Alterius Omega wrote: ↑Thu Oct 19, 2023 12:52 pm
So did you find a relatively simple formula for what taper angle is required to make a single reed instrument play in tune in both octaves for a given length?
I didn't even try. As pointed out early in this script, much depends on the reed, so I'm not confident we can draw general conclusions.
Alterius Omega wrote: ↑Thu Oct 19, 2023 1:36 pm
What didn't make sense to me is how a closed-end conical instrument can play the even harmonics and my current understanding is that it actually can't. It still can only play the odd harmonics (that is, it can make the half-waveform shape of an open-end flute but only every other one). The only difference is that the taper of the bore gives a different equation for the actual wavelength/frequency produced due to the cross-sectional area increasing linearly.
What I think this means is that while a sax for example can still only produce every other hamonic waveform, the equation is different such that the actual frequencies of the pitches of those waveforms has been "squashed" closer together so instead of the 3rd harmonic producing a 12th, it actually produces an ocatve. Am I on the right track?
You're more-or-less on track, but we need to distinguish between
modes of resonance, which define the resonant frequencies of a particular system, and
harmonics, which are strict multiples of the fundamental resonance that may not correspond to modes. See
https://newt.phys.unsw.edu.au/jw/inharm ... ances.html for more explanation.
In a flute or whistle, the resonant modes are where the tube contains a standing wave of one half cycle, two half cycles, three half cycles, .... In a cylinder, the frequencies of these resonances are pretty close to one, two, three, ... times the frequency of the lowest resonance, which are all the harmonics of the lowest resonance.
In a reed instrument, the resonant modes are where the tube contains a standing wave of 1/4, 3/4, 5/4, ... cycles. In a cylinder, the resonant frequencies are pretty close to 1, 3, 5, ... times the frequency of the lowest resonance, which are the odd harmonics of the lowest resonance. Moving to a cone, as the bottom end of the cone gets bigger, all of the resonant frequencies increase, while the ratio between those frequencies decreases. Eventually, as the cone gets wide enough (and long enough), the second mode falls at twice the frequency of the first mode, which makes it the second harmonic. (Getting the higher modes to line up with the higher harmonics can take some fancy adjustments to the bore profile.)