How harmonic are 'harmonics'?

describe What are harmonics? How do they differ from resonances? How harmonic are musical resonances? What is the difference between the harmonics in the spectrum of a note and the 'harmonics' played on a string or wind instrument?

A note from most musical instruments usually has a harmonic (or nearly harmonic) spectrum, meaning it has frequency components at (approximately) harmonic ratios: if the fundamental frequency is f, there are peaks in the spectrum at f, and also at (about) 2f, 3f, 4f, etc. For many instruments (bowed strings and wind), but not all (exceptions including plucked strings and percussion), these ratios are very close to exact: the second frequency component is at 2.000 times f, the third at 3.000 times f, etc. But why is it so? or rather, why is it sometimes so? The short answer is that there is an important difference between instruments that use non-linear vibrations (voice, bowed strings, woodwinds and brass) and those that use linear superposition (plucked strings and most percussion).

Consider a note with frequency f (measured in hertz, Hz, or cycles per second); the duration of one cycle of the vibration is its period T (where f = 1/T). The vibration is periodic if successive cycles are identical. The vibrations produced in a singing voice, a bowed string instrument, a woodwind or a brass instrument are almost exactly periodic. This is due to non-linear behaviour in the sound production. The specific non-linear mechanisms involved are all complicated, different and interesting - and much studied in our lab. See voice, bowed string, clarinet, flute and brass instrument.

On our page on the sound spectrum, we show that an exactly periodic vibration (with frequency f) has an exactly harmonic spectrum. That is, it can be considered as the sum of simple (sinusoidal) vibrations with frequencies f, 2f, 3f ... and nf, where n is an integer. Each of these frequency components is a harmonic: the nth harmonic is the one with frequency nf. The lowest harmonic is called the fundamental; it has frequency f (sometimes called fo, where o stands of oscillation.) The other harmonics are sometimes called overtones.

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But now let's look at what string players do when they play 'harmonics' (and I'll use the scare quotes when I'm referring to this sense). In our page on strings, we see that the fundamental string vibration (frequency f) has maximum motion at midpoint, zero motion at the nut, and almost zero at the bridge. (This means that its wavelength on the string is twice as long as the working string length between nut and bridge.) But the sketch shows what happens if the player lightly touches the string at precisely 1/4 of its length. (Note: touching very lightly, not pushing against the fingerboard.) It vibrates with four loops instead of one (its wavelength is four times shorter) and it has the frequency 4f, which is two octaves above the fundamental. This is approximately true for a dozen or so harmonics: the nth harmonic plays at n times the frequency of the fundamental. See more about the harmonic series on Waves in strings, reflections, standing waves and harmonics.

When we play these 'harmonics', we might notice that, particularly for thick strings, notes above the first dozen or so are successively more out of tune: e.g. the twelfth 'harmonic' is sharp compared with two octaves and a fifth above the fundamental. This is because the strings are not infinitely flexible (they are not a physicist's 'ideal string').

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An experiment. Take a stringed instrument, and play the different 'harmonics' on one of the open strings - perhaps the lowest. Provided that the strings are reasonably new, the first several notes you play will be nearly exactly a harmonic series. And, when you pluck the open string in a normal way, you hear a musical note with a clear pitch.

Now take a very small piece of adhesive tape or putty. Wind the tape or putty carefully around the string at the midpoint. Pluck it again. You will probably notice that the pitch is a bit lower, but also less clear, and that the note sounds different: it may sound a bit like a bad quality bell. (If it doesn't sound weird, add more mass.) Then play the 'harmonics' (i.e. sound the string resonances). You will find that the odd numbered resonances (which require motion at the midpoint) are all flat, but that the even ones (for which the midpoint is a node or stationary point) have hardly changed. So this is no longer a harmonic series. Further, you can vary the departure from harmonicity by varying the size of the mass in the midpoint of the string. The two effects are related: when the series is harmonic, we hear a clear pitch; when it is far from harmonic, we usually do not hear a clear pitch, and the sound is more like that of a percussion instrument.

For the analogous resonances of some wind instruments, see open vs closed pipes.

I mentioned above that a periodic vibration has a harmonic spectrum. The converse is also true: the sum of harmonic vibrations is a periodic vibration. These make up the two halves of Fourier's theorem. The second is easy to see. Let the fundamental frequency f have a period T. The second harmonic with frequency 2f has a period of T/2, so, after one vibration of the fundamental (after time T), the second harmonic has had exactly two vibrations, so the two waves are ready to start again with exactly the same relative position to each other, so they will produce the same combination that they did for the first cycle of the fundamental. The same is true for each harmonic nf, where n is a whole number. After time T, exactly n cycles of the nth harmonic have passed, and so all the harmonics are ready to start again for a new cycle. This is explained in more detail, and with diagrams, in What is a Sound Spectrum? The harmonic series is special because any combination of its vibrations produces a periodic or repeated vibration at the fundamental frequency f.

The resonances of strings and pipes are not inherently harmonic. An ideal, homogeneous, infinitely thin or infinitely flexible string has exactly harmonic modes of vibration. So does an ideal, homogeneous, infinitely thin pipe. Real strings and pipes do not. We saw in the experiment that adding a mass - making the string inhomogeneous - makes the string inharmonic. (By the way, worn or dirty strings are also inharmonic and harder to tune. Washing them can help.)

Real pipes have inharmonic resonances because of their finite diameter: the end effects are frequency dependent. The pipes of musical instruments are complicated by departures from cylindrical or conical shape (valves and tone holes). Some of these complications are there to improve the harmonicity, but the results are rarely perfect. For any one note, however, the lip or the reed performs the same (strongly non-linear) role as the bow: the lip or reed undergoes periodic vibration and so produces a harmonic spectrum. Again, the operating mode of a brass or woodwind instrument playing a steady* note is a compromise among the tunings of all of the (slightly inharmonic) pipe resonances (mode locking again).

* "steady" here means over a very long time. Measurements of frequency are ultimately limited by the Musician's Uncertainty Principle (which is almost the same as Heisenberg's Uncertainty Principle, see this explanation). If you play a note for m seconds, the frequency of its harmonics cannot be measured with an accuracy greater than about 1/m Hz. If your spectrum analyser measures over only k seconds, it cannot measure much more accurately than very roughly 1/k Hz.

An interesting point about winds: the sound spectra of clarinets tend to have strong odd harmonics (fundamental, 3rd, 5th etc) and weak even harmonics (2nd, 4th etc), in their lowest register (but not in high registers). This effect is discussed in "pipes and harmonics" and "flutes vs clarinets".

Real strings also have inharmonic resonances because they are not infinitely thin or flexible, and so do not bend perfectly easily at the bridge and the nut. This bending stiffness affects the higher modes more than the lower, so the 'harmonics' are stretched, compared with harmonics. Solid strings are less idealy than wound strings, steel strings are less ideal than others, pianos - especially little pianos - are less ideal than harps. The inharmonicity disappears when the strings are bowed, but is more noticeable when they are plucked or struck. Because the bow's stick-slip action is periodic, it drives all of the resonances of the string at exactly harmonic ratios, even if it has to drive them slightly off their natural frequency. Thus the operating mode of a bowed string playing a steady note is a compromise among the tunings of all of the (slightly inharmonic) string resonances. This phenomenon is due to the strong non-linearity of the stick-slip action. It is called mode locking.)
animation of stick-slip motion

Real strings also have inharmonic resonances because they are not infinitely thin or flexible, and so do not bend perfectly easily at the bridge and the nut. This bending stiffness affects the higher modes more than the lower, so the 'harmonics' are stretched, compared with harmonics. Solid strings are less ideal than wound strings, steel strings are less ideal than others, pianos - especially little pianos - are less ideal than harps.

(We could continue the experiment mentioned above, where a small added mass makes a string inharmonic. Now take a bow and bow the non-harmonic string. If the mass is not too large, and particularly if you have good bow control, you will be able to produce a good musical note with a clear pitch.)

The inharmonicity can be made to disappear when the strings are bowed, but is present and often noticeable when they are plucked or struck. The animation at right comes from our page on the bow-string interaction. Because the bow's stick-slip action is periodic, it drives all of the resonances of the string at exactly harmonic ratios, even if it has to drive them slightly off their natural frequency. Thus the operating mode of a bowed string playing a steady* note is a compromise among the tunings of all of the (slightly inharmonic) string resonances. This phenomenon is due to the strong non-linearity of the stick-slip action. It is called mode locking. (One important result of mode locking concerns the inharmonic torsional modes of a bowed string.)

One final remark: the almost harmonic spectra of plucked or struck strings are a recent phenomenon. Periodic vibrations from nonlinear interactions have been here for millions of years (and perhaps billions of years if we consider whistling wind). But strings uniform enough to produce nearly harmonic spectra when plucked ... strings like that probably that had to wait for humans to make instruments - perhaps the lyres of Ur, about 4500 years ago.

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