The reed is a valve, not a vibrator

A tube that skips every other partial says the note is decided by where the ends are. A cone is not a cylinder says it again and sharpens it: two instruments with the same kind of reed overblow at different intervals, so the reed is not the variable. The tube ends after it ends refines the length and never mentions the reed at all.

Three rungs of this ladder have established what the reed does not do. None of them has said what it does, and the answer is not a detail of tone. It is the reason there is a note.

The energy has to come from somewhere #

A standing wave in a tube loses energy continuously. Some leaves at the open end, which is what makes the instrument audible; some is lost to viscous and thermal effects at the walls; some leaves through the open tone holes below the first one. Left alone the wave dies in a few tenths of a second, which is what a tube sounds like when it is tapped.

For a sustained note something has to put the same energy back in, once per cycle, in phase. That is a strong requirement — energy delivered out of phase takes energy out — and the mechanism that satisfies it is the whole of what a reed is for.

What the reed actually does with the air #

The reed is a valve on a stream of air, and the geometry produces the crucial behaviour by itself.

Blow, and two things happen at once. The pressure difference across the slot drives air through it, and it also pushes the reed towards the lay, which narrows the slot. Flow through an opening goes as the square root of the pressure across it and linearly with the width of the opening, so the two effects fight.

Everything to the right of that peak is the argument. The flow falls as the pressure rises. A device whose current falls when its voltage rises has a negative resistance, and a negative resistance in a resonant circuit is an oscillator.

The rest is bookkeeping. Over each cycle of the standing wave the pressure at the mouthpiece swings above and below the player’s steady blowing pressure. If the working point is on the falling part of the curve, then the moments of high pressure are the moments of low flow, and the reed is admitting air preferentially when the tube’s pressure is low — which is delivering energy in phase.

Where a note starts, which is a pressure #

The oscillation begins when the energy the reed supplies per cycle exceeds what the tube loses. That is a condition on the slope of the curve rather than on the flow, and it gives a threshold.

Two things about that are audible immediately.

A wind note begins rather than fades in. Below the threshold there is no sound at all — not a quiet sound, none — and above it there is a note at a definite amplitude. That is a very different onset from a struck or plucked string, whose amplitude is whatever the player put into it, and it is why the envelope of a reed note has the shape it has.

And a leakier tube needs a harder blow. That is the practical content of the threshold figure and it is what a player experiences constantly: a note with a register hole open, a badly seated pad or a cracked reed all take more pressure to speak — and one hole is doing a dozen jobs, so a leak is never in one place only. The differences drawn here are between two and four thousand pascals, which is a factor of two in something a player controls directly.

The reed’s own frequency is not in the answer #

The obvious objection to all of this is that a reed is a stiff object with a resonance of its own, and it is: a clarinet reed’s own natural frequency, clamped as it is played, is somewhere around two thousand hertz.

A reed is a stiff object with a resonance of its own — a clarinet reed clamped as it is played sits somewhere around two thousand hertz — and the tube it is fitted to has resonances of its own at about 140, 430 and 710 hertz and upward. A system driven far below its driver’s resonance is one in which the driver’s resonance does not set the frequency. The lowest mode is below the reed by a factor of fourteen, which is far enough for the tube to win outright; the sixth is below it by only 1.27, which is not, and the text below computes where the crossover falls. So the reed’s own frequency is genuinely absent from the answer over most of the instrument and starts to intrude near the top of it, which is exactly where players report that the high register behaves differently.

That relation is the same one a bow has, whose stick–slip cycle runs at the string’s frequency rather than at any of its own, and a hammer has, whose contact time is short compared with the period it excites. The reed follows the tube because it is being asked to move slowly compared with what it can do, and at those rates its response is a stiffness rather than a resonance.

The factor is three or more only at the bottom of the figure, and the figure says so. A sixty-centimetre stopped cylinder has modes at 143, 429, 715, 1,000, 1,286 and 1,572 hertz, so against a reed at two thousand the six ratios are 14.0, 4.7, 2.8, 2.0, 1.6 and 1.27. Two of the six clear three. The sixth is twenty-seven per cent away, which is not far below a resonance by any reading.

Taking it across the instrument rather than across one fingering makes the shape of the problem plain:

At the top of a clarinet’s range the reed’s own resonance is within seven per cent of the note being played. The separation the whole argument rests on is a fact about the chalumeau and the lower clarion, it is gone by the top of the clarion, and in the altissimo the two systems are at the same frequency.

Which recasts the squeak. The section above treats it as an exception reached by pushing the pressure and the embouchure — something a player does by accident and could avoid. On these numbers it is not an exception at all: it is what the top of the range is, and a player in the altissimo is working in the register where the reed and the tube are competing rather than one following the other. The instrument is hardest to control exactly where the model stops being separable, and the difficulty of the altissimo is usually attributed to embouchure and to the weakness of the upper impedance peaks.

So the quasi-static model’s neglected term is neglected everywhere the instrument is easy and nowhere it is hard, which is a poor place for a neglected term to be. What it buys is the threshold arithmetic above, which is exact and is about the low register; what it costs is any account of the register the term was neglected in.

The exception is instructive. Push the pressure and the embouchure until the reed’s own resonance is engaged and the instrument does not play a note of the tube at all — it produces a squeak, at the reed’s frequency, and the tube is irrelevant. Every clarinettist has done this and it is the demonstration that the two systems are separate.

The model drawn above cannot produce a squeak, which is worth saying plainly. It is quasi-static: it assumes the reed’s position follows the pressure instantaneously, with no mass and no damping of its own. That assumption is exactly what makes the threshold arithmetic simple, and it is exactly what removes the reed’s resonance from the model. A squeak is what the neglected term does.

The control case, which has no tube #

If the tube sets the frequency and the reed does not, then a reed with no tube should have no note at all — or, more precisely, should sound at its own frequency instead.

That instrument exists, in quantity, and it is common on every continent. A free reed vibrates in a close-fitting slot with no resonator to speak of: the harmonica, the accordion, the harmonium, the sheng, the khaen, the reed organ.

A free reed’s pitch is the reed’s. Tuning a harmonica means filing the reed — removing metal from the tip to raise it, from the base to lower it — and there is no tube to shorten. One reed makes one note, which is why a chromatic instrument of this kind needs one reed per note and why an accordion is the size it is.

A beating reed with a resonator is the opposite object. One reed serves the whole instrument, the note is chosen by the length of the air column, and the reed is a valve rather than a source. The clarinet’s single reed plays every note the instrument has; the harmonica has a reed for each.

That is as clean a control as an argument of this kind gets. Same class of object, one variable removed, and the frequency changes hands.

The organ has both, side by side and named as such. A reed stop has a beating reed with a resonator tube, and the tube’s length is what the builder adjusts; a free reed stop has no effective resonator, and the reed’s own tuning wire is what is adjusted. Nineteenth-century organ builders installed both and knew the difference in exactly these terms.

The same valve, in a throat #

There is a second control case and this collection has just built it.

And yet the frequency is set differently. In the voice, the tract’s resonances are five hundred hertz and up while the fundamental is one or two hundred, so the resonator is far above the source and cannot control it; the folds run at their own rate and the tract filters the result. In the clarinet, the resonator is far below the reed and does control it.

The same valve, with the resonator on one side of it or the other, gives a source–filter instrument or a tube instrument. That is the cleanest statement this ladder has produced about what a reed is, and neither essay could have made it alone.

What the valve does to the spectrum #

The valve explains the frequency and it also explains the tone, by the same route the voice’s source does: a flow that spends part of each period at zero is a flow with a corner in it, and a corner is a full harmonic series.

The agreement between those two accounts is not a coincidence and it is not two explanations of one fact. The tube decides which partials can exist; the valve decides how strong they are, by how abruptly it shuts. A player blowing harder drives the reed further into the closing regime, the closure sharpens, and the upper partials rise faster than the fundamental — which is why a wind instrument gets brighter as it gets louder and not merely bigger.

That is a testable claim with an audible failure condition: an instrument whose loudness came from a linear amplification would keep its spectral balance across a crescendo, and none of them do.

Inward and outward, which decides the sign #

One more distinction is worth having because it separates the two families of wind instrument and it goes back to Helmholtz.

A reed can close as the pressure behind it rises — a blown-closed or inward-striking reed, which is what the clarinet, oboe and bassoon have — or it can open, which is a blown-open or outward-striking reed, as in some organ reeds and the harmonium.

The sign of that response decides where the instrument plays relative to the reed’s own resonance: a blown-closed reed sounds below its own frequency and a blown-open one above it. The curve drawn at the top of this essay is a blown-closed reed, and the falling branch — the negative resistance — exists because closing is what rising pressure does.

Brass players’ lips are the interesting case, because they are neither cleanly: a pair of lips is a mass with a sideways and an outward degree of freedom, and the effective behaviour changes across the register. That is part of why a brass instrument’s pitch is so much more under the player’s control than a clarinet’s, and it is why the same instrument can be lipped a semitone in either direction.

The pressure profile, which is where the valve has to sit #

One thing decides whether any of this works, and it is a matter of position rather than of physics.

A reed is at the closed end of the tube by necessity, not by convenience: it is the only place where the standing wave has enough pressure swing to modulate it. That is the same fact the clarinet essay used to explain the missing even partials, read for a different purpose — the reed end counts as closed because the reed is there and mostly shut, and it has to be there because that is where the pressure is.

The same reasoning says where a flute’s excitation cannot be. A flute has no valve and no closed end; it is excited by an air jet at an open end, which is a velocity antinode rather than a pressure one, and the whole regenerative mechanism is a different one — a jet whose deflection is driven by the flow rather than by the pressure. That is why a flute has a complete harmonic series and a clarinet does not, and it is why a flute’s threshold behaviour is far less abrupt.

Whose reeds, and when #

The valve description is Helmholtz’s, from Die Lehre von den Tonempfindungen in 1863, where the inward- and outward-striking distinction is set out and the reed is explicitly compared with the vocal folds.

The quantitative version is twentieth-century. Backus measured the pressure–flow characteristic of clarinet mouthpieces in the 1960s; Wilson and Beavers set out the quasi-static theory and its threshold condition in 1974; and the whole apparatus — including the reason a squeak needs the reed’s inertia — is standard in Fletcher and Rossing’s Physics of Musical Instruments.

What is worth noticing about that history is the order of it. The correct qualitative description arrived a century before the numbers, and the wrong one — the reed as a vibrating source that the tube resonates — remained in circulation the whole time, because it is the description that a reed’s appearance suggests and because it gets the tone-colour vocabulary approximately right while getting the mechanism exactly backwards.

What the picture cannot show #

There is no reed inertia in this model and so no squeak, no attack transient and no register instability. Everything computed here assumes the reed’s position follows the pressure with no lag. That is a good approximation for a note being sustained and a bad one for a note being started, which is where most of the interesting behaviour of a real instrument lives.

The conductance figures are stand-ins. The three horizontal lines on the threshold plot are round numbers chosen to bracket the reed’s own slope range, not measurements of particular tubes. What is computed is the shape — that the threshold rises with loss, and that nothing at all sounds below the turnover.

The pressures are for a single reed of stated stiffness. A closing pressure of five thousand pascals is a reasonable value for a clarinet reed of medium strength and the slider covers the range; a double reed, a lip and a free reed all have different numbers and, in the free reed’s case, a different mechanism.

And the flow is quasi-static Bernoulli flow. Real flow through a reed channel separates into a jet, the jet does not reattach, and the losses on the way out are not in the model. The turnover at a third of the closing pressure survives all of that; the absolute flow rates do not.

The ladder from here #

The tube’s length sets the note, and the length is fixed. What is not fixed is the speed of sound inside it, which goes as the square root of absolute temperature — so the next rung finds a wind instrument going sharp by more than a Pythagorean comma within ten minutes of being played, while the strings beside it go flat.