Imagine the following situation: You decide to make a set of tuned chimes in the form of a number of aluminum tubes of graduated lengths. You start by making your first chime at some reasonable length. When you mount it and strike it with a suitable mallet, it sounds fine, making a nice, clear note. You want to extend the range downward, so you cut another tube of the same diameter but a bit longer for a lower note, and it too sounds pretty good. But as you continue cutting longer tubes of the same diameter, you soon notice that in the longer tubes, although the notes are getting lower as expected, they aren’t as clear and strong as before. In addition, a higher note is becoming increasingly audible in the tone. As you continue downward with longer chimes, that unwanted higher note increasingly comes to dominate tone, while the intended lower note becomes more obscure and less recognizable. Somewhere along the line, upper note comes to dominate the ear’s pitch-sense and the ear ceases to recognize the lower tone as the defining pitch. Your intent to extend the range downward has hit a lower limit in which, in these longer bars, the expected lower note has been replaced in the ear’s perception by a higher overtone within the tone — one that may always have been present, but apparently was to high in pitch or too quiet to take much note of in the shorter bars.

I expect that many makers of experimental musical instruments have either run into this very problem with chimes, or have run into a closely related version of the same problem as it manifests itself in other instrument types. The situation essentially is this: even though we think of a sounding body such as an aluminum chime as having a single note, such bodies actually produce several frequencies corresponding to the multiple modes of vibration that the vibrating body is capable of. Through an unconscious process of acoustical analysis, our ears focus on one of those modes — usually the lowest, called the fundamental — as the defining frequency on which our sense of pitch for the sound is based. But if the frequency associated with some other mode happens to be much more prominent than the fundamental, then the pitch-sense can become confused, and may shift.

(I’ve discussed other aspects of these phenomena here.)

To continue with the aluminum chimes example just given: typically the lower of the two tones we’ve noted in the chime — the one that came through just fine in the shorter chimes but became increasingly obscure in the longer chimes — is the tone associated with the tube’s fundamental mode of vibration. The higher, unwanted tone that takes over in the longer bars is that of the next higher mode. This happens for a couple of reasons. The main one is that the chime is not very good at projecting lower frequencies out into the air, even if those frequencies remain present in its vibration. (For radiating low frequencies, it helps to have a large and wide surface area. The long, thin form of the chime does not do so well in this regard.) So as you get into those longer chimes, the chime does an excellent job of radiating the next higher partial while doing a poorer job of radiating the fundamental. The ear, as a result, tunes in less to the lower tone and starts to focus on the higher tone.

As the chimes get still longer and the fundamental mode still lower in pitch, an additional factor comes into play due to the fact that the ear is less sensitive to tones near the bottom of the hearing range. Carrying things further, at a certain length (although you will probably have given up on making longer and longer chimes before you get to this point), the frequency of the fundamental finally drops below the lower limit of your hearing range. With the fundamental thus disappeared, the ear would naturally focus on the audible upper partial instead — except: by that point it might be that some still higher partial, one that was too high to notice in the short bars, will have joined the fray by descending into the heart of the hearing range, and may even have begun to dominate the perceived tone just as the earlier one did (another usurper!). And you can imagine that with ever longer tubes, frequencies associated with ever higher modes of vibration, modes that would have been so high as to be unrecognizable in the shorter tubes will descend into the hearing range, becoming increasingly noticeable in the overall sound tone before themselves dropping into oblivion in tubes that are longer still. 

Now the title of this article should start to make sense. “The window of audiblility” refers to the fact that when a complexly vibrating object produces frequencies over a very wide range (as many vibrating objects do) a listener will only hear those frequencies that happen to fall within the human hearing range. And, depending on the vibrating object, it will also often happen that those near the middle of the range will come through strongest, for reasons having to do both with the efficiency of sound radiation from the vibrating object at different frequencies and with the ear’s regions of greatest sensitivity. If you start with a particular type of vibrating object, such as the tubular chimes described above, and attempt to extend its range musically by lengthening, you may find that instead of expanding the range, you are merely shifting which of its partials dominate the perceived pitch. Often you’ll find yourself in the confusing situation that a longer the tube (or whatever), instead of producing the expected lower pitch, seems instead to produce a higher pitch as the ear focuses on a higher partial as the defining tone for the longer tube.

A note on terminology: in large carillon bells, it is often the case that the frequency heard as the defining pitch is not the lowest frequency audible in the tone. Bell makers sometimes use the term prime tone for this defining frequency. Similarly, outside of the world of bells this can be a useful term for a complex tone’s perceived pitch-defining frequency. It’s particularly apropos in cases where that frequency is not the lowest frequency present, since in such cases the other commonly used word, fundamental, isn’t quite right and might cause confusion.

Here’s a related demonstration that I’ve occasionally bored entertained people with. Unlike the previous example in which we talked about trying to get lower and lower pitches from a vibrating body, this one looks at what happens when you progressively raise the pitch. I have an instrument called What-a-Shame [add link] which amounts to a mechanism for continuously varying the length of a narrow rod of spring steel held rigidly at one end. It’s analogous to a single tine on a kalimba, but variable in length from a maximum of close to three feet down to less than an inch. If you start with set this rod at full length and pluck the end — well, you can’t really pluck the end because at that length it’s too bendy and floppy for a pluck-like motion. But if you press the end downward and release, then it will start to swing up and down. This is basically the same thing that any other kalimba tine does when you pluck (press down and release), but in this case the frequency is very low — countably slow, in fact; maybe two or three cycles per second. Of course that’s well below the hearing range; much too low to hear. But if instead you use a fingernail to pluck nearer the mounting point, then you do hear something: a gong-like tone with several frequencies present. Maybe your ear will hear one of the frequencies as the defining pitch, giving you something to latch on to as the prime tone, or perhaps the blend of tones will strike you as being a gong-like sound with no one defining pitch. The frequencies that make up this gong tone arise from several modes of vibration from among the many that are present in the rod. They are those which happen, at this particular rod length, to be near the heart of the hearing range and strong enough to dominate the listener’s perception.

Suppose you now start repeatedly plucking near the mounting point while at the same time gradually shortening the rod. The gong-like tone you originally heard starts to rise in pitch as the rod gets shorter — that is, all the frequencies associated with the several contributing modes of vibration proceed to get higher. But then an interesting thing happens. At the start your ear naturally tracks that original gong-tone as it rises in pitch. But soon, at the edge of your consciousness, a lower tone begins to make its presence felt. As you continue to make the rod shorter and all the pitches continue to rise, this lower tone becomes increasingly noticeable, until at some point your ear no longer tracks the original tone as its focal point for pitch, but turns its attention instead to this new one as it moves into the heart of the hearing range. The switch to this new prime is made unconsciously, but if you listen well, carefully tracking your own perceptual processes, you can catch it happening. This new prime is, of course, the pitch associated with another of the rod’s many modes of vibration, rising from subsonic obscurity into the the range of audibility. As you continue repeatedly plucking while you shortening the rod, this process — that of the ear shifting its focus to new lower modes as one after another rises into prominence– happens repeatedly. Eventually, as the rod is shortened to just a few inches long, the lowest of the modes comes into play, rises through the range and finally disappears skyward as the rod shortens to an inch or less. With no further lower modes to be heard, the game is over. This last, lowest mode is the one commonly called the fundamental; it’s the same mode, in fact, that was swinging inaudibly at two or three cycles per second at the start of the process.  

Back once again to the title of this essay: what you’ve just heard is each of a series of progressively lower modes passing through the window of audibility as the rod is shortened causing all of the modes to rise.

Most musical instruments are based upon vibrating bodies of relatively simple and uniform shapes, such as strings, bars, rods or tubes, as well as conical or cylindrical air columns. In these cases, the relationships between the frequencies produced by the different modes of vibration tend to follow predictable patterns. The patterns vary from one form of vibrating body to another (uniformly made stretched strings, for instance, produce the pattern known as a harmonic overtone series, while free bars and kalimba-style tines typically produce inharmonic series). But in all of these cases, the intervals between the partials get smaller as you go up the series. For example, in the harmonic series (the characteristic overtone progression that we see in strings and most wind instruments), the interval between the frequencies of the first and second modes is an octave; between modes two and three is a fifth; between modes three and four is fourth, between four and five a major third, and so forth through an ongoing series of progressively smaller intervals. By the time you get to, say, the thirty-first and thirty second modes, the intervening musical interval is microtonally small. This has implications for the “window of audibility” question: If you are dealing with a vibrating body whose fundamental mode is far below the hearing range, then the modes you’ll actually be able to hear will be well up in the vibrating form’s overtone series, and are likely to be quite close together in pitch. The perceived effect of such closely spaced higher modes is quite different from the effect of an audible fundamental with only a few widely spaced modes in the audible range above it. In the case of the audible fundamental colored by a few widely spaced overtones above, it’s fairly natural and easy for the ear to recognize the individual pitches, and to focus in particular on the fundamental. The tone will tend to sound more open and clear, and it’s likely that the ear will recognize a single identifiable pitch. But up higher in the series, with a crowd of more closely spaced tones audible, the effect will be much more ambiguous pitch-wise. If you find yourself not too high in the series, the relationships between the pitches may sound chordal or, depending on the intervals between them, they my just sound exotic and strange. If you’re quite far up in the series, then the effect will be be more cluster-like, with many pitches crowded close together and no clear chordal effect.

An interesting possibility sometimes arises in these situations. As described earlier, you may find yourself in a circumstance in which, with the fundamental too low or too weakly radiated to be easily heard and recognized, some other mode functions as the prime tone, coming to the fore to dominate the tone and stand in as the prime tone. Furthermore, it may by chance be the case that one or more of the other modes audible above the prime fortuitously show up in musically coherent relationships to the prime. This may happen in either of two ways. One is if they chance to add up to a musically pleasing chord. The other way that a musically fortuitous relationship may arise between the audible modes is if some higher modes, in their relationship to the prime, happen to mimic the familiar and coherent relationships of the harmonic series. For instance, they may appear close to the octave, twelfth, or double-octave above the prime tone. This will contribute to a much clearer pitch-sense than a bunch of random overtone intervals would. If this is the case, it has the potential to make the instrument more musically useful for melodic or chordal playing. There are at least a couple of moderately well known instruments that work this way. One is the classic toy piano. This instrument uses very narrow tines in which the perceived prime is the sound of the one of the higher modes, with another prominent mode contributing the 12th above. (I wish I could tell you which modes these are, but I’ve found it difficult to determine because I haven’t been able to see how what I’m hearing reflects the expected overtone pattern for this type of vibrating body. My guess is that they’re the second and third modes, but they may be the fourth and fifth.) Another such instrument is the orchestral tubular bells. In these, according to sources I’ve seen, the fourth mode functions as the prime aided by a couple of seemingly harmonically related modes sounding above.

Something to keep in mind should you become enamored of the idea of making an instrument like this (that is, an instrument which takes advantage of a fortuitous set of overtone relationships you’ve discovered somewhere up in the sounding elements’ overtone series): Often the set of overtone relationships that happen to work so nicely at one pitch level is quickly lost when you attempt to extend the range, as the tones associated with unwanted other modes become more prominent at different pitch levels, burying the desirable overtone pitch relationships you seek to duplicate under less appealing ones. In my experience you’re lucky if you can get as much as an octave in range before other unwanted modes and relationships start taking over. Toy piano and tubular bells are and carefully crafted exceptions.       

To conclude this article on a practical note: One of the subtopics of this article has been how the frequencies associated with different modes become more or less prominent in an instrument’s sound as you seek to extend the range by making the vibrating elements longer or shorter, larger or smaller. One thing I haven’t yet addressed is: is this inevitable, or, if you want more consistency across a larger range, are there things you can do to manage the effect? Going back to our very first example, if you’re adding to a set of aluminum chimes by making longer and longer ones in hopes of getting lower tones, but you find that as the chimes get longer the intended fundamental tone is gradually lost behind the sounds of increasingly prominent higher modes … is there anything you can do to mitigate that? In this particular case, is there any hope of keeping the fundamental front and center even as you add longer chimes in order to descend further in pitch? Or, if you’re exploring an instrument idea in which you’re trying to highlight some mode other than the fundamental as the prime, are there ways to extend the range over which the desired tone comes through clearly?

Here are a few options. Often the need is to find ways to increasingly bring out the lower frequencies as you attempt to extend the range downward. The hope is that the frequencies that are prominent in the blend that you like at one pitch level will remain prominent in the blend as you go lower. (And  hopefully whatever approach to this question you come up with will be applicable in reverse if your intent is to go higher.)

In many cases the most useful trick is, if possible, to make your vibrating elements thicker as you make them longer in search of lower notes. This helps counteract the fact that most vibrating bodies, when they are long and thin, will more easily set up vibrations in the higher modes and won’t do as well with the lower modes. This is true with air columns and strings just as it is for idiophonic elements like rods and bars. Note that there’s a certain irony here: for many types of transversely vibrating bodies, making them thicker and thus stiffer will raise their frequency, not lower it. This means that if you follow this advice, your lower-pitched elements will have to be even longer than they would otherwise need to be. Admittedly, a potentially massive increase in overall size is not always practical, and other factors may make you decide not to do it.

But that leads to another consideration: even if you don’t make your lower-pitched elements thicker all around, it can be very helpful to try to make them in such a way as to have more surface area. This allows more effective radiating of lower frequencies. And, depending on the instrument, there are other ways to enhance lower frequency radiation, making it less likely that lower modes will be overwhelmed by higher.For instance, in instruments with soundboards and/or resonant air chambers, larger boards and chambers contribute to lower frequency radiation. In particular, it’s quite noticeably true that air chambers will give you more low-end bang for your buck than a soundboard alone. As another example, adding air-resonator tubes below free bars such as marimba bars can make a big difference in bringing out the low end.

How the sounding elements are excited makes a big difference in which modes predominate in the tone. In orchestral chimes and toy pianos mentioned above, the choice of striking point is crucial in bringing out the desired modes preferentially. In those cases the intent is to bring out selected higher modes and while de-emphasizing the fundamental; in other cases choice of striking point can be helpful in bringing out the fundamental. The character of the beater or plectrum also matters, with harder, smaller or lighter beaters tending to favor the highs while softer, heavier or larger favors lows.

One more trick: as we’ve mentioned earlier, typically the lower modes that may have been getting lost in the lower parts of the range are still present in the vibration, but are just not being radiated into the air efficiently enough — often because the main vibrating body doesn’t have enough surface area to project the lower frequencies well. In such cases, if you give the instrument a pickup, the pickup may respond to the those lower frequencies much more generously than they were radiating on their own. Assuming your amp and speaker are reasonably capable of handling the low end, those lower frequencies will then be much more audible in the amplified sound. This is particularly true with magnetic pickups (the type used electric guitars — but keep in mind, these only work when the vibrating bodies they’re monitoring are made of magnetic materials like steel). Magnetic pickups simply monitor the movement of the initial vibrator, and if the lower frequencies are there, the pickup will get them even if they are not transmitting well acoustically to soundboard, air chamber or surrounding air. Thus, the mix of vibratory modes that was audible in the higher tones but is no longer acoustically audible in lower tones may reappear with the aid of a magnetic pickup to help reinforce the lower modes. Magnetic pickups may be a lifesaver, for instance, in bringing out the fundamental that might otherwise be lost in the low notes in kalimbas.

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