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Hindawi Publishing Corporation
EURASIP Journal on Advances in Signal Processing
Volume 2011, Article ID 151436, 13 pages
doi:10.1155/2011/151436
Research Article
Trombone Synthesis by Model and Measurement
Tamara Smyth and Frederick S. Scott
School of Computing Science, Simon Fraser University, Surrey, BC, Canada V3T 0A3
Correspondence should be addressed to Tamara Smyth, tamaras@cs.sfu.ca
Received 29 August 2010; Revised 15 December 2010; Accepted 24 January 2011
Academic Editor: Vesa Valimaki
Copyright © 2011 T. Smyth and F. S. Scott. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
A physics-based synthesis model of a trombone is developed using filter elements that are both theoretically-based and estimated
from measurement. The model consists of two trombone instrument transfer functions: one at the position of the mouthpiece
enabling coupling to a lip-valve model and one at the outside of the bell for sound production. The focus of this work is on
extending a previously presented measurement technique used to obtain acoustic characterizations of waveguide elements for
cylindrical and conical elements, with further development allowing for the estimation of the flared trombone bell reflection and
transmission functions for which no one-parameter traveling wave solution exists. A one-dimensional bell model is developed
providing an approximate theoretical expectation to which estimation results may be compared. Dynamic trombone model
elements, such as those dependent on the bore length, are theoretically and parametrically modeled. As a result, the trombone
model focuses on accuracy, interactivity, and efficiency, making it suitable for a number of real-time computer music applications.
1. Introduction
Instrument synthesis involving real-time interactive sound
often faces trade-offs between accuracy and computational
efficiency to provide both parametric control and quality
sound production. It is often the case that a more playable
model, one that is more responsive to human gestural
input, is a better sounding virtual musical instrument than
one that prioritizes acoustic precision. That is, the more
the sound production can be effectively controlled in the
hands of a musician employing phrasing, nuances, and other
musical subtleties, the more the perceived sound quality will
approach that of an actual acoustic instrument. Nevertheless,
producing a model that is as acoustically accurate as
resources will allow requires knowledge of the instrument’s
acoustic characteristics, properties that may be effectively
obtained by measurement.
Acoustic accuracy becomes increasingly important if the
focus of the model’s application is less on interactive sound
production and more on model validation, inverse modeling,
and parameter estimation. For example, if the goal is to
extract physical parameter values from an instrument during
performance, the model’s produced sound when played with
proper parameter values will likely require a higher degree
of actual (over perceived) similarity to the instrument being
modeled. In this application, which is gaining increasing
attention in the physical modeling community [1–8], the
virtual model must often also account for the frequency
characteristics of any variables involved in the acquisition
of data, such as instrument radiation, mic placement, or
inclusion of any measurement device/apparatus that may
also alter the acoustic behaviour of the instrument being
modeled.
In this work, a physics-based synthesis model of a trom-
bone is presented, suitable for applications mentioned
above. That is, the aim is for high-quality real-time sound
production, with highly intuitive and interactive control
parameters, yet with a sufficiently accurate acoustic model
of the trombone instrument that its inverse transfer function
may be applied during real-time performance. Deconvolving
the effects of the bore and bell in the instrument’s produced
sound allows for the estimation of the dynamic effects of the
lip-valve signal, a signal related to the valve’s volume velocity
holding the primary sound control information for most
wind instruments such as blowing pressure, embouchure,
and more advanced playing techniques. The instrument’s
2 EURASIP Journal on Advances in Signal Processing
produced sound may be seen as the coupling of the pressure
input from the lips (the product of the volume velocity
and the bore opening’s characteristic impedance) with the
instrument bore and bell—a convolution of the lip-valve
signal and the trombone impulse response. With this view,
isolation of the lip-valve signal is a matter of obtaining and
deconvolving the instrument’s transfer function, as well as
any measurement variables, from the instrument’s produced
sound.
The focus herein, therefore, is on obtaining a parametric
model of the trombone’s transfer function in two positions:
one tapped at the position of the mouth and the other outside
the bell. The former may be coupled to a lip-valve model,
providing feedback of bore resonances and the pressure
difference across the lip valve (required for dynamic models
in which the bore pressure influences the behaviour of the
vibrating lips [9]), while the latter may be convolvedwith the
lip-valve signal to provide the instrument’s produced sound.
In both cases, a low-latency convolution operation must be
used, with the delay being less than one round-trip from
mouthpiece to bell and back, in order to yield the accuracy
of its waveguide counterpart when coupling the lip valve
tothebore[10,11]. Both transfer functions are expressed
parametrically, allowing for any necessary change in their
frequency response during performance.
To this end, the instrument transfer functions are gener-
alized to be valid for three physically assembled instrument
structures having incremental resemblance to the final
complete trombone: (1) a two-meter long cylindrical tube
with a closed end (I1 in Section 3), (2) the same cylindrical
tube with an open end (I2 in Section 3), and (3) the same
cylindrical tube with a trombone bell (I4 in Section 3). The
mouthpiece is omitted because its structure is not constant
during performance (as is the assumption made of the
trombone bore and bell). The vibrating lips create a time-
varying volume and opening in the mouthpiece, making it
difficult to characterize using the methods described here.
The model employs a measurement and a processing
technique from previous work [12], whereby waveguide
elements are estimated from several measurements of the
system’s impulse response, with the system having incremen-
tally varying boundary conditions to allow for the isolation
and estimation of filter transfer functions. The method
as described in [12] is explored using simple cylindrical
and conical tube structures, as these are well described
theoretically and provide a basis for validating measured and
estimated data. The method is extended and applied here
to measure instrument structures that are more difficult to
account for theoretically, and which are not expected to
change during performance, such as the reflection and trans-
mission of the trombone’s bell, with comparison to classic
modeling techniques showing consistency in the results.
2. Trombone Instrument Model
It is well known that wave propagation in wind instrument
bores may be modeled in one dimension using the waveguide
structure shown in Figure 1, with a bidirectional delay line of
Pressure input
X(z)
+
R0(z)+Y0(z)
λ(z)z−M
z−Mλ(z)
RL(z)
TL(z)
YL(z)
Instrument signal
Figure 1: Waveguide model of a cylindrical tube with commuted
propagation loss filters λ(z) and open-end terminating reflection
and transmission filters RL(z)andTL(z), respectively, and a
reflection filter R0(z)atthe(effectively) closed-end termination
corresponding to the position of the mouthpiece. Two instrument
transfer functions (1)and(2) are developed for observation points
yielding Y0(z)andYL(z), corresponding to the bore base and the
instrument output, respectively, in response to input pressure X(z).
length Msamples accounting for the acoustic propagation
delay in the cylindrical and/or conical tube section of a
given length and filter elements λ(z), R0(z), RL(z), and
TL(z), accounting for the propagation loss, reflection at
the mouthpiece, and open-end reflection and transmission
occurring at the position of the bell, respectively, all of which
may contain delays, poles, or “long-memory” information
on the acoustics of the noncylindrical/nonconical bore
section [13,14].
The bidirectional delay line is suitable for modeling the
left and right travelling waves in both cylindrical and conical
tube structures, for which exact analytic solutions are known
and can be derived by solving the wave equation. When
tube contours depart from these simple shapes, however,
the model must account for the continuous reflection and
transmission that is distributed along the length of the
changing cross-sectional area. For the flared opening of many
brass instruments, such as the trombone bell, it has been
shown that no one-parameter traveling wave solution exists
[15–17]. Nevertheless, computationally efficient approxima-
tions may be made using one-dimensional models, such
as those using Webster’s equation for adapted coordinates
[15,17], or a piecewise connection of several cylindri-
cal segments (discontinuous approximation [13]) and/or
conical segments (continuous approximation [18,19]), or
“constant curved” (C1-regular approximation [20]) corre-
sponding to the instrument’s profile. As wave propagation
within horns involves an admixture of higher-order and/or
evanescent modes, a one-dimensional model cannot capture
thecompletebehaviour.Another approach therefore, shown
in Figure 1, is to separate a horn into its cylindrical/conical
and flared sections, with lumped filters accounting for the
reflection and transmission of the flared bell. It has been
repeatedly observed that these filters tend to be minimum-
phase [12,21],andthusitishypothesized that these filters
possess this property, making them, by definition, well posed
in terms of causality and stability.
Inthecaseofthelumpedmodel,reflectionandtrans-
mission filter properties of the bell, which contribute signif-
icantly to the instrument model’s characteristic resonances,
EURASIP Journal on Advances in Signal Processing 3
may be obtained theoretically or by acoustic measurement
[12,22–26]. The latter is particularly beneficial if the section
need not be parametric because it is not expected to change
during instrument performance. Because a measured fre-
quency response is not limited by theoretical approximations
for propagation losses or those occurring at the open-end
boundary, it is expected to provide greater accuracy for
waveguide elements such as the bore and bell—provided the
method for obtaining their frequency response is valid.
As is mentioned in Section 1, three structures bearing
incremental resemblance to the complete trombone instru-
ment are constructed, all of which may be modeled using the
waveguide structure shown in Figure 1, with the characteris-
tic waveguide elements being changed according to tube/bore
width and length and the conditions at the boundaries
(observation points at the position of the mouthpiece and
outside the bell). The incremental measurements allow for
the isolation and estimation of each waveguide element,
including variables in the measurement setup, such as the
speaker transmission and reflection functions.
As mentioned above, depending on the method being
usedformodelingthelipvalve,itisusefultotapthesignal
flow diagram in Figure 1 at two different positions: at the
bore base, yielding pressure Y0(z), and at the instrument
output, yielding transmitted pressure YL(z), both in response
to input pressure X(z) at the bore base (position of the
mouthpiece). This yields two separate instrument transfer
functions for both coupling to the lip valve and producing
the instrument’s sound output. Ignoring the time-varying
component in the mouthpiece, computing the global transfer
function H=Y0/X at the bore base (position of the mouth-
piece) in the z-domain, yields
H(z)=Y0(z)
X(z)=1+λ2(z)RL(z)z−2M
1−λ2(z)RL(z)R0(z)z−2M,(1)
where λ(z) is the propagation loss and R0(z)andRL(z)are
the reflection functions describing the boundaries at the
position of mouthpiece and bell, respectively. Similarly, com-
puting the global transfer function G=YL/X at the instru-
ment output in the z-domain yields
G(z)=YL(z)
X(z)=TL(z)λ(z)z−M
1−λ2(z)RL(z)R0(z)z−2M.(2)
For all z,suchthat|z|<1, |λ2(z)RL(z)R0(z)|<1, a series
expansion of (1)and(2),
Y0(z)=X(z)1+λ2(z)RL(z)z−2M
×1+R0(z)RL(z)λ2(z)z−2M
+R2
0(z)R2
L(z)λ4(z)z−4M+···,
YL(z)=X(z)TL(z)λ(z)z−M
×1+R0(z)RL(z)λ2(z)z−2M
+R2
0(z)R2
L(z)λ4(z)z−4M+···,
(3)
makes a standard train-wave decomposition appear.
Blowing
pressure
pm(t)
Reed model Z0U(t)
g(t)
h(t)
yL(t)
y0(t)
Figure 2: A convolutional synthesis approach to the signal flow
diagram shown in Figure 1, with signals h(t)andg(t)beingthe
impulse responses of the instrument tapped at the positions y0(t)
and yL(t), the inverse transforms of (1)and(2), respectively. The
input pressure is the product of the characteristic (wave) impedance
Z0at the mouthpiece and the volume flow U(t), a signal generated
by a reed model in response to a blowing pressure pm(t).
Expressing the instrument model in this way conve-
niently allows the outputs shown in Figure 1 to be simulated
as the convolution of the input and the instrument transfer
functions at these two positions, as shown in Figure 2,
where h(t)andg(t) are the inverse Fourier transforms
of the frequency responses corresponding to (1)and(2),
respectively.
3. Measurement Setup
Filters for the equivalent digital waveguide and convolutional
synthesis models (Figures 1and 2) described by instrument
transfer functions (1)and(2) are estimated from measure-
ment using a technique introduced in [12,27]andlater
applied in [28]. The technique uses a method similar to [29,
30] for extracting reflection functions using train-wave-like
decomposition. In [29,30], reflection functions are obtained
for a complete instrument (combined bore and bell) by
measuring a B-flat trumpet and then made parametric by
altering the delay time between arrivals to simulate a change
in pitch—simulating a slide trumpet. In the method here,
however, individual waveguide elements corresponding to
those in Figure 1 are estimated from arrivals in several
instances of the system’s measured impulse response hav-
ing incrementally varying terminating/boundary conditions.
Sincethebellmayberemovedfromthetrombone,its
reflection and transmission functions may be estimated
using the measurement system described herein.
3.1. Measurement System Specifications. A test signal, a log-
arithmic swept sinusoid of sufficient length (20 seconds) to
ensure a sufficiently large signal-to-noise ratio (SNR) [31],
drives a speaker (CUI Inc. CMS020KLX) that seals one end
of a two-meter-long cylindrical acoustic tube. Audio (24
bit/44.1 kHz) input/output is done using a MOTU Traveler
mk3.
A microphone (JL-061C) is press fit into a hole in the
tube wall, as close to the speaker as possible and flush with
the tube interior wall (see Figure 3), to capture the signal
y0(t)—the pressure consisting of the sum of right and left
travelling waves—in response to the test signal transmitted
from the speaker. A second microphone (same model) is
4 EURASIP Journal on Advances in Signal Processing
Figure 3: A speaker seals one end of a two-meter long cylindrical
tube, with a colocated microphone press fit into the tube wall, flush
with the tube’s interior wall. A second microphone is placed 7 cm
outside the instrument/tube’s open end, on the tube/bore axis. This
setupmaybemodeledusingthesignalflowdiagraminFigure 4.
placed 7 cm outside of instrument’s open end, on axis
with the tube, to capture the signal yL(t). The peak level
for y0(t) measured approximately 90dB above the noise
floor standard deviation, whereas the SNR for yL(t)was
approximately 70 dB, both for a 20-second-long test signal.
As described in [31], the recorded signal is linearly
deconvolved to separate nonlinear harmonic distortion
(caused by the speaker) from the desired linear impulse
response—the response of the system had it been excited
with a pulse only one sample long (an ideal condition that
would otherwise be physically unrealistic in band-limited
systems). Experiments showed that by using this technique,
measurements taken with varying levels of speaker distortion
(including no distortion at all) have no visible effects on the
results presented here.
The measurement setup is applied to the four incremen-
tally constructed “instruments” I1, I2, I3, and I4, which may
be modeled following Figure 4 and/or Figure 11, allowing
for the confirmation of measurement consistency and the
isolation and estimation of waveguide model elements that
may be validated through comparison with theoretical
expectation. All instruments consist of a two-meter-long
cylindrical tube (PVC pipe) with a diameter of 2 cm and are
terminated at one end with a speaker, with the opposite end
having the following terminations:
I1: closed, producing a “perfect” reflection,
I2: open, producing a reflection and transmission for
open cylinders,
I3: shorter cylinder affixed, for calculating cylindrical
reflection and transmission in the presence of a 2-
port scattering junction between cylindrical sections,
I4: trombone bell affixed, producing a reflection and
transmission having the bell’s acoustic characteris-
tics.
Speaker
output
σ(z)+
ρ(z)+
Signal at interior microphone
Y0(z)
λ(z)z−M
z−Mλ(z)
R(z)
T(z)
YL(z)
Signal at exterior
microphone
Figure 4: Waveguide model of a cylindrical tube adapted from
Figure 1, with microphone capturing signal response Y0(z)atthe
bore base, and a colocated speaker having a transmission function
of σ(z) and a reflection function of ρ(z). At the opposite end, there
are three possible terminating conditions relating to instruments I1,
I2, and I4: (1) I1: R(z) =1andT(z)=0 for a perfectly closed
tube, (2) I2: R(z)=RL(z) and amplitude complementary T(z)=
TL(z)=1+RL(z) for an open cylinder, and (3) I4: R(z)=Rb(z)
and T(z)=Tb(z) for the appended trombone bell. Instrument I3 is
described by the diagram in Figure 11.
454035302520151050
Time (milliseconds)
−0.2
0
0.2
0.4
0.6
0.8
1
Amplitude
Closed cylinder arrival responses (adjacent speaker)
c1(t)=F−1{σ(ω)}
c2(t)=F−1{σ(ω)λ2(ω)(1 + ρ(ω))}
c3(t)=F−1{σ(ω)λ4(ω)ρ(ω)(1 + ρ(ω))}
Figure 5: Arrival responses for closed cylinder, showing evenly
spaced pulses as inverse transforms of combinations of transfer
functions for the speaker σ(ω), propagation losses λ(ω), and speaker
reflection ρ(ω).
The tube must be long enough to ensure that each arrival in
the measured impulse responses decays before the onset of
the next. As shown in Figures 5,7,and9,thetwo-metertube
is sufficiently long to satisfy these criteria, allowing arrival
echoes to be windowed for extraction from the measured
impulse response with negligible loss of information. A
rectangle window with a taper at the tail provided by a raised
cosine further ensures the complete decay of each echo.
The measurement technique applied to instruments I1
and I2 is described in this section, summarizing the measure-
ment technique in [12] (but using current measurements),
as well as providing a foundation and validation, as these
structures are well described theoretically. Beginning each
measurement session with measurements of I1 and I2 may
be viewed as a calibration step to ensure consistency of mea-
surements. Responses of I1 allow for the estimation of system
variables, the speaker transmission ρ(ω), estimated reflection
offthe speaker ρ(ω),andtheestimatedpropagationlosses
λ(ω)—all of which are necessary to estimate elements from
subsequent measurements. Measurement of I2 (along with
elements estimated from I1) allows for the estimation of
the open-end reflection RLand transmission TL.Both
EURASIP Journal on Advances in Signal Processing 5
101
100
10−1
Frequency (kHz)
−25
−20
−15
−10
−5
0
Amplitude (dB)
Estimated (blue) and theoretical (green) propagation loss
Figure 6: Theoretically modeled propagation loss (green smooth
curve) [21] and estimated propagation loss
λ(ω)(blue)showaclose
fit, in spite of a far-from-ideal speaker reflection response ρ(ω)(red
high-pass curve).
454035302520151050
Time (milliseconds)
−0.5
0
0.5
1
Amplitude
Open cylinder arrival responses (adjacent speaker)
o0,1(t)=F−1{σ(ω)}
o0,2(t)=F−1{σ(ω)λ2(ω)RL(ω)(1 + ρ(ω))}
(a)
454035302520151050
Time (milliseconds)
−0.5
0
0.5
1
Amplitude
Open cylinder arrival responses (7 cm from open end)
oL,1(t)=F−1{σ(ω)λ(ω)TL(ω)}
(b)
Figure 7: Arrival responses of the open cylinder at speaker (a) and
open end (b), showing evenly spaced pulses as inverse transforms
of combinations of transfer functions seen in Figure 5,withthe
addition of the open-end reflection RL(ω) (a) and transmission
TL(ω) (b). Notice the alternating polarity (a) as compared to
Figure 7, due to the negative reflection of an open end.
propagation losses λ(ω), reflection RL, and transmission TL
are compared with theory to ensure measurement accuracy.
Measurement of I4, formed by appending a trombone
bell to the end of the two-meter tube, allows for the
estimation of the trombone bell reflection and transmission
functions using either the method used for estimating
the open-end reflection for I2 or an improved method
that accounts for the junction introduced by appending
an acoustic object with a different cross-section than the
measurement tube, as described in Section 4.Instrument
I3, also described in Section 4,isanintermediarystepused
to validate the method of accounting for the junction by
comparing the open cylinder reflection estimated from this
measurement to that estimated from measurement of I2.
3.2. Estimating Speaker Transmission/Reflection and Propaga-
tion Losses from I1. In instrument I1, the two-meter-long
tube is closed at both ends, with the speaker at one end and
a rigid termination (a piece of plastic) producing a perfect
reflection at the other. As seen in Figure 5,themeasured
response to the test signal at the inside microphone yields
a sequence of uniformly spaced echoes or arrivals, separated
by the time it takes the sound to travel the length of the tube
and back, each one decaying almost completely before the
onset of the next. The first three arrivals, each having ample
SNR, are sufficient for estimating the speaker output σ(ω),
the speaker reflection ρ(ω), and the propagation loss λ(ω)
transfer functions.
Assuming both a microphone magnitude response that
is flat in the band of interest and no prior circulating energy
in the tube, the first arrival is simply the speaker output, the
speaker transfer function, given by
C1(ω)=σ(ω).(4)
Assuming sufficient decay of c1(t) before the onset of c2(t),
as observed in Figure 5, the second arrival is the sum of
the incoming reflection from the closed end, C−
2(ω)=
σ(ω)λ2(ω), and the simultaneous reflection from the speaker,
C+
2(ω)=σ(ω)λ2(ω)ρ(ω), and is given by
C2(ω)=C+
2(ω)+C−
2(ω)=σ(ω)λ2(ω)1+ρ(ω).(5)
Like the second arrival, each subsequent arrival consists of
the previous arrival with round-trip wall losses λ2(ω)anda
speaker reflection ρ(ω), yielding a third arrival given by
C3(ω)=σ(ω)λ4(ω)ρ(ω)1+ρ(ω).(6)
An intermediate variable,
ζ(ω)=C1(ω)C3(ω)
(C2(ω))2=ρ(ω)
1+ρ(ω),(7)
is then defined to yield estimates for the speaker reflection
transfer function
ρ(ω)=ζ(ω)
1−ζ(ω)(8)
and finally the round-trip propagation loss
λ2(ω)=C3(ω)
ρ(ω)C2(ω).(9)
As shown in Figure 6, one-way estimated losses obtained
using the square root of (9) show a very good fit to those
modeled from theory as described in [21,32].
3.3. Estimating Cylindrical Open-End Reflection and Trans-
mission from I2. Taking another measurement after remov-
ingtherigidterminationandleavingthetubeopenyields
the sequence of arrivals seen in Figure 7,withthetop
showing the arrivals o0,n(t) at the mic adjacent to the
speaker (corresponding to y0(t)) and the bottom showing
the arrivals oL,n(t) at the mic placed outside the tube, 7 cm
6 EURASIP Journal on Advances in Signal Processing
from its open end (corresponding to yL(t)). Using these
measurements along with those made of the closed tube,
allows for the estimation of the open-end reflection and
transmission functions, RL(ω)andTL(ω), respectively.
The first arrival at the speaker location O0,1(ω), like
C1(ω), is the speaker transfer function, and the comparison
of the two is a useful way of ensuring consistency from
onemeasurementtothenext.ThesecondarrivalO0,2(ω)
contains the same elements as C2(ω) but with the additional
contribution of the open-end reflection, RL(ω):
O0,2(ω)=σ(ω)λ2(ω)RL(ω)1+ρ(ω).(10)
Third and subsequent arrivals at this location follow in
the same fashion, consisting of the previous arrival with
additional accumulated round-trip losses as it propagates
from the microphone to the open end and back again.
The open-end reflection may be estimated by taking the
ratio of the second arrival spectra from the open and closed
tubes:
RL(ω)=O0,2(ω)
C2(ω)=σ(ω)λ2(ω)RL(ω)1+ρ(ω)
σ(ω)λ2(ω)1+ρ(ω).(11)
As shown in Figure 8(a),theestimatein(11) very closely
matches the theoretical expectation described in [12].
The first arrival at the mic placed outside the tube, 7 cm
from the open end is given by
OL,1(ω)=σ(ω)λ(ω)TL(ω).(12)
Dividing this first arrival spectrum by the product of the
speaker transmission (4) and wall losses estimated using the
square root of (9) yields the estimated transmission from an
open cylinder
TL(ω)=OL,1(ω)
C1(ω)
λ(ω).(13)
As seen in Figure 8(c), the estimated transmission (13)shows
a very close match to the amplitude complement of the
theoretical reflection (the expected relationship between
reflection and transmission for a cylindrical opening), vali-
dating the measurement taken from outside the tube’s open
end.
This section set up the measurement technique and
showed impulse response measurements made from instru-
ments I1 and I2, consisting of evenly spaced arrivals,
from which the speaker transmission σ(ω)andreflection
ρ(ω), cylindrical propagation loss λ(ω), open-end reflection
RL(ω), and TL(ω), corresponding to waveguide elements in
Figure 4, could be estimated. The method is validated by
comparing propagation loss λ(ω) to theory described in
[21] and the reflection/transmission to theory described in
[12,33]. Though comparison is made using magnitudes of
the frequency response, it should be noted that phase is also
important. It has been found that these elements tend to be
minimum phase, or approximately so, so that by definition
they are stable and causal, and a match in magnitude would
yield a match in phase as one is related to the other through
101
100
10−1
Frequency (kHz)
−12
−10
−8
−6
−4
−2
0
2
Magnitude (dB)
Estimated (blue) and theoretical (green) reflection magnitude
(a)
101
100
10−1
Frequency (kHz)
0.5
1
1.5
2
2.5
3
3.5
Phase (radians)
Estimated (blue) and theoretical (green) reflection phase
(b)
101
100
10−1
Frequency (kHz)
−40
−30
−20
−10
0
10
Magnitude (dB)
Estimated (blue) and theoretical (green) transmission magnitude
(c)
Figure 8: Estimated (blue) and theoretical (green, smooth curve)
open-end cylinder reflection magnitude (a) and phase (b). Theory
is based on the Levine and Schwinger approximation [33]with
suitable expressions found in [12]. The estimated transmission
magnitude (blue) is plotted with the amplitude complement (this
assumption is valid for cylinders only) of the theoretical (green,
smooth curve) reflection (c).
the Hilbert Transform. As the minimum-phase property
is very desirable for synthesis, in part because it ensures
invertibility, it is one that is imposed on any filters fit to the
measurements made here.
4. Appending the Trombone Bell
A trombone bell is attached to the open end of the tube,
with measurement yielding the arrival sequence shown in
Figure 9. The arrivals follow the same pattern as those for
the open tube, but with the bell reflection replacing the open
cylindrical reflection, RL(ω)=Rb(ω).
The bell reflection function may be estimated by taking
the spectral ratio of the second arrival B0,2(ω) to that of the
closed tube C2(ω), as was done for the cylindrical open end
EURASIP Journal on Advances in Signal Processing 7
454035302520151050
Time (milliseconds)
−0.5
0
0.5
1
Amplitude
Cylinder + bell arrival responses (adjacent speaker)
b0,1(t)=F−1{σ(ω)}
b0,2(t)=F−1{σ(ω)λ2(ω)RB(ω)(1 + ρ(ω))}
(a)
454035302520151050
Time (milliseconds)
−0.5
0
0.5
1
Amplitude
Cylinder + bell arrival responses (7 cm from bell)
bL,1(t)=F−1{σ(ω)λ(ω)TB(ω)}
(b)
Figure 9: Arrival responses of the cylinder and bell at speaker (a)
andoutsidethebell(b).Noticethelow-amplitudeeffects of the
junction reflection just before the higher-amplitude bell reflection
in the second arrival ((a), underlined at approximately 12–14
milliseconds).
in Section 3.3. That is, the estimated bell reflection
Rb(ω)is
theoretically given by
Rb(ω)=B0,2(ω)
C2(ω)=σ(ω)λ2(ω)Rb(ω)1+ρ(ω)
σ(ω)λ2(ω)1+ρ(ω).(14)
This would be the case if a discontinuity were not created
by appending the bell to cylinder. That is, though the radii
of the cylinder and bell provide a good nested fit (there
is no leak created at their junction) the bell radius at the
bell’s small end is not precisely flush with cylinder radius,
with the difference being sufficient to create a reflection and
transmission at their junction. The bell reflection estimated
using the ratio of second arrivals in (14) would therefore also
include the effects of this junction. This may be seen in the
second arrival of the impulse response b(t)0,2,whichshowsa
downward followed by an upward pulse before the onset of
the actual effects of the bell (see b(t)0,2 of Figure 9(a)).
Though the effects of the junction likely have very
little perceptible consequence, it is worthwhile to develop a
method that accounts for the junction so that other acoustic
objects/bells having a possibly greater mismatch to the
tube’s cross-sectional area may also be appended and more
precisely estimated. The processing method described below
is applied to measurements of another tube structure,
constructed by appending a larger, but shorter, cylindrical
tube section to the end to the two-meter tube (shown in
Figure 10). As the junction between two cylinders and the
open-end cylindrical reflection is well described theoretically,
a comparison between theory and estimated results is again
made to validate the processing technique.
Figure 10: Instrument I3 is created to verify results that account
for the junction created by appending a second tube object to
the original measurement tube. It has a cylindrical open-end
reflection which may, like that of instrument I2, be validated
through comparison with theoretical expectation.
4.1. Estimating Open-End Reflectance in the Presence of a Junc-
tion Using Instrument I3. To isolate the open-end reflection
RL(ω) and transmission TL(ω) in the presence of a junction,
instrument transfer functions (1)and(2)aremodifiedto
account for the reflection and transmission occurring at
the discontinuity between two adjacent cylindrical segments
having cross-sectional area S1and S2, respectively. As shown
in Figure 11, a reflection and transmission will occur, with
the reflection coefficient approximated by
k=S1−S2
S1+S2.(15)
Assuming a sufficiently large Mto isolate the second
arrival, that is, that once again the second arrival decays
sufficiently before the onset of the third, the transfer
function of the second arrival to the input Hk,2 =Y0,2/X
at the instrument base, accounting for junction reflection
coefficient k, may be expressed in the z-domain as
Hk,2(z)=Y0,2(z)
X(z)
=(1+R0)λ2
Mk+RL1−k2λ2
Nz−2N
1+RLkλ2
Nz−2N,
(16)
and an estimated open-end reflection function given by
RL,k(z)=−
H2
H2k−(1−k2)(1+R0)λ2
Mλ2
Nz−2N,(17)
where
H2=Hk,2 −(1+R0)λ2
Mk,(18)
and R0,RL,λM,andλNare all functions of z(omitted for
brevity). Figure 12 shows how closely using the ratio of sec-
ond arrivals (14) compares with theory and how accounting
for the junction using (17) can improve the match.
The transmission function may be similarly estimated
using the first arrival only, which arrives at the microphone
(placed 7 cm outside the second tube) after a delay corre-
sponding to the time taken to travel the length of the first
and second tubes. The transfer function of the first arrival to
the input Gk,1 =YL,1/X at the instrument output, accounting
8 EURASIP Journal on Advances in Signal Processing
X(z)
+
Y+
0(z)
+
R0(z)Y0(z)
λM(z)
λM(z)
z−M
z−M
Y−
0(z)λN(z)
z−N
z−N
λN(z)
YL(z)
TL(z)
RL(z)
k−k
1+k
1−k
+
+
Figure 11: A signal flow diagram of two adjoined cylindrical tubes (instrument I3), with a two-port scattering junction due to the difference
in the tube’s cross-sectional area. With the junction removed (or equivalently by setting k=0), the diagram is equivalent to the instrument
waveguide model in Figure 1.
101
100
10−1
Frequency (kHz)
−20
−15
−10
−5
0
5
Magnitude (dB)
Reflection estimate (blue) using ratio of second arrivals
(a)
101
100
10−1
Frequency (kHz)
−20
−15
−10
−5
0
5
Magnitude (dB)
Reflection estimate (blue) accounting for junction
(b)
Figure 12: Estimated and theoretical open-end cylinder reflection
magnitude as calculated using the ratio of second arrivals (a)
according to (14) and by accounting for the discontinuity created
by appending a larger radius tube (b) using (17). Notice the latter
produces a better fit to the theoretical reflection.
for junction reflection coefficient k,isgiveninthez-domain
by
Gk,1(z)=YL,1(z)
X(z)
=TL(1+k)λM(z)λN(z)z−(M+N)
1+RL(z)kλ2
N(z)z−2N,
(19)
yielding the estimated transmission transfer function
TL,k(z)=Gk,1(z)+Gk,1(z)RL(z)kλ2
N(z)z−2N
(1+k)z−(M+N).(20)
As shown in Figure 13,theeffects of the junction are
less significant when estimating the transmission function,
with both methods (13)and(20) producing very similar
results. This is not surprising since the recursive effects of
101
100
10−1
Frequency (kHz)
−50
−40
−30
−20
−10
0
10
Magnitude (dB)
Transmission estimate (blue) assuming a uniform cylinder
(a)
101
100
10−1
Frequency (kHz)
−50
−40
−30
−20
−10
0
10
Magnitude (dB)
Transmission estimate (blue) accounting for junction
(b)
Figure 13: Estimated and theoretical open-end cylinder transmis-
sion magnitude as calculated using (13), (a), and by accounting
for the junction created by appending a larger radius tube using
(20), (b). Notice there is less of a discrepancy between the two
methods than was seen for the reflection shown in Figure 12.A
slight measurement error is observable at approximately 5 kHz,
when the signal is higher than 0 dB.
the junction (already minimal in this case) in the first arrival
transmitted from the bell YL,1 are limited to the pressure
circulating in the second tube section, while the effects of
the junction inside the tube at the mouthpiece Y0,2 include
pressure that has circulated in both tube sections, as well as
the round-trip propagation from the speaker end to the open
end and back.
It should be noted that complete models were also
developed for transfer functions Hkand Gk. It was found,
however, that estimating
RL,kand
TL,kfrom Hk,2 and Gk,1,
respectively, yielded slightly better results, likely because the
first and second arrivals are not subject to the measurement
noise that would result as the SNR deteriorates with each
subsequent arrival in the entire impulse response.
EURASIP Journal on Advances in Signal Processing 9
Figure 14: The profile of the bell shows a close fit to the equation for
theBesselhorn,givenin(21), using the parameters from Tabl e 1 .
4.2. Trombone Bell Model and Measurement. The close match
achieved between theory and measurement for the adjoined
cylinders described above provides confidence that the same
measurement and processing technique may also be applied
to acoustic elements having more complex shapes such as the
trombone bell studied here. That is, the bell reflection and
transmission can be obtained using (17)and(20).
Though it is more difficult to validate the bell mea-
surement with theory, as the theory becomes increasingly
approximate for shapes departing from the purely cylindrical
or conical, it is nonetheless interesting and worthwhile to
observe consistencies in behaviour between the measure-
ment/estimation obtained here and approximate theoretical
expectation. To provide this comparison, a one-dimensional
model is developed from which bell reflection and transmis-
sion characteristics may be obtained. This is then compared
with the estimated reflection and transmission obtained
using (17)and(20).
For computational efficiency, musical horns are fre-
quently modeled in one dimension, with models either based
on Webster’s equation [16,34,35] or a piecewise connection
of cylindrical or conical sections corresponding to the
bell’s profile [15,23]. Since both piecewise conical and
horn function methods allow for the modeling of spherical
wavefronts, they are expected to provide better accuracy, and
asshownin[16], both methods produce very similar results.
Here, the bell is modeled using a piecewise connection of
conical segments.
As shown in Figure 14, the profile of the bell modeled
here is well described by the so-called Bessel horn,
a(x)=b(x+x0)−γ,(21)
where x0is the position of the mouth of the horn, xis the
distance from the horn mouth, and a(x) is the radius over the
length of the bell. The variables band x0are chosen to give
the correct radii at the small and large ends, while γdefines
the rate of flare [36]. Bessel horn parameters used here are
provided in Tab l e 1 , with the resulting curve showing a good
match to the trombone bell profile in Figure 14.
The bell is modeled using a piecewise connection of
Nconical sections (or frustums) where, assuming both
constant pressure and incoming volume velocities at the
Tab le 1: Parameters for Bessel horn described by (21) best fitting
the trombone bell.
Quantity Variable Value
Length of the bell (m) .502
Radius at bell mouth (m) .108
Radius at small end (m) .01
Bell flare constant γ.7
Position of the bell mouth (m) x0.0174
Fitting parameter b0.0063
junctions, the relationship among traveling waves in adjacent
sections may be written in matrix form as
⎡
⎣p+
n
p−
n⎤
⎦=An⎡
⎣p+
n+1
p−
n+1⎤
⎦, (22)
where the scattering matrix is given by
An=⎡
⎢
⎢
⎢
⎢
⎣
Zn
Zn+1
Zn+1 +Z∗
n
Zn+Z∗
n
ejkLn+1 Zn
Z∗
n+1
Z∗
n+1 −Z∗
n
Zn+Z∗
n
e−jkLn+1
Z∗
n
Zn+1
Zn+1 −Zn
Zn+Z∗
n
ejkLn+1 Z∗
n
Z∗
n+1
Z∗
n+1 +Zn
Zn+Z∗
n
e−jkLn+1
⎤
⎥
⎥
⎥
⎥
⎦,
(23)
for a section of length Lnand with complex wave impedance
Zn. For a model having Nsections, N−1scatteringmatrices
are multiplied,
⎡
⎣p+
1
p−
1⎤
⎦=A1A2···AN−1⎡
⎣p+
N
p−
N⎤
⎦, (24)
to yield the model’s single final scattering matrix
P=
N−1
n=1
An,(25)
relating the bell input and output traveling pressure waves.
The expression for the reflection function of the bell may
be formed from the above model by taking the ratio of the
wave reflected by the bell p−
1to the bell input wave p+
1,
RB=p−
1
p+
1
=λ2(ω)p+
NP2,1 +p−
NP2,2
p+
NP1,1 +p−
NP1,2
=λ2(ω)P2,1 +P2,2RL(ω)
P1,1 +P1,2RL(ω),
(26)
where the final expression is obtained by incorporating an
open-end reflection at the termination of the Nth section by
substituting p−
N=p+
NRL(ω) and by commuting round-trip
propagation losses λ2(ω).
Similarly, the bell transmission is given by the ratio of the
wave radiated out the bell p+
NTL(ω), where TLω)istheopen-
end transmission function, to the bell’s input p+
1,
TB(ω)=p+
Nλ(ω)TL(ω)
p+
1
=λ(ω)TL(ω)
P1,1 +P1,2RL(ω).(27)
10 EURASIP Journal on Advances in Signal Processing
0.50.450.40.350.30.250.20.150.10.050
Bell length (meters)
0
0.02
0.04
0.06
0.08
0.1
Bell radius (meters)
Figure 15: Geometry of bell (solid line) showing radii of eight
piecewise conical/frustum segments and resulting modeled profile
(dashed line).
101
100
10−1
Frequency (kHz)
−40
−30
−20
−10
0
Magnitude (dB)
Estimated (green) and modeled (blue) bell reflection
(a)
101
100
10−1
Frequency (kHz)
−40
−30
−20
−10
0
Magnitude (dB)
Estimated (greed) and modeled (blue) bell transmission
(b)
Figure 16: Estimated and modeled bell reflection (a) and transmis-
sion (b) magnitudes.
As in Section 3.3, suitable expressions for RL(ω)andTL(ω)
may be found in [12].
The trombone bell is modeled using eight conical sec-
tions, with geometry and dimensions determined from the
Bessel horn function (21)andTa b le 1 , with profile plotted in
Figure 15. The commuted propagation losses are modeled as
described in [21], using the median of the bell profile radii
and the bell length (from Tab l e 2 ) as input parameters.
As mentioned above, it is not expected that the estimated
and modeled bell reflection and transmission will match to
the same degree as with the open cylinder, largely because the
1-D model does not take into account higher-order and/or
evanescent modes. Indeed, if the match were too good, it
would negate the need to measure the bell reflection and
transmission for improved accuracy. Nevertheless, as shown
in Figure 16, the magnitude of the modeled bell reflection
(26) and transmission (27) functions displays very similar
behaviour to that estimated from measurement, (17)and
(20), respectively, providing confidence in the accuracy of the
measurement and estimation technique.
Tab le 2: Trombone tubular sections (numbers correspond to parts
in Figures 17 and 18) and dimensions, including top (t.) and bottom
(b.) inner and outer slides, retracted and extended (ext.).
Part Length (cm) Radius (cm)
t. inner slide (1) 70.8 0.69
t. outer slide, ext. (2) 53 0.72
slide crook (3) 17.7 0.74
b. outer slide, ext. (4) 53 0.72
b. inner slide (5) 71.1 0.69
Gooseneck (6) 24.1 0.71
Tuning slide (7) 25.4 0.75, 1.07
Bell flare (8) 56.7 1, 10.8
5. Trombone Model and Measurement
With the trombone bell measurement providing both reflec-
tion and transmission transfer functions and theoretical
propagation losses for cylinders with a different cross
section, it is possible to assemble results to complete the
instrument transfer functions described in (1)and(2)for
the trombone. The only value yet unknown is the reflection
at the mouthpiece R0(ω). As this is expected to change
during performance with the vibrating lips changing both
the mouthpiece volume and the opening to the bore, it is
not suitably obtained using the methods described here, but
rather is left for work currently in progress, whereby the
generalized valve model [9] is configured and coupled to the
trombone instrument model presented here.
A complete trombone (mouthpiece omitted) is shown
in Figure 18, with corresponding trombone components and
dimensions provided in Tab l e 2.Figure 17 shows an interior
view of the complete trombone in both retracted and
extended positions, producing bores with effective lengths of
209.1 cm and 315.1 cm, respectively, with asterisks showing
possible cylindrical junctions that may or may not be consid-
ered depending on the desired level of accuracy. Trombone
components 1–7 in Figure 17 are modeled as cylindrical
waveguide sections, following dimensions in Ta b l e 2 for
appropriate delay length and radius, parameters used for the
propagation loss model described in [21].
In future work, it would be very interesting to compare
the results of the trombone model presented here, in
retracted, extended, and intermediate positions, with an
accurately obtained input impedance of the complete trom-
bone. In the meantime, however, the validation of the model
and measurement of the trombone’s composite elements
provides confidence that their assembly into the real-time
model, described by (1)and(2), contributes an accurate,
real-time interactive virtual instrument suitable for control
parameter estimation and, when combined with a lip-
valve model, interactive sound production, both significant
applications in real-time music performance.
6. Conclusions
The trombone model presented here consists of two trom-
bone instrument transfer functions, one taken at the position
EURASIP Journal on Advances in Signal Processing 11
Retracted
76 5
13
8
(a)
Extended
76
8
5
1
4
23
(b)
Figure 17: Interior view of trombone, in both fully retracted and
fully extended positions, showing assembly of components from
Tabl e 2 .
65
43
2
1
Figure 18: Trombone tubular components corresponding to
Tabl e 2 .
of the mouthpiece for dynamic coupling to a lip-valve
model and one taken outside the bell for the production
of the transmitted sound and for inverse modeling applica-
tions. Model transfer functions consist of several dynamic
and static unknown acoustic waveguide elements due to
frequency-dependent propagation and boundary conditions.
Those expected to change during performance, such as the
effective bore length with the moving hand slide, are made
suitably parametric, with theoretical propagation losses for
cylinders being dependent on both tube length and radius.
Elements not expected to change during performance, such
as bell reflection and transmission, are obtained by extending
a previously introduced measurement technique shown to
be accurate for cylindrical and conical sections, but further
developed here for the estimation of more complex acoustic
tube structures.
The model elements estimated from measurement are
validated by comparison with theoretical expectation. For
the cylindrical case, which is well described theoretically,
the propagation losses and open-end cylindrical reflec-
tion/transmission are shown to produce a very close match
to theory. This both validates the measurement/estimation
technique and provides confidence that theoretically mod-
eled propagation losses may be used for cylindrical/conical
sections in lieu of measured data (should measurements
be difficult to obtain), with little consequence to synthesis
accuracy. For efficiency and interactivity in real-time perfor-
mance, in fact, it is likely preferable to use a theoretically
based parametric filter for propagation losses such as that
described in [21].
In the case of the trombone bell, the flared opening
departs considerably from the purely cylindrical or conical
sections of the bore and has no one-parameter traveling wave
solution. The acoustic behaviour of such structures cannot
be completely described using a one-dimensional model, as
such a model would not capture higher-order and/or evanes-
cent modes. Nevertheless, bell reflection and transmission
functions are estimated from measurement and then com-
pared with a one-dimensional bell model developed using a
piecewise connection of conical segments corresponding to
the bell profile. The bell model is expressed in the frequency
domain, using a matrix notation from which expressions
for bell reflection and transmission transfer functions are
obtained algebraically. Though it is not expected that the
bell model and measurement comparison will yield the close
match seen for the unflanged open-end boundary condition
at the end of a cylindrical pipe, the comparison does show
consistent behaviour. Considering the prior validation of
the measurement technique for cylindrical structures better
described theoretically, the response of the bell based on
measurement is expected to be more accurate than that
described by the one-dimensional model.
Though it is the magnitude of the frequency response
that is presented here, this in no way implies that the
phase is not important. Rather, it has been found that filter
elements possess the minimum-phase property, or at least
approximately so, making plots of their phase redundant to
the magnitude, as they are related by a shift of 90 degrees. A
match in magnitude, therefore, implies a match in phase as
well. This desirable property also assures that, by definition,
the elements are invertible and well posed in terms of
causality and stability.
In this work, a parametric synthesis model of the
trombone is developed, focusing both on accuracy and
efficiency, making it suitable for real-time computer music
applications. In particular, it is currently being used to
determine suitable input parameters for the generalized
pressure-controlled valve model, described in [9], with the
trombone model here providing an instrument body for the
“blown open” configuration of the valve. It is also being
used in developing inverse modeling strategies to extract
12 EURASIP Journal on Advances in Signal Processing
a lip-valve signal—a signal corresponding to the pressure
input into the bore at the mouthpiece—using only the signal
recorded from the trombone during real-time performance.
This will allow trombone performers to generate computer
input parameters (perhaps to another synthesis model) by
making subtle changes to embouchure and input pressure,
allowing them either to extend their own instrument through
effects processing or perhaps to control another virtual
instrument altogether. The trombone is a particularly well-
suited instrument for this goal because, unlike other wind
instruments having toneholes and a myriad of possible
fingerings, the trombone is well described by a relatively
simple waveguide with only a length that changes with
a movement of the slide during performance. Trombone
synthesis by model and measurement provides a parametric
model that is accurate, efficient, and interactive, making it
suitable for both interactive sound production and inverse-
modeling (parameter estimation), both having considerable
significance to applications in real-time music performance.
Acknowledgments
The authors would like to sincerely thank the anonymous
reviewers of this article, whose diligence and attention to
detail have greatly strengthened the presentation of this
work. Jonathan Abel continues to impact this research
through ever-illuminating discussion and insight. In addi-
tion, the authors would like to acknowledge the support of
the Natural Sciences and Engineering Research Council of
Canada (NSERC).
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