On the cutoff frequency of conical woodwind instruments
Jean KERGOMARD(1) , Erik Alan PETERSEN(1) , Tom COLINOT(1) , Philippe GUILLEMAIN(1) , Michael JOUSSERAND(2)(1) Aix Marseille Univ, CNRS, Centrale Marseille, LMA, UMR 7031, Marseille, France, kergomard@lma.cnrs-mrs.fr(2) Buffet Crampon, 5 rue Maurice Berteaux, 78211 Mantes la Valle, FranceAbstract
Since Arthur Benade, the cutoff frequency of woodwind instruments is a well-known concept that is a result of theacoustic regularity of the tonehole lattice. The distinction between the global cutoff frequency and the local cutoff
frequencies was recently introduced. The first can be approximately defined from the input impedance curve, while
the second is related to the different cells of the lattice and can be theoretically determined from their individualgeometries. From the local cutoff frequencies, it is possible to check the acoustic regularity of an instrument, as wasdone for the clarinet, even if the geometry is irregular. The present work aims at finding a generalization for conicalinstruments. The existence of a global cutoff is evident in the impedance curves of oboes, bassoons, and saxophones
published by Benade. However, the existence of periodicity is not obvious for a conical geometry because the cells
are a priori very different, due to the taper. The present work uses a change of variables to investigate an acoustic
periodicity for conical resonators and derives an equation for quantifying the acoustical regularity of a given conicalresonator. This method can be applied to the case of a saxophone, for which the local cutoff frequencies are calculated
from the geometry of the instrument.
Keywords: Musical acoustics, saxophone, cutoff frequency

1Introduction

Physical modeling of woodwind instruments typically divides the problem into two parts: the reed as a nonlinear
excitation mechanism, and the body as a passive resonator that facilitates the auto-oscillation of the reed inaddition to radiating sound. The acoustical response of a resonator is often characterized by its input impedance.
If the resonator has a series of open toneholes, the input impedance shows low frequency and high frequency
characteristics due to the periodicity of the lattice of toneholes, and the transition occurs at the tonehole lattice
cutoff frequency [1].
In general, a global cutoff frequency can be estimated from measurements of the input impedance. In the caseof cylindrical woodwind instruments, such as the clarinet, a local cutoff frequency can be defined from thegeometry of the instrument by calculating the theoretical Helmholtz resonance of adjacent tonehole cells [2].
This shows that, regardless of cell dimensions, an acoustic regularity can be found such that if each cell has
the same local cutoff frequency, the input impedance will display the same global cutoff frequency as any given
cell. In the current work, we seek to generalize the concept of acoustic regularity to conical instruments.
The transfer matrices, including a change of variables, that describe wave propagation in a cone with toneholes [3] are presented in Section 2. Acoustic regularity of a conical lattice of toneholes and its application is
discussed in Section 3, followed by conclusions and perspectives in Section 4.

2Simple formula for the cutoff frequency of conical tubes

2.1 Basic equations
Consider a truncated cone of length L, input radius R1 , output radius R2 . The longitudinal coordinate is r.
At the abscissa r, the radius is R = rtan(ϑ ), where ϑ is the half-angle at the apex. The length L is equal tor2 − r1 = (R2 − R1 )/tan(ϑ ). The area of the cross section is S = πR2 . In the frequency domain, the Helmholtz
equation for the acoustic pressure P(r) is written as:

6452

d 2 (rP)+ k2 (rP) = 0(1)dr2Losses are ignored, k = ω/c is the wavenumber, where ω is the angular fequency, and c the sound speed. The
acoustic velocity V is derived by using the Euler equation, ∂r P = − jkρcV , where ρ is the air density.
Between the two abscissae r1 and r2 , the following transfer matrix equation can be written as:




PrPr
coskL
jρcsinkL=.
(2)
PPV r − jωρV r − jωρj(ρc)−1 sinkL
coskL
12

When a tonehole is present, we assume that its effect is that of a shunt acoustic mass, m = ρh/s, where s isthe cross section area of the hole, and h its chimney height that includes inner and outer length correction. Thesimplest model for the hole is considered, with the continuity of the planar wave pressure. At the location ofthe hole, the volume velocity relationship is written as:
VL = VR + P/( jωmS).

(3)

The subscripts L and R correspond to the left and right sides of the hole, respectively. Therefore the transfer
matrix from the left to the right can be written as
PrPr
1 0=(4)PP
V r − jωρV r − jωρ
Y 1LRwhereY=1.
jωmS(5)

We use the notations ML and Mh for the matrices defined by Eq. (2) and (4): they are the matrices for the wave
propagation over the distance L and through the hole n, respectively. If we build a “conical” lattice with theproduct (ML Mh ML )N , the lattice corresponds to the repetition of N cells of tube segments of length 2L with ahole at the middle. If the length 2Ln and the (specific) admittance Yn of the hole is the same for all cells, theresulting lattice is a periodic medium, despite the fact that the tube is a cone, and the geometry is not periodic.
We notice that the model is the same for a cylindrical tube, but here the characteristic quantity of a hole is1/Yn = jωmn Sn , where Sn varies from one cell to the next one. Another difference with a cylindrical tube liesin the term −P/jωρ, which has to be taken into account at the extremities of the lattice. It can be ignored at
the open end because the radiation impedance is small but, at the input, it is responsible for a big qualitative
difference between the behaviors of cylindrical and conical tubes. This difference is analyzed in many papers.
2.2 Local cutoff frequency
Formally, the analysis of a conical lattice is identical to that of a cylindrical lattice. We do not detail here theexplanation given in [2] concerning the distinction between global and local cutoff frequencies.
The local cutoff frequency for a cell of the lattice is calculated by deriving the coefficients (classically denoted
A, B ,C , D) of the transfer matrix of one cell, ML Mh ML , and equating A = D = ±1. This leads to the followingequation:jρccot(kL) = 2 jωmS.
(6)
At low frequencies, this leads to the value of the local cutoff fc
fc 'c
1c
1pp
=2πL 2mS/ρL + 1/3 2π 2mSL/ρ

(7)

6453

The first expression is found by retaining two terms of the expansion of the cotangent function in kL, while thesecond retains only one term. The first is slightly more accurate than the second. This gives the definition of
the local cutoff frequency.
In contrast to the cylindrical case, this local cutoff frequency is not the eigenfrequency of a cell closed at its
extremity (see [2, 4]). For a cylindrical geometry, the term 2SL defines the volume of the cell, and a simple
interpretation is to treat fc as the Helmholtz resonance of an equivalent cell. This is not valid for a conicalresonator because 2SL is not the volume of a truncated cone if S is at the midpoint.

3Analysis of the regularity of a lattice

3.1 Perfect acoustic regularity
In [2] it has been shown that if the (local) cutoff frequencies of the cells are identical, the cutoff frequency
seen in the input impedance curve, called the global cutoff frequency, are equal to the local frequencies. The
equality of the local cutoff frequencies can be obtained with various dimensions of the holes. Equation (7)provides the condition for acoustical regularity: 2LhS/s is independent of the hole. Because this analysis isfor conical resonators, it is important to remember that the area S is the cross section of the main tube at thelocation of the hole. This analysis shows that despite an irregular geometry, an acoustic regularity can be found.
3.2 Analysis of the regularity of an alto saxophone
We measured the geometry of an alto saxophone Senzo, produced by Buffet Crampon. In [2] the aim was toanalyse the regularity of a clarinet. It was shown that it is possible to divide the instrument into asymmetriccells with equal local cutoff frequencies. Two methods were used, but an approximate approach was also
proposed in order to examine the degree of regularity without requiring the division of the instrument into
discrete cells, which is not a trivial task. This later approach is now used for the investigation of a saxophone.
Equation (7) is modified as follows:
Sn 1Sn1= 2Ln hn + = 2Ln hn
2kcn
sn 3sn(8)If the cutoff is identical for two adjacent cells (n and n + 1), we can write the geometric relationship between
adjacent cells asLn mn Sn = Ln+1 mn+1 Sn+1 =

ρ.
2kc2

(9)

The distance between hole n and n + 1 is d = Ln + Ln+1 , so a characteristic cutoff frequency is defined bycfch =2πsρ2d

11+mn Sn mn+1 Sn+1(10)If the cutoff frequencies of the adjacent cells are equal, this expression is valid. Therefore, if all pairs ofadjacent cells have a similar frequency calculated by Equation (10), the resonator is acoustically regular. Otherwise, if this frequency varies with the considered pair of cells, the variation can be viewed as a measure ofirregularity.
Applying this definition of a characteristic frequency to the geometry of a saxophone and comparing with theglobal cutoff frequency from a measured impedance is continuing as work in progress.

6454

4Conclusions and perspectives

The global cutoff frequency and a local cutoff frequency defined by the tonehole lattice geometry is treatedfor cylindrical instruments in Moers et al [2]. Here, we attempt a generalization from the specific case of acylindrical tonehole lattice that conical case. We derive a local cutoff frequency that can be applied to individualcells. However, this result is not sufficient to analyze a conical tonehole lattice because there is no simple way
to divide the lattice into cells. Therefore, we derive a characteristic frequency relating the acoustic masses of
two adjacent cells and the length of cone between them. This local value is viewed as a measure of acoustic
regularity for a conical tonehole lattice.
In “Fundamentals of Musical Acoustics,” Benade provides input impedance measurements to demonstrate thetonehole lattice global cutoff frequency for many instruments including the clarinet, oboe, basson, and even lesscommon instruments such as the tarogato. However, he does not provide similar figures relating to the cutoff
frequencies of a saxophone. Although the cutoff frequency is apparent in input impedance measurements (forexample the website of Wolfe [5, 6]), the phenomena is not as clear as it is for other instruments. The current
work shows that simple eigenfrequency interpretation that works for cylindrical instruments can not be applied
to the saxophone, but that a second method of evaluating acoustic regularity shows promise.

ACKNOWLEDGEMENTS
This work has been partly supported by the french Agence Nationale de la Recherche (ANR16-LCV2-0007-01
Liamfi project). The authors thank Thierry Mialet of Buffet Crampon for his participation in this work. The
authors would also like to acknowledge Patrick Sanchez for supplying the saxophone geometry.

REFERENCES[1] Benade, A. H., Fundamentals of Musical Acoustics Oxford University Press, London (1976).
[2] Moers, E.; Kergomard, J., “On the cutoff frequency of clarinet-like instruments. Geometrical versus acoustical regularity,” Acta Acustica United with Acustica 97, 984-996 (2011).
[3] Benade, A. H., “Equivalent circuits for conical waveguides,” J. Acoust. Soc. Am 83 (5), 1764-1769 (1987).
[4] Chaigne, A.; Kergomard, J. Acoustics of musical instruments. Springer (2016)[5] Wolfe, J., Website regarding musical acoustics, http://newt.phys.unsw.edu.au/music/saxophone/soprano/E4.html.
http://newt.phys.unsw.edu.au/music/saxophone/soprano/E4.html[6] Chen, J. M., Smith, J., Wolfe, J., “Saxophone acoustics: Introducing a compendium of impedance and soundspectra,” Acoustics Australia 37 (1), 18-23 (2009).

6455