Is there a formula for calculating the length of tubing needed to produce a note on an instrument (for those fairly well versed in brass instruments - and possibly others, but I don't know much about them - I'm referring to notes within a single partial). Would it just be as simple as calculating the wavelength from the frequency of each cent, or is there something else I'm missing?

The tube will be 'played' with a trumpet/french horn mouthpiece, if that would help.


EDIT: I've heard that a trombone slides out about 8cm per semitone, but I have also heard that this varies from partial to partial, so was uncertain if this would be constant.


It wouldn't be complicated to derive such a formula, but I think there are good reasons not to do that. In short: the black box would be too big because the number of adjustable parameters is not sufficiently low.

The straightforward basic steps would be:

  1. Choose a suitable model for your horn profile
  2. Get the frequency from the tubing length (including various corrections)
  3. Transform the frequency in herz to properly scaled log-scale.

Easy, right? But you must consider these facts (based on the brass instrument motivation):

  • Analytical solutions of horn equations are complicated and usually correct only to limited order of error. Numerical solutions does not provide generality you need to derive the final formula. Keywords for further reading: horns, waveguides of variable cross-section, Webster equation...

    • Calculation of mouthpiece cavity connected to the tube does not make the solutions any easier and it must be taken in account

    • If you want a cent-precision then the speed of sound must be considered temperature dependent which gives us an additional parameter.

    • Without perfect knowledge of driving signal you wouldn't be sure that the eigenfrequency is the exact sounding frequency (usually they slightly differ).

    • You need to choose a reference frequency to get an absolute pitch in tones and cents. But there are multiple possible reference frequencies.

Nevertheless: you can achieve some sort of estimation by asking the question: How would a change of length affect a change of pitch? That would lead to some simple formulas but you can't expect that to be cent-precise.

  • $\begingroup$ Brilliant - thank you (and sorry replying took so long). This is great, thank you! $\endgroup$ – DoublyNegative Jul 23 '17 at 19:33

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