Basic Physics of Stringed Instruments
In this section, an overview of the physics of stringed instruments is provided, covering such topics as standing waves, harmonics, and harmonic analysis. Other physical aspects of music are also mentioned, including sound waves and resonance. The reader is not assumed to have had any substantial background in mathematics nor physics--only a fleeting acquaintance with functions of sine and cosine. For those with a stronger mathematical background, some equations are included, though the analysis is by no means rigorous. The goal of this section is to provide a foundation for analyzing the sound and music of the kora.
Waves
When looking at musical instruments, the most important physical concept is that of a wave. In general, a wave is any disturbance that propagates through a medium. (Light, a strange time-changing crossed electric and magnetic field with both wave-like and particle-like properties, is a wave that requires no medium to travel through. The ones considered here, though, all do.) There are many familiar examples of waves: waves on the surface of a pond, sound waves, fans in sports arenas 'doing the wave.' All of these transmit energy through the medium without the medium itself having to move. The two types of waves we will focus on here, though, are standing waves and sound waves.
Standing Waves
What happens when a musician plucks a string? How can we describe this physically? To answer these questions, it is useful to consider a simpler case: a hand shaking a length of rope up and down from its left end. When the hand moves up, the left end of the rope moves with it, which pulls on the rope farther to the right, causing it to move up, and so on. Likewise when the hand moves down. The parts of the rope, like the hand, only move up and down. However, the wave pattern, or just the wave, moves to the right. If we take a snapshot of the wave at a certain time, it will have the shape seen to the right.
At some later time, the rope will have the same shape, but the wave will have progressed along the rope. The time it takes for any part of the rope to complete a cycle (i.e., end up where it began) is called the period of the wave. |
Let us denote the vertical axis y and the horizontal axis x. Note that the height, y, depends on both the position x and the time t. Mathematically, we can describe this wave as follows:
Note that the speed of the wave is the amount of time it takes for one wavelength to pass. This amount of time is precisely the period of the wave. Hence, the speed of the wave is the wavelength divided by the period. We can also define the frequency, f, of the wave as the inverse of the period and equivalently express the wave speed in terms of this quantity. (Shown on right.)
What does this have to do with standing waves--wave patterns that themselves do not move, though each part of the wave oscillates up and down, like a plucked string? Well, it turns out that a sinusoidal standing wave is the sum of two traveling sinusoidal waves--the ones that we just described. One that moves in the + x direction (to the right), and one that moves in the - x direction (to the left). The derivation of this is shown on the right.
What does this physically mean? When a player plucks a string, they send out two sinusoidal waves along the string in opposite directions. These waves get reflected at the end of the strings and add together, and the result is called a standing wave. The standing wave (SW) equation is |
Let's examine this equation: When a string is plucked, there is no motion at the endpoints of the string (unless you have a bum kora). If we let the string start at x = 0 and end at x = L (the string has some length L), then this means that the height of the standing wave, y(x,t), is always zero at these points, no matter what the time t is. So, the first sine term involving x must be zero at these points. The point x = 0 is not a problem, since sin(0) = 0. It is the point x = L that warrants further consideration. Recall that the sine function is zero at any integer multiple of pi. This means that we must have
Thus, for standing waves on a given string of length L, there are only certain wavelengths that are possible. Correspondingly, since the speed v of the wave is fixed, there are only certain frequencies that are possible. Recall that the speed of the wave is the wavelength times the frequency. So, we can also write
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The frequency corresponding to n = 1 is called the fundamental frequency of the string. The higher frequencies are called the nth harmonic frequencies. (For example, for n = 3, this is called the third harmonic frequency, or just the third harmonic. Confusingly, this is also called the second overtone by musicians.)
The short video on the left demonstrates the possible wavelengths (and frequencies) of standing waves. These guys have white lab coats on, so the stuff they say must be (and is) legitimate. |
The Sound of the Kora
When a musician plucks a stringed instrument, the wave doesn't look anything like the standing waves produced in the above video. How can we explain this? We showed that there are only certain possible wavelengths for standing waves; if the wave doesn't look any of the ones in the video, then it must be a combination of all of the ones in the video! This principle is called the superposition principle. We already used it once to add together two sinusoidal waves to form a standing wave.
The mathematical process that is used to model the shape of a plucked string (as well as many other things) is called a Fourier analysis, which is often called harmonic analysis when applied to musical instruments. In our case, it involves adding together the waves of the possible frequencies in a sort of weighted average.
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If you find yourself asking these questions, here is some justification, although it may be a little confusing if you are unfamiliar with calculus.
The properties of all waves can be succinctly stated in the following partial differential equation (ie, an equation involving derivatives, or rates of change): This equation is accordingly called the (one dimensional) wave equation; it is one of the most important equations in all of physics. All waves "obey" it. If you are familiar with calculus, check that the sinusoidal wave and standing wave equations satisfy this differential equation. If you aren't familiar with calculus, don't worry! The superposition principle is just a mathematical property that arises from this equation. Basically, if any two functions obey this equation, then the sum of the functions solves this equation (specifically, any linear combination of two solutions is also a solution). Thus, we can add together wave functions that obey the same wave equation to get a new wave function that still obeys the same wave equation!
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Sound Waves
Now that we know how musical strings vibrate, it is natural to consider how we actually hear the sound. The sound of plucked strings themselves is very quiet--try strumming an electric guitar that is not plugged into an amplifier, for example. To remedy this, acoustic instruments use a resonating chamber, or just resonator, to amplify and project the sound. In this resonating chamber, the air vibrates as the instrument vibrates, creating regions of lower and higher pressure. It is this pressure difference that our ears and brains interpret as sound.
When the resonator is vibrating, all of the vibrating strings contribute. The vibration of the resonator is the sum of the vibration of the strings--another example of the superposition principle. This statement really just says that we hear all the strings at once. Now, while the wave pattern of a single string is fairly simple, the wave pattern of the resonator can get very complex (refer back to the opening picture of "Jarabi's" wave form). The adjacent strings of a kora are tuned in thirds (Aning 1982, 175). Below are two graphs of the total wave form of multiple strings vibrating. The vertical axis is the total vibration pattern, and the horizontal axis is time, normalized in terms of the period of the note of lowest frequency. Both graphs were created using vPython.
When the resonator is vibrating, all of the vibrating strings contribute. The vibration of the resonator is the sum of the vibration of the strings--another example of the superposition principle. This statement really just says that we hear all the strings at once. Now, while the wave pattern of a single string is fairly simple, the wave pattern of the resonator can get very complex (refer back to the opening picture of "Jarabi's" wave form). The adjacent strings of a kora are tuned in thirds (Aning 1982, 175). Below are two graphs of the total wave form of multiple strings vibrating. The vertical axis is the total vibration pattern, and the horizontal axis is time, normalized in terms of the period of the note of lowest frequency. Both graphs were created using vPython.
Note that the tempered scale was used for these notes, with the frequency of A4 set to 440 Hz. These frequencies are probably not the same on a kora, depending on whose you have. For a fuller discussion of this, see the discussion of tuning at Music of the Kora.
Just considering the fundamental frequencies with two notes and constant amplitude (force of plucking the string), the wave forms already become quite complex. When a kora virtuoso like Diabate plays, plucking numerous strings at once with varying strength, with frequencies ranging from the bass to the higher registers, a wave pattern like the one seen at the top of this page is formed. However complex, though, it can still be analyzed in theory by the same principles used above!
Just considering the fundamental frequencies with two notes and constant amplitude (force of plucking the string), the wave forms already become quite complex. When a kora virtuoso like Diabate plays, plucking numerous strings at once with varying strength, with frequencies ranging from the bass to the higher registers, a wave pattern like the one seen at the top of this page is formed. However complex, though, it can still be analyzed in theory by the same principles used above!
Construction of the Kora
The sound of an instrument is intrinsically wound up in its construction. Here, a brief overview of the kora's build is presented. For a fuller discussion on this topic, see Noah's page.
The calabash gourd as a resonator is one of many unique, defining characteristics of the kora. The gourd, 40-50 cm in diameter, is cut in half and covered in cowhide; it has a hole cut out near the right hand-grip, and it is usually decorated by the player. The neck of the instrument is made of African rosewood, and it is about 120-130 cm long. The strings fan out in two sections--one for either hand. The player, usually seated on the ground, grabs the hand grips and plucks the strings with only thumbs and forefingers. The strings are now made of nylon, though before mid 1900s they were made of rawhide. The rings along the neck (collars or braided tuning thongs) are made of iron--adjusting their position along the neck is how the instrument is tuned (Hale 1998, 154-155). The acoustic properties of wood is a very active area of research. Unfortunately for the kora, most researchers are concerned with, well, wood; there is little to no information on the acoustic/resonant properties of a half-calabash gourd covered in cowhide. Rosewood is a very sought-after material for acoustic instruments, but the typical kind is Brazilian rosewood, which is listed as a wood best-suited for stringed instruments among Norway spruce, Sitka spruce, Mulberry, Japanese maple, and others (Yoshikawa 2007, 570). Suffice it to say that the unique materials and construction of the kora heavily contribute to its unique sound. |
Now that we have a firm(er) footing on the physics of sound and stringed instruments, as well as the kora's build, we are ready to look at the music of the kora. The things discussed on this page are invaluable for understanding the musical concepts that we will look at: The relationship between frequency of a string and the speed of the wave--which depends on the string's tension, seen in the next section--is crucial to understand for the tuning of the kora. Harmonic analysis and the presence of higher harmonics explains the timbre of the kora, and so does its construction. The construction of the kora--particularly its separated sections of strings--also allows for the unique sound and playing style that make the kora such a recognizable instrument.