Sitar price is very costly but if you have that type of budget then you can easily afford it. In this article we will also cover the topic of sitar strings . Some peoples also search about sitar indian on the internet. I am an Indian and I have an experience about sitar instrument for 5 years. And I know how to make a sitar and what improvements have been made in it from time to time. If you are not believing me, then read this article at once. In the table of contents click on a particular topic and, you can directly access that topic.
1. Introduction - Sitar
The sitar is an ancient Indian string instrument that features heavily in Indian classical music. This is usually accompanied by a Tambura, a similar drone-type instrument that is used to set the tonic of the piece being performed.
The origin of the sitar can be dated back as far as the Middle Ages and is usually found in the northern part of India. It does not feature at all in southern Indian classical music.
Indian Instrument Sitar
The sitar became popular in the western world through the music of Pandit Ravi Shankar during the 1950s and George Harrison of the Beatles in the 1960s (Park 2008).
It is known for its unique timbral quality, which is attributed to its sympathetic strings, the construction of its bridge, long hollow neck, and its resonating chamber.
It is usually played by balancing the instrument between the player's left foot and right knee. This position then allows the player's hands to move freely around the instrument's neck without supporting its weight.
2. Mechanics of Sitar
The sitar has a very unique and distinguishable body. On the neck of the instrument, all the frets are moveable, allowing for fine-tuning and the use of microtones.
There are generally around 14 of those depending on the type of sitar. They are also suspended off the neck, allowing the sympathetic strings to run underneath and resonate freely.
Usually, the sitar has about 21 strings, most of these being sympathetic. These sympathetic strings are also known as tarb. These strings are never really ever touched as they are just meant to vibrate sympathetically.
Although, sometimes, you may hear a player strum all of these at once for effect. Along with the sympathetic strings, you have the main six or seven strings. Three of these, called chikaari, provide the drone while the rest are used to play the melody (Courtney 2010).
Badaa goraa and Chota goraa of a Sitar (Courtney 2010)
The most important parts of the sitar are the two bridges. There is the large bridge called the badaa goraa for holding the drone and melody strings in place, and then there is the smaller bridge for the sympathetic strings called the chota goraa.
These bridges are collectively known as jawari and are generally made of camel bone. The shape of the jawari are like slopes, and it is the way the string interacts with these slopes when plucked that give the sitar its particular timbre.
Initially, when the sitar string is plucked is, there are a shortening and lengthening of the string relative that is relative to the slope, which leads to the string generating overtones.
This particular process is explained in better detail further on in this dissertation.
The resonator of the sitar is called the Kadu. These are very delicate and usually are just made of a gourd.
On some sitars, there are two resonators, the other one being at the top of the neck. The gourds may also sometimes have strings inside of them that are there to resonate sympathetically.
3. Bridge Structure
As mentioned before, it is the sitars sloped bridge construction and its relationship to the strings that give it its specific buzzing sound. What happens is explicitly a type of nonlinear distortion occurs when the string is plucked due to the interaction between the camel bone and the string.
This nonlinear distortion gives rise to additional overtones, somewhat similar to what happens when amplitude clipping occurs. Due to the square wave-like properties forced upon by the clipping, it creates odd harmonics that are not present in the original signal (Park 2008).
The previous section mentioned that a shortening and lengthening of the string relative to the jawari occurs. This also lends to the buzzing timbre and affects the string's pitch ever so slightly since its length is changing.
Concerning the actual model, this requires that there be a dynamically evolving delay line.
How this was implemented is explained further in the documentation.
Further to how the nonlinear distortion occurs due to the friction between the sting and jawari, the transverse waves traveling along the string interact with the jawari just before they reach the point of termination.
Generally, in a stringed instrument with a typical style bridge such as that of a western classical guitar, this termination point is where the waves usually flip over and travel in the opposite direction. However, in the sitar, before this happens, the larger amplitude transverse waves in the string interact with the jawari earlier than the smaller ones, altering the shape of the strings and causing it to bulge.
The transverse waves are not terminated but interact with the jawari, unlike the smaller ones, which mainly reflect. This dramatically increases the higher partial content at large wave amplitudes but not at smaller wave amplitudes.
This interaction between the string and jawari reduces gain substantially, as the energy is transferred to the louder, higher partials. The imprecise termination point of the sitar is akin to the fretless electric bass.
4. Sitar Physical Model
The wave equation is a fundamental second-order partial differential equation that describes the propagation of waves with speed v. Initially, the idea was to use the Karplus-Strong algorithm for the main and sympathetic string synthesis because of its low computational costs.
Still, on further research, investigation, and testing, it was decided to use the bi-directional digital waveguide approach instead. The Karplus-Strong algorithm was reserved for another modeling approach.
The bi-directional digital waveguide approach is a much more realistic model of how a one-dimensional string vibrates as it takes into account two acoustic waves are traveling in opposite directions. It is known that the vibration of an ideal string can be described as the sum of two traveling waves going in the opposite
directions (D’Alembert 1747).
Another modeling approach that is unique to this particular model concerns the non-linear bridge structure or jawari as it is officially called. The jawari, because of its design, requires that there be a dynamically changing delay line. The amount of delay length modulation that occurs in this delay line is all relative to how much energy is in the plucked string.
The length of the string changes more rapidly at the attack portion of the signal, gradually becoming less random and settling into a more periodic pattern as the energy dissipates through the termini. This particular problem of non-linearity was solved using the Karplus-Strong algorithm and a feedback loop from the main sitar string itself.
This technique is explained with more clarity further on in this document.
This model also makes use of fractional delay filtering. Fractional delay filtering is a modeling technique that allows for the accurate cancellation and dampening of musical tones (Lehtonen et al. 2008).
Usually, delay lines in these particular types of models could only be of an integer sample length, causing the physically modeled instrument to be slightly out of tune, but by using fractional delay filtering, this can be avoided.
The other modeling techniques used in this particular model are all techniques that have been used for the modeling of the western classical guitar, but they have been adapted to the sitar. These techniques include the sympathetic resonance of strings, comb filtering, all-pass filtering, and body resonance filtering.
5. A Brief History About Sitar
The first use of physically-based models to synthesize sound was by John Kelly and Carol Lochbaum (Kelly and Lochbaum 1962). They implemented a simplified model of the human vocal tract as a one-dimensional acoustic tube of varying cross-sections.
The most widely heard example of physical modeling for many years is Stanley Kubrick's 2001: A Space Odyssey. Most of the early work on physical modeling of musical instruments was focused on vibrating strings.
This was due to them being computationally efficient to calculate. It was Pierre Ruiz in 1970 that was the first person to synthesize a musical instrument using a physical model.
It was then Ruiz, and Lejaren Hiller discovered that the quality of a vibrating string sound was mainly defined by how the string loses energy (Hiller and Ruiz 1971a, 1971b). Similar to Ruiz and Lejaren Hiller's publication by McIntyre and Woodhouse, some approaches would describe theoretical results to a realistically lossy vibrating string equation (McIntyre and Woodhouse 1979).
These techniques were then to be followed by the Karplus-Strong algorithm (Karplus and Strong 1983). The Karplus-Strong algorithm was discovered as an effortless computational technique that arose from work being conducted on wavetable synthesis.
It works by feeding a burst of white noise into a feedback loop of length L samples. On each loop, the white noise is filtered over and over again by a simple averaging filter. The frequency-dependent decay of the white noise created for the first time on a computer sounded very string-like.
What made this algorithm so successful was that the realistic string timbres produced with great ease were very computationally efficient. This was very relevant at the time as modern-day standards would have limited processing power.
Seemingly this technique had nothing to do with physics, and it wasn't until David Jaffe and Julius O. Smith did further work with it and showed a clearer understanding of it about the physics of a plucked string (Smith 1983; Jaffe and Smith 1983).
After this, Julius Smith introduced the theory of digital waveguides and generalized the underlying ideas of the Karplus-Strong algorithm (Smith 1987). Karjalainen says that digital waveguides are physically relevant abstractions yet computationally efficient
models, not only for plucked strings but also for various one-, two-, and three-dimensional acoustic systems (Karjalainen et al. 1998).
These digital waveguides proved to be an efficient model of many linear and physical systems such as strings and acoustic tubes. One of the advantages of these waveguides over analytical methods was introducing non-linearity into models, just like those that would have to be considered when modeling the sitar (Smith 1987).
This enabled researchers to produce a variety of different realistic instrumental sounds. To this day, digital waveguides are still a crucial modern research topic concerning the field of physical modeling.
They are still used extensively in many commercial synthesis systems, whether it is hardware or software.
The first commercially available systems to include digital waveguides were at the beginning of the 1990s. These were Bontempi-Farfisas MARS
in 1992 and then this was followed by Yamahas VL1 in 1993
6. A Taxonomy
Below the image is a taxonomy of the different types of physical modeling synthesis techniques used. Only DWGs (Digital Waveguides) will be covered in this
dissertation. This figure has been given so the reader can see where this technique is derived from and how it relates to other methods.
7. Simple Physical Model of Sitar
7.1. What is a Model ?
Model-Building is a fundamental human activity. For our purposes, the model can be defined as all forms of calculations that predict the behavior of physical objects or phenomena based on the initial state and the power of "input."
Our first successful model occurred in our head. It is effectively constructing a simplified abstract view of what usually may be a very complex system.
Gaining an understanding of a complex natural system such as a musical instrument is generally accomplished by combining or building upon more straightforward and more basic models.
I say we were to look at a guitar. The guitar is comprised of many mechanical parts such as the strings, resonator, and bridge. Each of these parts is the building block to the overall complex model.
For virtual musical instruments and audio effects, the model replaces the real thing, allowing us to understand how it works (Smith 2010).
7.2. Basic Vibrating String Model
The basic model of a vibrating string is based on Newtonian principles. The vibrations in the string are transverse waves or, in this case, transverse acoustic waves.
To derive the equations governing small transverse vibrations of an elastic string, which is stretched to length L, you have to make simplifying assumptions so that the resulting equation does not become too complex.
First of all, place the string along the x-axis, stretch it to length L, and fix it at the ends x = 0 and x = L. The string is then distorted instantly, say t = 0; it is then released and allowed to vibrate.
The problem is to determine the vibrations of the string, that is, to find its deflection at u(x; t) at any point x and where t > 0. To do this, the following can be assumed.
- The mass of the string per unit length is a constant. The string is perfectly elastic and doesn't offer any resistance to bending.
- The tension caused by stretching the string before fixing it at the endpoints is so significant that the action of the gravitational force on the string can be neglected.
- The motion of the string is a slight transverse vibration in a vertical plane; that is, each particle of the string moves strictly vertically.
T is the string tension in the string and p is the string's linear mass density.
The derivation of this equation is beyond the scope of this dissertation but can be found in any applied mathematics textbook (Kreyzieg 1999). This PDE is the starting point for both digital waveguide models and finite difference schemes.
7.3. Mass-Spring System
This section discusses the principles behind a basic mechanical system's motions, a mass on an elastic spring. The string of a musical instrument is a mass-spring system. If you are to take an ordinary spring and suspend it vertically from support and then at the other end attach a body of mass m.
Here you are to assume that m is so large you can disregard the mass of the spring. Then pull the body down a certain distance and
then release it. You will notice that it undergoes a motion. This motion is governed by Newton's second law:
Mass x Acceleration = myn= Force
Where Force is the resultant of all the forces acting on the body. Here, yn= d2y / dt2, where y(t) is the displacement of the body and t is time. At first the string is unstretched, but then when the body is attached, the body stretches the string by an amount s0.
This causes and upward force F0 in the spring. It has been experimentally shown that this restoring force F0 is relative to stretch, say,
F0 = -ks0
This is known as Hookes law. k is called the spring constant. Where the larger the value for k, the more stiff the spring is, hence giving a smaller s0. s0 being the amount of displacement. The extension of s0 is such that F0 balances the weight W = mg. Consequently F0 + W = -ks0 + mg = 0.
These forces do not affect the motion. The entire system is at rest, this is what is called the static equilibrium of the system. The position of the body at the static equilibrium position is y = 0. We measure the displacement of the body from the static equilibrium position as y(t).
The main point is that F0 is the restoring force. It has the tendency to restore the system back to its static equilibrium position y = 0. With this understanding of a how a mass-spring system works, it brings us on to damped and undamped mass-spring systems.
Every system has damping otherwise it would just keep moving forever. It would be like if a string was plucked and it kept vibrating forever. Although, to explain the next point we are going to look at an undamped system first.
Let’s take for an example an iron weight on the end of a spring. In this situation F1 is
the only force Mass x Acceleration = myn= Force causing the motion. Hence, making
myn = -ky from Mass x Acceleration = myn= Force. This means that the model for the mass-spring system without damping becomes:
my'' + ky = 0
By finding the complex roots of this equation we get the general solution
The corresponding motion to this equation is called a harmonic oscillation. These harmonic oscillations are similar to the waves that occur when a string is plucked. When
the string is plucked or in this case when the iron weight is displaced the spring makes these harmonic oscillations. By applying the addition formula for cos, this equation can be written as
y(t) = C cos(w0t - g)
And, since the period of the trigonometric function y(t) = C cos(w0t - g) is 2 x 22/7 (PI) /W0 the body executes at W0 /2 x 22/7 (PI) cycles per second. This quantity is called the frequency of the oscillation and is measured in Hertz.
In the case where the system has been damped which is more likely to be the situation. We connect the mass to a dashpot to demonstrate its properties. By looking at the equation governing the system we can derive three different cases. The damp ed system equation being
m'' + cy’ + ky = 0
Where -cy’ = F2, this being the force imposed by the dashpot. The three different cases are, overdamping, critical damping and underdamping. It is the roots of equation my'' + cy’ + ky = 0 that determine this.
Case 1: In the overdamping case the body does not oscillate since the damping takes the energy from the the system and there is no external force that keeps the motion going. The equation myn + cy’ + ky = 0 has distinct real roots _1, _2 in this case
Case 2: The critical case marks the border between the non-oscillatory motions and oscillations; this explains its name ”critical case”. It has to do with the fact equation my'' + cy’ + ky = 0 has a real double root.
Case 3: Underdamping is the most interesting case. Underdamping occurs when the roots of the equation are complex conjugate roots.
Underdamping would be similar to the case in most strings on an instrument. When the string is initially plucked it settles into a periodic behaviour corresponding to a harmonic oscillation. These three cases are illustrated in
There is a particular modeling technique based solely on this mass-spring paradigm, as mentioned before (Hiller and Ruiz 1979). As can be seen, it requires a detailed description of all the physical characteristics of the vibrating objects.
Furthermore, it requires that you stipulate the boundary conditions for the PDE of the one-dimensional wave equation. It also requires the physical description of the excitation mechanism. The difference equations presented earlier are the equations that are then used to compute the resulting sound output (Bianchini and Cipriani 2008).
7.4. D’Alemberts Solution of the Wave Equation
With D’Alemberts travelling wave solution it can be shown that the vibration of an ideal string can be described as the sum of two travelling waves going in opposite directions using the wave equation. We will start with the wave equation
If we are to denote the right travelling waves and the left travelling waves by the following equations:
v = x + ct, z = x - ct
Then u becomes a function of v and z. The derivates of the wave equation in
can now be expressed in terms of the derivatives with respect to v and z by the use of the chain rule. This becomes
ux = uvvx + uzzx = uv + uz
We now apply the chain rule to the right side of the equation giving us
uxx = (uv + uz)x = (uv + uz)vvx + (uv + uz)zzx = uvv + 2uvz + uzz
Now we transform the other derivative in
giving
utt = c2(uvv2uvz + uzz)
By inserting the two results into
we get
This resulting equation can now be solved by two successive integrations with respect to z.
where h(v) is an arbitrary function of v. Integrating this with respect to v gives
where u(z) is an arbitrary function of z. Since the integral is a function of v, say, 0(v), the solution u is of the form u = 0(v) + u(z). Then because of my'' + ky = 0 We get
This is known as D’Alemberts solution of the wave equation. The traveling-wave solution of the wave equation was first published by d’Alembert in 1747 (D’Alembert 1747)(Kreyszieg 1999). The bi-directional digital waveguide is based on this very principle and will be discussed further on the dissertation.
7.5. Sampled Traveling-Wave Solution
In order to use the traveling wave solution in the ”digital domain” it is neccesary that you sample the traveling-wave amplitudes at intervals of T s econds. The continuous traveling-wave solution to the wave equation given in (3.16) can be sampled to give
where x = cT denotes the spatial sampling interval in meters, T denotes the time sampling interval in seconds, and y+ and y-are defined for notational convenience (Smith 2010) .
8. Digital Waveguides
Here the Karplus-Strong algorithm and the extended version of it will be explained in detail. This chapter will also introduce the bi-directional digital waveguide; this is the modeling technique that is central to the modeling of the sitar strings.
8.1 Karplus-Strong algorithm
The Karplus-Strong algorithm was discovered by two men around 1980. Their names being Alan Karplus and Kevin Strong. The paper on this algorithm was published in1983.
It was Alex Strong in December of 1978 that conceived its most straightforward modification and called it the Plucked-String algorithm. How it works is by merely averaging two successive samples (Karplus and Strong 1983). This can be written mathematically as
Karplus-Strong Algorithm
It was discovered that this averaging process produced a slow decay of whatever waveform was being computed by it. This algorithm made a pitch of period p+12 samples and sounded similar to the decline of a plucked string.
What was so remarkable about this algorithm was that there was no multiplication required. Making it exceptionally computationally efficient.
Back then, they did not have anywhere near the same microprocessing power that we have nowadays, so this would have been fast and easy to implement considering the limitations (Karplus and Strong 1983).
Strong says the naturalness of the sound derives largely from differing decay rates for the various harmonics. No matter what initial spectrum a tone has, it decays to an almost pure sine wave, eventually decaying to a constant value (silence) (Karplus and Strong 1983).
The actual excitation of the algorithm requires that a noise burst be fed into the system. How Strong initially did this was by feeding the algorithm with a wavetable filled with random values.
Every time, the use of a different random wavetable had the advantage of giving each repetition of the same pitch a slightly different harmonic structure.
This gave each note its character, sort of like a real instrument. Usually, what would be used to excite the system would be a burst of pink or white noise (Karplus and Strong 1983).
Once the noise burst is fed into the system, it is immediately output and then fed back into a delay line of L samples long. The result of this delay line is then fed into the averaging filter as described already.
This is usually a first-order low pass filter. Also, the filter gain must always be less than one, or else the signal will never decay and could make the system unstable.
The output of the averaging filter is then output and at the same time sent back into the delay line. This process keeps repeating until the signal is averaged out to silence (Karplus and Strong 1983).
The length L in samples of the delay line determines the fundamental pitch of the note being played. L is determined by the equation L = Fs=F1, where Fs is the sampling frequency. The overall effect of the algorithm is quite realistic and very similar to a plucked string sound considering it is such a simplistic procedure.
It may not have a natural-sounding guitar string tone, but various extensions can be applied to help this, which will be discussed next. Alan Karplus conceived a simple variation of the algorithm for drum timbres. Since we are only interested in strings, this will not be discussed.
8.2. Karplus-Strong Extended
Around the same time that the paper about the original Karplus-Strong algorithm was published, David A. Jaffe and Julius O. Smith published an article about various extensions to the original algorithm.
The need to implement these extensions came from the musical needs that arose out of the composition of May All Your Children Be Acrobats (1981) and Silicon Valley Breakdown (1982), both by David Jaffe (Jaffe and
Smith, 1983).
One of the first modifications made was about the tuning. The fact that the delay line length L had to be an integer caused tuning problems. The tuning problems occurred at high frequencies.
The fundamental frequency f1 = fs / (N+1/2 ) meant that the pitches were rounded off. This was barely noticeable for low pitches (large N), but as the pitch increased, it becomes more and more off-sounding (Jaffe and Smith, 1983).
The solution to this problem was fractional delay filtering. It can be shown experimentally that by using a fractional delay filter, there is a more accurate cancellation and dampening of musical tone partials (Lehtonen et al. 2008).
What was needed was introducing a filter into the feedback loop, which would delay the signal slightly without altering the loop gain. The filter that was raised was an all-pass filter.
It ensured there was no change to the growth of the signal. The equation for this filter and its transfer function is as follows.
The only thing that the all-pass filter affected was the phase of the signal (Jaffe and
Smith, 1983).
Another problem with the algorithm was decay-time; the difference between the decay times for a low pitch and a high pitch was drastically different. The ability to control decay time is very important if you want to have a realistic realization of a plucked string.
Consequently, Jaffe and Smith found methods that could be used to control decay time. One of the ways was to introduce a loss factor p. Where equation
Where |p| <= 1 if the string is to be stable. Essentially what decay shortening does is
produce a damped version of the Karplus-Strong algorithm. Where low-pitched notes are comparable to low notes on real strings.
Another technique that was employed was decay stretching. This was done by changing the feedback average Ha to a two-point weighted average. This reduces the amount of energy loss at high frequencies.
For the greatest control it is said both the uniform loss method and two-point-averaging method should be used together (Jaffe and Smith, 1983).
Dynamics was another issue that was dealt with.
Where the output of the system was directly related to the noise burst being input into the system. What enabled this to work was, since the strings that were plucked hard had more energy in the higher partials than the strings plucked lightly, a one-pole low pass filter could be used to attenuate these higher partials before they were fed into the system.
This allowed the user to be able to set if the string was to sound muted when it was plucked or alternatively sound like an open string. All that the user had to do was adjust the cut-off point of the one-pole low pass filter and you could get varying excitation timbres (Jaffe and Smith, 1983).
Some of the other extensions had to do with pick position and pick a direction. Pick position involved implementing a comb filter just after the noise burst.
Depending on the comb filter's delay length, you can choose the string at various positions, allowing you to suppress specific harmonics.
Pick direction can then also be controlled by lowpass filtering the noise burst before it is fed into the delay line or using a rich harmonic spectrum instead of a noise burst. Another way to affect the noise burst is to change the duration of the noise burst.
To model sympathetic string vibration, Jaffe and Smith sent a small percentage of the string output from a plucked string to another string that had been tuned to a different pitch.
Since the sympathetic string was tuned to a different pitch, all the partials of the plucked string that did not coincide with the sympathetic string would have been attenuated (Jaffe and Smith, 1983).
There will be a further discussion about sympathetic strings in this dissertation, as it is central to the sitar model. It can be seen here that these extensions it can make the fundamental algorithm much more expressive and realistic sounding.
Usually, although very similar to a plucked string, the Karplus-Strong algorithm does have a very artificial sound.
8.3. Bi-directional Digital Waveguides
A bi-directional digital waveguide is essentially a bi-directional delay line at some wave impedance. This is also considered a lossless digital waveguide. Wave impedance is the ratio between the force of a wave to the velocity of a wave.
For linear time-invariant systems, impedance may vary with angular frequency (w). The bi-directional waveguide works because each delay line contains a sampled acoustic traveling wave (Smith 2010).
Since it is a bi-directional waveguide, this means that there is a sampled acoustic wave traveling from left to right and right to left in each of the delay lines.
In this model, d’Alemberts are traveling wave solutions whereby it can be shown that the vibration of an ideal string can be described as the sum of two traveling waves going in the opposite direction (d’Alembert 1747).
The type of bi-directional digital waveguide we will be dealing with in this dissertation is rigid terminations. If we terminate a length L ideal string at x = 0 and x = L, we then have the boundary conditions
y(t, 0) = 0 y(t, L) = 0
How this system works is, the excitation is fed into the system at an arbitrary point given in the image below. The acoustic traveling waves proceed to travel around the bi-directional waveguide being delayed by N/2 samples by the delay lines. It can be seen in the diagram
That there are the two termination points as mentioned before. These would typically be the nut and bridge of, say, a guitar. The reader may also notice the -1 at each of these termination points.
The -1 is there to invert the phase of the acoustic wave, just like how an acoustic wave would flip over and change direction in the real physical world if it were to meet the termination point.
This is a far more realistic simulation of a traveling acoustic wave than the single delay line technique formulated by Karplus and Strong. The example that has been discussed here is for only a one-dimensional waveguide.
This technique can be extended to two and three-dimensional waveguides and model drum skins using digital waveguide meshes.
A number of the various extensions discussed in the previous section can be applied to the bi-directional digital waveguide, such as fractional delay filtering and excitation.
Matti Karjalainen et al. have looked at the possibilities of this in another paper. They employed two different models, one where the bi-directional digital waveguide had a bridge output and the other where it had a pick-up output (Karjalainen et al. 1998).
The sitar model demonstrated in this dissertation was loosely based around this.
9. Sitar Model
9.1. Introduction
The sitar model, as mentioned before, was developed and tested in MaxMSP. The entire patch consists of three main parts.
There is the poly~ abstraction of the main strings, the sub patch for the sympathetic strings, and then there is a bank of digital filters being used to model the resonator.
All these components fit together to model the sitar how the patch is controlled either by an external MIDI device or by the kslider object in MaxMSP.
The most important part of the patch is the poly~ abstraction. The main strings are modeled within this abstraction, and it also gives the patch its seven-note polyphony.
Since this is the most important part of the model, it is the first part that will be discussed in detail.
9.2. Main Strings - poly~ abstraction
This part of the patch contains the main digital waveguides, the excitation mechanism, and various objects to make sure the poly~ functions correctly.
These can be seen in the Second Spectral Analysis chart. The first group of objects in the patch working from left to right is there to receive the pitch and velocity to be used in the waveguide sub patches; there is also a thispoly~object to decide whether that instance of poly~is busy or not.
The next group of objects in the Second Spectral Analysis chart) are there to excite the strings. There is a linear ramp generator there to create a pink noise envelope.
Pink noise is used because all the frequencies present are of equal amplitude and random nature, meaning that no two excitations will be the same. The original idea was to use a recording of a sitar impulse response and use the commuted waveguide synthesis technique.
Still, as explained before, due to the sitar's non-linear model, this would not have been effective. There is also a comb filter set up just before the excitation is sent to the strings.
This comb filter is there to adjust the pick position with the slider in the main patch. This a Karplus-Strong extended algorithm concept, as discussed earlier in this document. The slider can be seen in the Second Spectral Analysis chart.
The next group of objects in the Second Spectral Analysis chart are the bi-directional digital waveguide sub patches. The reason there are two of these is because of string coupling.
One of these is the string vibrating in the horizontal plane, and the other is vibrating in the vertical plane. This gives the string a more realistic sound. Generally, if just one waveguide is used, it sounds very static.
These are both then summed together to provide the overall string sound. They are also scaled since they are being added together. The contents of the digital waveguide sub patches will be discussed further in this section.
After the waveguides have been summed, there is a group of objects to test to see if the gain of the strings is less than 0.001 and if so, it mutes the poly~instance it is in and sets its status to being not busy. This was implemented to make poly~more effective.
9.3. Bi-directional Digital Waveguide sub-patches
This is the most important part of the whole patch and can be seen in the Second Spectral Analysis chart. This
particular sub-patch is broken into two parts.
On the left side, you have the bi-directional digital waveguide of the string, and then on the right, you have a Karplus-Strong algorithm implementation.
This KS algorithm is fundamental in giving the string its non-linear distortion and its characteristic buzzing timbre.
First of all, the bi-directional digital waveguide part of the sub-patch will be explained, and then the KS algorithm implementation will be tied in.
Bi-directional Digital Waveguide
The bi-directional digital waveguide that has been implemented in this patch also uses some of the Karplus-Strong extended algorithm concepts.
It makes use of the tuning allpass filter, dynamic-level lowpass filter, and string damping lowpass filter, as well as a few implementations that were necessary for the string to sound like a sitar string.
As soon as the string is excited, it is passed through a one-pole lowpass filter; this is the dynamic-level filter. The value for this filter is controlled from the main patch, and there is one for each dimension of the string.
This filter controls the timbre of the string each time it is plucked. It is used to make the string sound muted if this is the desired effect.
After the one-pole filter, the excitation enters the bi-directional digital waveguide. It can be seen in the Second Spectral Analysis chart that there are four tapin~ and tapout~ objects. These objects are effectively the delay lines. These are responsible for the pitch of the string.
If you were to unwind the waveguide and have the two *~-1 multipliers as your termination points of the string, you would see that each delay line is effectively divided in two by the two tapin~tapout~pairs.
This is because excitation of the string has to be at least in the center of the delay line and be fed into the circuit at the same position in each direction of the delay line.
This makes sense since if you were to pluck a string in
an existing physical system, you could only do so at one position at any given time.
If you look at the delay sub-patch within the patch in the Second Spectral Analysis chart, you will see that this is the mechanism that controls the delay time for tapin~ tapout~.
This works because the MIDI value received is converted into the frequency of the note being played.
Since frequency is measured in Hertz and Hertz means cycles per second, the frequency value is divided into 1000 to give the delay time in milliseconds.
The reader may also notice that this is then fed into a mstosamps~ object, one is subtracted from it, and then a sampstoms~ object is used to convert back again. The mstosamps~ and sampstoms~ are used to convert from milliseconds to samples.
One is subtracted because the creators of MaxMSP have designed the tapin~ tapout~ to have a minimum delay of one vector size, which needs to be compensated for.
Once the excitation is in the waveguide, it moves through it like a transverse wave would in an actual physical system. The *~-1 multipliers are there to reverse the wave phase every time it passes through them. The same way a wave flips over when it reaches its termination point in an actual physical system.
This is why the two *~-1 objects are considered the termination points. The string damping dials on the main patch control the damping lowpass filters featured in the delay loop; these can be seen in the First Spectral Analysis chart.
The velocity of the note being played is mapped to the MIDI values 100-127, and then these are converted to frequency values for the lowpass filters. This is how the string damping mechanism works.
A clip ~ then follows the string damping to normalize the signal going through the digital waveguide. This is just in case the model becomes unstable. A multiplier then follows this; the multiplier is used to set the strings decay rate.
The velocity of the note being played is mapped to the values of each of the multipliers. It works on the principle that the larger the velocity, the longer it will take for the strings to decay.
The last object left to discuss in the digital waveguide is the all-pass filter. This is the most important part of the waveguide as it is the part that gives the delay line a fractional delay and also dynamically changes the delay length giving the sitar its characteristic timbre.
The middle inlet for the all-pass object is what controls the delay time of the filter. Two different processes modulate this value.
The main one is the Karplus- Strong algorithm that is to the bi-directional digital waveguide's right, and the other is by a sub-patch called delay all pass.
Within the delay all pass sub-patch, you have a mechanism to create a slight vibrato to form the overall beating effect between the strings. The amount of beating that occurs is relative to the velocity of the note being played.
At any one time that a string is being played, all the other strings that can be activated through the poly~ object are receiving a very slight signal, which is being modulated by the delay all pass sub-patch. This is to help model sympatheti
c resonance between the main strings.
Karplus-Strong Algorithm
The Karplus-Strong algorithm, in this case, is not being used to generate sound but as a way to control the delay length of the bi-directional waveguide dynamically.
This was implemented because it was felt that the best way to control the decay rate of the dynamically changing delay length of the bi-directional waveguide was by using something similar to the waveguide.
The KS algorithm was chosen because it is inexpensive
and it would naturally complement it. The KS algorithm receives the same excitation and pitch values as the bi-directional waveguide, so that decay rates of the two are somewhat similar.
The velocity of the note being played also affects how much the KS algorithm modulates the decay rate of the dynamically changing delay length of the bi-directional waveguide.
It can be seen in the patch that the receiving object known as this bridge length controls this. This takes the velocity of each note being played and maps it to suitable bridge modulation parameters.
The note's velocity affects the dynamic delay length is
modeled on how it works for a real sitar. The output of the KS algorithm is then fed into a sub-patch that smoothes out the changes in delay length.
If this were not implemented, it would drastically affect how the sitar sounded due to the sudden changes in delay length and cause glitches in the audio output.
Some of the output of the bi-directional digital waveguide is also fed back into the KS algorithm. This keeps the energy in the KS algorithm relative to the energy in the bidirectional delay line.
The rate of change of the dynamic delay length changes more randomly in the attack portion of the signal; over time, it becomes less random and then settles into a more periodic pattern as the strings waveform itself becomes more periodic.
Eventually, this approaches zero. It could not be found anywhere in all of the literature reviewed or on the Internet; this approach to non-linear distortion is implemented and unique to this attempt to physically model the sitar.
It is hoped that this approach to the modeling of this type of non-linear distortion is considered for other instruments.
9.4. Sympathetic Strings (Tarafdar)
The sympathetic strings of the sitar are on a different bridge to the main strings. The bridge has the same shape as the main bridge, so the sympathetic strings were implemented similarly to the main ones.
However, there are a few differences. One of them is that each of the individual sympathetic strings can be tuned to whatever the user desires to the western musical scale.
The other difference is that there is no string coupling; this is due to the CPU's limitations. The sympathetic string sub-patch works because all the energy that comes from the leading-strings is scaled and fed into each of the individual sympathetic strings.
It is scaled due to the energy lost in the energy traveling from one bridge to another. This amount of scaling was determined by trial and error.
The damping on each of the strings is higher than on the main strings; this because these strings aren't being plucked but are only resonating to the main strings.
By default, the sympathetic strings in the sub-patch turned to what they would typically be tuned to in Indian classical music. However, this does vary significantly to the raga being played.
Finally, the output of all the strings is summed and scaled again. Having these strings adds greater depth to the sound of the model. The instrument sounds very dry when it is turned off.
9.5. Resonator (Kaddu)
Originally the plan was to use commuted synthesis to model the resonator as mentioned already. Further investigation and research determined that this technique is unsuitable for a sitar due to its non-linear model.
This technique only works for linear time-invariant systems. Commuted synthesis is where you take an impulse response recording of the resonator of the instrument being modeled and convolve this recording with the excitation mechanism in the model.
This is the basic idea behind commuted synthesis, and it dramatically reduces the complexity of stringed instrument implementations since the body filter is replaced by an inexpensive lookup table (Smith 1993).
Instead, the implementation used in this model is a bank of bandpass filters set to different frequencies and Q values.
Unfortunately, an actual sitar was not obtainable when this implementation was being developed, so an analysis of the actual body resonances of a sitar was not performed.
The resonances used were similar to those of a Martin D-28 guitar (Fletcher and Rossing, 2005). Fletcher and Rossing's books weren't used, they were used mainly as a guideline, and a lot of trial and error was involved in getting it to sound correct.
There is a massive contrast between the sound of the sitar with the resonator and it not having it. It was the fffb~ object that was used for the filter bank.
The fffb~ object is a MaxMSP implementation of a bank of bandpass filter objects. It is much more efficient to use this instead of a group of reson~ objects.
9.6. Conclusion
As the reader can see, the approach used to model the sitar was to develop each part separately and tie them together.
It can also be seen that the dynamic delay line is very important in giving the sitar its characteristic timbre. This is not to say that the resonator and sympathetic strings are not as important.
As mentioned previously, without these, the instrument would sound very artificial and not have any natural-sounding qualities to it.
It is also hoped that the unique approach to the dynamic delay line that was implemented has made the model more natural sounding.
10. Results and Analysis of Sitar Model
The analysis was approached by recording a real sitar playing the note F4; then, this note was played a few different times on the modeled sitar and recorded.
Audacity was then used to perform spectral analysis on each of the recordings. The type of analysis done was Fourier analysis with a Hanning window and a window size of 512.
Since every note played by the sitar will be different every time due to the random nature of every pink noise burst excitation, not every modeled sitar pluck analysis will be the same.
Once the first recording was made, analyzed, and then compared, a second recording was made with several adjustments, which will be discussed later. The results are as follows:
We see in the first analysis chart; the modeled sitar is very close to the real sitar up until roughly 10,000 Hz. We see the fundamental is close, and many of the partials
are the same, but the modeled sitar lacks a lot of energy in the higher partials.
Third Spectral Analysis. Red = Modelled Sitar, Blue = Real Sitar
This could either be due to an incorrect tuning of the body resonances, or it could be due to a lack of energy being supplied to sympathetic strings. Before the second recording was made, a few adjustments were made to the sitar model.
One of the body resonances was slightly changed, and also, the amount of energy being sent to the sympathetic strings was increased slightly. It can be seen straight away Second Spectral Analysis chart that there is a difference.
It seems very similar to the real sitar up until 12,500 Hz, and then it begins to taper off, but at the same time, the difference between the upper partials isn't as severe. This could be due to the tuning of the sitar's sympathetic strings.
A third recording was made, but this time the sitar's sympathetic strings were tuned up a whole octave. It can be seen in Second Spectral Analysis the results were a lot more satisfactory. There wasn’t as big a difference between the higher partials.
Second Spectral Analysis. Red = Modelled Sitar, Blue = Real Sitar
10.1. Efficiency
The model was tested on a MacBook Pro with a 2.26 GHz Intel Core 2 Duo processor. It was found that the most amount of CPU power that was used was 62%.
Considering the number of different strings that were modeled, this is very efficient. In the model, you have a seven-note polyphony poly~ abstraction and thirteen sympathetic strings, all being used simultaneously.
Although, when it was tested using string coupling in three different dimensions, it would max out the CPU, and distortion occurred. The third dimension being the longitudinal dimension.
Third Spectral Analysis. Red = Modelled Sitar, Blue = Real Sitar
This third dimension could have been used to create a more realistic tone. The model could have been made even more efficient if the Karplus-Strong algorithm was used for the sympathetic strings, although there may have been a loss in sound quality.
10.2. Implementation Issues
All of the implementation issues to do with the model had to do with MaxMSP. A lot of them were about the CPU.
It would have been more efficient to have implemented this model in C++ as MaxMSP is has a lot of its processes running when you are using the patch, but time constraints would not allow this.
It was also originally planned to use a guitar with MIDI pickups as the interface for the model, but due to the instability of the delay lines in MaxMSP, this wasn't feasible.
The ability to bend the strings of the MIDI guitar would have been a nice touch to the sitar and would have made it more expressive since there is a lot of string bending in actual sitar playing.
The other MaxMSP issue that was encountered had to do with signs (Signal Vector. Size).
The higher the signs, the more accurate the high notes would sound. The reader may notice that the notes at the higher end of the kslider sound slightly out of tune when using the patch.
This is because the signs could only be set to 8. If it is set any smaller, it causes the audio to distort due to the CPU being overloaded.
Conclusion
The goal of this dissertation was to physically model an instrument that hasn't been developed that much in the physical modeling sense.
As mentioned before, a lot of the research about physical modeling has been focused on the western classical guitar. The reason why the sitar may have been overlooked so much is maybe because of its complex design.
There were many more factors to be considered when it came to modeling this particular instrument. When this dissertation was initially started, it was assumed that the modeling process would be relatively simple, and the implementation would take a lot less time than predicted.
The reason why the modeling process took so long was because of the nonlinear bridge structure. It took a lot of testing and re-evaluation of parameters before the desirable sitar tone was achieved.
During the development of the sitar model, as mentioned before, a new and unique modeling approach was taken about the sitar's nonlinear bridge structure.
The Karplus-Strong algorithm controlled the dynamically changing delay line that had to be implemented due to the bridge shape. The Karplus-Strong algorithm being chosen to control this parameter because of how computationally efficient it is and how likened it is to how an actual string decays.
Immediately after this was implemented, the difference in how realistic the timbre of the sitar became was noticeable. We believe this modeling approach warrants further investigation as it has never been implemented before and is a new and innovative approach to this kind of modeling problem.
By looking at the spectral analysis of the sitar versus the real sitar, it could be said the model was quite successful. Although there are still a few bugs in the model, one must regard the tuning, particularly at the higher pitches.
It would also be nice to implement the ability to pitch bend the notes. This was attempted, but it was unsuccessful as MaxMSP kept distorting. Future research would also be exciting to model amplitude limitations for the strings at the frets since the sitar has such unique frets.
This is similar to what to the nonlinear distortion that occurs in the slap bass technique. It would also be interesting to see what the model sounds like if there was a string coupling effect applied to the sympathetic strings. It was due to CPU limitations that this couldn't be achieved.
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