Fletcher and Rossing in _The Physics of Musical Instruments_ give you more analysis of flares of musical instruments than a musician (and most physics teachers) would want. He talks about the Bessel flare and flare parameters, radiation patterns, intonation, anharmonicity of overtones, etc. It's almost reductionistic beyond artistry. It does help you understand why professional level instruments that play well are so expensive. All the individual parameters (mouthpiece volume and taper, tubing diameter, flare of bell, tubing bend radii, etc) interact, so constructing a well-playing wind-powered instrument is a work of artistry and experience. In the woodwinds, you have the edge shaping of the tone holes to consider also; the edge shape affects intonations and strengths of overtones.
A fun exercise is to get a brass mouthpiece and some straight tubing on the order of a meter and try to play it. Then fashion a bell (or use a medium sized funnel) on the end and play again. Check the ease of playing and the overtone structure of each setup and compare. It's amazing what the flared bell does.
Bill Nettles
Union University
-----Original Message-----
From: phys-l-bounces@carnot.physics.buffalo.edu [mailto:phys-l-bounces@carnot.physics.buffalo.edu] On Behalf Of Ricca, Bernard P
Sent: Wednesday, April 07, 2010 11:26 AM
To: phys-l@carnot.physics.buffalo.edu
Subject: Re: [Phys-l] musical instruments
Mike,
---- snip ----
Not having built any brass instruments myself, I am just guessing that early instrument manufactures had to experimentally determine the correct flare to make an instrument that could be played in tune. At instrument manufacturing/design companies today, I do not know if the proper flare is theoretically calculated, or if it is experimentally determined. Does anyone on this list know?
---- snip ----
The answer is "both". The flare serves more than one purpose, and, IIRC (my notes are not easily accessible, and it has been several years since I paid attention to this) you can determine some theoretical shapes (e.g., a "Bessel" flare) that, to a first approximation, are pretty good for the generation of a harmonic series. These are then tuned experimentally to account for curved tubing, desired tone, etc.
Barney
Bernard Ricca, Ph.D.
Director, Graduate Program in Math, Science and Technology Education Department of Mathematical and Computing Sciences Saint John Fisher College 3690 East Avenue Rochester, NY 14618
585-899-3866 (Office)
585-899-3872 (FAX)
bricca@sjfc.edu