First sax Build

I'm researching in order to build my own sax.

 

As a backpacker I have chosen PVC as the material as it is light, cheap and durable. The tone is lovely also, as I have seen online.

 

I have 2 main dilemmas:

 

1. As my pet peeve is instruments that cannot play all 12 notes, I assume I will need to construct key/pad mechanisms, unlike the PVC sax I have seen online which merely has drilled holes. My priorities are 1.Sax is chromatic, and 2. Sax is small :

- what is the smallest sax I can get one full chromatic octave on?

- what is the smallest sax I can get one full chromatic octave, and a higher register on?

- an alto appears to have a regular and higher register, however appear to be chromatic, is this correct?

 

2. A mathematical formulae for plotting my tone holes; the parameters for the tone holes in relation to the parameters of the tube. (I'm aware that register holes compromise precision for ease). Knowing this can be the difference between making 3 prototypes before a final instrument, rather than drilling random guessed holes into 100 prototypes.

 

I've been fortunate in finding abudant information on the physics behind the woodwind instrument family, but it's been hard to find exactly what would help me to craft a sax.

Thank you so much if you can help me in any way! :)

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