In Which I Rant About The Overtone Series

GAH. I'm reading through Hindemith's "The Craft of Musical Composition", and like every other book written about music, no introduction is complete without a discussion of the overtone series. And somehow, maybe because they're musicians rather than physicists, it's ALWAYS unsatisfactory. I've NEVER seen the overtone series explained correctly except in the Griffiths Intro to Quantum Mechanics book. Yeah, QM, not music theory. It's ridiculous. The overtone series isn't what it claims to be, and explanations of other phenomena involving it are just wrong. I don't know why. So here we go.

There is no such thing as the overtone series. There is AN overtone series, but not THE overtone series, except in mathematics. That description makes a lot of sense to someone who has studied Fourier analysis and less sense to someone who hasn't, but it's worth a try. Pretend that there is a perfect string with constant density and infinitesimal width and that it has some finite length L and is held taut and fast at both ends. This is an idealization, remember. Now, pull the string in some shape with very small deviations from the resting state. Regardless of what that shape is, and this is the magic of it, you can write it as a sum of sine functions. This is called a Fourier decomposition, or a Fourier series. Those sine functions will look like your normal sine wave, but only those sine waves that have value 0 at the endpoints will be in the series -- so sin(pi*L), sin(2pi*L), sin(3pi*L), ..., sin(n*pi*L), for any integer n. For your perfect idealized string, the relationship between wavelength and frequency is such that if the whole string (sin(pi*L)) vibrates at a particular frequency (which depends only on the string's length, tension, and density), half the string (sin(2pi*L)) will vibrate at double the frequency, a third of the string (sin(3pi*L)) will vibrate at triple the frequency, and so on. So WHATEVER you do to the string at the start (assuming your stretching is very small), it will be vibrating in some weird pattern, but that weird pattern will be a sum of the frequencies of the whole string, half, one third, one fourth, and so on. If the whole string vibrates at a frequency of 64 Hz, after you make your weird initial condition, you will hear the 64 Hz, twice that (128 Hz), three times that (192 Hz), and so on. You'll actually hear all those notes. That is THE overtone series. It's described as C', C, G, c, e, g, bb, c', ..., with some exact pitches, where the G's are a little tiny bit sharp, the E's are somewhat flat, the Bb's are very flat, and so on, and authors make various claims as to its usefulness in various situations and as various explanations.

Of course, what I described is the perfect string. Real strings don't work like that, though they come close. Piano strings especially don't work like that, and if you look at them, you'll see that they aren't just plain strings, since piano makers try to correct those effects. The perfect string, where the overtones are integer multiples of the fundamental, is a linear medium. Real instruments aren't. The overtone series of a crotale, for instance, which is a round metallic pitched plate, is nothing like a string's -- since it's a circle, you can't break down a perturbation into sines; you have to use a different set of functions at different frequencies. Non-pitched instruments are even more nonlinear. Strings come close to being linear, but aren't exactly. Not even the human ear is exactly linear in its response. Vibrating air columns aren't. Reeds certainly aren't, and they respond differently at different frequencies.

This perfect string is used to describe the sounds that you hear from an instrument. An instrument produces, along with the fundamental that you hear clearly, a series of overtones at varying strengths. An oboe has strong overtones, for instance, which explains its rich sound, while the clarinet has weak ones, which explains it's dark quality. But since no instrument is really linear, this overtone series isn't exactly the one described. The set of frequencies sounded is called a spectrum, and it doesn't necessarily look like the overtone series described. It's just not perfectly in tune.

The overtone series is considered the basis of all everything. It's annoying, because, well, it isn't. Only the octave and fifth are actually important. The overtone third isn't the same third as the one in our triads, and this is obvious when we consider the minor triad -- it's not more dissonant than the major triad, but the minor third is much farther away from the "natural" third of the harmonic series. The third in our triads is, I believe, psychologically conditioned in us rather than an aural phenomenon like the octave and fifth. When those are in tune, mathematical cancellations in the sound waves (and the sound waves in the air themselves ARE linear, unlike the ones on the string) are audible, and the interval is perceived as "open". If the third is tuned correctly, it can also sound "open", and an in-tune major ninth can also sound "open". That is a DIFFERENT characteristic from tonal character. We can see from the scales of non-Western cultures that thirds can actually vary widely in pitch, and that major doesn't necessarily mean happy, and so on. The third isn't really a function of the overtone scale. The other notes aren't, either! At least not the way we do it. We take a fifth -- actually, something very close to the fifth -- and stack it repeatedly until we get to the original note (assuming everything is octave-reduced, of course), and THAT is what our system of harmony is based on, not the overtone series -- or rather, only indirectly the overtone series. Luckily, 3^12 = 531441 is very close to 2^19 = 524288, a difference of 1.36% (or 1.35%, depending on how you count it), so the perfect fifth needs to be fudged only a tiny bit to make it fit the octaves. There are twelve of them, so we get 12 evenly spaced tones in an octave, each therefore having a frequency of 2^(1/12) the previous pitch. This is NOT the overtone series! The overtone series does affect consonance and dissonance, to some extent, but only in the "beats" between notes of similar pitch -- play a C and a C#, and the waves will interfere and sound somewhat ugly. Play a C and a B, and it won't be so bad, but there will be beats between the B and the first overtone of C to create the dissonance. The dissonance can be hidden, say by playing C E G B, in which case the ear will hear a C major chord and an E minor chord rather than a strident major seventh interval. However, only the first few overtones (essentially up to the perfect fifth, not even so far as the "perfect" third) actually have this aural property to a reasonable strength. The higher terms in the overtone series are meaningless in this respect.

Finally, people try to say that the most stable voicings for chords have a spacing like that of the overtone series, wide at the bottom and tight at the top. THIS IS A CROCK OF BULLSHIT. The reason for this has nothing to do with the overtone series, except perhaps that some of the basic principles are the same. Two notes in the lower register are MUCH closer in frequency than the same two notes in a higher register due to the logarithmic nature of the scale. Two low C's are 32 Hz apart, from 32 Hz to 64 Hz, whereas two high C's are, say, 4096 Hz apart, from 4096 Hz to 8192 Hz. It's 8 octaves from the 32 Hz to the 8192 Hz, by the way. Part of the ear's pitch resolution is based on linear frequency, not the logarithm of it, which is what musical pitch is based on. This changes for the very high notes, of course, at the threshhold of hearing, but it's still very clear in, say, the piccolo range. So at the upper end of things, close pitches can be resolved by the ear much more easily than at the lower end of things. Furthermore, the overtone series DOES play a role, but it's only that part of the reason for the dissonance of low pitches is that the overtone series clash. They shouldn't clash much, but remember that the instruments are nonlinear, so they end up clashing quite a bit, especially on the piano. On a string or electric bass, which produces rather pure pitches, the clash is due to the bad resolution rather than the overtones, since the overtones are mostly absent.

OK, I think I've exhausted my rant. Feel free to disagree, of course.