Inharmonic resonances in wind instruments 

A key observation is that the trombone is the only instrument of the four that uses these high resonances to play, i.e. the trombone plays notes with a fundamental quite close to the frequency of a high resonance. The shape of the trombone's bell, the length and angle of the flare and the mouthpiece volume and constriction, are all chosen (or have ‘evolved’, in the engineering sense) to achieve something close to harmonicity. We’d expect only close, not exact harmonicity: the trombonist doesn’t play exactly at the frequency of a resonance, but a little above the impedance maximum. (See this paper for details.) Call this difference δ_n for the nth peak, so the trombonist plays at 2f+δ₂, 3f+δ₃, etc, where f is the implied 'fundamental'. But the widths of the resonances do not increase in proportion with n, and δ_n is not proportional to n. So, a hypothetical trombone with exactly harmonic resonances would play flat in the high range. Consequently, we’d expect instead that the higher resonances of a good instrument would lie successively higher above nf, which is indeed what we see in the graph above.

The discussion now turns to woodwinds, beginning with the clarinet. I’ll use ‘register’ to mean notes having the same mode of vibration for their fundamental, so the 2nd register on the clarinet starts on (written) B4, a 12th above the start of the 1st on E3.

The clarinet bore is often compared with a cylinder. Writing speed of sound c, an ideal cylindrical pipe of length L has impedance peaks (resonances in this context) at frequencies very close to f = c/4L, 3f, 5f etc – the odd members of the harmonic series. (Because of a small end effect that increases slightly with increasing f, those expressions are slight overestimates and the resonances are slightly inharmonic.)

Let's continue considering only the complete bore with all holes closed (as for the bottom note E3), because a musician might ask: why are the clarinet resonances so inharmonic? Why hasn’t ‘evolution’ driven the clarinet towards cylindrical shape, and so to more closely harmonic higher resonances? After all, it would be easier for a maker to produce a cylindrical bore. Here's one obvious difference: in contrast with the trombone, the higher resonances of this clarinet fingering are not used for higher notes. To play the next mode (the note B4, a twelfth higher), we also use the left thumb to open a register hole. Opening this register hole weakens and detunes (raises the frequency of) the first resonance. It also raises the frequency of the second and higher resonances, though to a smaller extent. For E4 (impedance graph here), the second impedance peak is about 1.8% or 30 cents flatter than the second impedance peak for B4 (graph here). A cylindrical clarinet might play that E3-B4 twelfth in tune if both notes were played without a register key (what players call ‘bugling’) but adding the register key would play the upper note sharp. The third and higher impedance peaks for fingerings based on E4 are not used for the third and fourth registers: all notes in those registers have fingerings with several of the lowest tone holes open - see here for examples.

The main ‘evolutionary’ pressure on the shape of the clarinet flare (with the bell) has been to achieve in-tune twelfths (frequency ratio of 3:1) between played notes in most of the first and second registers, usually with some of the bottom tone holes open. (The bell also has a different role: its job is to act as a high-pass filter that radiates high frequencies well. This is needed because a series of open tone holes acts like this, and the bell makes the lowest notes on the clarinet sound more like those of notes with tone holes open. See here for further explanation.)

On the clarinet, the thumb register key is used throughout the second register, but the third and fourth registers use pinched or open tone holes functioning as register keys. This complicates things, but similar effects apply. We’ll come back to clarinet harmonicity, but first let’s look at the sax.

The saxophone. In part because it is made from rolling up a flat sheet of metal, the (soprano) sax is quite close to being a truncated cone. For an ideal cone, if the truncated section is replaced by an equal volume with compact geometry, then, in an approximation that works at low frequencies, the resonances lie quite close to the complete harmonic series. The saxophone mouthpiece volume plus the effective volume of the reed is close to that of the truncation. See Saxophone acoustics for further explanation. So, at low frequencies, we’d expect the first several resonances of the complete bore and mouthpiece to be nearly harmonic, as the graph shows (blue points; graph repeated here). Here, ‘low frequency’ means that the length of the truncation is much less than the wavelength. Because the truncation is substantial, this condition is not maintained at high f, so the blue points rise above the harmonic line at high f.

The sax has operational differences that change the evolutionary pressures. First, unlike the case of the clarinet, the sax fingerings for the four lowest notes in the first register are not used with a register hole (octave key) for the second register. So, all standard fingerings in the 2nd register have several tone holes open. Further, while the first register hole on the clarinet doubles as a tone hole (for the note Bb4), the saxophone’s register holes are much smaller and have smaller effects on the frequencies of the impedance peaks, though they still weaken the first peak enough to stop that register playing. (The third and fourth registers of the saxophone are more complicated and involve a lot of vocal tract influence.) (Impedance plots and sound spectra are here.) (And if you are wondering why the clarinet doesn't have a pair of dedicated and automated register holes like the saxophone, instead of a single thumb key that also doubles as a tone hole, then you and I are not the first. Arthur Benade mentions it in Fundamentals of music acoustics.)

Resonance harmonicity and harmonics

Consider a note played with frequency f_p. If its nth harmonic (at frequency nf_p) falls close to an impedance peak, then we expect the bore to act as an impedance matching transformer at that frequency and that it will therefore radiate strongly. That resonance-harmonic proximity may also contribute to sustaining the auto-oscillation of the reed or lips. Creating harmonic resonances for these two advantages is called the Bouasse-Benade prescription. However, remember that, in the graph above, the trombone impedance peaks lie a little above the harmonicity line, by an amount that increases with frequency, which assists intonation, as we argued abov . So, for the trombone playing a lowish note, with its fundamental slightly above an impedance peak, the harmonics nf_p lie close to resonances, which act to radiate higher harmonics well and contribute to the brassy loudness and timbre. So, one might ask, shouldn’t higher the fourth and higher impedance peaks of the clarinet E3 fingering lie close to harmonic ratios to coincide with the (odd) harmonics of that note? They would on a cylindrical clarinet, but not on a real one: as we’ve seen above, the clarinet is not a cylinder, so the resonances of E3 are not harmonic. So, compare the sound spectrum and the impedance spectrum on the bottom note, shown on this page.

On that page, we see that the 1st and 3rd harmonics lie near to the 1st and 2nd impedance peaks, and the 5th and 7th harmonics lie successively less close to the 3rd and 5th impedance peaks. In the sound file, we see that the 1st, 3rd and 5th harmonics are larger than their even neighbours, but after that, the even-odd relation breaks down: the higher resonances are strongly inharmonic and the 7th harmonic is weaker than its neighbours. This would not happen in a hypothetical cylindrical clarinet: we’d expect it to have stronger high odd harmonics and a brighter sound. But that instrument wouldn’t play the E3-B4 twelfth in tune (or F3-C5 and some others). Further, remember that cylindrical shape can’t be achieved for the notes with open tone holes, so this hypothetical harmonicity of resonances and stronger high harmonics for the lowest note or two might give us timbre that was brigher than that of higher notes, as well as cause the bad intonation mentioned above.

Finally, many of the notes on clarinets and other woodwind instruments are played using just a single resonance near the fundamental: notes in the upper second and in higher registers usually don’t have any systematic pattern of harmonic-to-mode matching. (Look around on the clarinet, saxophone or flute sites.) I like to illustrate this observation with a riddle. Here are the sound pressure spectra of the note D5 played by a flute and a clarinet, measured in each case near the first open tone hole. Can you tell which is the flute and which is the clarinet? (This is Figure 2 from this scientific paper. ) Answer below.

So, do the inharmonic resonances for E3 and nearby notes make the clarinet sound soft with dark timbre for these notes? The reed vibration of a steady note is periodic, so the sound spectrum is exactly harmonic (limited only by the Musician's Uncertainty Principle). Harmonics are generated by the non-linearity of the reed vibration (especially the odd harmonics, when the reed motion and the air flow are clipped). The fundamental radiates poorly, because of an impedance mismatch at the bell, while the higher harmonics are relatively well radiated by the bell. When a bore resonance is available (and has high Z and appropriate phase), the relevant harmonic ‘benefits’ from that resonance acting as a further impedance matcher. The fifth and especially higher harmonics of E3 don’t get this help. But the fifth harmonic is around 800 Hz, so it is already getting into a range where the ear is sensitive, so it manages pretty well without a tuned bore resonance acting as a (further) impedance matcher. A somewhat similar story applies for saxophones and flutes, though they have small or no bells. For the trombone's low notes, several harmonics do fall close to resonances, and that is important to sound radiation at those frequencies, and thus to timbre. But for very high notes, the higher harmonics fall in a range were there are no more resonances. For this reason, and also because the lip vibration is more sinusoidal, the output sound has much weaker high harmonics.

See Flute Acoustics, Clarinet Acoustics, Saxophone Acoustics, Double Reed Acoustics and Brass Acoustics for more about each instrument. See also Euphonics, the site where this question was explicitly raised. In the figure closest above, the top spectrum is the flute. The flute has a little more broadband noise than the clarinet. By tradition rather than for physical reasons, the flute is played with more vibrato than the clarinet, and this widens the harmonic peaks a little. The weak second harmonic on the flute is real but not systematic. I chose this note because it gave a (slightly) weaker second harmonic on the flute and I varied the embouchure a little to enhance it.

Strings gives an introduction to the resonances of strings, and how this gives rise to almost harmonic spectra for plucked string instruments. Bows and Strings explains how bowed string instruments have exactly harmonic spectra, even with inharmonic strings.