Acoustics & Music Theory
why it sounds like that
Series

Tone holes — the series

7 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Where each family's tone-hole lattice stops reflecting. The cutoff frequency of an open tone-hole lattice, from Benade's formula, for four woodwind geometries: clarinet 1824 Hz, oboe 2990 Hz, flute 1690 Hz, bassoon 506 Hz. Below its cutoff a note's wave turns round at the first open hole and the instrument is a tube of that length; above it the wave passes through the whole lattice and radiates from the far end, so the upper part of every note's spectrum leaves the instrument from the same place whichever note is fingered. That is what gives a family one recognisable voice across its range.

    Above a certain note the holes stop working

    A row of open tone holes reflects the wave and makes the tube shorter — up to a frequency. Above it the wave runs straight past the whole lattice and leaves from the bell, so the top of every note's spectrum radiates from the same place whichever note is fingered. That cutoff is computable, it differs by family, and it is most of what makes an oboe sound like an oboe.

    part 1 · instruments
  2. One register vent, 12 fingerings. Where the second mode's pressure node sits for each fingering of a stopped tube, against a single register hole drilled 12 cm from the mouthpiece. The node is a third of the way along the sounding length, so it moves every time a hole is opened, and the vent's error runs from -38 to -1 cents across the range. A perfect register system would need one hole per fingering. The number of holes actually fitted is one, and the leftover is a design decision rather than a fault.

    One hole doing a dozen jobs

    A register key works by forcing a pressure node where the second mode already has one, which kills the fundamental and leaves the mode above. The node sits a fixed fraction along the sounding length — and the sounding length changes with every fingering, while the hole stays where it was drilled. The leftover error is computable, and it is why the throat notes are the ones players complain about.

    part 2 · instruments
  3. One key, nineteen notes, one right answer. A register hole disturbs a mode in proportion to the square of that mode's pressure at the hole, so the place that spoils the fundamental and leaves the third harmonic alone is the third harmonic's own pressure node — a third of the way down whatever length is sounding. The length changes with every fingering and the key does not move, so the two curves cross at one note. At A3 the key sits almost exactly on the node and the twelfth speaks cleanly; at A♭4 it is at 62 per cent of the tube and disturbs the third harmonic by 96 per cent of what an antinode would, which is the throat of the instrument and is exactly where players say the notes are worst.

    The hole that spoils a note

    A tone hole shortens the tube. A register hole does the opposite job: it is small enough to shorten nothing and is placed where it will wreck the fundamental's resonance and leave the third harmonic's alone, so the note jumps a twelfth instead of retuning. The place that does both is a pressure node of the harmonic being kept — a third of the way along whatever length is sounding — and the length changes with every fingering while the key does not. One key is at the right place for exactly one note, and the note it is worst for is in the throat of the instrument, which is where players say the instrument is worst.

    part 3 · instruments
  4. One vent on a cone, over the notes it has to serve. A cone's modes are a full harmonic series, so the mode a register vent has to keep is the octave rather than the twelfth, and its pressure node sits at half the sounding length measured from the virtual apex. The apex is a fixed point of the instrument and the bell end is not, so every fingering moves the ideal position and there is one hole. The upper line is how much the vent spoils the fundamental, which is what it is for; the lower is how much it damages the octave, which is the cost. Their ratio runs from 451.2 at F♯4 down to 1.9 at B♭3 — a factor of 235 across a single register. The vent is placed at 37 per cent of the longest sounding length, which is where the worst fingering is least badly served.

    The vent a cone cannot place

    A clarinet's register key has to spoil a fundamental and leave a twelfth. A saxophone's has to spoil a fundamental and leave an octave, whose pressure node sits at half the sounding length from the virtual apex — and the apex is a fixed point while the bell end is not. Over one register the ideal position moves by a factor of two, and one hole is right for one note.

    part 4 · instruments
  5. Seven holes that all sound 196 hertz, and none of them agrees about the twelfth. Each dot is a hole radius, placed at the station that makes the first resonance 196 hertz. The stations run from 411 millimetres for a 7.5-millimetre hole to 307 for a 1.4-millimetre one, which is a fifth of the tube. Up the axis is what the second resonance does: a cylinder's should be three times the first, and it is -2 cents from it for the widest hole and -453 for the narrowest. The hole's inertance rises with frequency, so a narrow hole lengthens the tube more for the twelfth than for the fundamental — and two holes that are interchangeable in the first register are a fourth apart in the second.

    A hole is a short tube

    Four earlier essays have treated an open tone hole as a point where the pressure is released. It is not: the air in a hole has mass, and a hole with mass does not end the bore, it loads it. Seven holes drilled at seven stations all sound the same G — and their twelfths are spread over a fourth. Cross-fingering falls out of the same arithmetic, and it is not made of what everybody says it is.

    part 5 · instruments
  6. A woodwind with holes graduated 12 mm to 6 mm, drilled so that every fingering is in tune. A cylindrical bore 567 millimetres of acoustic length and 15 across, with twelve tone holes through a 4-millimetre wall. Opening them one at a time from the far end takes it up a chromatic scale from D3 to D4. The stations are not copied from a maker's drawing: each was solved so that its own fingering sounds its equal-tempered note in this model, one hole at a time down the tube with every hole below it already open, which is what a reamer and a tuning fork do. The worst fingering is 11.9 cents out. The diameters run 12.0 millimetres at the bell end to 6.0 at the top, and the spacings close from 30 millimetres to 19.

    The cutoff that is a list

    Five earlier essays have quoted one number for a woodwind's cutoff — 1,824 hertz for a clarinet — from a formula written for an infinite lattice of identical holes. Solve a whole twelve-hole chart instead and the number is eleven different numbers, running from 2,193 hertz down to 1,574, which is 574 cents. The lowest fingering has no cutoff at all, and which way the list runs turns out to be a design decision rather than a fact about woodwinds.

    part 6 · instruments
  7. Where a woodwind's A♭3 leaves it, below its corner and above it. The same fingering — 6 holes open on a 15-millimetre bore 567 millimetres long, sounding A♭3 at 207 hertz — drawn twice, with each opening's circle scaled by the share of the radiated power that leaves through it. At 400 hertz 78 per cent of it leaves through the first open hole, the bell takes 0 per cent, and the number of apertures really doing the radiating is 1.6; At 2600 hertz 6 per cent of it leaves through the first open hole, the bell takes 49 per cent, and the number of apertures really doing the radiating is 3.3. The lower frequency is below this fingering's corner and the higher one above it: below the corner the instrument is a short tube with one opening at the end of it, and above the corner it is the whole lattice at once. The power-weighted station — where a listener would say the sound is coming from — moves from 392 millimetres to 515.

    Where a woodwind actually sounds from

    Every number so far is read at the mouthpiece, and the corner's whole musical meaning is at the other end. Run the same solver forwards and it gives the flow leaving every hole — from which a clarinet's radiating aperture turns out to be a function of fingering and of frequency, but not the way it was predicted to: the fingering sets how far the aperture opens, almost exactly to the number of open holes, and barely moves the frequency at which it does.

    part 7 · instruments

All series