Above a certain note the holes stop working

The first three essays in this field treated an instrument as a tube with two ends. An instrument is a tube with a row of holes in it, and the usual account of what the holes do — each one shortens the tube, so opening a hole raises the note — is true for the fundamental and false for most of the spectrum above it.

The cutoff is a property of the row of holes rather than of any one of them, and it barely moves as the player changes note. Everything below it belongs to the fingered length; everything above it belongs to the instrument as a whole.

Why a hole is not simply a shorter tube #

An open tone hole is a short side-branch to the outside air. Whether it reflects the wave depends on how its acoustic impedance compares with the bore’s, and that comparison depends on frequency.

At low frequencies the air in the hole has little inertia to overcome and the hole is effectively a hole: the pressure there is forced to atmospheric and the wave turns round. As the frequency rises, the mass of air in the hole matters more — the hole behaves increasingly like a small inductance rather than an open door — and less of the wave is reflected. Somewhere the reflection fails and the wave continues down the tube, past the open hole, past the next one, and out of the end.

With a row of holes the transition is much sharper than it would be for one, because a periodic lattice of side branches behaves like a filter. Below its cutoff it stops the wave; above it, it passes it. Arthur Benade worked this out in the 1960s and the cutoff frequency comes to

$$f_c = \frac{c}{2\pi}\,\frac{b}{a}\,\frac{1}{\sqrt{t_e\,s}}$$

with $b$ the hole radius, $a$ the bore radius, $s$ the spacing between holes and $t_e$ the hole’s effective height — its chimney plus the end corrections at each end of it, which is the same correction the previous essay was about applied to a much shorter tube.

Large holes, close together, in a narrow bore with thin walls give a high cutoff. Small holes far apart in a thick-walled instrument give a low one. Those are recognisable descriptions of a flute and a bassoon respectively, and the formula returns about 1,690 Hz and about 510 Hz for them.

What a cutoff does to a family’s sound #

Here is the consequence that makes this worth an essay rather than a footnote. Below the cutoff, each note’s spectrum is shaped by the tube from the mouthpiece to the first open hole — a different length for every note. Above the cutoff, every note’s spectrum is shaped by the whole instrument, which does not change.

So an instrument has two regimes stacked inside every single note. The lower partials belong to the fingering. The upper partials belong to the instrument, and they leave it from the same place, through the same bell, with the same radiation characteristics, whatever is being fingered.

Every woodwind has one and they are an octave and a half apart. Below its cutoff a note is defined by the first open hole; above it the wave runs past the lattice and the tube behaves as though the holes were not there — which is the boundary this essay is about, stated as one number per instrument.

That is a source-filter arrangement, and it is the same argument that identifies a vowel. A vowel is recognisable across a wide range of pitches because its formants are a property of the mouth and not of the note; an oboe is recognisable across its range for the same reason, and the fixed thing is the lattice cutoff plus the bell’s radiation above it.

This is why an instrument has a voice rather than a tone. A synthesiser that models a woodwind by moving a fixed spectrum up and down gets it audibly wrong, and the reason is not subtlety of the model — it is that the real instrument’s upper spectrum does not move with the note at all.

Reading the four families off the picture #

The four cutoffs in the first figure are worth walking through, because each one is a recognisable sound.

A flute has large holes close together in a relatively wide, thin-walled tube, giving a cutoff near 1,690 Hz. That is not especially high, and what matters is what is available above it: a flute’s jet produces little high-frequency energy, so there is not much there to pass the lattice in the first place. The result is a sound with a strong fundamental and not much else, which is why a flute is the closest orchestral instrument to a pure tone and why its waveform is nearly a sine.

An oboe has small holes in a narrow bore, and its cutoff comes out near 3,000 Hz — high, but its reed produces a great deal of energy up there, so what passes the lattice is substantial. The characteristic oboe sound is a strong band of upper partials radiating from the bell.

A clarinet sits near 1,800 Hz, low enough that a large part of its spectrum is in the “instrument” regime while its odd-partial low register is in the “fingering” regime. Two mechanisms, one instrument.

A bassoon has small holes, wide spacing and very thick walls — its tone holes are drilled at an angle through several centimetres of maple — which puts its cutoff around 510 Hz. Almost everything above the second or third partial of a bassoon’s note is in the lattice-transparent regime, which is why a bassoon’s sound is so nearly independent of register and so immediately identifiable.

The four knobs, and what a maker does with them #

The formula has exactly four things in it that a maker controls, and each one is a lever with a known direction.

Hole radius. Bigger holes raise the cutoff, brighten the instrument and make it louder. They also require pads that a hand cannot cover, which is what keywork is for — the Boehm system exists because acoustically correct holes are larger and further apart than fingers can manage.

Bore radius. A wider bore lowers the cutoff, because the cutoff depends on the ratio of hole to bore. So widening a bore for loudness darkens the instrument, and worsens its intonation across the range at the same time. Three consequences from one dimension, pulling in different directions.

Spacing. Holes further apart lower the cutoff. Spacing is not free: the holes must be where the notes are, which is set by the scale and the sounding length, so this is the least adjustable of the four.

Wall thickness. A thicker wall means a longer chimney, a larger effective height and a lower cutoff. This is why undercutting a hole — reaming the chimney out into a bevel — brightens an instrument as well as adjusting its pitch, and why a bassoon with tone holes drilled at an angle through several centimetres of maple is at the dark end of the family by construction rather than by choice.

The baroque-against-modern comparison is the one worth dwelling on. Two instruments of the same length, playing the same notes, with the same fingers on them; the difference in their voices is substantially a difference in one computed frequency, and that frequency follows from hole size and wall thickness rather than from anything a listener would name.

What the cutoff does to consonance #

There is a consequence for harmony that is easy to miss and follows directly from the site’s earlier work.

Roughness between two notes is computed from the interaction of their partials, and the wells in the roughness curve — the intervals that come out smooth — depend on which partials each note has. A lattice cutoff decides which partials survive to be radiated, so it decides where the wells are.

An odd-partial spectrum’s roughness minima are not the same set as a full harmonic spectrum’s, which matters here because above the cutoff a cylindrical instrument’s radiated spectrum changes shape. The note is not merely differently tuned above the boundary; it is differently constituted, and any interval it makes is being scored against a different curve.

Hole size is the single lever a maker has over both quantities at once: wide holes raise the cutoff and kill the cross-fingerings, narrow ones lower it and make them strong. A baroque flute and a Boehm flute are the two ends of that trade, and neither is a mistake.

This is not a licence for a general claim that instruments have their own consonances — the effect is small against the effect of the ratios themselves, and no orchestrator writes by it. What is true, and is worth stating carefully, is that consonance is a property of a pair of spectra rather than of a pair of numbers, and an instrument’s construction sets its spectrum. The number in this essay is one of the places where a piece of woodwork reaches into a question that looks purely arithmetic.

The cutoff is also why the register break sounds different #

An instrument’s cutoff is fixed, and its notes move. At the bottom of the range a note’s fundamental is far below the cutoff and it has many partials in the fingered regime; at the top of the range the fundamental itself may be approaching the cutoff, and the note has almost nothing below it.

This is a second mechanism, distinct from the end correction’s growing error, by which the top of a wind instrument’s range is a different object from the bottom. The intonation changes for one reason and the timbre changes for another, and both are consequences of a fixed quantity against a rising note.

It also puts a hard ceiling on the useful range. Once the fundamental is above the cutoff, the tube stops behaving as a tube of the fingered length at all: there is no reflection to define a mode, and the instrument stops speaking. Every woodwind’s top note is somewhere near that limit, and where it is depends on the same four numbers as the cutoff.

Working one out, with the numbers #

The formula is worth evaluating once by hand, because it makes clear how little of it is adjustable.

Take a clarinet. The bore is about 15 mm across, so $a = 7.5$ mm. A tone hole is around 9 mm across, so $b = 4.5$ mm. The wall is about 4 mm thick, and the effective height adds roughly one and a half hole radii of end correction to that, giving $t_e \approx 10.8$ mm. The holes in the working part of the lattice sit about 30 mm apart.

Then $b/a = 0.6$, $\sqrt{t_e s} = \sqrt{0.0108 \times 0.030} = 0.018$ m, and

$$f_c = \frac{343}{2\pi} \times 0.6 \times \frac{1}{0.018} \approx 1{,}820\ \text{Hz}.$$

That is a little above A6, and the whole of a clarinet’s written range lies below it — so every fundamental the instrument plays is in the fingered regime and everything from about the fourth partial upward, on a low note, is in the instrument regime.

Now change one thing. Reaming the holes out to 11 mm across takes $b/a$ to 0.73 and the cutoff to about 2,200 Hz, a difference of nearly three semitones in where the instrument’s own voice begins. Nothing else about the instrument has moved: same length, same holes in the same places, same notes. That is the sensitivity a maker is working inside, and it is why a hole is reamed a fraction of a millimetre at a time.

What the picture cannot show #

The formula is a lumped approximation of a periodic structure. It treats the lattice as uniform, and a real instrument’s holes are neither equally sized nor equally spaced — they are placed to put notes where they belong, which is a different constraint. The cutoff of a real instrument is therefore a band rather than a frequency, and the number the figure prints is the middle of it.

Which holes are open changes the lattice, and the section on what a finite lattice does prices it. The corner really does move far less than the note; what the caveat that used to sit here missed is that the corner is not the quantity fingering changes.

Nothing here is a claim about loudness or projection. How much of what passes the lattice actually reaches a listener is a radiation question — how big the opening is against the wavelength — and the answer depends on where the listener is standing, not only on what leaves the instrument.

And the cutoff says nothing about the reed. What is available to be filtered comes from the excitation, and an oboe reed and a flute’s jet produce very different amounts of high-frequency energy. The lattice decides what survives; something else decides what there was.

What a lattice of two holes does #

Benade’s formula is for an infinite periodic lattice, and no fingering has one. A note near the bottom of the range has almost every hole closed and a lattice two or three holes long; a note near the top has ten or twelve open. Cascading the real thing — each open hole a shunt inertance, each span between holes a lossy cylinder, terminated in the bore’s own impedance so what is measured is the lattice and not a standing wave behind it — says how much that matters, and the answer has two halves that point opposite ways.

The corner is genuinely stable. From two open holes to twelve it sits between 1,805 and 1,955 Hz — a spread of about a semitone and a half, against a playing range of three and a half octaves. So the essay’s central claim survives its own worst case: the frequency that divides the two regimes belongs to the instrument and not to the fingering, and one number for a family is fair.

The wall behind the corner is not stable at all. The same two-hole lattice attenuates by seven and a half decibels where the twelve-hole lattice attenuates by seventy-two, and seven and a half decibels is not a wall — it is a slight preference. A low note, with everything open below the first closed hole, has a genuine two-regime spectrum. A high note has a corner in the same place with almost nothing behind it, so its upper partials are neither reflected nor cleanly passed.

That is the correction the two-regime picture needs, and it is not a small one: the regimes are sharp at the bottom of the range and blurred at the top, which is a second reason, distinct from the two already given, why the top of a woodwind’s range is a different object.

It also settles the ceiling paragraph, which as written cannot be the mechanism. Once a fundamental approaches the cutoff the instrument is being fingered with nearly everything closed, so there is scarcely a lattice for that fundamental to pass — at one open hole the clarinet’s lattice attenuates 2.3 decibels and blocks nothing worth the name. The top note is not where the lattice stops reflecting; it is where the tube, the reed and the register mechanism run out between them, and the cutoff is a bystander.

The bassoon is the instrument this changes most. Even fully open its lattice reaches only 36 decibels against the clarinet’s 73, because its holes are small against its bore and each side branch reflects weakly. So a bassoon does not have two regimes with a wall between them; it has a long soft transition, which is a better description of the sound than the binary account and is what its four numbers say when they are asked the question rather than reduced to one.

Whose instruments, and when #

The theory is Arthur Benade’s, developed through the 1960s and set out in Fundamentals of Musical Acoustics (1976), and it is one of the few results in this field that changed how instruments are designed rather than only how they are explained. Benade’s own instruments and his collaborations with makers used the cutoff explicitly as a design parameter.

The claim about family voices is a claim about Western woodwinds with keyed lattices — the Boehm flute and clarinet, the conservatoire oboe, the Heckel bassoon — as they settled between about 1830 and 1900. It is not a general claim about wind instruments: a ney or a xiao has a handful of holes and no lattice in this sense, and the argument simply does not apply. Nor does it apply to brass, which has no tone holes at all and gets its fixed upper-spectrum character from a bell and a mouthpiece instead.

Where this goes #

Immediately next is the other thing a hole can be. A register vent is not a tone hole: it is a small hole whose job is to destroy the fundamental rather than to shorten the tube, and it has to sit exactly where the second mode’s pressure node is. That node moves with every fingering and there is one hole, which makes the register mechanism a compromise with an error that can be computed note by note.

Sideways, into the field this site opened last: the two-regime structure above is a filter, and the ear does something similar with a filter of its own. The critical band is a fixed width in the cochlea against a note that moves, which is why the same interval is rough in the bass and smooth in the treble — the same shape of argument as this essay’s, on the receiving end of the same sound.