I very much enjoy unusual musical tone qualities, including ones in which the pitch-sense of the tone is ambiguous or fickle. In a companion essay  to this one I spoke about being attracted to sounds in which elements of identifiable pitch are blended with pitchless noise elements. The resulting tone qualities may sometimes be a bit tiresome or irritating, but at other times they are oddly evocative; sometimes even (to someone in the right frame of mind) beautiful. In this essay I’ll talk about another sort of tonal ambiguity: the sort that can occur in sounds in which there are multiple frequencies present, but the relationships between the frequencies that make up the sound are odd or irregular. [Since this article was written I’ve uploaded one more essay on this topic. Find it here.]

Most musical tones contain several frequencies happening at once. The listener’s ear typically hears one of the frequencies present – usually the lowest one – as the defining frequency, giving the tone its perceived pitch. For instance, imagine a stretched string vibrating at 196Hz (196 cycles per second). That happens to be the tone that musicians call G below middle C. Due to the characteristic vibration patterns of strings, that same string vibration most likely includes an additional frequency very close to 392Hz, twice the lower frequency, plus another around 588Hz – triple the original frequency – and so forth upwards in multiples of the original frequency. Each of the higher frequencies is a little quieter than the one before. The ear normally doesn’t hear the additional frequencies separately and doesn’t normally recognize them as separate pitches; instead they blend into an overall composite tone quality with that low G as the defining pitch. That frequency is considered to be the fundamental, while the frequencies above can be seen as overtones.

The several frequencies come from the fact that the vibrating string actually engages in several patterns of vibration simultaneously, with different frequencies associated with the different patterns of vibratory movement. Because a musical string under tension is a relatively simple and uniform vibrating system, these multiple frequencies follow a predictable pattern in their relationships to one another. In the case of strings, this is the pattern just described in which if the lowest frequency is f1, the next higher one will be 2 f1, then 3 f1, 4 f1, 5 f1, and so forth. The listener’s ear and brain, synthesizing these things without conscious awareness, effortlessly interprets the multiple frequencies in this simple relationship as a single unified tone.

Tone qualities in which the overtones show this particular pattern – that is, tones in which the overtone frequencies are integral multiples of the fundamental frequency – can be called harmonic. Try to think of this term as having not so much a musical meaning as a mathematical one, referring to the simple arithmetic relationship between the frequencies in the series. Some form of this relationship comes up in many places in nature, and of course it comes up in other musical instruments as well. For instance, the overtones in well made wind instruments are harmonic. So too are those in a few other specialized musical vibrating systems, such as the interesting but unusual case of metal rods which have been excited in such a way that the vibration is not side-to-side, but traveling longways in the rod.

Other musical vibrating systems produce overtones too, but the overtone frequencies typically don’t line up in the same harmonic pattern of integral multiples. These inharmonic instruments include drums and most idiophones. In these instruments the overtones may appear in a limitless variety of other relationships to each other and to the fundamental. For instance, the most prominent overtones in a rectangular marimba bar (one which has not been reshaped by undercutting) typically appear at frequencies around 2.76, 5.40 and 8.93 times the fundamental.

It’s sometimes said that it’s the pattern of the overtones that gives the sound of each instrument its distinctive and recognizable character – e.g., a clarinet sounds like a clarinet because the positioning and relative strength of the overtones are uniquely characteristic of clarinets. This isn’t entirely true; there are other factors that contribute substantially to the characteristic sound of an instrument. Still, the configuration of the overtones most certainly is a primary factor. In a nice turn of phrase, the late musical acoustician Arthur H. Benade referred to the characteristic pattern of any instrument as its overtone recipe. Since Benade, the phrase has been widely adopted. It carries implicitly the intriguing suggestion that you could create any particular tone quality in the same way you’d prepare a dish of Beef Strogonoff, by following the recipe – that is, by adding the right overtone ingredients in the right amounts.

The distinction between tone qualities featuring harmonic overtones and those with inharmonic overtones is an intriguing one. It happens that sounds in which the overtones are harmonic and those in which they’re inharmonic produce very different effects on the ears. It’s kind of mysterious: Why should this one specific pattern of overtone relationships – the one we refer to as “harmonic” – be so uniquely different from all others in its sound-effect and our response to it? There are physical and physiological reasons that might partly explain this, but the very pronounced difference in musical effect remains quite remarkable. To illustrate this difference in psychoacoustic response, I have occasionally done a little experiment (it works nicely in lecture-demonstration situations). The experiment uses a guitar or any other string instrument with reasonably long strings. To start, pluck a string, let it ring in the normal fashion and let your ears take in its sound, which is a typical string sound with its harmonic overtones. Next, attach a very small weight* to the string almost adjacent to the bridge, like a sixteenth of an inch away, and pluck again. Listen very closely: has the tone altered? The answer is yes, but with the weight this close to the bridge, the change may be too subtle to recognize. Move the weight a tiny bit further from the bridge, perhaps an eighth of an inch now, and pluck again. The tone may be beginning to sound just the tiniest bit odd. Repeat the process, moving the weight by small increments progressively farther from the bridge, plucking and listening at each step. W find that as the weight is moved farther from the bridge the tone becomes increasingly peculiar. It ceases to sound like a string, and begins to sound gong-like. What’s happening is, the presence of the weight alters the patterns of vibration in the string in such a way that the formerly harmonic configuration of overtones becomes increasingly inharmonic – that is, the frequencies of the overtones are no longer integral multiples of the fundamental. By the time the weight is some modest distance from the bridge, we’re now hearing a set of overtone pitches that bear little recognizable relationship to the fundamental; they’re just a bunch of arbitrary, seemingly unrelated pitches.

[*Finding something suitable to serve as the weight is a bit of a challenge. It should be quite small and reasonably hard, and it should be possible to fix it securely in place, yet still movable to new locations where it can be again fixed in place. One item that works fairly well is the type of fishing line sinker called a “split shot sinker.” I won’t go into the details of how these things work – if you’re not already familiar with them you can figure it out once you get your hands on them. One of these sinkers will be good for several repositionings. When it begins to get too malformed for further use, you can replace it with a new one, since they’re normally purchased in packets containing many of them. The recommended size for this purpose is quite small, such as the size designated as BB.]

Recall that it is characteristic of harmonic overtones that they blend very closely with each other and with the fundamental in the listener’s perception; so much so that it’s difficult to hear them as separate pitches at all. With this close blend they strongly reinforce the tone’s pitch sense, allowing us effortlessly to identify the blend as a single clear pitch at the frequency of the fundamental. But move those overtones out of position, and the sense of unified blend disappears, and with it the sense of a single well defined pitch. It becomes easier to recognize the different tones within the blend, almost as if you were hearing a chord, and it’s less clear which if any of these tones should be heard as the defining tone. The impression of a gong-like quality reflects the fact that gongs typically have similar properties. Namely, they have many prominent sustaining tones in inharmonic and seemingly arbitrary relationships.

I mentioned earlier that unlike most strings and winds, many other musical instruments produce inharmonic overtones. An interesting example is the one just touched on a moment ago: wooden bar percussion instruments such as marimbas. Now, as a matter of fact, most professionally made western marimbas have their bars reshaped in such a way that the most prominent of the overtones that would otherwise be inharmonic are realigned to match harmonics of the fundamental. This gives the bars a coherent tone and a nice, clear pitch sense. But in many nonwestern marimba-like instruments, as well as many homemade marimba-like instruments in the west, this hasn’t been done. And yet, in contrast to what I said earlier about inharmonic weighted strings, these instruments have easily recognized pitch. What’s the difference here? The situation with wooden bar instruments which haven’t been overtone-tuned is that the overtones are very widely spaced. There’s a prominent fundamental at the bottom; the next overtone is well over an octave above that, the next quite a bit higher still. In theory there are many more overtones beyond those, but they have relatively little effect on the pitch-sense because they’re so very high and also because they are quiet and very brief in duration while the fundamental sustains. The ear tends to seek pitch-definition through the lowest and also the loudest and most sustaining of the frequencies present in the tone, a task made easier in this case by the wide spacing of the overtones. Put all these things together and we find that, in our ears’ unconscious processing of the available pitch information, we are able to recognize a fundamental in the inharmonic bar tone. The case is very similar with lamellaphones (that’s the generic term for mbira and kalimba-like instruments). They too have widely spaced inharmonic overtones of relatively short duration, and likewise with them we are able recognize the fundamental. But notice that these two instrument types come across as having very distinctive sound qualities, with a certain quaint or colorful quality to the sound. That’s a product of the inharmonic overtone placement. It’s also worth noting that in very dense musical contexts the sound of these instruments can be confusing to the ear. Particularly on the low notes where the prominent overtones fall closer to the heart of the musical range, you may find your ear inadvertently tracking that first overtone rather than the fundamental, following a melody at some odd inharmonic pitch level rather than following the intended lower tones. In spite of this, the fact that we can usually recognize a fundamental pitch in the marimba bars and kalimba tines contrasts with the gongs and de-harmonicized strings mentioned earlier, in which the ear may have a harder time deciding on and sticking with any one frequency as the fundamental. The greater ambiguity in these harder-to-interpret instruments stems from the facts that in them the inharmonic overtones tend to be not so widely spaced, they all sustain about equally well, and they are more nearly equal in volume.

But this distinction is not clear cut. Consider the extremes: In some sounds, particularly those with harmonic overtone recipes, the ear easily recognizes a single defining pitch. In other sounds, particularly those with prominent, closely spaced inharmonic overtones in the heart of the range, the ear has difficulty focusing on any single defining pitch. Between those extremes lies a lot of ambiguous territory of tones in which the pitch sense may be to varying degrees fickle, shifting, inconsistent or vague.

Another example: compared to marimba bars and lamellaphone times, drums are an ambiguous case. Circular drum membranes naturally tend to produce many patterns of vibration, yielding many different frequencies that can appear in the heart of the range, not always widely spaced. We don’t normally think of drums as having pitch, but often enough the ear does recognize some pitch in drum tones, though as through a glass darkly. An interesting special case is that of tablas. In these Indian drums, the drumhead is given special treatment to bring out the fundamental more strongly. (The treatment involves adding a thick paste to the drumhead to weight the center.) The result is that tablas are among the most clearly pitched of drums.

Also interesting to note: The instruments just cited for their interestingly inharmonic tone qualities – wooden bar percussion, lamellaphones and drums – are the emblematic African instruments. Perhaps one could speculate that in African music cultures there’s a culturally based interest in these more complex and piquant tone qualities. Classical Western instruments, by way of contrast, for the most part favor harmonic tone qualities weighted toward the lower parts of the harmonic series – winds and strings designed to bring out the fullness of the lower harmonics while de-emphasizing the brighter high harmonics. Asian instruments, for their part, often feature harmonic tone qualities highlighting the very high harmonics – think of some of wide-open Eastern double reed instruments with their extremely brilliant or edgy sound qualities, the several Indian string instruments with jawari bridges designed to bring out the very high string overtones, and the mirlitons added to some Chinese flutes, once again enhancing the high harmonics.

As I said at the start of this essay, I’m interested in tones that encompass odd frequency blends, and those in which the pitch sense is on the borderline of recognizablilty. To explore these things a bit more, I decided to experiment further with the weighted strings. I realized that the weighted strings provide a nice opportunity to play around with different overtone recipes, because shifting the weight alters the overtone relationships. The process isn’t like manipulating overtones using a synthesizer or computer, because you can’t prescribe particular overtone relationships and then try them out in a systematic fashion; the weighted strings don’t admit of that kind of control. The process is more about placing the weight on the string and then moving it about, trying it in different locations and listening to the resulting tones in search of sounds that are interesting or attractive. In many sorts of exploration (not just with weighted strings) this kind of randomness sometimes yields more interesting results than controlled and prescriptive methods. Yet the process with the weighted strings doesn’t have to be quite as random as I’ve just made it sound. It is possible to at least roughly control the resulting pitch relationships for some of the overtones, as I’ll describe later. You can see some of my weighted string instruments here.

One thing I explored in my weighted string experiments was true randomness. I created a zither-like instrument with eighteen strings, and applied weights haphazardly to the strings, each string with its weight placed differently from its neighbors in an un-premeditated fashion. This meant that each string would have its own unique overtone recipe due its particular weight placement, and the different strings would have differing tone qualities as a result. This is quite different from the situation in most musical instruments: typically we look for uniformity and consistency of tone color across all the notes on an instrument; here, all the notes are deliberately made to sound different from each other. The string’s fundamentals are strong enough, and the lower overtones widely enough spaced, that the ear can usually, if sometimes ambiguously, recognize the fundamental and hear is as a defining pitch. So I was able to tune the strings to a recognizable scale, and you can play recognizable melodies on the instrument. The overall effect is definitely odd – both because of the inharmonic overtones and their resulting in the gong-like tone quality, and because of the variation in timbre from string to string. It’s a pretty cool effect.

A moment ago I said that despite the odd overtone mixes, the ear still recognizes the randomly weighted string tone as having a defined pitch, making it possible to tune the instrument for scales and to play melodies. But remember that in instruments like this we are in that odd territory where the pitch sense may not be dependable. In this case the ear recognizes pitches pretty well when the instrument plays alone and not too many notes are played at once. But In crowded musical contexts it becomes more difficult to recognize fundamental pitches, making it harder to follow melodies or harmonies played on the strings. In especially crowded contexts the string tones may come through only as noise – very interesting and unusual noise perhaps, but without melodic or harmonic meaning. This is something we find frequently in tones with prominent inharmonic overtones: the pitch sense is dependent on musical context.

In this work with weighted strings I also explored more deliberate overtone tunings, achieved through more deliberate weight placements. Here’s how this works: It turns out that, mixed in with the various harmonically-skewed overtones that are part of the weighted string tone, there are often a couple of tones associated with the shorter string lengths from each bridge to the weight. These tones sound as part of the overtone blend when the string is plucked, but they tend to be fairly prominent. And they are, within limits, tunable: by shifting the weight to one side or the other, you can control these tones in a fairly predictable manner (though, for reasons I won’t go into, the relationship to the fundamental is not as simple as you might at first imagine). This means that you can deliberately tune one of these overtones relative to the fundamental, albeit a bit roughly. You can, for instance, place it two octaves above the fundamental. The result in that case is an overtone recipe in which there’s a clear fundamental with a fairly strong overtone at the two-octave level, plus a mess of other overtones which are randomly inharmonic. With the octave reinforcement, the pitch-sense of such a tone is quite clear, but those strong inharmonics definitely give it an unusual character. I made a lyre-like instrument with this weighted overtone tuning, the strings tuned to a standard scale. It can give you clear melodies but in a most unusual tone quality.

You can also locate the weights in ways that bring out other tunings for that prominent overtone. For instance, you can place the weight so as to place the overtone a twelfth above the fundamental. The pitch sense in this case is not quite as clear as with the two-octave placement; none-the-less this relationship of the twelfth does tend to reinforce the fundamental in the ear’s interpretation, so you get reasonably strong pitch-sense in a somewhat more peculiar tone quality. For this to work, the fundamental needs to project reasonably strongly: if the twelfth is louder than the fundamental it’s potentially confusing to the ear.

You can experiment with any number of other odd overtone tunings this way. You’ll find that while the most obvious relationships such as the two-octave positioning or the twelfth are meaningful to the ear, this clarity quickly diminishes as you get into less obvious relationships. That’s not to say that the resulting timbres aren’t interesting; just that they tend to feel more random, and the clarity of pitch diminishes accordingly.

So much exploring to do; so little time!


For another essay discussing topics related to this one, see here.


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