The tube ends after it ends

Both of the previous two essays computed a tube’s modes by dividing the speed of sound by its length. That is the right calculation and the length in it is wrong, by an amount that is small, fixed, and — because it is fixed — increasingly serious the higher the instrument plays.

Fifty-two cents is half a semitone. Nothing in the instrument has changed except which hole is open.

What is actually happening at the opening #

At an open end the air inside the tube is continuous with the air in the room, and the standard boundary condition says the pressure there equals atmospheric — a pressure node. That is not quite true, and the reason it is not quite true is that the air just outside the opening is still being pushed around.

A column of air oscillating in a tube has momentum. When it reaches the opening it does not stop; it carries on out into the room, spreading as it goes, and the reflection happens a short distance beyond the physical end. The node is therefore outside the tube, and the tube behaves as though it were longer.

How much longer is a genuinely hard problem in radiation theory, and it was solved for the unflanged case by Harold Levine and Julian Schwinger in 1948. The answer is

$$\Delta L = 0.6133\,a$$

with $a$ the bore radius. A flanged opening — one that emerges into a flat plate rather than into free space — gives about $0.82\,a$. Both are proportional to the radius and neither depends on the frequency, which is the fact this essay is about.

Why “fixed” is the dangerous word #

An instrument plays a range of notes by changing its sounding length. The correction does not change with it.

At the bottom of a register the sounding length might be 60 cm and the correction 4.6 mm — under one per cent, worth 13 cents, and a maker simply builds the tube a little short to absorb it. At the top of the same register the sounding length is 15 cm and the same 4.6 mm is three per cent, worth 52 cents. A single compensation cannot serve both, because it is a fixed length against a length that has shrunk by a factor of four.

This is the general shape of every intonation problem in wind instruments and it is worth stating as a rule: an error that is a fixed length is an error that grows as the notes rise. Everything in the rest of this essay is a response to it.

A wide bore is a loud bore and an out-of-tune one #

The correction is proportional to the radius, so a wide instrument has a larger one everywhere and a larger spread across its range. The handle on the first figure makes this concrete: at 2 mm radius the whole range is within a few cents, and at 17 mm the top of the range is a quarter-tone out before anything else has gone wrong.

A wide bore is desirable for other reasons. It radiates more efficiently, it has a lower cutoff for the tone-hole lattice — which is what decides the family’s voice — and it takes more air, which means more power. So the bore diameter is a straight trade between loudness and intonation, and different traditions have settled it differently: a classical clarinet has a narrower bore than a modern German one, and the tonal and intonational differences between the two schools of instrument are largely this parameter.

How big is it against the things this site already measures #

Thirteen cents at the bottom of a register and fifty-two at the top are numbers that mean nothing without something to hold them against, and this site has spent four phases assembling the comparisons.

Against the ear. The smallest frequency difference a listener can reliably hear is about five cents in the middle of the range. Both ends of the end-correction error clear that comfortably, so this is not a subtlety detectable only by instruments — it is audible everywhere in an instrument’s range, and the top of the range is audibly wrong by a wide margin.

Against the commas. The Pythagorean comma is 23.5 cents and the syntonic 21.5. The whole apparatus of historical temperament exists to distribute quantities of that size. The end correction at the top of a two-octave register is larger than either, on a single note, from a mechanism that has nothing to do with tuning systems at all.

Against a category. A major third can be about seventeen cents away from just and remain the same chord. Fifty-two cents is well outside that: a note that far out is not a differently tuned version of the intended note, it is on its way to being the neighbouring one.

What the responses are #

Four devices in the woodwind and brass shop are all responses to this one fact.

A tuning slide is the crude one, and it is worth being exact about what it is crude at, because the obvious account of it is backwards. Moving a joint changes every sounding length by the same number of millimetres, and the end correction is the same number of millimetres at every note — so the slide has precisely the right shape, and shortening the tube by 4.60 mm removes the entire error at every length in the range:

That is not a coincidence and it is this essay’s own closing rule read in the direction it is usually not: a correction that is a length against a thing that is also a length is a constant fraction and can be absorbed once. The end correction is the absorbable case, which is why makers absorb it and why no player thinks about it.

What a slide cannot absorb is an error stated in cents, and that is where the familiar complaint comes from. Pull out far enough to flatten the note at 30 cm by ten cents — 1.74 mm — and the same 1.74 mm delivers 5.0 cents at 60 cm and 19.9 at 15. The residual runs from five cents sharp at the bottom to ten flat at the top, across two octaves, from a slide that is exactly right in the middle. So pulling out to flatten a sharp top note leaves the bottom of the instrument flat is a true observation about a real problem, and the problem is a change of pitch standard, or a hot hall, or a player’s embouchure — anything whose error is a ratio. It is not the end correction, which the slide had already dealt with.

The residual the remaining three devices exist for is the one neither the slide nor any single number can reach: the correction is not one length, because the openings are not one size. It is proportional to the radius of whatever is open, and a tone hole is very much smaller than the bore’s end. Taking the same low-frequency formula as an order-of-magnitude guide — which is all it is for a hole, and the limitations below say why — a 2 mm hole carries about 1.2 mm of correction against the bore end’s 4.6, and the 3.4 mm difference is worth 13 cents at a 45 cm sounding length and 38 at 15 cm.

That error is a different length at every hole, so no single adjustment anywhere can take it out, and every note carries its own share of it. Which is exactly the description of the tool that addresses it: undercutting is applied hole by hole, by hand, with a reamer, and it is the reason two instruments from one bench are not the same instrument.

Undercutting a tone hole — reaming the inside of the hole into a bevel — changes the effective end correction at that hole, note by note. This is the fine adjustment, it is applied by hand, and it is the reason two instruments from the same maker are not identical.

A register vent is placed with the correction included in the arithmetic, which is one of the several reasons it cannot be right for every fingering at once.

A parabolic head joint, on a flute, is a deliberate departure from a cylinder introduced by Theobald Boehm in the 1840s specifically to move the upper modes relative to the lower ones. It is the same problem solved by changing the bore’s shape rather than its length.

It also re-reads the fifty-two cents this essay opens with. That number is what an uncompensated tube would be flat by at the top of its range, and no instrument is uncompensated: the maker took the millimetres off before the tube left the bench, and the figure is a picture of a design decision rather than of a defect anybody hears. What survives compensation is the hole-to-hole spread, which is smaller — tens of cents rather than fifty — and is the part that is genuinely hard, because it is not one number.

The four responses have one thing in common that is worth naming: none of them removes the error. A slide moves it, undercutting redistributes it, a vent absorbs part of it and a tapered head joint trades one distribution for another. What a maker is doing is choosing where the residual sits — which is the same activity as choosing where to hide the comma in a tuning system, arrived at from a completely different direction, and with the same conclusion that the choice cannot be avoided.

The correction is also a radiation efficiency #

There is a second consequence of the same physics and it runs in the opposite direction, which is why bore design is a trade rather than an optimisation.

The reason the wave does not turn round cleanly at the opening is that some of it escapes into the room. That escape is the instrument’s sound: an opening that reflected perfectly would be silent. So the end correction and the radiated power are two faces of one process, and the parameter that controls both is the size of the opening relative to the wavelength — the quantity $ka$, which decides how directional a source is as well.

The quantity controlling both is the size of the opening relative to the wavelength — $ka$, which decides how directional a source is as well. Below $ka = 1$ an opening fills the room and reflects nearly everything back down the tube; above it, it beams and it radiates.

The same arithmetic, in a place nobody expects it #

The pattern — a fixed quantity against a shrinking one — is not confined to wind instruments, and two of its other appearances are already on this site under other names.

A piano’s stiffness. A real string is not perfectly flexible, and its stiffness adds a term that pushes each partial sharp by an amount growing as the square of the partial number. That is not a fixed length, but it has the same structure: a small constant of the object multiplying a quantity that grows with the note, so a defect invisible in the bass is the whole reason a piano’s octaves are stretched at the treble end.

A room’s modes. A rectangular room’s lowest modes are separated by intervals large enough to be individually audible, and its high ones are packed too densely to distinguish. The dividing line is the Schroeder frequency, and it exists because the mode count in a band grows as the cube of frequency while the band’s width does not — again a fixed quantity against a growing one.

Naming the pattern is worth more than any of the three instances. When a correction is a length and the thing being corrected is also a length, the correction is a constant fraction and can be absorbed once. When the correction is a length and the thing being corrected is a frequency, it cannot. Almost every intonation problem in instrument design is of the second kind.

What the picture cannot show #

The correction is not exactly frequency-independent. It is very nearly so below the cutoff of the opening, and it falls off at high frequencies where the opening starts to radiate efficiently — which is precisely where the $ka$ argument above takes over. The figures use the low-frequency limit, which is the right approximation across an instrument’s playing range and is not the whole function.

A tone hole is not a simple open end. The correction at an open hole partway along a tube depends on the hole’s diameter, its height, the bore’s diameter and what is downstream of it — the whole lattice. The single-number correction here describes the end of a tube. The hole case is genuinely harder and is what makes woodwind design a craft.

The tone-hole numbers above are illustrative and the paragraph after them says why. They apply the unflanged end formula to an opening it was not derived for, so they show that the correction varies with the size of the opening and roughly by how much; they are not a computation of a tone hole’s correction, which depends on the hole’s height, the bore, and the lattice below it.

The single number hides a shape. Levine and Schwinger’s 0.6133 is the limit for an unflanged pipe at low frequency, and real openings are neither perfectly unflanged nor perfectly flanged. A recorder’s window, a flute’s embouchure hole with a lip partly across it, and a clarinet’s bell are three geometries whose effective corrections are different from each other and from both textbook values, and none of them is a number that can be looked up. What is dependable is the proportionality to the radius and the independence of frequency, which is what the argument here rests on; the constant in front is a measurement per opening.

Nothing here says which note a player will actually produce. A wind instrument’s note is set by the reed or the jet interacting with the resonator, and a player has substantial control over it — a clarinettist can bend a note by a semitone with embouchure alone. The corrections above describe the resonator. A player is a compensating system sitting on top of it, and the reason wind intonation is a skill rather than a specification is that everything in this essay has to be corrected in real time by ear.

Whose instruments, and when #

The Levine and Schwinger result is 1948 and is the rigorous version; Rayleigh had the flanged case in the 1870s and the empirical figure of “about six-tenths of the radius” was in use by makers long before either. The interesting historical point is that the instruments came first: Boehm’s flute, with its parabolic head joint and its hole positions computed rather than copied, was designed in the 1830s and 40s on the basis of measurement and reasoning that predates the theory by a century.

The claim about tuning slides is a claim about Western orchestral wind instruments as they have been built since the nineteenth century. It is not universal — a shakuhachi has no slide and no keys and is tuned by the player’s head position and finger shading, which is the same compensation performed continuously rather than built in.

Where this goes #

The three essays so far have treated the tube as a tube with an end. An instrument is a tube with a row of holes in it, and the holes are not simply ways of making it shorter: above a certain frequency the wave stops noticing them and runs the length of the instrument, which is what gives a whole family one voice.

Two rungs further on, the same reasoning about a fixed error against a shrinking length reappears in a completely different instrument. A guitar’s saddle compensation is a fixed length added at the bridge, and it cannot be right at every fret for exactly the reason a tuning slide cannot be right at every note.