· (1) How do instruments make sounds?
· (2) Why don’t drums create pitched sound?
· A wave is a pattern of displacements from some ‘neutral’ or ‘at rest’ position, traveling through a region of space in particular directions, at a particular speed. The leading edge of the wave is called the wave front.
· Displacement, which may be distance, pressure, or some other quantity, is defined at each point of the region in which the wave is moving.
· Directed displacements (such as distance) can be parallel to wave direction (longitudinal wave) or perpendicular to wave direction (transverse wave). View animations of transverse, and longitudinal traveling waves.
· Some waves (mechanical waves) move through an elastic medium while others don’t.
· For acoustic music, we are primarily concerned with mechanical waves moving in an elastic medium, especially strings, membranes, air, and other materials such as metal or wood. Examples of mechanical waves include waves on strings, in air, in water, on membranes, in solid bodies of wood, metal, ceramic, or other substances.
· When we say a medium is elastic, we mean that it acts as if composed of tiny masses connected by tiny springs. If an impulse is applied to any point, a ripple or pulse wave is generated, which propagates through the medium (for example, if you tap one end of a taut rope). If a series of impulses is applied, a wave train is generated. It turns out that the wave speed is determined almost entirely by properties of the medium alone – density and elasticity (the size of the ‘tiny masses’, and the resiliency of the ‘tiny springs’).
· Electromagnetic radiation (including light) is not a mechanical wave – there is no medium (though 19th century scientists found this hard to believe and wrongly postulated an ether through which light moves).
· Multiple waves may pass through one another. When they cross, they simply add. This means that the displacement at any point is the sum of the displacements of each of the component waves crossing at that point. This property is called the superposition principle. Here is an example of two pulses passing through one another.
· When a wave moves in a bounded region of space, it may be reflected off the boundary, as in the case of this pulse reflection on a string with a fixed end. In the case of a wave train, the reflected wave may add to the incident wave.
· Waves may be effectively one-dimensional (e.g. on a string, or in a narrow cylinder), two-dimensional (on a pond or drum head), or three-dimensional (sound in open air). The wave front is correspondingly zero, one, or two-dimensional.
· Much of our sensory data about the world comes via biological wave receptors (eyes and ears).
· We’ll consider only one dimensional periodic waves (but try to generalize to two or three dimensional waves)
· A periodic wave repeats, cycling in time and space, as in the case of this transverse wave.
· At any instant in time, take a photo of the periodic wave. You’ll find that it repeats over a distance L, called the wavelength. This is the length of one wave cycle in space.
· At any position in space, measure the periodic wave as it passes. You’ll find that it repeats over a time T, called the period. This is the length of one wave cycle in time.
· The inverse of the period is called the frequency, and is measured in cycles per second, Hertz (abbreviated Hz). We will call the frequency f. By definition, f = 1/T.
· For instance, if the period is half a second (the wave repeats every half second), then it repeats twice every second: the frequency is 2 Hz.
· What is the speed of a periodic wave? By definition, with wavelength L and period T, we know that a length L of wave must pass any given point in time T. This means the wave is traveling at the speed of L/T, or L*f. To restate: the speed of a periodic wave is the wavelength times the frequency.
· For example, if a sound wave has a frequency of 440 Hz (the note A in the middle of the piano keyboard), and a wavelength of 2.25 feet, then its speed must be 440 * 2.25 = 990 feet per second, because 2.25 feet of wave are passing any fixed point 440 times per second. In fact, this is roughly the true speed of sound in air (the precise speed depends on air pressure and temperature, which together determine density and elasticity of the elastic medium).
· The amplitude of a periodic wave is its maximum displacement.
· All waves travel, but sometimes the sum of multiple waves appears to be stationary.
· In particular, when a one-dimensional periodic wave reflects off a boundary, and is superimposed upon the incident wave, a standing wave develops, as a result of superposition. Here is an animated illustration of how this happens.
· If the space within which this wave moves is bounded on the other side, then it will again reflect and again. In this case the standing wave will be preserved if and only if all reflections reinforce one another. This condition means that for each bounded medium only certain frequencies will generate standing waves. These frequencies are called resonant frequencies of the medium, and they depend on the medium’s physical properties and dimensions.
· The result may be transverse standing waves (as on a string), or longitudinal standing waves (as in the air column in a pipe).
· Note that in one dimension the standing wave is characterized by equally spaced nodes (which remain motionless - undisplaced, at the “rest” position), and between them antinodes (which move maximally). Adjacent antinodes are always oppositely displaced.
· When a system is excited at its resonant frequencies, energy is stored in the standing wave rather than lost at boundaries, because there is little destructive interference. As energy is pumped in at these frequencies, the amplitude of the standing wave will increase, even to the point of destroying the system.
· Think of a swing which you pump at the right rate (frequency) to go higher and higher. Or singing at just the right pitch (frequency) in a bathroom, so as to make your voice appear louder and louder, or making waves in a tub that increase until the water spills out.
· Mathematically, resonances are determined by the medium’s physical properties and dimensions. By superposition, sums of resonant frequencies also determine possible resonances. The simplest resonances are called normal modes of vibration, and are associated with definite frequencies.
· Thus a system can vibrate at one specific mode, or (more commonly) a complex sum of modes (by superposition).
· However the sum of modes need not be periodic! To foreshadow: for a one dimensional medium, the sum is always periodic; for a two or three dimensional medium, the sum need not be periodic.
· A sound wave is a wave propagating through air. Unconfined, sounds waves are three-dimensional.
· Audible sounds correspond to sound waves in the frequency range 20 Hz (lower hearing threshold) to 20,000 Hz (upper hearing threshold).
· Approximately, pitched sounds correspond to periodic sound waves. Pitch is directly related to frequency (as frequency increases, pitch goes up). In fact, a constant increase in pitch corresponds to a constant ratio in frequency: If sounds A and B sound is if they are the same pitch distance apart as sounds C and D, then the ratio freq(A)/freq(B) = freq(C)/freq(D).
· Loudness is directly related to average wave amplitude.
· Timbre means “sound quality”.
· Timbre is a kind of catchall qualitative category: the way you differentiate two sounds of identical pitch and loudness. For instance, a trumpet and a violin could produce sounds of identical pitches and loudnesses. The difference between them would be timbral.
· Timbre is determined by the wave as a whole. Two properties are particularly important: envelope, and waveshape (for periodic waves).
· The relative loudness curve from start to stop is an important characteristic for identifying timbre, and is called the envelope.
· The envelope includes periods of attack, sustain, and decay. The relative lengths of these periods are important in identifying timbre.
· Thus a gong sound may build gradually, and decay much more gradually, whereas a xylophone sound starts and stops suddenly (“dry” envelope).
· For periodic waves, sound timbre is closely related to waveshape: the shape of one repeating wave cycle.
· Waveshape in turn depends on the so-called Fourier components of a periodic wave. Periodic waves with complex waveshapes result from adding simple sounds at frequencies which are multiples of a single “fundamental” frequency. Your ear hears fuses these components, hearing this sound as a single pitch at the fundamental, with a particular timbre.
· Timbre and Harmony: When Fourier components are completely blended together, the ear hears a single pitch with a distinctive timbre. When they are not completely blended, the ear hears harmony.
· Experiment with the Fourier synthesis module to create periodic waves of identical pitch, but varying timbre. Note how the waveshape changes.
· Can you cause your ears to analyze the sound as harmony, or hear it “whole” as timbre? [Hint: First add frequencies gradually one by one, so that a choir effect appears: harmony (you are able to hear separate voices). Now mute the sound (by selecting “audio off”, wait a minute, and then turn it back on (“audio on”). Do you still hear the harmony, or have the Fourier components fused into a single timbre?]
· In music, the basic properties of sounds are: pitch, loudness, and timbre.
· We now understand roughly how these properties are related to sound waves.
· It remains to answer our opening questions: “how do musical instruments create sound?” and “Why don’t drums usually produce pitched sound?”.
· (1) An energy source (usually human powered) which excites and thus transmits energy to a
· (2) sound generator: bounded medium (air column, string, membrane, solid body) within which standing waves develop (one instrument may possess more than one sound generator). The sound generator is string(s) for chordophones; air column for aerophones; membrane for membranophones; other for idiophones.
· (3) For instruments other than aerophones, the standing waves in the sound generator radiate sound waves which are transmitted to a resonating chamber (partially bounded air chamber) which develops its own standing waves, amplifying and filtering the sound coming from the sound generator, and radiates a traveling sound wave to your ear. For aerophones, standing waves in the sound generator are radiated to your ear directly.
· (As a bounded airspace, the concert hall can also develop standing waves, shaping the sound of instruments playing within it – in a sense, the hall is part of the instruments it contains!)
· All the energy which reaches your ear is a small fraction of the energy the human performer put into the instrument in the first place!
· Energy source: human body. You move your hand to pluck or strum the strings
· Sound generators: Strings (one dimensional – like a line). Waves move on the strings, reflecting off the fixed ends, and producing standing waves
· Resonating chamber: Body. Strings produce traveling sound waves, which excite standing waves within the guitar’s body. Traveling waves radiate to your ears from holes in the guitar body.
· Energy source: your breath. Your lungs compress to blow a stream of air into one end of the nay.
· Sound generator: the air column inside the instrument (essentially one dimensional, if narrow). Your breath excites standing waves inside the tube. Traveling waves radiate to your ears from the nay’s holes.
· Energy source: human body. You strike the drum head with a mallet.
· Sound generator: the drum head, a plastic membrane (two dimensional – a surface). The strike sets up standing waves in the head.
· Resonating chamber: the drum body. Standing waves in the head produce standing waves in the air cavity contained in the drum body. From here traveling waves radiate to your ears.
· Energy source: human body. You strike the gong with a mallet.
· Sound generator: the gong itself (three dimensional – a body). The strike sets up standing waves in the gong.
· Resonating chamber: some gongs partially enclose an airspace within which standing waves may develop. Traveling waves radiate to your ears.
· The most famous system for classifying musical instruments (Hornbostel-Sachs) is a taxonomy, whose first division depends on a classifying the sound generator: idiophones (3d body), membranophones (2d membrane), chordophones (string), aerophones (column of air).
· The second division depends on mode of excitation of sound generator for idiophones and membranophones (percussion).
· We consider here the sound generator component.
· Sound is pitched when the standing waves on the sound generator are periodic.
· We argued that the simplest standing waves are the normal modes, each of which is associated with a definite frequency, i.e. is periodic.
· Any complex sum of these normal modes is also a possible mode of vibration of the sound generator.
· Therefore the sound produced by an instrument is pitched when the complex sum is also periodic.
· It is not hard to demonstrate that a complex sum is periodic if and only if each frequency in the sum is an integral multiple of a single number; this is the same thing as saying that each period must be an integral division of a single number- such a condition is required for all the periodic waves to “fit” one inside the other.
· SO: sound is pitched when all the normal modes of the sound generator excited by the energy source have frequencies that are integral multiples of one number (as long as that number falls within the range of hearing). That number is often called the fundamental frequency.
· When each frequency in a series of frequencies is a multiple of one number, we say that the series is harmonic.
· If you play around with the Fourier synthesis module, you will intuitively see why adding frequencies that are integral multiples of one number preserves periodicity. All frequencies are here multiples of one fundamental frequency. No matter how you combine them, you always end up with a periodic wave.
· Example of pitched sound: three normal modes excited at frequencies 100 Hz, 200 Hz, 300 Hz (all are multiples of 100 Hz)
· Example of pitched sound: three normal modes excited at frequencies 150 Hz, 300 Hz, and 375 Hz (all are multiples of 75 Hz)
· Example of unpitched sound: three normal modes excited at frequencies 150 Hz, Ö2 * 150 Hz, and p * 150 Hz.
· We’ll observe this in what follows.
· Standing waves may be created when string is excited by energy source (bowed, struck, etc.). These may be normal modes, or complex sums of normal modes.
· Velocity of a wave on a string: It turns out that the velocity of a wave on a string = v = sqrt (tension of string/density of string).
· This formula makes intuitive sense, since it properly accounts for the medium’s density and elasticity. Tenser strings, and lighter strings produce faster waves. Think about it.
· There are various “normal modes” of standing wave vibration on a string. These are mathematically the simplest. Let’s explore them first.
· Examine again the transverse standing waves.
· Each mode is characterized by fixed point, or nodes (zero-dimensional) separated by points of maximal movement (antinodes). The string curves smoothly from node to antinode, without any complexity.
· Nodes are equally spaced across the string.
· Between nodes, adjacent segments of the string display opposite displacements. These segments oscillate up and down with time.
· What is the wavelength? The standing wave always repeats after two nodes, so the wavelength equals twice the distance between two adjacent nodes (convince yourself by observing the transverse standing waves).
· Where are the nodes? If the ends of string are fixed, then they must be nodes.
· However it will be simpler to count only the right hand end a node in what follows.
· Assume a string of length S with n nodes.
· The distance between adjacent nodes is simply S/n
· SO: wavelength = twice S/n = 2S/n
· BUT: frequency = velocity /wavelength
· THEREFORE: frequency = v/(2S/n) = nv/2S
· The frequency of the first normal mode is called the fundamental frequency = F = v/(2S)
· The frequency of the second normal mode is 2v/2S = v/S, which is twice the fundamental frequency F.
· The frequency of the nth normal mode = nv/(2S) = n * F
· The following chart summarizes.
|
Normal mode # |
Frequency |
Ratio to fundamental |
|
1 |
v/(2S) = F |
1 |
|
2 |
v/S = 2F |
2 |
|
3 |
3v/(2S) = 3F |
3 |
|
4 |
2v/S = 4F |
4 |
· Ratios show that frequencies of normal modes are all multiples of F. This series of frequencies is thus harmonic.
· Therefore any complex sum of normal modes is also periodic. Sound waves generated by a string are pitched.
· Inserting the formula for v, the fundamental frequency F is = sqrt (tension of string/density of string)/(2S).
· Convince yourself that this makes sense. What happens to frequency (pitch) as you increase tension? Density? String length? Stringed instruments vary pitch by varying tension, density, and string length. Can you give examples?
· Here nodes are points where pressure does not change, and waves are longitudinal rather than transverse. Normal modes look something like this. Another slight difference arises since the ends of the air column need not be nodes (when they are open, they’re nodes; when closed, they aren’t).
· But the argument used for strings more or less applies to air columns as well, because the air column is one-dimensional.
· The conclusion: complex standing wave vibrations of an air column are periodic, hence produce the sensation of pitch.
· Control of pitch is somewhat different, however.
· Unlike string instruments, wind instrument players cannot vary density and elasticity of the medium (air). But they can vary pitch by changing the length of the vibrating medium (air column), by opening or closing holes to vary its effective length.
· They can also do something stringed instrument players rarely do (through a technique called playing harmonics): they emphasize different normal modes from the same column length.
· Thus a flute player can play more than one note, without changing which holes are covered, by exciting different normal modes through breath control.
· We consider here the case of a drum membrane tensioned across a circular frame. The frame provides a fixed boundary analogous to the fixed ends of a string.
· Drum is usually struck (occasionally rubbed or plucked).
· The membrane is displaced from its rest position, but (being elastic) rebounds. Two-dimensional waves travel outwards, and reflect from the fixed boundary to make standing waves. The membrane vibrates up and down.
· Only rarely is pitched sound created – why?
· Vibrations of membranophones are much more complex because the wave generator is two-dimensional.
· We first examine the normal modes of vibration for a circular membrane.
· On the one-dimensional string, we saw that nodes were points (zero dimensional).
· On the two-dimensional membrane, nodes are lines and circles (one dimensional). Lines are diameters (passing through the center of the membrance), and circles all center on the membrane’s center.
· These lines and circles do not oscillate up and down, but stay at the at-rest membrane plane.
· The angles between lines are equal, and the distance between concentric circles is equal (just as the distance between nodes on a string was equal).
· Qualitatively, we can describe these modes using the notation (m,n), where m is number of diameters, and n is number of concentric circles.
· [These ideas are illustrated in this diagram showing membrane modes; note that regions marked + and – always oscillate in opposite directions.]
· Note that the drum frame boundary counts as one nodal circle (like the fixed end of the string), so n is always greater than zero.
· These diameters and circles divide the drumhead into segments, which oscillate up and down. Adjacent segments have opposite displacement.
· Try to understand these concepts by observing these animations of normal modes on a circular membrance.
· Here is a beautiful animation for the mode (2,2) – can you see the nodal circles and lines?
· Assume an infinitely thin, uniform membrane of radius R, vibrating in the normal mode corresponding to m nodal diameters, and n nodal circles, notated (m,n).
· Each normal mode is periodic.
· It turns out that the frequency of normal mode (m,n) = f(m,n)= c*j(m,n)/ (2pR) where j(m,n) is nth root of mth Bessel function (mathematical constant), and c = sqrt (tension of drum head/density of drum head) (another constant related to velocity) (it is hard to derive this; let’s take it on faith)
· [This diagram of membrane pitches shows pitches of normal modes on a musical staff.]
· This information is summarized in the following table:
|
Normal mode |
Frequency |
Ratio to lowest mode (0,1) |
|
(0,1) |
c*j(0,1)/ (2pR) = 2.40 |
1.000 |
|
(1,1) |
c*j(1,1)/ (2pR) = 3.83 |
1.596…=j(1,1)/j(0,1) |
|
(2,1) |
c*j(2,1)/ (2pR) = 5.14 |
2.142…=j(2,1)/j(0,1) |
|
(0,2) |
c*j(0,2)/ (2pR) = 5.52 |
2.300…=j(0,2)/j(0,1) |
· The main point is that these Bessel function constants are not integral multiples of a single number!!
· SO: Each normal mode of vibration is periodic and pitched.
· BUT: frequency ratios show that frequencies of normal modes are not integral multiples of a single number. The series is not harmonic.
· Real standing waves on the drumhead consist of a complex sum of normal modes (like the sums you created with the Fourier synthesis module); it is very hard to excite just one mode.
· Therefore, although individual normal modes are periodic, their sums need not be.
· Therefore sound from a vibrating drumhead is generally not periodic, hence not pitched.
· The sum of normal modes will however produce an impression of “high drum sound” or “low drum sound”, as well as a distinctive timbre.
· f(m,n)= 1/(2 pi * r) * sqrt (tension/density) j(m,n)
· where: r=radius, j(m,n) is “nth root of mth Bessel function”
· Thus frequency varies with radius, tension, and density as expected. Making the drumhead bigger, looser, or heavier produces a lower sound. Making it smaller, tighter, or lighter produces a higher sound.
· Though radius and density are ordinarily fixed, some drums (e.g. Nigerian pressure drum) allow control of tension as a means of varying timbre. Compound drums (sets of drums) enable the drummer to vary radius and density as well (by striking different drums in the “drum kit”).
· If drums generally do not created pitched sound, why do some drums (“tuned drums”) create a sensation of pitch?
· Everything we’ve explained depended on an idealized drum (infinitely thin head uniformly stretched across a circular frame).
· “Tuning” a drum requires damping of particular partials, or modifying normal mode frequencies by deviating from the “idealized” case.
· [Examine once again the diagram of membrane pitches]
· Most energy tends to be in (1,1), then (2,1), (3,1) modes (1,2,3 diameters); also (4,1), (5,1). (1,1) is often heard as the basic pitch of a drum.
· With more than one nodal circle [(m,n) for n>1], decay is more rapid, and so these are less audible.
· Thus if we can make the series (1,1), (2,1), (3,1), (4,1), (5,1) harmonic (or nearly so) we can create the impression of pitch.
· For instance, orchestral timpani (kettle drums) are tuned and pitch change is possible with a foot pedal. Timpani can perform melodies.
· How is this possible? It is believed that due to its large head, air damping tends to lower the frequencies of lower normal modes, turning the series (1,1), (2,1), (3,1), (4,1), (5,1) into a nearly harmonic series.
· [Observe the effects air damping on this series, via musical notation.]
· The tabla (and several other south Asian drums) are tuned by applying a paste to the drum head.
· Thus weighted, a subset of harmonic partials (perhaps only one) dominates, and creating the sensation of pitch.
· Pitch also depends on strike position (see below).
· For each performance, the tabla tunes up with the other instruments, using movable blocks to adjust tension in the drumhead.
· Even for an untuned drum, “pitchedness” may vary depending on where you strike it.
· There is a simple principle here: modes with nodal lines and circles passing through the strike position will be very weak, because the strike position will tend to move (whereas nodal lines and circles don’t move).
· The (0,n) no-diameter modes tend to decay more rapidly, and thus don’t produce sense of pitch.
· Hitting drum at center prevents diameter modes ((m,1) modes for m>0) (which are usually stronger and may nearly create harmonic series)
· With only the (0,n) modes left, one often hears a pitchless thump.
· 2-circle modes (1,2), (2,2), (3,2) if excited tend to create non-harmonic sound. These modes are characterized by a nodal circle, about midway between center and rim.
· To increase pitchedness one wants to reduce the amplitude of these 2-circle modes.
· This can be accomplished by striking the drum near the circle midway between center and rim.
· Such a strike often produces the most tonal sound (tone on the conga)
· Tends to excite many modes at once, particularly higher frequency ones. The sound tends to decay quickly.
· Therefore gives a sharp, high, but unpitched sound.
· Drums play a limited melodic role in much music. Sound is classified as high and low, but only rarely as a precise point on the linear tonal continuum.
· Most drum sound is tonally “non-linear qualitative”: drums produce a set of sounds varying by timbral character rather than quantitatively along a linear dimension (pitch).
· Drums are much freer than other instruments to participate in a wide array of ensembles. Whereas melodic instruments tend to be locked into a particular tonal system, drums are not. They need not align with other instruments “quantitatively” (in terms of pitch) but only “qualitatively” (in terms of timbre).
· Thus most drums can be played together.
· As a consequence, the percussion section of orchestras and bands can be quite diverse, and “drum circles” can include a wide variety of different kinds of drums (recall the film World Drums).
· The non-linear property of drum sound is beneficial for world music fusions, seeking to create a synthesis of different musical traditions (not surprisingly such fusions are often termed “world beat”).
