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Editorial Reviews. Review. "Deals with the principles of acoustics and their relationship to Musical Instrument Design: Practical Information for Instrument Making - Kindle edition by Bart Hopkin, John Scoville. Download it once and read it on.
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- Musical Instrument Design: Practical Information for Instrument Making by Bart Hopkin
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- Musical Instrument Design : Practical Information for Instrument Making
Your points will be added to your account once your order is shipped. Click on the cover image above to read some pages of this book! This is an encyclopedic, large-format book containing hundreds of illustrations. While not geared toward making conventional instruments, Musical Instrument Design provides all the information that anyone amateur or professional should ever need to construct an amazingly wide variety of percussion, string, and wind instruments. Includes many designs along with parts lists and detailed construction instructions.
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Musical Instrument Design: Practical Information for Instrument Making by Bart Hopkin
As an example, consider a plucked string sound. The tone begins quite suddenly at the moment of plucking, reaching its greatest volume almost immediately; it can be said to have a sharp attack. There follows a longish period of decay in which the volume gradually diminishes until it becomes inaudible. The attack is further marked by a distinct plucking sound an unpitched noise of very short duration which is quite different from the ensuing string tone.
The purer string tone that follows is likely to start out relatively rich in high-frequency overtones; these decay more rapidly than the lower-pitched components, and so the overall tone quality becomes a bit less brilliant during the course of the decay. Other instrumental sounds can also be described in terms of attack and decay and related factors. Although I've spent less time discussing them here, these time-change elements contribute as much as does timbre to the overall impression of tone quality. Directional factors also contribute to perceived effect.
Their results play out differently for high- and low-frequency sounds. Low-frequency sounds tend to have a room-filling quality, primarily because they spread out from their sources over a wider periphery, and find their way around obstacles more effectively than high frequencies. The fact that the human head is one such obstacle means that it is harder to locate low-frequency sounds directionally, since the sound is more likely to reach both ears equally. Higher frequencies spread over a narrower angle and don't find their way around obstacles as well, allowing more for the differential effects that the two ears use in sound location.
Through this chapter I have used the word "noise" to describe sounds that lack definite pitch. The term should not be regarded as pejorative. Steady oscillation and the harmonic series are not gospel truth. Music making has always involved plenty of pitchless sounds, from maracas and snare drums to the muffled bass drum. I encourage you to cultivate an appreciative ear for these things.
As you learn to listen closely enough to make conscious what your ears already hear, you will discover that the ears' analytic capacities are great. This sort of listening awareness is valuable for anyone interested in musical instrument design and construction. It combines the vibrating patterns of a tuning fork and a simple tube chime, giving rise to a complex array of frequencies.
You can tune the frequencies as you make the chime albeit roughly , and bring out different frequencies in the tone by how and where you strike. All you need to make fork chimes is metal tubing. Aluminum or brass tubing will work well, but you can also make a fine instrument with the same steel conduit recommended for the bar gong described in Sidebar Get enough tubing to make several chimes.
The drawing shows the design of the fork chime, with typical dimensions. At this point you have a simple chime; it will produce a nice tone when suspended and struck with a beater. Next, cut the slit in the opposite end using a hacksaw or sabre saw. Try to make it straight and even, but don't worry if it isn't perfect.
Support the tube by the cord, and strike near the split end. A new tone emerges. It comes from the two halves of the split end engaging in the pattern of vibration normally associated with tuning forks. The chime tone is still there though slightly raised in pitch due to the cutting , and you can hear it by striking at the center. Several other slightly quieter tones appear when you strike in different places, most of them being overtones of the basic chime and fork tones.
Spend a few moments striking the fork chime in different ways, and listen to the tones you can get. Do you like the musical relationships you hear? If not, you can change them. Cut the slits longer to lower the fork tone. Raise the chime tone by cutting a small amount of tubing from the opposite end. Experiment until you find a set of tones that form an attractive blend. When you have made several fork chimes with pleasing relationships both within themselves and among one another, hang them some place where you can occasionally play them as you pass by.
We will be studying the vibrating patterns of chimes and tuning forks in Chapter 4, "Idiophones. In the last chapter I talked about how people perceive sounds. We now turn to objects in the external world, the actual sound makers that are the subject of this book, and begin to investigate some underlying principles of musical instrument acoustics.
In the process I will refer to various instruments in order to illustrate ideas, but remember that fuller information on specific instrument types appears in subsequent chapters. This may seem like a rather specialized requirement, but steady-frequency oscillation occurs in nature all the time. There are three factors in most natural oscillations: displacement, restoring force, and something working in opposition to the restoring force, usually inertia. For the present, we can define inertia as the tendency of an object in motion to remain in motion in the absence of any force to stop it.
To understand these factors, think of a pendulum. Pendulums move much more slowly than sound vibrations, but their principles are the same. Displace a pendulum to one side, and a restoring force gravity causes it to swing back toward its center rest position. But its inertia causes it to overshoot, leading to displacement in the other direction. The restoring force of gravity enters the picture again; it gradually overcomes the inertia, causing the pendulum to slow and then reverse, swinging back toward center, and then overshoot once again.
Eventually, if nothing occurs to inject more energy into the system, the pendulum gradually dissipates its energy and resettles at equilibrium position, but not before having undergone a series of steady oscillations. In sounds, the restoring force usually comes not from gravity, but from the springiness or resilience of the materials involved.
In the prong of a kalimba African thumb piano , for instance, the rigidity of the prong causes the prong to spring back after displacement. With strings it is the resilience of the string and its mountings that makes it return after being pulled to one side; for drumhead membranes it is much the same. The mechanisms at work are harder to picture for wind instruments, but air, too, has innate springiness, especially when it is in a partially enclosed chamber like a bottle or wind instrument tube.
The mass of the air, minuscule though it may be, plays its role in inertia. The rules that determine frequency of vibration vary for different types of vibrating objects. But, broadly speaking, frequency depends upon two things: the strength of the restoring force, and the strength of the influences working contrary to the restoring force, primarily inertia.
Since inertia is a function of mass, we'll focus on mass. Here is the general rule: Greater restoring force leads to higher vibrational frequency. Greater effective mass leads to lower vibrational frequency. I have used the phrase "effective mass" to reflect the fact that the location of a given mass may make a difference because of leverage effects.
An increase in mass at the end of a prong or the center of a string, for instance, lowers frequency more than the same increase near the mounting. Figure shows how the two factors play out in various combinations, using a simple prong fixed at one end as a model. Not every oscillation in the world is a product of the forces described here.
In designing loudspeaker cones, for instance, engineers must try to suppress the natural vibration factors in order to create something that will perform as intended, under outside controlling forces. Likewise, it is possible to build musical instruments whose initial vibrations are to some degree independent of the forces described. But the vast majority of natural oscillations do follow these principles, and there are very few are the acoustic musical instruments in which they don't play a defining role.
The description given here of displacement, restoring force and inertia at work provides sufficient understanding for a lot of musical instrument design work. But we can deepen our understanding by looking at the situation from another perspective. In a more sophisticated view of the same events, the simple back and forth motion of vibration can be seen as a wave of displacement running rapidly through the vibrating medium or object.
Consider a rope lying along the ground. Someone holds one end and gives an abrupt vertical shake.
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The curved hump that scoots from the shaker's hand on down the rope is a traveling wave. In classic wave-like fashion, each point on the rope engages in only a small movement, up and then down again, as the hump passes. But this series of localized movements gives rise to the progressive movement of the wave as a whole. There are two important differences between the rope and the musical wave media that we are interested in.
One is that the rope's wave occurs on a large scale and progresses relatively slowly, making the wave observable. The other is that the wave in the rope dissipates all its energy along the way and at its end. If the rope were anchored rigidly at both ends and held in the air under tension, the energy would not dissipate so fast.
Instead, the wave would reflect bounce back at the far end, and would then be seen running back in the opposite direction along the rope, only to reflect again at the near end, continuing in a back-and-forth movement until all its energy is dissipated. Sidebar , "Watching Waves" describes further experiments along these lines with observable waves. This same sort of motion occurs in musical strings.
It is also closely analogous though harder to picture to what happens in drumheads, music box prongs, marimba bars, wind instrument air columns. A traveling wave is somehow instigated, and it repeatedly reflects back upon itself each time it reaches a boundary. And here is the link between traveling waves and vibration: The vibration of any given point can be seen as a result of the passage at that point of the series of reflected waves, causing the point to move back and forth with the arrival of each wave front.
The round trip time for each wave is constant, giving the resulting vibration a steady frequency. Vibration patterns of this sort can be called standing waves. Standing waves are the steady-state vibrations that arise as a result of traveling waves reflecting back and forth in a string, air column, or other medium of finite dimensions. The interacting wave fronts reinforce or cancel one another to varying degrees all along the medium at each instant to create the standing wave form.
It is not intuitively obvious, but the cumulative effects of the multiple reflections can account for all the complexity of the vibration patterns we see in musical instruments, including the presence of multiple frequencies fundamentals and overtones in the vibrating object. These natural frequencies are the frequencies at which the system will oscillate if given some sort of initial impulse and then left to vibrate on its own, as, for example, a guitar string will vibrate at a certain frequency each time it is plucked.
Resonance is a function of this property. To illustrate, consider a tube that has one stopped end and one open end. The air in that tube has a springy quality: if it is momentarily compressed into the tube by some inward movement at the open end, increased pressure within will cause it to surge back out; in doing so it overshoots slightly, creating a relative vacuum that pulls it back in again.
As this continues an oscillation of the air results just as if one had compressed and then released a coil spring. Now let's add to this system a driver something that will repeatedly force air in and out of the open end of the tube, driving the enclosed air at some specific frequency. As an example, we'll use a piece of wood mounted so as to act as a marimba bar. Marimba bars flex up and down at the center when struck. If one is placed over the opening of a tube a short distance away, it has the effect of pushing small amounts of air in and out as it vibrates.
Like most vibrating systems, the bar has a natural frequency at which it "wants" to vibrate, and so does the air column below. What happens with the air in the tube now if someone strikes the bar? The answer depends on the relationship between the natural frequency of the bar and the natural frequency of the air column.
In the likely case that the two frequencies are different, the air responds minimally. The bar is trying to drive the air at a frequency the air doesn't want to go at. If the frequencies are roughly the same, however, the air joins the bar in oscillating at the frequency they now share. The intensity of the tube's response can easily be heard in the resulting sound: the bar tone is greatly enhanced and augmented by the air sound from the tube. Creating a good coupling between bar and resonance tube is one of the joys of instrument making.
This is a good example of resonance response. The word resonance refers to an oscillating system's enhanced response to a driving force at or near any of its natural frequencies. It comes about because of the fact that if the driving frequency matches the oscillator's own natural frequency, the driver can consistently impart energy at just the right time and in the right direction to maximize its effect. A classic analogy is that of pushing a child on a swing.
The child-plus-swing, like all pendulums, follows the rules for oscillation described earlier, and has its own natural frequency of oscillation. That frequency, of course, is well below the hearing range. The driver for the system is the person doing the pushing. If the pusher times the pushes so that the direction of the pushes always contributes to the swing's natural movement, the imparted energy will accumulate and the swing will go higher and higher.
This the pusher does instinctively by adjusting the pushing frequency to match the natural frequency of the child and swing, one push per swing cycle. If the pusher pushes at the wrong times pushing forward, for instance, in the middle of the swing's back swing the two forces cancel. Then, rather than building up, the swing's motion diminishes. This inevitably happens at least some of the time if the pushes consistently come at a frequency very different from the natural frequency of the swing. If the pusher's frequency is close but not identical to the swing's natural frequency say, just a little too fast then some accommodation will be reached in which each swing cycle is slightly foreshortened by the tooearly push.
It will oscillate a little faster, but with less amplitude, as part of each push will be wasted in counteracting rather than contributing to the swing's natural movement. We can sum this up by saying that the swing shows an enhanced resonance response when driven by the pusher at the swing's natural frequency; the response diminishes rapidly if the pusher's driving frequency departs slightly from the swing's natural oscillation frequency, and drops very nearly to nothing for any pusher so witless as to push at some completely unrelated frequency.
The same sort of interaction happens with the marimba bar driver and the tube resonator. After the bar's first downward flex has initiated an oscillation in the air at the mouth of the tube, then if the driver frequency and the natural frequency of the tube are close to matching each subsequent downward flex comes at about the right time to reinforce the air's next inward movement, one vibratory cycle later. The recurring reinforcement perpetuates and increases the tube vibration.
To aid in visualizing the resonance patterns of different oscillators, one can plot resonance response curves. These are graphs of resonance response against frequency. For the marimba resonator tube, such a curve might look something like that shown in Figure A.
The tube shows a very high, narrow peak at its fundamental frequency, plus several more very pronounced peaks corresponding to prominent overtones. The peaks indicate the enclosed air's strong resonance response for drivers within certain narrow frequency bands, while the wide valleys between show that the tube scarcely responds at other frequencies. Many other vibrating systems used in musical instruments, such as strings, most wind instrument air columns, marimba bars and kalimba tines, show similar curves, having narrow, welldefined and well-separated peaks with very little resonance response in between them.
Some, such as globular flutes and blown jugs, show just one pronounced peak, indicating that they scarcely resonate overtones. But the resonance response curves for many other vibrating systems look quite different. For instance, rather than showing one or more well-defined narrow peaks, they may appear as a hilly landscape. Humps in the curve may be broad rather than tall and narrow.
Translated into acoustic results, a broad peak indicates that the system in question shows a somewhat enhanced resonance response over the general range of driving frequencies covered by the hump. The response is not as selective, nor is it as pronounced as a sharp spike. Somewhat similar for practical purposes is a curve that is spiky, but with so many overlapping spikes representing many overlapping resonances that virtually any input frequency will be covered at least to some degree Figure B.
This sort of response is typical of string instrument sound boards. For instance, Figure C is a typical resonance response curve for a violin body shown in general outline, with smaller spikes smoothed out. The curve reveals that the violin has good general response to a wide range of frequencies, but has an enhanced response over two broad peaks in what, for the violin, comprises the lower part of its frequency spectrum.
To see how this plays out, consider the violin string as driver, and the body as the resonance system being driven. The reasonably good response of the body over a broad frequency range means that the body will adequately resonate, and thus project, pretty much any pitch that the string dictates. The two peaks, meanwhile, have a profound effect in coloring the sound, and play an essential role in giving the violin its characteristic tone. They are formants discussed in the previous chapter , emphasizing those frequencies injected by the string that happen to fall in their general area.
The acoustical causes of these two peaks in violin bodies have been extensively studied; we will learn more about the factors at play in string instrument bodies later on. These two types of resonance response patterns those with well-defined, separated and very pronounced peaks and those with broad, generalized frequency response have distinct roles in musical instrument design. You need the well-defined frequency responses typical of strings, wind instrument tubes and the like, to ensure mat instruments dependably and unambiguously produce their intended pitches.
The violin string wouldn't do its job as a driver of precise and controllable frequency if it weren't frequency-specific. On the other hand, you need the more generalized response typical of soundboards in cases where you want to pick up, resonate and amplify not a single frequency, but a broad range of frequencies.
A soundboard whose resonance response shows isolated spikes separated by deep valleys means trouble: the pronounced frequency biases will scarcely allow some pitches injected by the driver to sound, while making others disproportionately strong, and distorting still others in tone or pitch. A gentle peakiness or a great many overlapping peaks in something serving as a resonator can be valuable, in that the peaks may lend character and an attractive color to the sound.
Acousticians use the term to refer to the rate at which a vibrating system dissipates energy. The more heavily damped a vibration is, the more rapidly it spends its energy, and as a result, the shorter its duration. A vibration with zero damping would sustain itself forever without needing more input energy, but this, of course, is beyond the capabilities of mortal instrument makers. Energy dissipation can take various forms, but the important distinction for us is between 1 dissipation through radiation or transmission of the vibration, and 2 dissipation through mechanisms like friction.
Radiation 1 above may be to surrounding air, or it may be to other solid bodies which can in turn radiate to the air. With musical instruments, the hoped-for result is that the vibration becomes audible, as when a string transmits its vibratory impulses to a soundboard, and thence to the air. Damping due to friction 2 above is energy lost without contributing to sound. It can occur within the original vibrating medium, or it can occur indirectly, as when the initial vibrating object spends itself in driving some other friction-laden object.
Simply stopping the strings of a guitar with one's hand works that way, providing so much frictional damping through the soft flesh that the string's vibrational energy is absorbed virtually immediately, stopping the sound. Damping, and the related question of transmission-vs. Let's summarize their effects: 1. Heavy damping corresponds to little sustain for a given vibration, if there is not continuous additional energy input into the system.
Thus, vibrations in heavily damped plucked or hammered strings die away quickly, as do vibrations in wooden vibrating bar instruments, plucked rods like kalimba tines, and so forth. On the other hand, wind instrument vibrating systems can be heavily damped without dying away, because the player continuously injects energy into the system by blowing.
The same is true of members of the violin family: the string vibration is heavily damped by the player's fleshy finger on the fingerboard witness the sound of violin pizzicato , but the player compensates by injecting energy continuously with the bow. When damping is heavy and energy loss is due primarily to internal friction, then a relatively small part of the total vibrational energy goes into audible sound in the atmosphere.
This makes for an inefficient sound maker, with relatively little sustain and most likely a small maximum volume level. That's why pillows make poor musical instruments: Hit a pillow as hard as you like; inject as much energy as you can. Most of that energy is lost to internal friction, and the sound is unimpressive.
When the damping is heavy because the system is radiating sound energy rapidly to the surrounding air, then the initial result is greater volume. This may be contrary to your intuitive sense of the word "damping". The greater volume is coupled with lesser sustain. In other words, if the vibration rapidly shoots off all its energy as sound radiation, the result will be a short, loud sound.
When damping is small, which is to say that the vibrating body releases energy only slowly, the result will be a longer-lasting sound of lesser volume. Musical instruments can be deliberately designed.
Musical Instrument Design : Practical Information for Instrument Making
Now we can talk about how such vibration patterns arise. This kind of acoustic behavior is easiest to describe in connection with musical strings, so we will start with them. But bear in mind that these phenomena have their parallels in membranes, winds, and solid vibrating materials as well.
Within the tone of a typical musical string, a sharp-eared listener can pick out the fundamental tone along with a few audible overtones above it. The fundamental is normally the most prominent and the easiest to focus upon, and its pitch is heard as the pitch of the overall sound. Now, these multiple tones must come from somewhere; they must somehow reflect patterns in the string's movement. The vibrating string, fastened at both ends and stretched tightly between its two anchors, is capable of several modes of vibration.
Each mode is a pattern of vibratory movement in the standing wave. The simplest and most prominent mode, labeled as the string's first mode of vibration, is the one in which the entire vibrating length of the string flexes from side to side, assuming at the extremes of each half cycle something like the curved shape of left and right parentheses, as shown in the top part of Figure A.
This first mode is responsible, it turns out, for the pitch that we hear as the fundamental in the string's tone. Additional modes of vibration involve subsections of the string engaged in smaller movements, as shown below in Figure A. The second mode takes the form, at the extremes, of shallow S shapes.
The two sides of the pattern work together at a single frequency, and so this second mode, despite its seeming dividedness, is responsible for a single pitch ideally, an octave above the fundamental. We can now highlight an important aspect of such vibrating patterns. In the second mode, as in all standing wave patterns, there are points of maximum and minimum movement. The center point in this second mode is virtually immobile; it merely pivots.
Points of maximum movement appear one quarter of the way from each end. These defining points have names. The point that does not move is a node, and the points of maximum movement are called antinodes. Remember the meanings of these words if you are not familiar with them already; you will encounter them many times before we get to the end of this book.
Normally those harmonics blend into the overall string timbre and do not stand out as separate tones. But there are ways to hear individual harmonics. The following instrument design takes advantage of a simple but elegant technique that isolates string harmonics over an extended range. It was developed by guitarist and composer Glenn Branca. The instrument is a simple six-string board zither, with a middle bridge dividing each string into two half-strings. Under the strings on one side there is an electromagnetic pickup such as electric guitars use.
The player plucks the strings on the no-pickup side, varying the pitch with a Hawaiian-guitar-style steel slide. The vibration on the no-pickup side is scarcely audible, since the board radiates sound poorly. But any tones matching the harmonics of the half-string length will start a sympathetic vibration on the other side, to be sensed by the pickup there and amplified.
Because this system does very well at isolating tones high up in the harmonic series, where the tones are close together, the player will find a vast array of pitches waiting to come forth, making for a very full musical palette. Just plucking the string and sliding the steel over its length produces a cascade of audible pitches. The tone, as it comes through the amplifier, is clean and lucid, with no plucking sound at all. Materials Remember to look to Appendix 1, "Tools and Materials," for sources. You can vary dimensions and the number of strings to suit your needs or taste. One six-foot 2x4 board.
One electric guitar pickup with cable and jack. An electric guitar amplifier or, alternatively, you can hook up to your stereo or whatever else will serve. Six tuning pins, and a tuning wrench or crescent wrench to turn them. Six 1-inch 8 roundhead wood screws. Music wire enough for 6 strings each about seven feet long. Use several different gauges, between about. A "slide," meaning a hard, reasonably heavy, hand-held object to press against the strings to stop the vibration at different points.
Glass and steel slides are manufactured and sold specifically for guitar players. But any hard, heavy object that feels comfortable in your hand will work. Construction Procedure Cut the 6-foot board. If necessary, carve out a seating with chisels or router to fit the pickup, so that the pickup with foam beneath will lie about!
Drill holes at one end for the tuning pins as shown in Figure B, small enough for the pins to fit firmly. For the most common zither pin size the appropriate hole size is "". Tap the pins part way in, then screw them the rest of the way in taking advantage of the fine threads on the pins. In the equivalent locations on the opposite side, place six strong wood screws, screwed down so that the heads are " from the board surface.
Put the smaller rods for the end bridges in place and secure them with screws, epoxy or any other effective means. Mount the pickup in its seating, with the foam rubber beneath cut to size as necessary. Use U-shaped fencing nails to fix the first few inches of cord to the 2x4, to prevent stressing the wires where they join the pickup. Attach the strings end to end, from screws to pins, as indicated in Figure D and its caption. Use the tuning pins to tighten them moderately tight, then wedge the larger rod that forms the center bridge beneath them at the midpoint.
You may devise your own tuning for the strings. Plug the jack from the pickup into an amplifier and turn it on. Playing Technique Position the board in front of you as shown. Pluck near the bridge to your right. Move the steel slowly along the strings as you pluck, in order to hear the rainbow of harmonics that appears over a relatively short sliding distance. There's no need to pluck hard or to apply great pressure with the steel.
When you pluck the string on the no-pickup side, any harmonics having a node at the point where you place the steel will be excited, and will sympathetically excite the matching harmonic in the half-string on the other side of the bridge. The pickup there will sense it, and it will be amplified. The first few harmonics the second through the eighth or so sound readily, and their nodal locations are not confusingly crowded together. The higher harmonics become increasingly difficult to locate, but with practice you will be able to control them as well.
To aid in locating the steel for the different harmonics, you can mark off the nodal positions on the board beneath the strings, creating a chart similar to that shown in Figure E. Acknowledgments The German guitarist Hans Reichel has developed a "Pick-behind-the-bridge Guitar" based upon the same principle as Glenn Branca's harmonics guitar described here.
While the beauty of the Branca instrument lies in its simplicity, Hans Reichel's instrument is more complex, and demanding both to make and to play but with huge musical potential amply demonstrated in Hans' own playing. In addition to the node at the center, the second string mode has a node at each end, since these points too are immobile.
The first mode, which we looked at earlier, has but one antinode point of maximum vibration at the center, and the two nodes at the ends. The third mode of vibration for a stretched string takes the form of three string segments, as shown in the third drawing in Figure A. This mode is responsible for the next higher overtone, ideally sounding at the interval of a 12th above the fundamental. You can see the form of the fourth mode in Figure A as well, and you can easily imagine what the higher modes will be. One could go on with this indefinitely, but in practice the strength of the vibrational modes diminishes as they get higher.
Overtones above about number 16 are significant in relatively few instruments, and in many only the first few overtones play an important role. Notice, by the way, that the string overtones form a harmonic series.
These modes of vibration don't normally occur in isolation; the string vibrates in many modes at once. It does this in a manner analogous to that described in connection with a vibrating particle of air, in Sidebar The forces at play in the several modes operate on the string in an additive manner, augmenting or counteracting one another in varying degrees at each point along the string and at each instant in time, to produce a single complex vibration pattern. When that pattern is transmitted to the ear via soundboard and air , the ear manages to extract the frequencies associated with the subsidiary modes that are present, and recognizes them as the fundamental and overtone pitches.
These same principles apply not only to strings, but to most instrument types. For instance, a rigid rod fixed at one end, as in a kalimba tine, manifests its first four modes of vibration as shown in Figure 25B. The resulting overtone pitches in this case are distinctly non-harmonic, with the second and third modes producing pitches at just under two octaves and a minor sixth, and just under four octaves and a major second above the fundamental.
Circular membranes, as in drum heads, show a set of modes which are somewhat analogous to string modes, but with more variations deriving from their twodimensional nature Figure C. The resulting overtone series once again is non-harmonic. Fuller information on these and vibrational modes for other instruments appears in the chapters devoted to the specific instrument types.
The modes of vibration just described can be called transverse modes. But relative to what? Relative to the direction of travel of the traveling wave in the vibrating medium. If you look back at Figure , you'll see this effect in a transversely vibrating string: as the traveling waves move lengthwise along the string, each point in the string accommodates the wave with a lateral movement.
As a practical matter, transverse vibrations are the most musically useful sort of oscillation in solid vibrating materials such as strings, bars, rods and membranes. In addition to the transverse modes of vibration, there are longitudinal modes. Longitudinal vibrations are most important in air columns. They can occur in strings, rods, and other long, thin media as well, but only rarely are they musically significant in these cases. Longitudinal modes are those in which the predominant vibratory motion of any given point is along the same axis as the motion of the traveling wave in the vibrating material.
They simultaneously manifest themselves in a complementary form, as pressure waves traveling the length of the medium, giving rise to constant fluctuations of pressure at any given point. I've looked through a few of the following books, and thought they were very helpful or inspirational. They all seem mighty interesting Some are geared to young children and others are suitable for adults.
And they range from the beginner instrument maker to the highly experienced. Most of these books are available at www. Banta, Christopher C. Cline, Dallas. Homemade Instruments. Cotton, Maggie. Fletcher, Neville and Thomas D. Physics of Musical Instruments.