# Woodwind Reed Susceptance

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Hello musicians and everyone!

About woodwind instruments, I mentioned sometimes the reed susceptance. Here are explanations.

This sketches a clarinet mouthpiece and reed. A double reed works similarly
https://en.wikipedia.org/wiki/Double_reed

The pressure oscillations of the air column move the flexible reed. When the pressure is lower, it closes the reed. When it's higher, the more open reed lets more breath pass. More throughput when the downstream pressure is bigger is the opposite of a usual obstacle or loss: it's a negative conductance that provides power to the oscillation of the air column. That's known.

The reed must be closed when the downstream pressure is low to sustain the oscillation. This needs that mainly its stiffness determines the position. Its inertia would close the reed when the downstream pressure is high, damping the oscillation. That is, the reed's resonance is higher than the instrument's notes. It can be heard as a high pitched hiss if the musician takes the mouthpiece much too far, and during articulation using an imperfect saxophone mouthpiece.

Similarity of acoustics with electricity lets replace Pa by V, m3/s by A, then we use ohm and siemens: S=ohm-1. For sine U and I of given frequency, Y=G+jB where Y is called admittance, G conductance and B susceptance, while Z=R+jX where Z is called impedance, R resistance and X reactance, with j2=-1. We think of sine sounds and linear reeds to understand but none is. Horribly common.

If a clarinettist blows 5L in 25s which the reed modulates by full 200µA peak while the pressure oscillates by 0.1bar peak, the reed's conductance is -20nS=-(50Mohm)-1. The air column of D=14.6mm has a wave impedance of 2.5Mohm and its losses are less than (50Mohm)-1 to oscillate with this reed, resulting from a strong resonance consistent with stiff intonation.

As the pressure oscillation moves the reed, the sweeping area displaces air. Here pressure lets absorb a current (air throughput) when it rises, not when it's low or high, and lets provides a current when it falls. Like air's compressibility does, the reed adds a susceptance representable by a capacitor. I haven't seen this in books nor research papers, and at least one instrument maker ignored it. But bassoonists know it by experience while clarinettists may ignore it, not to mention organists.

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A Bb clarinet mouthpiece can have 18mm facing length and 1.1mm tip opening, which the clarinettist reduces to a variable amount as he presses the reed with his jaw covered by the lip. Let's take 12mm width, remaining 15mm length and parabolic 0.8mm opening: the 38mm3 displaced by 0.05bar make 7.6pF, as compliant as 1.1cm3 of air, equivalent to 6.4mm length of air column; for a low note it's 1.2% of the length or 0.2 half-note. This is consistent with how much a clarinettist can pull the pitch with his embouchure.

A bassoon's double reed can be 3mm open for low notes, mean 10mm wide and move over 10mm length. 0.02bar moving most of the 67mm3 make 33pF like 4.7cm3 of air. At the bassoon's narrow D/L=0.02 cone, it's as much as 0.36m from the apex or 12% of the length. Consistently, a bassoonist can pull the pitch by half a tone with his embouchure, and tunes the instrument by cutting the reed.

When sounding a double reed alone, or a single reed on the naked mouthpiece, the reed's capacitance resonates with the outlet's inductance. Where the bassoon's bocal fits, L=20mm D=3mm make 3.5kH that resonate at 470Hz, not too bad estimate, and much lower than the cane's flexural resonance.

At a double reed, the embouchure varies the tip opening and the mobile width, but the mobile length little. At a single reed, it varies the mobile length and tip opening but not the width. The equivalent capacitance can diminish much more with a double reed, possibly because the opening can reduce to nothing; one can pull the resonance of a naked double reed much more than of a single reed.

The embouchure influences also the resonance mode of the air column - the register. By the reed's conductance or the susceptance, which are about as big? I'm not quite sure. The reed and mouthpiece must fit an instrument to sound its full range. At a bassoon, the better controlled reed lets play all the range without the lone register key, which wouldn't suffice for the range, is built little efficient and renamed "whisper key". As opposed, a clarinet would be unplayable without its speaker key.

Marc Schaefer, aka Enthalpy