Getting the impedance relationships right: for many types of musical instruments, this is one of the less appreciated, less understood, and more important aspects of instrument making.

Acoustic impedance is a technical subject and you can find articles on the web and elsewhere discussing the mathematics of it in some detail. This article is not mathematical or technical; instead it aims to provide a more intuitive understanding of how impedance relationships play out in the sorts of musical instruments one might put together in a home workshop. I, of course, have no PhD in acoustics, nor am I an expert in the underlying math, but I find that a generalist’s sense of what’s going on impedance-wise is often crucial in success or non-success in the making of an instrument. This is particularly relevant for instruments other than familiar and standardized types: In building standard instrument types you can follow the model of existing instruments and probably get things reasonably well in balance, but with new designs you have to find your own way.

So what are we referring to with this word? Very roughly speaking, impedance is a measure either of how much concentrated force is present in a vibrating body, or of how much concentrated force it would take to get a vibration going in the body. Consider the situation, common in musical instruments, in which you have an initial vibrating body, such as a musical string, and for the instrument to work that initial vibrator needs to communicate it vibration to another body, such as a soundboard. In this situation, the string in itself scarcely moves any air when it vibrates; to be heard well it needs to communicate its vibration to the board which, having much more surface area, does a better job of radiating the vibration out into the air. Now, for the sake of discussion, imagine a slightly absurd situation in which the material you’ve chosen for the soundboard is extremely massive and rigid — say, a piece of heavy hard wood which has a soundboard-like amount of surface area, but which is two inches thick. As you can probably imagine, in this case the vibration will not be very effectively transmitted from the string, through the soundboard and into the air, because the soundboard is too heavy and rigid to be driven effectively by the string. There isn’t enough energy and force contained in the string’s movement to drive such a massive board. This is an impedance mismatch. The impedance of the string vibration at the bridge where it should be driving the soundboard is too low; the impedance of the massive board is too high.

And now imagine the opposite case: instead of the overly heavy and rigid board, you set up the string to drive a thin balloon membrane stretched across the top of a box or pan. The way to do this would probably be to make some kind of a supporting body to serve as a string carrier taking most of the string’s tension, and configure things so that the string crosses a minimal bridge of some sort pressing down on the balloon membrane. This too will provide not-great results, but of a different sort from the too-heavy board. Now the problem is not that the string cannot transmit its energy to its board; it’s that it transmits too much too quickly. The string vibration will be almost immediately damped out, as in its first few excursions it overdrives the balloon membrane, dissipating its energy in an exaggerated motion and in heat. The impedance of the stretchy membrane is too low for the string.

A valuable way to see these two situations is as follows. Start with the idea that the ongoing vibration in a string comes about when travelling waves of displacement in the string reflect back and forth from one end of the string to the other. The interaction of the reflected waves creates the standing wave that is the string’s vibration. With the too-heavy board, when a wave front in the string hits the bridge, it scarcely moves the bridge and board, so very nearly all of its movement and energy reflects back into the string in the form of a wave going back the other way, which in turn can be reflected back at the other end. Because of the efficiency of the reflection, the conservation of energy, and the minimal amount of damping, the string will vibrate very well and with excellent sustain. But since it fails to move the soundboard appreciably, it will scarcely be audible. Contrast this with the string on the balloon membrane: when the traveling wave in the string hits the bridge on the membrane, the yieldingness of the membrane means that it accommodates the string’s movement with a generous movement of its own, to such a degree that very little of the string’s movement and energy is reflected back into the string. Thus, scarcely any reverse travelling wave arises in the string, with the result that the string does a poor job of setting up a standing wave and ongoing vibration. So while you may hear a dull thump as the membrane responds to the string’s initial movement, you won’t hear much after that. 

For a successful instrument, we seek a middle ground between these extremes. The impedance relationship between the string and the board should be such that the string can drive the board enough to be heard, yet not so much that you don’t get enough reflection to set up an ongoing standing wave in the string. For this, the impedance balance between the string and the board is the key. Of course, there’s no one correct balance; it’s a matter of how you choose to manage the trade-off. A relatively light-weight (low impedance) string on a relatively heavy board favors a slower transmission of energy and lots of reflection; this will tend to yield good sustain but less volume. A heavier string on a lighter, more compliant (low impedance) board may yield more initial volume but less sustain. Of course there are many more considerations playing into the final result, but that, in its crudest form, is the impedance trade-off.

Impedance questions play out particularly dramatically in lamellaphones (kalimba-like instruments and their relatives). Most lamellaphones the tines, being quite rigid, function as relatively high impedance drivers — higher than would be typical in our previous example, strings. Thus, if you tried to mount any but a very light and flexible lamellaphone tine on a typical guitar soundboard, it wouldn’t work well. The soundboard would not provide a solid enough base for the tine; it would not provide sufficient counterpoise to the tine motion; in short, the soundboard’s acoustic impedance would be too low for the much higher impedance of the rigid tine. The result would be that there would be insufficient reflection at the point where the tines are mounted to the board and, as with the string on the flimsy membrane, the tine would not set up clear ongoing vibrations. For this reason, most kalimbas require heavier bridge-and-soundboard assemblies than most strings do.

To transmit from the high-impedance tines ultimately through to the low-impedance air, some traditional lamallaphones go through a series of impedance step-downs that has proven to work very well. (We should note that of all the vibrating media in question here, it is the final medium, the air itself, that has the lowest impedance.) In these lamellaphones — taking as an excellent example the Shona mbira — the tines are mounted on a fairly small but quite thick, hard and heavy board. It provides good counterpoise for the tines but, being fairly small and not terribly responsive or resonant, does not project very well into the surrounding air. For playing situations that call for a fuller sound, these boards are lodged firmly in a very large hollow half gourd shell which, being light and responsive but with lots of surface area, does an excellent job of communicating to the surrounding atmosphere. 

Worth briefly noting here: Situations sometimes arise in which you’d like to have a fairly large, light, low-impedance soundboard, but the drivers (e.g., tines) are too heavy for it. A workable solution in cases like this may be to couple the relatively light soundboard with a relatively heavy, rigid bridge to support and provide sufficient counterpoise for the initial drivers. This intermediary step can provide the necessary balances between the components.

Another way to think about impedance questions is to consider how “concentrated” the acoustic energy is in a vibrating body. In airborne soundwaves, however much energy there may be in total, it is typically very widely dispersed; at no point in space is a great force of energy concentrated. That is a low-impedance vibration.The above-mentioned kalimba tine is the opposite: even if the tine itself is a small thing and the vibration is neither widespread in area nor terribly large amplitude, a rather forceful vibration and a lot of vibratory energy may be concentrated in the tine, especially near its base. The late Francois Baschet, French instrument designer, thought a lot about these questions. In his most famous instrument, the Baschet Cristal (look it up if you don’t know it), several steps of acoustic energy transmission depended on good impedance balances to get the sound from the initial vibrators out into the air. Francois came up with this easy and intuitive rule of thumb: for practical purposes in instrument making, a low impedance vibration is one which you can easily stop by touching with your hand. A high impedance vibration is one that is strong and concentrated enough to carry on in spite of your hand-damping. Example: If you touch the ends of the vibrating tines of a typical tuning fork, your hand will fairly easily kill their vibration: the vibration near the ends of the tines is moderately low in impedance. Yet you can hold the base of the fork in your hand, and the vibration in that portion of the fork is not killed. This is evidenced by the fact that if you press the end of the fork against a table top, you can hear that the table top now acts as as a soundboard, projecting the vibration that it picks up from the end of the fork. Thus we know that the vibration in the base of the fork persists even though you’re holding it. This is a classic high-impedance vibration, not large in amplitude but forceful enough to withstand the hand-damping and to drive a rather heavy table-top to boot. This highlights something worth keeping in mind: that the vibration within a single vibrating body may be higher in impedance in one area, and lower in another. Thus, a guitar string may be relatively low in impedance at its midpoint, where the amplitude is greater but it is easily damped, and higher in impedance quite close to the bridge where the same force is concentrated into a smaller amplitude — a good thing, because the greater impedance is needed there to drive the soundboard.

Impedance questions play out very differently in different instrument types. In almost all, these questions are integral to the functioning of the instrument, but not always in ways that present difficult questions or dilemmas to the builder. In winds for instance, impedance questions may seem trivial: the initial vibration is in the air to begin with, so the matter of transmitting it out into the surrounding atmosphere through suitable impedance balances might seem unproblematic. And, indeed, the wind instrument builder may make wonderful instruments without ever giving a thought to these things. But in fact, impedance differences are crucial to the functioning of such instruments. This shows up primarily as impedance differences between the restricted air within the instrument and the open air without. It’s crucial, for instance, at the far end of the air column, where an impedance change sets up the balance between how much of the wave energy propagates out into the larger atmosphere to be heard, and how much reflects back into the instrument to perpetuate the standing wave. 

Here’s another example of impedance relationships in play in musical instrument design, this example pertaining to an instrument I was just recently working on. The heart of this instrument, which I call Miago Trod, is a set of threaded rods ranging in length from about 24 to 40 inches long. The rods are mounted vertically through a horizontal bar in such a way that about 30% of each rod’s length extends above the horizontal mounting bar, while the remaining 70% extends downward. The upper and lower segments, when struck with a suitable beater, will vibrate quasi-independently, producing different tones and tone qualities. Why these particular percentages? Because it happens that in rods of the given thickness and in this general length range, that length relationship can give you a nice octave agreement between the most prominent of the vibratory modes in the upper and lower portions. (In rods of this length and thickness, you can expect that the fundamental tone, corresponding to the lowest mode of vibration, will be inaudible or nearly so. The ear focuses instead on the pitches of one or more of the higher modes.) In my idealized conception for the instrument, not only would the upper and lower rod segments provide different pitch ranges and tone qualities and octave apart, but they would also enrich one another’s sounds. The enrichment would happen as, when the upper segment of any given rod is struck, there would naturally be some transmission between its upper and lower segments, creating sympathetic vibration between the two. The same would be true in reverse when the lower segment is struck. The fuller and more complex sound that this would create would, in the ideal, have an added element of coherence due to the octave relationship.

But here’s where the impedance question comes in: For the transmission and resulting sympathetic vibration to come into play, the horizontal mounting bar cannot be too heavy and rigid. That is to say, it should not possess an extremely high acoustic impedance. If it does, then a vibration in one segment will be so firmly anchored and immobilized at its mounting point that it will scarcely transmit anything through the mounting bar to its opposite segment. Thus, scarcely any sympathetic vibration will be set up. On the other hand, if the mounting bar is too light, another problem arises: the rod sections above and below will not have firm enough mounting to vibrate well; the mounting bar will not provide sufficient counterpoise; the travelling waves within the rods will not reflect well enough at the base to create a clear and well sustained vibration. This is not an all-or-nothing proposition though; it is a matter of degree: a somewhat flimsy base may yet be solid enough to allow some vibration, but perhaps with a less-clear tone and with less sustain.

So in making the instrument I experimented with a few different thicknesses for the mounting bar, and a few different thicknesses for the rods. In the end, I settled on a relatively heavy mounting bar, at 3/16” x 2”, with medium-weight threaded rods in the size designated as 10-24. This choice amounted to saying “With a lighter bar, the rod segments don’t vibrate as well as I’d like — not so clear in tone, and too little sustain. So instead I will use a fairly heavy mounting bar, and in doing so I will accept a bit less sympathetic vibration in order to keep a bit more sustain and clarity in the tone.”

In closing, let me encourage all readers to contribute generously to the Institute of Salubrious Impedance Relationships via our donations page at Salimprel.com, credit cards accepted just kidding. 

 

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