Changing the cone angle of a bore changes the mode relationships continuously. So Why doesn the cone angle affect the overblowing interval on woodwinds?
Let’s begin with the didjeridu, which is often approximately a truncated cone. The angle varies considerably among instruments. If you use a plastic pipe as a didjeridu, the hoot note (the first overblown note) of this cylindrical instrument is nearly a twelfth above the drone. Most real didjeridus are somewhat flared, however, as mentioned above, and they overblow typically a tenth. So the question is a good one: how is it that the narrow angle of a bassoon (0.8) and the wide angle of a soprano sax (3.5) overblow octaves? There’s a related puzzle: the wavelength of the note that these instruments play is not twice the length of the instrument, but rather about twice the length of the complete cone, i.e. (about twice) the length obtained by extrapolating the cone to a point, well beyond the mouthpiece. (See Pipes and harmonics for the sounding frequencies of conical and cylindrical pipes.) The explanation is that woodwind instruments are not simply truncated cones. The inside the saxop
