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So, hopefully this question doesn't sound too vague, maybe I can completely explain it here.

I have been trying to invent a theoretical instrument that has three main parts to it. The middle piece is just a linear tube with keys like any other woodwind instrument. The first piece is the mouth piece, I'm not sure whether I should make the mouth piece an open or closed hole (like a flute has an open mouth piece hole and the clarinet has a closed mouth piece hole). The end is a closed spherical chamber attached to the tube, like a cor anglais's bell, except it is closed and is a larger chamber. I don't know if this makes sense at all, I apologize.

I just want to know, how would this affect the sound of the instrument, having that spherical chamber at the end? How would it affect the pitch? If the instrument was closed at both ends, would it even produce a tone? I want to know the specific mathematics behind all of this, if that is even possible. I am speaking theoretically, so I have no specific measurements on these things.

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This question has a long ,complex, and somewhat incomplete answer. All the items you mentioned matter. Trivial example, open and stopped organ pipes. An open pipe generates a fundamental wavelength equal the length of the pipe; there is a node at each end (that is, the air is still at each end.) A stopped pipe generates a fundamental with wavelength half again as long as the pipe; there is a node at one end and an anti-node at the other (at one end the air is still, at the other it has maximum motion.) Flutes, oboes, bassoons, and saxophones act like open pipes; clarinets act as closed pipes. The "break" in fingering illustrates this.

Small variations in shape affect the tone quality. The shape is more important in general than the material. With high quality instruments, wooden and plastic clarinets sound similar.

A good place to find out about such things, one should consult texts and papers on things like "woodwind acoustics" or "woodwind construction" or "musical acoustics." A few references are given below.

http://kellerphysics.com/acoustics/Lapp.pdf (An overview of the physics of music. Covers strings, brass, woodwinds, voice, percussion, temperament, etc.)

http://www.woodwindforum.com/forums/ (A discussion forum for all things woodwind and more.)

http://www.acoustics.ed.ac.uk/wp-content/uploads/Theses/Logie_Shona__PhDThesis_UniversityOfEdinburgh_2012.pdf (It's about brass but some knowledge of heavy metal may be useful.)

http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=5200DCB4503DD126958B53A9FC01B290?doi=10.1.1.46.4096&rep=rep1&type=pdf (Thesis about using waveguide techniques to design saxophone mouthpieces.)

http://www.amazon.ca/Acoustical-Aspects-Woodwind-Instruments-Revised/dp/0875805779 (A text on the mathematics of woodwinds.)

http://speech.di.uoa.gr/sppages/spppdf/acoustica2004_physicalmodeling.pdf (Paper on modelling woodwinds for synthesizers.)

  • You could also mention the shape of the bore (how the diameter changes along its length) affects the overtone series. – Carl Witthoft Jan 8 '16 at 12:25
  • Amazing links, exactly what I was looking for! Thank you so much! I knew that the answer was going to be incomplete, but I need to understand the physics of woodwinds more before I can actually figure this out. – Sam Jan 9 '16 at 17:13

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