While studying the sound made various length tubes with bells on the end last summer (2016). Another teacher and I "discovered" a change in the harmonic pattern as we kept making the tube section longer for the same bell. At first we thought it was just caused by the rough nature of our instrument. We were using laser cut plywood rings glued together with a bell made out of laser cut rings of varying diameter (see the laser cut djembe I have posted for a rough idea of the technique). In order to study this more carefully I created 8 different cone/cylinder shapes using Creo Parametric that could be played by placing them directly against one's lips trumpet style or by putting a 3D printed "saxophone" mouthpiece on. We printed these on our AirWolf printer. The size of the printer determined the maximum length for the tubes. The mouthpiece I designed was based off a mouthpiece I found on Thingiverse that I modified to fit my tubes. The sound of the various shapes definitely changes fundamental frequency and harmonics (tone). It is also interesting to compare the sound from a single reed (saxophone) versus a lip reed (trumpet-style) for the various shapes.
With a little further research, we found that others had also looked at this. The image showing the frequency as a function of cylinder versus cone % comes from The Physics of Musical Instruments by Neville Fletcher and Thomas Rossing. (This is a great book if you want to go much deeper into the physics and math behind musical instruments.) We will be teaching a physics of sound course next year and intend to use these tubes as part of our discussion about various woodwind instrument designs and the tone of the instrument.
Please let me know if you try out these tubes and have any recommendations for improving them.