the science of the horn
A tube of one length can only sound a ladder of notes — the harmonic series. The slide exists to fill the biggest gap in that ladder by lengthening the tube, one semitone per position. Here is the whole idea, from a bugle call to the twelfth root of two, at every skill level.
start here — a ruler you can hear
Every other instrument in the band is built out of steps. A piano has 88 fixed keys and nothing in between. A guitar has frets. A trumpet has valves that bolt on fixed lengths of pipe. The trombone has none of that — just a tube that slides, smoothly, through every length in between. So it can play every pitch in between too. It is the one instrument in the room that is a number line rather than a set of dots, and that single fact is worth a term of mathematics.
A trombone is one long tube with a bend you can pull. The note is decided by the DISTANCE the air travels from your lips to the bell — and the slide changes that distance smoothly, through every length in between.
The glowing dashes are the air, moving at the same speed the whole time. Pull the slide out and it simply has further to go — and the note drops.
watch one puff of air make the trip
Sound races through air at about 343 metres a second. So a puff of your breath crosses the whole trombone in less time than a housefly's wingbeat. Press the button and we'll slow it down 300× so you can actually see it go.
time = distance ÷ speed
1st position: 2.74 m ÷ 343 m/s = 8.0 ms
7th position: 3.87 m ÷ 343 m/s = 11.3 ms
41% further to travel, 41% longer to get there, and a note six semitones lower.
longer tube = lower note
Seven bars, seven lengths, seven notes. Nothing else changes. To drop the note one semitone you multiply the tube by 2^(1/12) ≈ 1.0595 — so each bar is 5.9% longer than the one above it.
Read down the bars: every bar is longer than the one above it, and every note is lower. That is the whole law of a brass instrument in one picture.
A piano can only land on its 88 fixed points. A trombone slides through every pitch in between — the whole number line, not a set of dots.
the piano's version — seven buttons, nothing between
These are the only seven pitches a keyed instrument would give you across the same stretch. Press them: you can hear the staircase.
what the mathematics calls this
This is also why a trombone section can play perfectly in tune with a choir and a piano cannot quite: the trombonist is not choosing from a list, they are choosing from everything.
Make the tube longer and the note goes lower — not by subtraction, by division. f = k / L is the reciprocal graph you drew in class, and a trombone is the machine that draws it.
where the 319 comes from
L1 = 2.74 m (tubing in 1st position, Bb tenor trombone)
f1 = 116.54 Hz (its everyday Bb — the 2nd note of the tube's ladder)
k = f1 × L1 = 116.54 × 2.74 = 319.32 Hz·m
so f = 319 / L
One honest footnote for anyone who checks: 116.54 Hz is the everyday Bb a player actually uses in 1st position, not the very bottom of the tube. The true bottom — the pedal Bb an octave below, 58.27 Hz — is real but rarely played. It makes no difference to the sum: f × L stays constant along the slide for whichever rung of the ladder you are on. Pick a different rung and you get a different k, and the same reciprocal curve.
This is the reciprocal graph — y = k/x, a hyperbola. Two quantities are in inverse proportion when their product stays fixed while each one changes: double the length and you halve the pitch, triple it and you third it. The dashed rectangle on the graph is that product drawn as an area, and it is the same size wherever the slide goes. That is the whole definition, and a trombonist proves it every time they play.
Seven positions, one semitone apart each — perfectly even in music. But measure them with a tape and the gaps grow every single time: 8.1, 8.6, 9.1, 9.7, 10.3, 10.9 cm. Arithmetic in the ear, geometric in the arm.
the slide, drawn to scale
Each tick is one semitone lower than the last — even, musically. Look at the spacing.
the measurements
Reach is computed straight from the formula extension = (L₁ · (2^((n−1)/12) − 1)) / 2. The ÷2 is there because the slide is a U-shape: push it out by one centimetre and the air travels two centimetres further — down the tube and back.
| position | note | tube × longer | hand travel (cm) | gap from previous | gap ÷ gap before |
|---|---|---|---|---|---|
| 1 | Bb | ×1.0000 | 0.0 | — | — |
| 2 | A | ×1.0595 | 8.1 | +8.1 | — |
| 3 | Ab | ×1.1225 | 16.8 | +8.6 | 1.0595 |
| 4 | G | ×1.1892 | 25.9 | +9.1 | 1.0595 |
| 5 | Gb | ×1.2599 | 35.6 | +9.7 | 1.0595 |
| 6 | F | ×1.3348 | 45.9 | +10.3 | 1.0595 |
| 7 | E | ×1.4142 | 56.7 | +10.9 | 1.0595 |
Read the last two columns together. The gaps are 8.1, 8.6, 9.1, 9.7, 10.3, 10.9 — never the same twice, always bigger. And every gap divided by the one before it gives 1.0595, the twelfth root of two, every single time. That is a geometric sequence hiding inside a piece of brass. The musical steps are an arithmetic sequence (−1, −1, −1 semitones); the physical steps are geometric(×1.0595 each). This is why 6th and 7th positions are a genuine stretch for a young player, and why teachers say the low positions “feel further apart” — they measurably are.
Hold the slide perfectly still and buzz tighter: the note jumps up a ladder of whole-number multiples — ×1, ×2, ×3, ×4. Multiplication you can hear.
1st position, slide fixed — the harmonic series
Same tube, same 2.74 metres, no movement at all. Only the lips change. Each rung is 116.54 × n hertz — the times table of one note.
The same 116.54 Hz is added between each rung, yet the musical jumps keep shrinking — octave, fifth, fourth, third. Equal steps in hertz are not equal steps in music. The slide exists to fill in everything the ladder skips.
Name any note on a trombone and a player can tell you exactly two things: where the slide was, and which rung of the ladder their lips were on. That's a coordinate pair — and the two axes are built from completely different kinds of number.
touch any square to hear that note
up the side: the lips · along the bottom: the slide
slide position — 1 (tucked in) to 7 (arm out)
along the bottom — a continuous axis
Press slide across on any row. The slide does not jump between the seven marks — it passes through every length in between, which is why a trombone can smear one note into the next. Position 3½ is a real place. This axis is the number line: no gaps, no smallest step.
up the side — whole numbers only
Press a number along the bottom and the horn climbs the ladder without the slide moving at all. But there is no rung 3½ — the tube will only sound whole-number multiples of its fundamental. This axis is the counting numbers, and the lips can only ever land on one of them.
For the classroom