← Trombone Positions Calculator
A tube resonates with standing waves at specific frequencies — its harmonic series. Choose a slide position and click "Play" to hear it.
A plain tube: each partial fits n loops between two ends.
A tube that flares into a bell: shade shows pressure (brighter = compressed, darker = rarefied), driven by the lips buzzing at the mouthpiece.
Lips can buzz at any frequency — drag to explore. A tall bar means the horn resists strongly there (a real resonance).
A tube of a fixed length resonates with standing waves only if an exact number of wave oscillations matches the length of the tube. If there is just one wave oscillation, that is called the fundamental frequency. If there are exactly two waves, the frequency is doubled, and the note is an octave higher (this result is sometimes attributed to Pythagoras though it has been discovered all over the world). Doubling the frequency again gives 4 waves, and the pitch is two octaves higher, and 8 waves is 3 octaves higher. The partials in between give notes within these octaves.
In 1st position, the notes formed are pedal B♭, then B♭, F, B♭, D, F, A♭ (a well-known flat one, not quite in tune), then B♭ again, three octaves above the fundamental.
Moving the slide out to a lower position lengthens the tube, and lowers the whole series with it. Building every combination of position and partial into a full set of notes is covered on Slide Position Choices and Calculations.
This simplified picture treats the air like a vibrating string. A real horn's air-pressure mechanics work a little differently:
Without the bell, a plain cylinder, closed at the mouthpiece and open at the bell, only resonates at odd multiples of the fundamental — the even harmonics are simply missing. The bell's gradual flare makes a reflection that lets the even harmonics back in, completing the full series. (If you try removing the bell and playing on just the slide of a trombone, you'll find that there are much bigger gaps between the partials you hear.) This behavior occurs naturally when buzzing into conical animal horns. The craftsmanship and trial and error that led to the design of modern brass instruments evolved for centuries, long before its mathematical explanation, which was developed independently in the 1700s and 1800s. These two threads came together during the 1900s, largely due to applications of horn theory to loudspeakers.
The horn equation for a tube of continuously changing cross-section was formalized by Webster in 1919 (Acoustical Impedance and the Theory of Horns and of the Phonograph). It's still the standard starting point for horn-loudspeaker design today (see this clear introduction to horn theory), and the same methods could be used to model brass instruments.
The simulation above demonstrates these fluid dynamics results in action. The coding was done collaboratively with Claude Code, and the solutions the code found are demonstrated above.
Because reflection and bore shape both vary with frequency, a real tube's partials aren't spaced in the perfectly exact integer ratios an idealized string produces — they have to be measured or modelled one by one, which is what the impedance curve above is doing (its own partial numbers aren't spaced exactly evenly either; the readout shows both the partial number and the expected distance).