Effect of bore coneangle and bell diameter in woodwind instruments
Saxophones, oboes, bassoons and many other woodwind instruments have bores that are approximately conical. The cone angle has important effects on the playing behaviour and the sound. (The small photos on this page link to larger versions)
For the saxophone, the half angles of the cones are 1.74° and 1.52° for soprano and tenor respectively. For oboe and bassoon, the half angles are 0.71° and 0.41° respectively. (The flute body and the top part of the clarinet are largely cylindrical.) For a given bore length, the relatively large angle of saxophones gives them a large output diameter: even the soprano saxophone has a rather larger end diameter than oboe, bassoon and clarinet.
Wind instruments rely on standing waves in the bore, and the standing waves require reflections at both ends (see this page for explanation). For a cone (we’ll neglect the modest bells for now), the reflection at the output is weaker for a large output diameter. This is particularly true for high frequencies, which are radiated better than low. Or, more formally, consider that the radiation impedance goes as f/r and that the characteristic impedance of a duct goes as 1/r^{2}; so that the ratio of radiation to characteristic impedance goes as rf (or r/λ). So reflection is weaker for large aperture and for high frequency. (Informally, one might think of this as going from a limited space into a large one, and the reflection depending on the difference.)
So waves are radiated better for a large output radius. This makes the instrument with a large cone angle louder – which was one objective Adolphe Sax had when he invented the saxophone. Further, higher frequencies are radiated better than low (for all angles).
But more radiation means less reflection for wide bore instruments. This lower reflection reduces the strength of the standing waves in the bore. In practice, the effect of the cone angle is complicated by the tonehole cutoff effect: the cutoff frequency is higher in the saxophones than in oboes and bassoons. An estimate of the cutoff frequency is
f_{c} = 0.11 (b/a)v/√(st).
where b is the radius of a typical tonehole, a the radius of the bore, v the speed of sound, s is half the typical spacing between tone holes and t the typical effective length of the tone hole, including end effects. See this page for more detail.
The combined behaviour of cone angle and cutoff can be observed in the impedance spectra measurements for soprano and tenor sax – and compared: the soprano has a larger angle. The effect of cone angle alone can be compared in the measurements of the (atypical) lowest note. A few example spectra are shown below.
The large bore angle is one of the important differences between the saxophone and the bassoon. Similarly, the bore angle is an important difference between the oboe and the soprano sax, with the tarogato as a compromise. (The single vs double reed is another, of course.)
The narrower cone means that the bassoon can play well over three octaves without using the vocal tract. (The Rite of Spring and Shostakovich IX both call for D5.) The tenor sax can comfortably match this upper range, but its low note is Ab2, compared with the bassoon’s Bb1. (Incidentally, for bassoon parts written in the tenor clef, a tenor sax can read the part at pitch, pretending it is treble clef, and the bassoon parts written in bass clef can likewise be read by baritone sax, which almost covers the low range of the bassoon.)
The graphs below come from a large collection impedance spectra measurements for soprano and tenor sax
