FUNDAMENTAL, HARMONICS, OVERTONES, PARTIALS, MODES

As many readers will already know, most musical sounds contain multiple frequencies. These frequencies arise from different modes of vibration — that is, different patterns of vibratory movement — coexisting in the physical body that is the source of the sound. Certain terms are commonly used in talking about how the frequencies blend and how the ear and brain interpret them to arrive at a sense of pitch and timbre. These include words like fundamental, overtones, partials, and harmonics. For many musical sounds this terminology works well in reflecting the ways that people perceive and respond to multi-frequency tones. But for many other sounds, the terminology doesn’t fit quite so well, in that it doesn’t correspond very well to either the way we perceive the sounds or the physical reality that underlies them. In this essay I’ll be talking mostly about the cases where the fit between terminology and reality isn’t so good. But first, let’s have a look at the cases in which the terminology does work well.

The good-fit cases are the ones where the tone is made up of steady frequencies which can be seen as lining up in a coherent series ranging from lowest to highest. For this to work, we also have to assume that the lowest tone of the series isn’t so low in frequency as to be below the hearing range, and that higher tones in the series aren’t confusing the ear by being far louder than the lowest. The best examples of this are the sounds of strings and tubular wind instruments. Many synthesized electronic instrument sounds also fit the bill nicely, as they are actually designed to work this way. For sounds such as these, the ear typically recognizes the lowest frequency as the defining pitch, commonly called the fundamental. The overtones above blend more or less imperceptibly into the overall tone, but at the same time they contribute to the perceived timbre or tone quality. The language of “fundamental plus overtones” generally fits both the perception and the underlying physical events. 

An important consideration in these cases has to do with the relationships between the overtone frequencies and the fundamental. If the frequencies of the overtones have a certain well known and simple arithmetic relationship to that of the fundamental, then they qualify as harmonics. (The basic definition: if we call the fundamental frequency f, then the overtones above it are harmonic if appear at frequencies 2f, 3f, 4f, and so forth.) If the overtones don’t happen to fall into this mathematically defined relationship, then they are inharmonic. We find that harmonic timbres are easiest for the ear to make sense of. The best examples of this again are most winds and string instruments, which typically meet fairly accurately the criteria for harmonicity. And, again, a lot of synthesized tones are deliberately built this way as well.

In these harmonic tone qualities, the ear easily recognizes the pitch in an instantaneous, effortless and unconscious process. By contrast, some other instruments produce inharmonic overtone series, and here the process of pitch recognition is a bit less dependable. With many inharmonic timbres, particularly those which happen to have a single series of widely spaced overtones, pitch recognition still happens pretty easily much of the time. The ear picks up on the lowest component as the fundamental which anchors the pitch-sense. However, when it comes to reinforcing this pitch sense, the inharmonic overtones don’t do as good a job as harmonics. Nor do they blend as closely: while they do contribute strongly to the characteristic tone color, they may to some degree seem to stand out as separate tones. Examples for these sorts of inharmonic tone qualities are kalimba tines and simple rectangular marimba bars (although, as we’ll see in a minute, things can get more complicated with marimba bars.)*

*[In this discussion we’re not considering the practice, common for makers of western orchestral marimbas, of reshaping the bar in ways that cause some overtones that would otherwise be inharmonic to align harmonically.]

In cases where there’s a coherent overtone series but it’s inharmonic, the ear may fail to recognize the fundamental if the inharmonic overtones are too strong and the fundamental is not sufficiently prominent the sound. The sound may come then across as a blend of pitches with no one of them standing out as the defining pitch. Or it may happen that the ear may focuses on some particularly loud overtone to provide pitch-sense.

To repeat, the cases I’ve just been describing are those in which there’s a single overtone series in which the lowest tone of the series stands out clearly as a fundamental. These are the cases where it usually makes sense to think in terms of a pitch-defining fundamental joined by overtones which may be harmonic or inharmonic. Let’s continue to look a little more closely at these examples, starting with strings. One of the reasons things turn out so neatly with strings is that musical strings represent a uniquely simple physical vibrating system. In the ideal, a musical string is in effect a one-dimensional body, its mass uniformly distributed over its length, with no inherent stiffness, held under high tension at perfectly immobile anchor points. Of course strings in the real world don’t live up to the ideal, but they come close enough to give us something very close to the desired result: the lowest of the frequencies present (the fundamental), is usually loudest, and assuming that the string is well made and uniform in shape, the overtones generally come out quite close to the harmonic ideal. As a result, in our perception the pitch sense is clear and unambiguous. The harmonic overtones blend into the whole so closely that the tone seems integrated and coherent. With a few caveats, the situation is similar for well made tubular wind instruments: although the physical basis is different, a relatively simple and orderly, more or less one-dimensional physical situation gives rise to a coherent sound with a clear fundamental and an orderly harmonic overtone series.

But as soon as you start to get into other sound sources, things get more complex. Consider, for instance, a flat disk gong. Like the string, this is in many ways a very simple form — in this case effectively two-dimensional, defined by a single radius value (which is to say, circular), uniform in thickness and rigidity, and so forth. But the flat gong turns out to be considerably more complex in its resulting frequency blend. The first thing to note is that the relationships between the frequencies produced by simple disk gongs are not harmonic; they’re pretty much all over the place and any seemingly musical frequency relationships you might be able to pick out are purely coincidental. Furthermore, being two dimensional, gongs don’t have a single series or progression of overtones going from low to high. A plausible analysis would be that they have two series, one corresponding to modes of vibration featuring concentric nodal rings, and one corresponding to modes featuring nodal diameter lines … but then, it’s more complicated than that, because there are also modes that combine the two. The ear knows none of this, of course; it just hears a whole lot of inharmonically related frequency components combining to create a gong-like tone quality. For some mallet strokes the ear may perceive one of the many frequencies present as the defining pitch, and it would be tempting to call that one the fundamental. But it won’t necessarily be the lowest frequency present, nor necessarily even the lowest of either of the two main types of vibration pattern. And in fact, two listeners hearing the same gong tone might perceive different frequencies as the fundamental; even the same listener might hear different frequencies as the defining pitch with different strokes. This, then, is a case where thinking in terms of a single pitch-defining fundamental, plus a supporting cast of higher overtones adding color to the tone, doesn’t work so well.

Consider another example. Rectangular bars, such as xylophone bars, are mostly played in a way that brings out one particular set of modes which can plausibly be seen as having a lowest “fundamental” tone followed by a series of (generally inharmonic) overtones. But those same bars have many other physical vibrating patterns as well, including separate series for transverse modes in the side-to-side direction, torsional modes (vibratory patterns based in twisting motions), and longitudinal modes (vibratory patterns involving pressure wave fronts running back and forth along the length of the bar.) All may affect the tone. The result is decidedly complex and irregular. For this reason, bar-percussion instrument makers and players do many things to tame the complexity and present something more coherent to the ear — things such as choosing the right mallet and striking only at the center of the top surface of the bar (not the sides or ends). Most importantly, as footnoted above, bar makers may reshape the bar in ways that alter the relationships between the frequencies of the different modes, bringing some of the most prominent ones into harmonic relationships. These things help to simplify and clarify the tone, but they don’t change the point here, which is that simple xylophone-style rectangular bars typically produce a mess of harmonically unrelated modes, and this seriously affects pitch perception.

Here’s another interesting case study: carillon bells. In some ways bells are similar to the gongs discussed earlier: they tend to produce many inharmonic modes involving combinations of nodal lines and nodal rings. Makers of large bells go to great ends to manage the relationships between these modes, with the idea of tuning the most prominent among them to relationships that will make musical sense to the ear and help the listener to focus on the frequency of one mode in particular as the defining pitch. Nonetheless, you may have had the experience of listening to a melody played on a set of carillon bells, and at some point realizing that your ears were tracking some unintended mode in the bell sound as the defining pitch. Interestingly, the mode that is intended and usually heard as the defining pitch is not the lowest of the frequencies present. There is a quieter but still audible lower frequency within the carillon bell tone known as the hum tone. The mode that gives the bell its defining pitch is variously known as the strike tone, the prime, or sometimes, confusingly, as the fundamental.

I have found this bell terminology to be useful in other contexts. Recently I made a set of disk gongs in which I managed to adjust the inharmonic relationships between three of the most prominent modes to where they aligned in much more coherent relationships. (Specifically, I got them pretty close to octave relationships. I did this through a combination of careful sizing of the gongs and careful hammering of the nipple at the center. You can read about it here.) Like the big bells, these tuned gongs have a clear defining pitch. Borrowing the carillon terminology, this pitch can be called the strike tone or prime. Then there’s a quieter lower pitch, which can be thought of as the hum tone. And of course there are a lot of other frequencies present. The tone perceived as the prime is one of the three prominent modes tuned to octave relationships; there’s another an octave above and another two octaves above. Together these reinforce very nicely the sense of pitch, while additional quieter inharmonics add a bit of color.

I could continue with further examples of increasingly complex or irregular shapes, pointing out their resulting complex and irregular frequency blends, but I’ll refrain and just jump to the main point of this article, which is:

Aside from specialized cases such as strings, carefully designed wind instruments, and the artificial timbres of many electronic instruments, most physical vibrating bodies are not well described by saying they have a single fundamental followed by a series of harmonics. The notion of a fundamental plus a single orderly series of overtones doesn’t fit all or even most sounds in the real world. 

I’m happy to use the terminology of fundamentals and overtones, harmonic or inharmonic, in those cases where it aptly describes what’s happening both physically and in our perceptions. 

In some other cases, as just discussed, it seems useful to borrow terminology from the carillon bells. Those are cases where we can speak of a prime tone as the defining pitch, without implying that that defining pitch is the lowest tone present, nor that it is the fundamental tone of a single overtone series.

And then there are the other cases, most common in nature even if they aren’t the most favored in musical contexts. These are the cases in which many modes of vibration are present, each contributing its corresponding audible frequency, but with ambiguity as to which mode, if any, is most naturally heard as a defining pitch. Words like fundamental, overtone, harmonic and inharmonic may not be the most useful words in these cases. I’ve come to feel that the best way to think and talk about and describe such cases is to avoid preconceptions about the functions of the different frequencies present implied in those terms. Instead, start by simply thinking of the vibrating body as having multiple modes of vibration producing different frequencies. Then, if you wish to delve deeper, you can think about which modes are most audible, what the relationships between them happen to be, whether the ear tends to focus on one more than others, and how the ear makes sense of the blend.

For another essay exploring ideas related to this one, see here.