A tube that skips every other partial

A flute and a clarinet are both tubes about two-thirds of a metre long with holes in them. The flute’s lowest note is middle C. The clarinet’s is the D more than an octave below it, and if a player overblows a flute the note goes up an octave while the clarinet’s goes up an octave and a fifth. Neither difference is about wood, or reeds, or the player. Both follow from which end of the tube is closed.

The interval refusing to move under the slider is the whole argument in one gesture. A twelfth is not a fact about 60 cm of tube. It is a fact about having a closed end.

Why a closed end forces the odd numbers #

A standing wave in a tube is a pattern of pressure that repeats. The tube’s ends impose conditions on it and only certain wavelengths satisfy them.

At an open end the air is continuous with the room, so the pressure there cannot depart from atmospheric: an open end is a pressure node. At a closed end the air has nowhere to go, so it is compressed and rarefied to the maximum the wave can manage: a closed end is a pressure antinode.

An open tube of length $L$ therefore supports wavelengths $\lambda = 2L/n$ for every whole $n$, giving frequencies $nc/2L$. A stopped tube supports $\lambda = 4L/(2k-1)$, giving frequencies $(2k-1)c/4L$: the odd multiples of a fundamental that is itself half the open tube’s. The speed of sound $c$ is about 343 m/s at room temperature and appears in both, which is why every wind instrument goes sharp as it warms up and why the arithmetic here is one division rather than a model.

Nothing in that derivation mentions what the tube is made of. A stopped glass tube, a stopped brass tube and a stopped tube of rolled cardboard all have odd partials, and they have them to the same precision. This is the sense in which an instrument’s sound is decided by where its ends are rather than by its material — the material decides how much is lost per reflection, which is a matter of how loud and how long, not of which frequencies.

The first consequence: a hollow sound #

A tone made of odd partials only is missing every even one, and the missing partials are exactly the octaves and the octave-plus-fifths. The second partial of a complete series is the octave; the fourth is two octaves; the sixth is two octaves and a fifth. Remove them and what remains is the fundamental, the twelfth, the seventeenth, the twenty-first.

The complete series for comparison is the partials of an ideal string, at exactly one, two, three and four times the fundamental — which is what the open cylinder above already draws. The clarinet’s spectrum is that series with the second, fourth and sixth removed, and the removal is done by the geometry rather than by anything about the reed.

This is why a clarinet’s low register sounds hollow rather than merely quiet in the upper partials. The perceptual consequence is worth being precise about: a note’s partials are what its timbre is, and a spectrum whose members are 1, 3, 5, 7 has a different pattern of roughness against a second note than one whose members are 1, 2, 3, 4.

The roughness curve of a stopped-pipe spectrum wells on the fifth and not on the fourth, which is a measurable difference in what intervals a clarinet blends with, and the site’s own gate asserts it across the whole family of spectra rather than at one setting. It is the clearest available demonstration that consonance is a property of a pair of spectra and not of a pair of ratios: change which partials a note has and the intervals it likes change with it, without a single frequency ratio being touched.

The word “hollow” is doing no work here and this essay is not going to lean on it. What is true is that four of the eight strongest partials are absent, and the four that remain are 3, 5, 7 and 9 times the fundamental rather than 2, 3, 4 and 5.

Three numbers put a size on that, and the first two run against the obvious reading. Under a falling envelope, deleting the even partials removes half the partials and a quarter of the energy, because most of the energy is in the fundamental and the fundamental is odd. And it moves the spectral centroid down, to 0.79 of where it was — an odd-partial spectrum is darker than a complete one at the same fundamental, not brighter, since the partials removed are on average lower than the ones kept.

The third is where the interesting correction is. Computing the roughness wells for three spectra rather than asserting them:

A spectrum of purely odd partials has no well at the fifth at all. The claim that a stopped pipe wells on the fifth and not on the fourth is true of a real clarinet and false of the idealisation this essay has been drawing, and the difference between the two is the small even partials a real instrument has — a second partial at four per cent of the fundamental and a fourth at three. Those are the partials the geometry says should not exist, and they are what buys the fifth its well.

Even with them the well is a shadow of the string’s: four per cent prominence against thirty-seven, a factor of nine, and the clarinet’s deepest well is the major sixth rather than the fifth. So the honest form of the consonance claim is not that a clarinet prefers different intervals but that it has much weaker preferences of any kind — its whole roughness curve is flatter, because a sparse spectrum has fewer coincidences to find, and the intervals it does prefer it prefers faintly.

That is a better fact than the one it replaces, and it points at the same place: what a note blends with is a property of which partials it has, and a tube that skips half of them has given up most of the machinery that makes an interval smooth.

Why the reed end counts as closed #

The derivation above assumed a closed end at the mouthpiece, and there is plainly a hole there — the player is blowing through it. The justification is that “closed” in this argument means high acoustic impedance: an end that resists air flowing in and out, so that pressure can build. A reed sitting against its lay is exactly that. It opens for a fraction of each cycle and is otherwise nearly shut, and the aperture even when open is a small fraction of the bore’s cross-section.

The same reasoning covers a brass player’s lips and, in the other direction, explains why a flute is open at both ends despite having a player attached to one of them: the flute’s embouchure hole is a large opening to the room, air passes freely across it, and the pressure there is atmospheric. The player is a source of a jet, not a stopper.

This is not a detail. It is why the argument is about geometry and termination rather than about what is making the noise, and it is what makes the next essay’s result possible: a reed at the small end of a cone gives the complete series, and a reed at the end of a cylinder gives half of it.

The second consequence: an octave of free length #

A stopped tube sounds an octave lower than an open tube of the same length. That is a substantial saving in an instrument that has to be carried, and it is the reason the clarinet reaches down to a written E below the treble staff from a tube that a flute would use for a middle-register instrument.

It is also the reason a stopped organ pipe is half the length of an open one at the same pitch, which is not a small matter when the pipe would otherwise be thirty-two feet. Organ builders have known this for six hundred years and have a name for it — a stopped rank is gedackt, covered — and the tonal consequence is the one above rather than a separate one.

The third consequence, and it is the interesting one #

An instrument’s fingering system exists to fill in the notes between its modes. The lowest note is the fundamental of the whole tube; opening holes shortens the tube and raises it; and at some point the player runs out of holes and has to jump to the next mode instead. The number of semitones the holes have to cover is therefore the interval to the next mode.

For a flute, an oboe, a saxophone or a bassoon, that is an octave: twelve semitones. For a clarinet it is a twelfth: nineteen.

That is what the holes on each instrument have to cover before the next mode arrives. An octave — twelve semitones — is the gap on a flute, an oboe, a saxophone and a bassoon, all of which overblow at the second mode. A clarinet’s next mode is a twelfth away, nineteen semitones, so its holes have to fill seven more notes than anybody else’s before the register key does anything at all.

Seven extra notes is not a small imposition. A flute’s left hand and right hand each cover part of one octave and the fingering repeats; a clarinet’s does not repeat, because the register above is not the same pattern transposed. The instrument acquires a set of “throat” notes — the ones at the top of the lowest register, filled in with holes near the mouthpiece — which every clarinettist knows as the weakest notes on the instrument, and which exist because the gap had to be closed somehow.

This is the clearest case on the site of an instrument’s interface being decided by an acoustic fact. Nobody chose a fingering system that does not repeat at the octave. The tube skips its even modes, and the fingering follows.

And the mechanism the gap forced has a cost of its own. Getting from the first register to the second needs a vent — a small hole that kills the fundamental and leaves the third partial — and the place that hole has to be is a moving target, because it must sit where the second mode’s pressure node is and that node slides with every fingering. One hole, a dozen positions, and a leftover error that can be computed.

The number that is not a constant #

Everything above divides the speed of sound by a length, and the speed of sound is not a property of the instrument. At 20 °C it is about 343 metres per second, and it rises with temperature roughly as the square root of absolute temperature — about 0.6 m/s per degree, or 0.17 per cent of itself.

A tube’s frequencies are proportional to it. So a wind instrument warming from a cold room to playing temperature — say ten degrees — goes sharp by about 1.7 per cent, which is nearly thirty cents. That is six times the smallest pitch difference a listener can hear and larger than either of the commas the tuning ladder is built around. It is why a wind player pulls out at the barrel or the head joint after the first few minutes, and why an orchestra retunes.

A string does not do this. A string’s frequency depends on its tension and its mass, and warming a steel string makes it flat rather than sharp, because the metal expands and the tension drops. The two families move in opposite directions with the same change in the room, which is a fact about materials rather than about music and is the reason a concert hall’s temperature is a setting somebody chooses rather than a matter of comfort.

What the missing partials cost the notation #

One further consequence is administrative and is worth a paragraph because it is so often presented as arbitrary. A clarinet in B flat sounds a tone below what is written, and there are also clarinets in A, in E flat and in C. Transposing instruments are usually explained as a convenience for players moving between family members with one fingering, which is true. What is less often said is why the family has so many members in the first place.

An instrument whose registers are a twelfth apart cannot simply be rescaled the way an open-tube family can. The written range, the break between registers and the awkward throat notes all sit at fixed places relative to the tube, so a passage that lies well on one instrument may straddle the break on another built a semitone away. Building several sizes and writing for them as though they were one instrument is the response to a problem that a flute family does not have to the same degree — and notation hides the fact so successfully that the awkwardness reads as a quirk of the instrument rather than as arithmetic.

What the picture cannot show #

Three things, and the third is the one that stops this essay from being a complete account of a clarinet.

A real bore is not a perfect cylinder. A clarinet’s is very nearly one over most of its length, which is why the model works, but it flares into a bell at the bottom and is subtly reamed elsewhere, both of which shift individual modes by a few cents. The figures above draw the ideal object.

The modes are not exactly odd multiples. They are close, and the departures are of the order of a few cents to a few tens of cents — enough that a maker adjusts them and not enough to change which mode is which. What the figure draws is where the boundary conditions put them, not where a measurement finds them.

A tube does not sound on its own. Everything above is a description of the resonator’s modes, which is a statement about which frequencies the tube prefers. What makes a note is a reed opening and closing at one of them, and the reed is a nonlinear valve rather than a passive component: it is what turns a steady stream of breath into an oscillation, and it decides which mode wins and what the resulting waveform looks like. That is a different subject, and the mode ladder is the part of it that is arithmetic.

Whose instruments, and when #

The clarinet is the case because it is the only orchestral woodwind with a cylindrical bore and a reed at one end. That combination is rarer than it sounds: a saxophone has a reed and is conical, a flute has no reed and is open at both ends, and an oboe and a bassoon are conical. Among instruments in common use the clarinet family and the stopped organ pipe are most of the list, along with the panpipe, which is a stopped cylinder with no holes at all and one note per pipe.

The claim about fingering is a claim about the Boehm-system clarinet as it settled in the nineteenth century and about the earlier chalumeau it grew from, both of which had the problem and solved it differently. It is not a claim about all reed instruments everywhere: a Chinese bawu and a Hungarian tárogató are reed instruments with different geometries and different answers.

Where this goes #

Two directions, and the first is the one that most needs the second essay: the cone. A saxophone has a reed at a closed end and overblows at the octave anyway, and if the argument above were about reeds that would be a contradiction. It is not, and the reason is what a cone does to a wave.

The other direction is downward, into the assumption this essay quietly made. A tube’s modes were computed from its length — but a tube does not end where it ends, and the correction is a fixed number of millimetres against a shrinking wavelength, which is a rounding error at the bottom of the range and most of a semitone at the top.

And one connection worth making explicitly, because it runs the other way through the site: a stopped pipe is the model the perception essays use for the ear canal, which is a tube about 25 mm long closed at the eardrum. Its quarter-wave resonance near 3 kHz is part of why the ear is most sensitive there, and it is the same division of a speed by a length that produced the clarinet’s twelfth.