If you have fooled around with the making of pitched percussion instruments from available materials, perhaps you’ve encountered this situation. You find something that produces a nice tone when you strike it – let’s say, a length of steel conduit which you’ve suspended in a way that allows it to vibrate well – and you decide to make a series of tubes of graduated lengths to create a tuned set. You cut a second section of tubing a bit longer than the first and, as expected, you get a sound that is similar in tone but lower in pitch. So you cut another that is a bit longer, and another a bit longer – but something unexpected happens: you soon find that the next longer pipe perversely seems to produce a higher note than your original tube.
What’s going on? To get a sense of what’s happening this, we can think about modes of vibration, or patterns of vibratory movement within the vibrating body. Most vibrating bodies, such as these pieces of tubing, are capable of multiple modes, each mode producing its own natural frequency. When you tested that first piece, you were able to hear it as producing a certain note: you were able to recognize a certain pitch in its sound; you could have hummed that same pitch back with your voice; if you wanted to you could have run to the piano and check what note it was; and you could imagine cutting a series of pipes of different lengths to produce a scale starting from that note. But looking at this situation with an eye to the pipe’s vibratory modes, we can speculate that your sense of pitch for that first pipe was based on just one of the tube’s many modes. With that in mind you could say something like “This piece of tubing is capable of many modes and it’s likely that by striking it I excited quite a few of them. Due to a variety of factors both physical and psycho-acoustic, one of those modes predominated in my perception, and I interpreted that mode as the defining pitch.” You could then further speculate: “When I started hearing a seemingly higher note coming from a longer tube, perhaps my ear had switched its focus to a different mode of vibration which has a higher frequency.”
Following this line of thought we can now see how we ended up with a longer pipe segment producing a higher note than a shorter one. As we went through the process of cutting progressively longer tubing segments, the acoustics of the segments – the relationships between the vibrational modes in frequency and volume – shifted in such a way that the mode that had dominated the pitch-sense in the original shorter tube was superseded by some other mode in the longer tube. If, as often happens, it was a higher mode that came to the fore in the longer tube, then it’s quite possible that its pitch would be turn out to be the higher of the two.
This points to a fascinating psycho-acoustic question: how does ear sort through the various frequencies present in a complicated tone to arrive at a defining pitch? Various considerations come into play, and I won’t venture deeply here, but I can mention a few factors that contribute to pitch-sense in cases like this. First, unsurprisingly, louder components are more likely to dominate than quieter ones. At the same time, lower frequency components tend to have foundational quality which may give them pride of place. Thirdly, there is the question of harmonicity. This one calls for a bit more explanation. The ear seems to search for harmonic relationships among the frequencies present: do some or all of the frequencies present fall into the pattern of a harmonic series? If the ear finds what seem to be harmonic relationships, even in cases where the series is far from complete and just a few frequencies suggest the pattern, then it will tend to favor the implied fundamental of that series in assigning a sense of pitch. The lower components of the series are most important in this regard. Incidentally, this virtually instantaneous bit of neural processing – searching for and interpreting harmonic relationships among the frequencies present – is an astounding piece of brainwork, yet it happens routinely, unconsciously, and almost instantaneously. Remarkable though it may be, research does support the proposition that something like this actually happens.
In the case of our series of metal tubes, the break from the expected length/pitch progression. surprising jump in perceived frequency is made more likely by the fact that tubes like this are by nature inharmonic in the frequency relationships of their modes. This means that as the tubes get longer there is no orderly downward progress of a series of harmonic overtones pointing to an implied fundamental below. Each new pipe tone is just a free-for-all of inharmonic modes, leaving the ear free to choose among them as it will. It’s fairly likely that in this case the ear will just go for the lowest one that is also nice and loud.
If loudness is an important factor, this then this leads to the question of which modes in the tube will project most efficiently into the surrounding air. Tubes typically are long and narrow, lacking the sort of broad surface area that can efficiently radiate low frequencies. This means that as the tube gets longer, then whatever mode originally dominated our perception is increasingly disadvantaged as its frequency drops and doesn’t project into the air as well. This in turn allows the next higher mode to take over in the spotlight. Eventually, the ear will switch its allegiance to the higher mode.
You can have some fun exploring this if you have a hand-holdable magnetic guitar pickup, such as one made as a sound hole pickup. If your tube is steel, the pickup can pick up the lower modes in a long-ish tube, including those that are not radiating efficiently into the air. It allows you to monitor lower modes that have become barely audible acoustically in the longer tubes. With very long tubes, you can also have fun placing the pickup at different points over the tube, corresponding to antinodes (points of greatest amplitude) for various modes of vibration, to get a more analytical sense of some of the many modes that likely are present.
Meanwhile, regarding the planned metal tube instrument, we find ourselves in a disappointing situation: we found a percussion tone quality that we liked, and we wanted to build an instrument around it; hopefully an instrument with a reasonably large and musically useful range. But when we went to extend the range downward by making progressively longer tubes, we found that instead of getting lower with each longer tube, the perceived pitch instead soon jumped to the tone of the next higher mode, sabotaging the instrument’s intended range. Is there anything to be done about this? Well, yes – no magic bullets, but there are some things you can do to bring out the intended mode even as it descends and, we hope, keep the ear focused on it as the defining pitch. One is to use tubes of larger diameter for the lower notes. The larger diameter will allow the longer tubes to do a better job of both vibrating strongly in the preferred mode and projecting that mode into the air at the lower pitch level. Another is to use magnetic pickups to capture the lower frequencies. Additional tricks to help bring out the lower mode might include the addition of air resonators, the use a suitably soft and heavy mallets for the lower notes, and an effort to consistently strike the tube in the best location for bringing out the desired mode.
The above techniques are all widely known and used for bar percussion instruments. But they don’t resolve all cases, especially if you get into more exotic forms. Although our discussion thus far has focused on metal tubing as our example, the psychoacoustic mode-jumping issue comes up as much or more in other forms, from bells to kalimba tines, and especially in odd and exotic shapes. It’s particularly likely in forms in which there are many prominent modes that are close in frequency. This is not the case with short metal tubes, in which the lowest modes are likely to be prominent; there’s an octave and half between the first and second modes for such tubes. But as you get into longer and skinnier tubes, those lowest modes become scarcely audible, perhaps even subsonic. Then what you hear are the higher modes, which tend to appear at much smaller musical intervals, creating that much more confusion for the ear in deciding which mode to tune in to.
In short, we have a lot of situations in which the mode-jumping issue is a challenge not easily overcome. Indeed, there are times when the best response may be simply acceptance: accept that the attractive tone-quality you’ve found works only over a limited range. If you intend to make an instrument around it, accept that it will be an instrument of limited compass. Or, if you choose to make a larger gamut of tones, accept that the instrument will produce different tone qualities and different perceived pitch-sense in different parts of the range. This latter idea can be fun because it means the player can play around in the oddly ambiguous middle areas where the ear can’t quite decide which mode to pay attention to.
Let me describe one of my instruments for which this sort of pitch-perception question was very much in play. The instrument – which is quite different in type from the examples we’ve touched on thus far – is called Waterfall. As you can see on the linked page, it consists of a set of long, very thin steel bands which are fixed at one end and free to flop about at the other. The bands are ¾” wide by .010″ thick, and the tuned set ranges in length from about 3′ to 9′. To accommodate that length, they are mounted on a tall frame, positioned with a small upward arc leading to the hanging length. They’re made of spring-tempered steel, which has low internal damping and so can sustain vibration well, even up into the very high modes. You sound it by striking the bands with light, hard mallets (I use ping pong ball mallets). The long, thin, floppy bands produce countless modes of vibration ranging from subsonic (slow enough that you can watch the band sway) to highs well up into the sizzle range. Yet I found that within a certain range of lengths, one mode seems to stand out in the listener’s pitch perception – which is to say, when you sound bands within this length range, you definitely hear a recognizable pitch through all the crash and sizzle. And within this range, the expected relationships of length and perceived pitch hold – shorter bands predictably yield higher pitches. As you get into bands that are either longer or shorter than this range, the pitch-sense falters. Either the pitch becomes increasingly ambiguous and unclear, or the ear tries to focus on some other frequency among the many to gain a sense of pitch, outside of the expected length/pitch progression. To restate this in terms of what we know: Each of bands produces many many modes of vibration and corresponding pitches, but within this limited range one mode dominates enough to provide coherent pitch sense. This makes it possible to make a recognizable scale by cutting a series of bands to graduated lengths with predictable results in perceived pitch. Outside of that range, that mode no longer dominates in the ear’s perception, and the pitch-sense deteriorates. The reasons that that particular mode dominates over that particular range presumably have to do with the dimensions of the bands, their effectiveness in radiating sound into the air at different frequencies, the mechanics of the bands’ vibration patterns, and a few other factors (but to create a detailed and specific analysis of these factors would be quite a chore).
The range over which this works, and accordingly the range of notes I made available on the instrument, is a twelfth from D5 to A6, with some deterioration in tone quality toward the extremes. Not a huge range! But the sizzly sound is unusual and interesting, and regardless of its limited compass, the instrument has proved worthwhile in performance and recording.