Since the interval of an octave is a doubling in frequency, then theoretically the dimensions of a flute would be halved to produce a piccolo in a key one octave higher. Expanding upon that with the knowledge that in an evenly tempered scale every half-step bears the same proportional relationship to its neighbors as every other. Thus, if I would multiply the dimensions of a flute by some factor to get a flute in a key one half-step higher, I would repeat this multiplication twelve times to accomplish an entire octave. The factor for one half-step is then 0.943874, the twelfth root of one half.
Multiply [or divide] all length dimensions by this factor repeatedly for the number of half-steps up [or down] you wish to transpose. Thus you would first make a flute out of whatever tubing or pipe you have in mind, find out what key it turns out to be in, then scale the lengths and make another flute in the key you desire. There will almost certainly be some trial and error here but if you're not shifting too far you can compensate by adjusting the tone hole diameters.
Personally, I only very rarely play in the third register since the strange fingerings are hardly intuitive and may vary from one flute to another. For my flutes, I prefer a lower ratio, usually between 20:1 and 25:1 instead. The larger the bore, the richer the lower register and the more difficult the upper registers.
Rick Miller
rdmiller3@gmail.com