I have no idea if there's a rule for this so i would try and start cutting...
Start with the lowest notes so that you won't waist any material. Meaning you should start cutting long pipes so that you can still use that pipe for higher notes in case you've cut too short. I'm guessing you will start seeing a pattern after cutting a few notes...
A wild guess, no idea if it stands true: Maybe half the length will weild an octave higher? And 1/4 of the length will weild one fifth above? Etc?
I'd imagine that the relation of the chords in the guitar, and violin are the same, length wise. so it makes sense that the same would apply for any kind of tubes as well...
Just a wild guess, ok?
And wiki might be of help, if you search for frequencies and strings, etc...
If you have more than enough tubes, just give it a try: Cut it in half and see if it goes an octave higher. If it does, the above assumption is correct. Get the dimensions of a string and the tablatures in a guitar, and go by that...
I really hope it helps, though I've no idea if it's right or not.
Cant tell you what length to start at, it depends on the shape size and, if you need to know, the Youngs Modulus of the section. however once you find a starting length doubling it gives you an octave down and halving it gives an octave up. An increase or decrease in the length by 1.059 changes the pitch by a semitone. 12 x 1.059 = 2, 12 semitones in an octave
If you need concert pitch you'll have to start by trial and error
Make the full pipe with no holes make a note.
Determine the note.
Determine the note frequency.
Now you have a starting point.
I suspect that length is more important than diameter.
A little math lie this:
My tube is 36 inches, and plays an A 440.
Ergo 18 inches will yield 880 cps. Etc.
The 1.05 measurment given above is 1.05 of the new length.
Like on a guitar fretboard.
You will need to do your own math with your real measurements.
BUT, you already have two known points.
The note it makes right now, and the octave at one half length.
I think the halving theory only hold's true if you're doing "True" pathagorean tuning. If you want "equal temperment" you have to alter the scale a little bit. I think I remember with Pathagoras you end up with like a 36 note octave before it's complete. In equal Temperment apparently certain notes are actually out of "perfect" tune. "F" is the worst one if I remember. I'm not sure what the mathematic formula is but you ideally want to create a fundamental - then go in 100 cent increments to create a 12 note equal tempered octave.
If you're wanting to sample it I suggest just getting close then pitch and time it or use Melodyne to fix it. That's WAY easier than creating an instrument!
I think the halving theory only hold's true if you're doing "True" pathagorean tuning. If you want "equal temperment" you have to alter the scale a little bit.
No the octave is the one interval which is exactly the same in equal temperament and Pythagorean tuning, and is exactly 2:1. (On the other hand, I've a vague feeling that to get a frequency ratio of 2:1 the length of pipe needed might not be quite 2:1. I seem to remember something from physics lessons about the behaviour of the air in the mouth of the tube being different from the rest, so that the oscillating 'column' wasn't quite the whole tube).
On the other hand, the twelfth root of two (which should be used for equally-tempered semitones ) is actually closer to 1.06 (1.0594) than 1.05