A concert flute and a Bb clarinet are almost identical in length, around 66 cm each. Both are woodwinds, both use similar fingering patterns, and a skilled player can cover comparable dynamic ranges on each. Yet put them through a spectrum analyzer and the clarinet shows strong peaks at harmonics 1, 3, 5, and 7, with nearly nothing at 2, 4, and 6. The flute fills all of them. The clarinet's lowest note also sits roughly one octave below the flute's, despite the near-identical tube length.
Neither fact has anything to do with the player, the reed, or the mouthpiece. Both follow directly from bore geometry, and the physics behind them is more precise than most engineers expect.
Two Tube Models
Acoustics textbooks describe two idealized resonating tubes.
The first is a cylindrical tube closed at one end and open at the other. The closed end forces a pressure antinode (maximum pressure variation), while the open end forces a pressure node (minimum). The only standing waves that fit this boundary condition are those where the tube length equals an odd multiple of one quarter-wavelength:
L = λ/4, 3λ/4, 5λ/4, ...Rearranging for frequency: f_n = n * c / (4L) where n = 1, 3, 5, 7 only. Even harmonics are not supported. At standard temperature (343 m/s speed of sound), a clarinet's effective air column around 58 cm resonates near 148 Hz, 444 Hz, 740 Hz, and so on, with gaps at 296 Hz, 592 Hz, and every other even-harmonic frequency.
The second model is a cylindrical tube open at both ends. Here both ends are pressure nodes, so the tube supports L = λ/2, λ, 3λ/2, ..., giving f_n = n * c / (2L) for n = 1, 2, 3, 4, all harmonics. This is the standard model for the flute, which is cylindrical and open at both ends despite having a lip plate that partially covers the embouchure hole.
Where the Clarinet Fits
The clarinet's main bore is cylindrical. The reed acts as a pressure-controlled valve that approximates a closed end. By the formula above, it should produce only odd harmonics, and that is exactly what happens. Acoustics research at UNSW and multiple spectral analyses confirm the clarinet's bore resonances sit at approximately 1:3:5 ratios. In the lower register, odd harmonics dominate the radiated spectrum, with even harmonics present but substantially attenuated compared to instruments with a complete harmonic series.
The practical consequence is audible and measurable. The missing even harmonics give the clarinet its characteristic "hollow" or "woody" timbre compared to instruments with a complete harmonic series. More importantly, it forces a quirk of fingering design: the clarinet overblows at the twelfth (an octave plus a fifth, or 19 semitones), not the octave.
Here is why. When a flutist overblows, the instrument jumps to the second resonance, which is twice the fundamental frequency, one octave up. The clarinet's second resonance is the 3rd harmonic, which is three times the fundamental, a musical interval of a perfect twelfth. The register key on a clarinet does not produce an octave key like every other woodwind. It suppresses the fundamental and lets the 3rd harmonic speak, pushing the pitch up by that awkward 19 semitones. This is why clarinets need a more complex key system to cover a two-octave range, and why the chalumeau register and the clarion register do not share fingering patterns the way saxophone registers do.
The Conical Bore Surprise
Saxophone bore is conical. The instrument narrows toward the mouthpiece and widens toward the bell. Adolphe Sax patented the full family of saxophones in Paris on June 28, 1846, after demonstrating early versions at the Brussels Industrial Exposition in 1841. His insight was to combine the single-reed mechanism of the clarinet with a conical bore, and the acoustic result is entirely different.
A cone open at the wide end and effectively closed at the narrow apex by the mouthpiece should, naively, behave like the closed-open cylindrical tube and produce only odd harmonics. It does not. A conical bore produces a full harmonic series.
The explanation requires thinking about wave geometry. In a cylindrical tube, pressure waves propagate as plane waves with constant amplitude across the cross-section. In a cone, the cross-sectional area increases with the square of the distance from the apex. The pressure waves must be spherical, not planar, and the acoustic pressure takes the form:
p(x,t) proportional to sin(nπx/L) / xwhere x is measured from the cone apex. This 1/x amplitude factor changes the boundary conditions. The mathematics works out to exactly the same resonance series as an open-open cylindrical tube: f_n = n * c / (2L) for all integer n. The saxophone overblows at the octave. Its harmonic spectrum is dense and continuous.
This also explains why the oboe, another conical instrument, overblows at the octave despite using a double reed that physically closes one end. The bore geometry overrules the mechanical end condition.
And it explains the flute-oboe coincidence: both instruments are roughly the same length and play similar bottom notes. A cylindrical open-open tube and a conical closed-open tube of the same length produce the same fundamental frequency. Joe Wolfe's acoustics resource at UNSW states this directly: "Closed conical bores have the same set of resonances as open cylindrical bores of the same length."
Implications for Digital Synthesis
If you are building a digital waveguide model of a wind instrument, bore shape is not a minor detail you can skip.
Modeling a cylindrical instrument is straightforward: two delay lines representing the forward and backward traveling wave, with a reed model at one end and a bell radiation filter at the other. Julius O. Smith III at Stanford's CCRMA developed digital waveguide synthesis in the 1980s with this cylindrical model as its core. In a 1991 ICMC paper, "Waveguide Simulation of Non-Cylindrical Acoustic Tubes," he tackled the harder case of conical bores.
Conical instruments require more work. The cone has a virtual apex, the missing tip of the cone that exists beyond the mouthpiece. The mouthpiece itself is designed to provide an air volume equivalent to the missing truncated section, so the instrument's fundamental frequency and harmonics align properly. To model this in code, you need to simulate the acoustic volume of the missing cone tip, typically with a short additional delay line or filter section representing the apex. Without it, the simulated fundamental will be flat relative to the harmonics.
Gary Scavone at McGill's Computational Acoustic Modeling Laboratory extended Smith's work to build real-time conical bore models, and documented the subtlety that even small changes in cone taper angle at a junction require careful filter design. A sudden change in taper creates an unstable reflection filter if not handled correctly. The conical bore model is, in simulation terms, harder to stabilize than the cylindrical one.
Listening with This Knowledge
Once you know the physics, certain things become predictable:
A low-cut on a clarinet track below around 260 Hz removes very little useful content, because the clarinet lacks a strong second harmonic that might otherwise fill the 130-260 Hz range. A low-cut on a saxophone track at the same frequency noticeably thins the sound because even-harmonic content is present there.
Subtractive synthesis patches intended to sound like clarinet need to use a waveform that emphasizes odd harmonics: a square wave is a reasonable approximation (its Fourier series contains only odd harmonics with amplitudes proportional to 1/n). Patches targeting saxophone character should use a sawtooth or other waveform with a full harmonic series.
And when a sound designer says a synth lead sounds "clarinetty" even without any attempt at physical modeling, the descriptor often means exactly one thing: the even harmonics are rolled off or absent, and the odd ones are strong. Bore geometry, simulated by accident.
Arthur Benade spent his career from 1952 to 1987 at Case Western Reserve University mapping these acoustic behaviors, and his book "Fundamentals of Musical Acoustics" (1976) remains the clearest account of how instrument geometry determines harmonic structure. The central insight has not changed: the shape of the air column determines which resonances survive, and the reed just provides energy. The bore decides the color.