How to calculate the position and number of holes to put into a woodwind so it is able to play a certain set of pitches? I'm in the process of building a bamboo flute, or perhaps a "saxophone" of sorts.
I don't know how to calculate and find out the possible finger hole schemes to achieve the range I'm looking for. I want to be able to play the entire 12-tone chromatic scale, and if possible with a range of around two scales.
Let's start with the simple data involving the semitones of the Western 12-tone equal temperament: each semitone raises the acoustic frequency of the previous note by a ratio of ¹²√2=1.05946309436... 
I play mostly guitar and not woodwinds (yet), but from what I understand at least with the flute you achieve a raise of an octave by overblowing, so I suppose whatever nominal range the finger holes enable is extended in effect by an additional octave. In regards to the saxophone, which is played using a reed, I have no idea. Physics probably make a difference here.
In regards to the basic pitch of the instrument I understand that, at least with the flute, the effective length of the pipe is longer than the actual length (thus it produces a lower pitch than predicted by simple calculation), and it's connected to the pipe's diameter. This needs to be accounted for and corrected when planning the instrument ("end correction"), but: how is the correction calculated? If the pipe has a non-uniform diameter, what value do you use to calculate the correction? And does end correction needs to be done for a woodwind that uses a reed and only has one open end, such as a saxophone?
Thanks.
 A: The overview here is that the physics is complicated, and if you want your instrument to be usable you should not reinvent the wheel. Just reuse someone else's design that works.
Here is some information on the acoustics:
https://ccrma.stanford.edu/CCRMA/Courses/152/woodwinds.html
The end correction is typically calculated as 0.4 times the diameter of each free end. The saxophone acts as an air column that is open at both ends; you can't infer just by looking at an instrument's ends whether it behaves acoustically as a symmetric or asymmetric air column. The sax's mouthpiece acts like an open end, while the clarinet's acts as a closed end.
The tone holes effectively act to shorten the length of the tube. Sound can radiate both through the open tone holes and the end of the tube. To some kind of first approximation, you can just use the distance to the first open tone hole as the effective length of the tube and calculate a frequency by taking that to be 1/2 or 1/4 of a wavelength. But in reality this is going to make an instrument that plays out of tune.
We don't have enough fingers to play a chromatic scale by having 12 tone holes. This is why cross-fingerings are used. Cross-fingerings are different for different instruments, and there may be more than one for a given note.
