A cone is not a cylinder

A stopped cylinder has only odd partials, and the consequence for the clarinet is a twelfth between its registers and a fingering system unlike any other woodwind’s. That argument was careful to be about the tube’s ends rather than about its reed, and here is the test it has to pass. A saxophone has a reed at the small end and is closed there in exactly the same sense. It overblows at the octave.

Two objects with the same boundary conditions in the crude sense — one closed end, one open end — and different mode sets. Whatever decides this, it is not a list of which ends are shut.

What is actually different #

In a cylinder the wave is plane: the pressure is the same across any cross-section, and its variation along the tube is a sine. In a cone it cannot be, because the cross-section grows. A wave spreading out from near the apex is spherical, and a spherical wave’s amplitude falls as one over the distance travelled.

Write $r$ for the distance from the apex. The pressure in a cone is

$$p(r) = A\,\frac{\sin(kr)}{r}$$

with $k = 2\pi f/c$ as usual. That is the plane-wave sine divided by $r$, and the division is the entire difference between this essay and the last one.

Now impose the ends. At the open end, $r = L$, the pressure must vanish, so $\sin(kL) = 0$ and $kL = n\pi$ for whole $n$ — which gives frequencies $nc/2L$, every whole multiple of a fundamental. That is an open tube’s series, and it is the answer.

The apex has imposed nothing. As $r \to 0$, $\sin(kr)/r \to k$, which is finite and non-zero for every $k$: the pressure at the apex is a maximum, exactly as at a closed end, and it is a maximum whatever the frequency. A closed cylinder end selects modes because it demands an antinode at a fixed place and only some wavelengths oblige. A cone’s apex has an antinode automatically and therefore selects nothing.

That is the whole result. A conical instrument closed at its apex behaves as though it were open at both ends, and it does so because the closed end has stopped being a constraint rather than because it has stopped being closed.

The consequences, which are the mirror image of the clarinet’s #

Everything the previous essay derived from the odd-partial series reverses.

The register gap is an octave, so the fingering repeats. A saxophone’s fingering in the second register is the first register’s plus an octave key, which is why a saxophone is famously quick to learn relative to a clarinet, and why saxophonists moving to clarinet find the break the hardest thing about it.

The tone is not hollow, because nothing is missing. A conical reed instrument’s spectrum has all its partials and its timbre problems are of a different kind — an oboe’s spectrum is shaped by its narrow bore and its tone-hole lattice rather than by an absence.

The tube is twice as long for the same note. A saxophone in B flat is about the same sounding length as a B-flat clarinet an octave higher, which is why a tenor saxophone is a large object and a clarinet in the same key is not.

The waveform makes the same point in the time domain, and makes it more bluntly.

Two cycles of each spectrum, drawn as pressure, would show the odd-partial wave as nearly square — a square wave is exactly odd partials at one over n — and the complete-series wave as nearly a sawtooth. Those are the two classic textbook shapes, and this is where they come from: not from a choice of waveform but from which end of the tube is closed.

The reed is not the variable, and two instruments prove it #

The cleanest evidence that the reed is irrelevant to this question is that the two kinds of reed are distributed across the two kinds of bore without any pattern at all.

A clarinet has a single reed on a cylinder and overblows at the twelfth. A soprano saxophone has a single reed — a very similar one, on a similar mouthpiece — on a cone, and overblows at the octave. An oboe has a double reed on a cone and overblows at the octave. A crumhorn has a double reed on a cylinder and overblows, when it can be made to at all, at the twelfth.

Two reeds, two bores, four combinations, and the register interval tracks the bore in every case. If the reed decided it, the clarinet and the soprano saxophone would agree and the oboe would not; instead the saxophone and the oboe agree, across the largest difference in excitation the woodwind family contains.

This is the kind of test the site tries to build every argument around: not a demonstration that the explanation works, but an arrangement in which a wrong explanation would have been caught.

The apex that is not there #

A real conical instrument does not come to a point. It is truncated — cut off at the small end — and a mouthpiece is fitted where the missing tip would be. This is not a minor liberty, because the missing tip is exactly the part of the cone where the interesting behaviour lives.

The standard solution is that the mouthpiece must have the same volume as the missing cone tip. Do that and the instrument behaves as a complete cone to a good approximation; get it wrong and the registers stop lining up, in a direction that depends on which way the error goes. This is why a saxophone mouthpiece cannot be replaced by an arbitrary chamber of the right opening, why the volume of the chamber is one of the things a mouthpiece is specified by, and why pushing a mouthpiece further onto the cork changes not just the overall pitch but the relationship between registers.

It is also the clearest instance in this field of a design constraint that is not about the note being played. Everything else in these essays fixes a frequency; this one fixes an octave relationship, and a player who tunes by ear at one note can be left with the second register a measurable amount out.

The arithmetic is worth doing once, because the numbers are less forgiving than the rule sounds. An alto saxophone’s bore is about 14 mm across where the mouthpiece meets it and its taper is a little over one and a half degrees of half-angle, so the missing tip runs roughly a quarter of a metre back from the joint — a substantial fraction of the instrument’s own length — and its volume is around eleven cubic centimetres. That is the volume a mouthpiece chamber has to supply, in a component a few centimetres long, and it is why a saxophone mouthpiece has the shape it has rather than being a small clarinet mouthpiece.

Move the mouthpiece 3 mm along the cork and the sounding length changes by about half a per cent, which is eight cents and is what a player is listening for; but the effective volume of the truncation changes as well, and the two registers do not move by the same amount. That is the mechanism behind the standard advice to tune a saxophone on a note in the middle of the second register rather than at the bottom, and it is a consequence of the missing tip rather than of anything a beginner is doing wrong.

“Get it wrong and the registers stop lining up” deserves a rate, and the rate is available from the same geometry. Treat the chamber as a lumped compliance closing the truncated cone — which is what a cavity small against a wavelength is — and the resonance condition becomes

$$k\cot(k\ell) = -\frac{1}{x_0} + \frac{k^2 V}{S_0},$$

with $\ell$ the physical tube, $x_0$ the missing tip and $S_0$ the bore area at the joint. Solve it for the first two modes as $V$ is varied and the octave moves at 0.66 cents for every one per cent of chamber volume, with a larger chamber giving a narrower octave.

Put that in the units a maker works in. The missing tip of the alto is 11 cubic centimetres, so one cubic centimetre of mouthpiece chamber is worth six cents of the register octave — which is why a chamber volume is a specification rather than a consequence of the outside shape, and why two mouthpieces that play the same note can put the second register in different places.

One honesty about that number. A lumped cavity is exact only as the tip becomes short against a wavelength, and here $kx_0$ is 0.63 rather than small, so the model carries a systematic offset — it puts the perfectly matched octave at about 1.2 times the geometric tip volume rather than at 1.0. The slope is what survives that and is what is quoted; the intercept is the approximation’s, not the instrument’s.

What the same argument does to a flute #

A flute is a cylinder open at both ends, so by the arithmetic of the previous essay it should have the complete series and octaves that line up exactly. It does not, and the correction is instructive because it is a deliberate departure from a shape rather than a compensation for one.

The problem is the end correction at the embouchure, plus the fact that the player’s lips cover part of the hole and change how much. Both effects are lengths added to a tube whose sounding length is shortening as the player goes up, so the octaves come out progressively wrong in a predictable direction — precisely the mechanism the next essay is about.

Theobald Boehm’s answer, in the 1840s, was to make the head joint not cylindrical: it tapers slightly toward the cork, a parabolic contraction over the first fifteen centimetres or so. That is a small conicity introduced into an open tube for the express purpose of moving the upper modes relative to the lower ones. It works, it is still how flutes are made, and it is an instance of the general fact that a bore’s shape is a set of controls over individual modes rather than one control over pitch.

Why an oboe and a recorder are not the same argument #

Two instruments complicate the picture in useful ways.

An oboe is conical and has a reed, so it belongs here — but its cone is very narrow, its truncation is proportionally large, and its tone-hole lattice does a great deal of the work of shaping the sound. The complete series is the reason its registers are an octave apart; almost nothing else about an oboe’s sound follows from it.

A recorder is a cylinder, and it overblows at the octave — though only just, and its second register is notoriously unstable in a way that a flute’s is not, because the end corrections at a window and a foot are large fractions of a short tube. That looks like a counterexample and is not: a recorder is open at both ends. The window in its head is a large opening to the room and the labium is a jet edge, not a valve, so both ends are pressure nodes and the complete series follows for the ordinary reason. It is the same conclusion by the other route, and the two routes are the two ways of not having a stopped cylinder.

The instruments that do have a stopped cylinder are few, and the previous essay listed them.

The cone is why a family can be built at all #

There is a further consequence, and it is about instrument families rather than about single instruments.

An open or conical bore has modes at 1, 2, 3, 4. Those are the ratios a harmonic series has, which means the modes of the tube and the partials of the sound it makes are the same set of numbers. A reed driving such a tube finds every mode reinforcing the same fundamental, so the regime of oscillation is stable and strongly established, and the note “locks in”.

A stopped cylinder’s modes are at 1, 3, 5, 7 — also a subset of a harmonic series, so the same cooperation happens, which is why a clarinet works at all. But the even partials of its sound have no mode to sit on. They are produced by the reed’s nonlinearity and radiated weakly, which is why they are present in a real clarinet’s spectrum at low level rather than absent as the idealisation says.

An ideal membrane makes the contrast starker still: its modes are the zeros of Bessel functions, at 1, 1.59, 2.14 and 2.30 times the lowest, so they land nowhere near the whole-number grid a string’s partials sit on. A cone and a cylinder disagree about which whole numbers; a drum head is not on the grid at all.

The instruments where this cooperation fails are the interesting ones. No set of a membrane’s modes reinforces a common fundamental, so the object rings rather than sounds a note, and the pitch a timpanist tunes has to be manufactured by changing the modes themselves. Whether a resonator’s modes are harmonically related is therefore not a detail of tone — it decides whether the object is a musical instrument in the ordinary sense.

What the picture cannot show #

A real bore is neither exactly conical nor exactly cylindrical. A saxophone’s taper changes along its length, and a modern flute’s head joint is deliberately not cylindrical — it is parabolic, a Boehm invention specifically to correct octave relationships that a pure cylinder gets wrong. The figures draw the two idealisations, and the whole craft of bore design lives in the difference between them and any real instrument.

The algebra above assumed the wave is spherical about the apex. In a wide cone it is nearly so; in a rapidly flaring bell it is not, and the mode positions depart from the simple result. This is why a brass instrument’s bell is a hard problem and a saxophone’s cone is an easy one.

The mode ladder is not the note. These figures draw the frequencies the tube prefers, and a real instrument’s spectrum is what the reed produces once it has locked onto one of them — which is a nonlinear problem with its own literature, and which is why an instrument’s second register does not simply sound like its first an octave up. What comes out is also shaped by the envelope of the attack, and a wind instrument’s attack is a substantial fraction of what identifies it.

Nothing here explains loudness. A cone radiates more efficiently than a cylinder of the same length because its open end is larger, which is most of why a saxophone is louder than a clarinet. That is a claim about how a source of a given size couples to the room, and it is a separate mechanism from the one above.

Whose instruments, and when #

The result is old and its attribution is awkward. Daniel Bernoulli had the conical case in the 1760s; the complete modern treatment of truncated cones belongs to the twentieth century, largely to Arthur Benade. The instrument-making tradition arrived at the answers empirically first: Adolphe Sax patented the saxophone in 1846 with a conical bore and a single reed, and the choice of a cone rather than a cylinder is the whole reason the instrument exists as a distinct thing rather than as a loud clarinet.

The claim about fingering systems is a claim about the European woodwind family as it settled in the nineteenth century. It is not a claim about conical instruments everywhere — a shawm, a zurna and a suona are conical reed instruments played with very different techniques, and in several of those traditions the second register is reached by a change of lip pressure rather than by a vent.

Where this goes #

The mode ladders in both essays were computed from a length, and no tube ends where it ends: the wave carries on into the room before it turns round, by about six-tenths of the bore radius. That is a fixed number of millimetres against a wavelength that shrinks as the note rises, so it is a rounding error at the bottom of an instrument’s range and most of a semitone at the top — and it is the reason tuning slides exist.

Further up the same ladder is the question this essay’s cone raises and does not answer: the modes were derived for a tube with no holes in it. An instrument has holes, and above a certain frequency they stop being holes — the wave runs straight past them and leaves from the bell, which is what gives a family of instruments one voice across its whole range.

And there is a connection running sideways into the perception field that is worth stating, because the arithmetic is identical. The ear’s own canal is a stopped cylinder about 25 mm long, and its quarter-wave resonance near 3 kHz is a large part of why hearing is most sensitive there. The ear is a clarinet with one note.