Physics 136 / Music 126 Duke University Fall 2012 Handout 16

Notes on Wind Instrument Shapes

In designing a wind instrument, we'd like to be able to sound notes based on its different normal modes (as with a bugle, for instance), and to fill in notes between available modes by changing the effective length (by lengthening the instrument with valves or a slide, or shortening it by opening tone holes).

So the modes we use must differ by musically appropriate intervals, even as the overall length is varied. These criteria may be stated formally as the following two rules:

I. The frequency ratios of the various normal modes must be independent of the length of the instrument.

II. The higher mode frequencies must be integral multiples of a fundamental mode frequency.

It can be shown [A. H. Benade, J. Acoust. Soc. Am. 31, 137 (1959)] that all shapes satisfying criterion I. are among the Bessel horns, i.e. their cross-sectional areas S(x) at each position x along their lengths can be expressed as S(x) = c(x-x_o)^e, where e is called a "flare parameter".

Each such shape has a "dual" with cross section S_d(x) such that the product SS_d is constant for all x. If the narrow end of the dual is closed, the two shapes will share the same set of normal mode frequencies (though not, in general, the same standing wave patterns or coupling to the air outside). Since S (proportional to x^e) and S_d (proportional to x^-e) have the same frequencies, then, we need consider only positive values of e.

flare parameter e	bore type	dual
0	open cylinder	(no dual)
0	semi-closed cylinder	self-dual (e = -e)
2	cone	S(x) proportional to 1/x^2

Bessel horn segments with 1 < e < 2 are used in wind instruments -- duals of this range (-1 > e > -2), especially, are used as bell shapes -- but not as primary bore shapes, because of their non-harmonically related normal modes.

Among the infinite number of Bessel horn shapes, only certain values of e turn out to satisfy criterion II. The smallest are: e = 0, 2, 7, 13, and 19.5 (and, of course, the corresponding duals).

flare parameter e	S(x)	shape	2nd register interval	nth mode frequency
0	indep. of x	cylinder, semi-closed cylinder, open	twelfth octave	(2n-1)v/4L nv/2L
2	proportional to x^2	cone	octave	nv/2L
7	proportional to x^7	(see drawing)	fifth	(n+1)v/2L

Consider examples of each of these shapes with lengths chosen so they share the same fundamental frequency.

flare parameter e	bore	example	length	maximum diameter if d = 1/8 in at x = 2 in
0	open cylinder	flute	2 ft
0	closed cylinder	clarinet	1 ft
2	cone	oboe	2 ft	1.5 in
7	impractical horn	(none)	4 ft	705 ft

[A one-foot-long e = 7 horn, sounding two octaves above the instruments in the table above, would still reach 66 inches in diameter for any reasonable "blowing end"!] e = 13, 19.5, et c., would have even more rapid flares!