Harmonic lengths of a set of tubular drums?
I have a 200 cm long piece of a clear plastic tube with 20 cm outside diameter that I would like to cut into three pieces to make a bigger diameter version of the tubular drums that exists today in smaller diameters (Octobans etc).
Is there any mathematical theory that could help me to predict what lengths I should cut the tube into?
I can of course tune the drum heads to different notes but I'm looking for a way of knowing how to cut the tube into three pieces that will resonate harmonically together by themselves and also produce three clearly distinguishable pitches.
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As the pitch is produced by the air in the tubes depending of the length you can choose the natural physical ratios of the length of the air wave known since Pythagoras.
If you want to have a triad (fifth, octave, third) 1/3 + 1/4 + 1/5 = 200cm = 20/60+15/60+12/60
that means the ratio has to be 20:15:12 (20+15+12=47)
You divide the 200cm:47=42.5cm (2.5mm rest)
5th = 20*42.5cm=85cm
8ve = 15*42.5cm=63.75cm (root)
3rd = 12*42.5cm=51cm
Sum = 199.75cm Rest difference = 2.5mm
(the rest 2.5mm you can use as fall out from cutting)
Another solution might be:
Root = 1/2
3rd = 2/5
5th = 1/3
(Common quotient of 2, 5 and 3 is 30)
that is 15/30 (root) 12/30 (3rd) 10/30 (5th)
(15+12+10)= 37 parts
200cm:37=5.4cm (2mm=rest)
Root=15*5.4cm=81cm
3rd=12*5.4cm=64.8cm
5th=10*5.4cm=54cm
Sum=199.8cm (2mm = rest for cutting)
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