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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.

Per Alenfelt
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  • I'd have thought a simple division by 6 would do it. One 1/6th, next 2/6 (1/3), last 3/6 (1/2), so allowing for mimimal kerf, 333mm, 666mm and 999mm. – Tim Apr 07 '20 at 15:06
  • @Tim: wouldn't 1/6 be the octave of 1/3? – Albrecht Hügli Apr 07 '20 at 16:08
  • how to cut the tube into three pieces that will resonate harmonically: what do you mean by this? which intervals? root, fifth, third? – Albrecht Hügli Apr 07 '20 at 16:09
  • Yes, I expect so, but what difference would that make? The tautness of the head will have as much bearing. – Tim Apr 07 '20 at 16:21
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    Google 'octobans' and dimensions are stated there on Wiki. The maths could be worked out from those dimensions. – Tim Apr 07 '20 at 16:26
  • Thanks a lot for all comments. What I mean by "harmonic resonance" is simply that they should sound good together. I have a pitch interval in my head that I have not defined yet. The tuning of the drum head will fix this but I had the idea that there would be preferable ratios between the tube lengths that maybe could be predicted (Helmholtz equation anyone?) but when I compare the different lengths of sets of tubular drums from different places it does not seem as important as I thought. – Per Alenfelt Apr 07 '20 at 21:03
  • I’ve tried to give 2 solutions of triads: 1.) so,do,mi 2.) do,mi,so. I’m not sure, whether my reflections are correct and my counting is true. I’ve calculated in the head. – Albrecht Hügli Apr 07 '20 at 21:40
  • @tim Is it possible to use 'tubular drums' without a drum head, beating them on the side as is done with tubular bells/chimes? Tho' I guess that wouldn't produce the sound type that the OP is looking for here. – Carl Witthoft Apr 08 '20 at 14:32
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    @CarlWitthoft - it's possible to play most instruments that way! Even cellos! Wonder if you've tried it..? Seriously, tubular bells et al, work best because the material they're made from - metal - has a better vibratory factor (maybe there's a better word?) than the plasic envisaged here.Miced up it could be interesting, though. – Tim Apr 08 '20 at 14:50
  • Ordinary toms in a drumkit are often tuned to 4ths or 5ths with each other. – Tim Apr 08 '20 at 14:51

2 Answers2

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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)

Albrecht Hügli
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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)

Albrecht Hügli
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  • if I would only cut the tube into two pieces and go for the root and 5th, can I then simply use the same mathematics but start with: 3/6 + 2/6 => 5 parts? – Per Alenfelt Apr 08 '20 at 07:51
  • If my assumptions are correct - in this case 1/5200=40 -> one part would be 40 and the 2 lengths 1/2:1/3 = 3/6:2/6 were 340 : 2*40= 120:80 you wouldn't have any rest for cutting but I think you could ignore it. Try this out with a cheap plastic tube ;) – Albrecht Hügli Apr 08 '20 at 08:02