechanical components, are used everyday. With the advance of fast cheap computers, physical modelling is becoming more accessible.

Yamaha, has just released, a new synthesiser, which works with a physical model of a pipe/string(both very similar), this can cater for all the ‘performance parameters’, producing change in timbre which are exact and predictable. This gives a very accurate, authentic sound, who’s characteristics change, depending on how the instrument is being played. The challenge is getting the model to work in real-time.

It was proposed to design and implement a model of a drum head. It is intended to start the with a basic model and then increase the complexity of it. This will be done to the point where implementation of a complex model becomes impossible within the time or hardware restraints.

It was proposed to concentrate on north Indian drum called Tabla. Tabla and other Indian drums are unique as the drum head is weighted. It is intended to investigate the difference that the weight makes to the sound and the physics of the drum. Also it is hoped to take into account the way the drum is hit. There are many ways tabla are hit, this coupled with the ‘weighting’ of the drum head, gives the tabla a unique sound.

As mentioned in the opening paragraphs, there is a new synthesiser on the market which, uses ‘physical modelling’ techniques. I wrote to Yamaha, the manufacturers of the product, asking for advice about my project. The letter sent and the reply is in the appendix, they couldn’t reveal much until the patents were cleared. Apart from recognising that it is incredible computationally intensive, and ‘how good’ the whole idea of physical modelling is, they didn’t give any new information. It mentioned in the letter however that Stanford(America), and IRCAM(France), were researching physical modelling. In a review of the VL1, it seemed the hardware could be based on the work on Julius Smith at Stanford university. The magazine gave a list of journal articles, about physical modelling and related subjects. Three such articles were ordered though the inter-library loans scheme.

The rest of the research was basically the reading of textbooks and journal articles on relevant subject matter. One paper was particularly good, but it was only tracked down in the closing weeks of the project, it is mentioned at several points in the text. It serves as a convenient ‘yard stick’ and a stimulus to the ‘Experiment to find the eigenvalues and eigenvectors’. The paper was ‘ Vibrations of Indian musical drums regarded as composite membranes’, by B.S. Ramakrishna and Man Mohan Sondhi.

The tabla (actually two separate drums - shown above), are the most important percussion instrument in North India. It accompanies vocalists, instrumentalists and dancers in every style of music from classical to folk. Legend has it that tabla were created by famous Pakhawaj player, Sidhar Khan Dhari in the 18^th century. The Pakhawaj is a barrel shaped drum used in classic music. After Sidhar Khan lost a music contest, an angry argument ensued and his Pakhawaj was chopped in half by a sword. The first tabla was (accidentally !) created.

The creation of tabla has also been accredited to Amir Khusrau (in the 13^th/14^th century). He wrote about his achievements but never mentioned tabla, so this seems doubtful.

The first pictorial evidence of tabla, in their present shape was first seen in the 17^th century, in paintings depicting court life.

The tabla consist of the dahina (Hindi for right) and the baya (Hindi for left). The dahina is the higher pitched of the two, its made of very dense wood, it is conical in shape (wider at the bottom).

The drum head (puri), is unique, and is made of goat skin, the lao. There is a weight in the middle, the syahi. The syahi is perfect circle, in the middle of the puri, it is a semi-permanent paste made of coal dust, iron fillings, and rice paste, its around one millimetre thick. Around the outside of the puri, is a ring of thicker skin, this is called the chanti, this is not attached to the lao. The puri is laced by buffalo skin straps, baddhi, and tensioned by round wooden ‘chocks’, called gittak.

The baya is larger, and tuned lower, its body is large, heavy, metal and bowl shaped. The puri is made of goat skin, and is laced by goat skin straps, or strong rope. If rope is used, it is threaded through metal rings, kara, at regular intervals. The syahi, is thicker, about two millimetres, and is placed off-centre.

The tabla are played, with the fingers, and the palm of the hand. There are many, many different strokes, in order obtain many different sounds.

The baya has two main ways of being hit. The position of the hand is the same for both - The heel of the hand is placed on the puri, on the widest part between the edge of the puri and the syahi. The hand is brought down and slaps the puri, and mutes the vibrations - this is called the KI stroke. The other stroke is called GHI. Here the drum is struck, on the far side of the syahi, with the index or middle finger, it is not damped and heel of the hand is slide away from the body and downwards, the pitch of the note is raised. This gives a ‘whoop’, type sound.

The are many strokes executed on the dahina. They are a combination, of where on the puri it is hit (three area’s - chanti, lao, and syahi), whether the stroke is muted or not. Also a technique is used where a finger is place on the edge of the syahi and is struck. This makes the puri vibrate, in higher harmonic modes, gives a higher ringing sound.

The basis of Indian music is the Raag, this consists of two parts. The raga which is the a particular music scale, on which the instrumentalists play. A raga however is not just a scale, its also a mood, and different raga’s have different times of day they should be played. Indian music concerts often take place at ‘odd times’ for the westerner, and early morning Raags played in concert, are a rare pleasure.

The other part of the Raag is the Taal, this is the rhythmic cycle, the piece is played over. The number of beats in one cycle can range from 2 to 108. The most common in teen tal, which is a sixteen beat cycle, the division of the 16 beats, is also variable and is dependent on the raga, and on the player, and the style and mood of the piece.

The most common division of teen tal is 4+4+4+4, which is the equivalent of a four bar phrase in ‘common time(four beats in the bar)’ , in western music. More interesting the division could be 7+9 or 3+2+5+2+4 or any combination. It is the job of the tabla player (in Hindustani music) to keep that cycle going, the most important beat which is stressed most is the first one, this is called the Sam, the other important beat is the khali, which is the ‘empty’ beat, which is not stressed and there is no hit on the baya.

Another important concept is the different parts of the raag, a typical raag might start with any alaap, which is a slow piece with no particular rhythm, the tabla player doesn’t accompany. The instrumentalist, explores the raga, and gradually progresses further up the raga, embellishing and improvising, the speed and complexity increases, this section which the alaap leads to is called the Jor. The next section could be optional, this is called the Gat, its a fixed composition, and the tabla accompanies, the instrumentalists. The final section is the Jhala, there is a rise in speed and excitement, and eventually it can end in great excitement. During this section several things can happen and there in generally a lot of interplay between the instrumentalist and the tabla player. This interaction, was not always part of the music, instrumentalists, used to prefer a more passive tabla, which just kept time. Half way through this century the tabla player, interacts and improvises along, with the instrumentalist. The interaction can take several forms - Jawabi sangat, the imitation by the tabla of the rhythm played by the instrumentalist, -Jawab sawal, the inter-play between the tabla and the instrumentalist, in a kind of ‘question and answer’ dialogue. Other forms which I have heard is the when the tabla try and follow the rhythm played be the instrumentalist, simultaneously this is very exciting, especially when played at speed, and of course the tabla solo.

There any many brilliant tabla players, it takes years of training to get really understand and to able to improvise many intricate rhythms.

Anything by Zakir Hussain or any group with him playing tabla.

Making Music by Zakir Hussain

Shakti with John Mc.Laughlin and Zakir Hussain

Soul Searcher By Lakshminarayana Shanker, featuring Zakir Hussain Master drummer of India, Sarwar Sabri (Often plays at Leeds collage of music)

1) The derivation of mathematical models of a drum, using differenttechniques. The models should be as general a possible, allowing for a non-homogeneous drum head, and enough initial conditions to allow the different strokes of the tabla to be implemented.

2) The implementation of the models, on appropriate hardware, and software.

3)Developing a way to verify a mathematical modal against a real tabla.

4)Choosing the best model, in terms of speed and accuracy. Address the problem of real-time implementation, modifying models as necessary. These are the points which need addressing, they appear, in a logical order of working. However they be covered concurrently.

The modelling most physical systems, mathematically result in equations involving rates of change, of two or more independent variables, such as displacement, angle, or time. These lead, depending on how the model is formulated, to a partial differential equations, or a set of ordinary differential equations.

The essence of the project will be to formulate these equations, solve and implement them as efficiently as possible. Each method will be treated with separately, and then the discuss the merits of each with regards to accuracy, and ease of possible implementation.

Matlab will be used for all the implementation done a PC. For the Finite element analysis, the Abaqus package which runs on the HP Apollo workstations, will be used.

State space is method of modelling a system, which lends itself very well to implement on digital hardware.

The state of a dynamic system is the smallest number of variables (called state variables), so that the initial state of these variables and knowledge of the input at , completely determines the output of the system.

Another way to think about it is, that an nth order differential equation can be written a 'n' coupled first order differential equations.

The diagram below shows a general multiple input/multiple output (MIMO) system.

The general form of a state-system is

The outputs of the system also need to be defined in terms of the state variables thus:-

Together Eq.’s 1&2 can be used to model as system. Notice that N the number of inputs and m the number of outputs need not be equal. Also matrices A,B,C and D can be written as functions of time, the state space method can be used in time varying and non-linear applications.

Any system has 'energy storage devices', in electrical circuits these are inductors and capacitors, in mechanical systems these are the variables associated with kinetic and potential energy, namely position, velocity and acceleration.

One way to find the A (state) matrix is to write explicitly the differential equations governing the system and to convert these into state variables. Another way for the experienced analyst is to choose the state variables first, and write down the state equations straight away, with prior knowledge of the defining differential equations.

Lets look at the two methods, on two common linear time-invariant systems.

Lets just take a simple mechanical system,

The general equation for system is

The force due to :- momentum + kinetic energy + Potential energy = input force.

Choosing other state variables as

Therefore we can re-write [Eq.3] as,

The output y of the system just wants to be in the position x of the mass, therefore

Writing Eq.’s 4&5, in matrix form then.

Now we have worked a few basic derivations, lets have a look at how we can start looking at a simple building blocks which could be used for the tabla.

The derivation follows closely that of Eq.’s 3-6,

We choose our state variables as :

can be found by resolving forces due to the springs. Where in general where e is the extension, and k the stiffness of the spring.

This holds only for small angles of deflection, which in the case of the puri, is a perfectly reasonable assumption.

Since sound is a change in pressure of the air, lets look at the attributable by the drum.

This ignores the phase shifts, between different points on the puri. This does not compromise the model, the ear is fairly insensitive to phase shifts. It would only really cause only cause at problem at high frequencies if the ‘mic’ was close to the puri. Also with the diameter of the puri being 0.13m, the relative phase shift between side of the puri and the other will only be small.

The system has no inputs, the outputs just depend on the initial states only.

The actual outputs in terms of state variables are, the even numbered states, the velocities. If the puri is simply hit, there by giving it a velocity at t=0, the velocity is the initial condition of the system. Describing how and where the puri is hit is a simply matter of giving the correct mass or masses a velocity as their initial conditions.

Since matlab is a matrix based language, it is very convenient for state space analysis

There is a command on matlab-initial which take arguments-

- is the initial states and t is an equally spaced time vector. I have implemented an 'n' mass system, which is based on the three mass system .

The file is called tabla.m, it uses matlab's ‘for’ loops and matrix indexing to make the matrices for an 13 mass system. It relies on the similarity of the middle masses, with only the two end masses, incorporating the boundary conditions being unique.

The masses of each mass were determined by the position of the mass.

Considering the puri, from above,

The approximation used is the mass1 represents the mass of and

mass 2 represents the mass of the semi-circular section, , and mass 3 equals mass of all the way mass 7((13+1)/2). Since the mass will be symmetrical around the centre mass, equations have been derived which start at the centre and work towards the edge. When it comes to implementing it I have filled the mass vector from the middle to the right hand end and then filled it again with the same vector from the middle to the left hand end. Of course by representing the tabla head in this way, we then are assuming that the puri vibrates in only circular modes. This is a very big generalisation but it's demonstrates a fairly complex model working

The ‘masses’ can be expressed as

In the model tabla.m, three different hits were implemented, one to the middle (tin), the edge (edge) and a damped hit(dmp).

The damped hit differs form the rest as its we use a modified A matrix called da, which is exactly the same as A expect the damping on mass 5, has been increased to simulate a finger on the puri at that point.