Thompson_Aaron_MP

SAE Oxford
Exploring the Acoustical Design of a Natural
Trumpet through Digital Waveguide Modelling
Aaron Thompson
Student number: 15351
AD1111
Date of submission: 14th
October
Word Count: 10906

2

Affidit:
I hereby declare that I wrote this thesis on my own and without the
use of any other than the cited sources and tools and all explanations
that I copied directly or in their sense are marked as such, as well as
that the thesis has not yet been handed in neither in this nor in equal
form at any other official commission.
……………………………………………………….
Aaron Thompson, SAE Oxford, (14/10/12)

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Preface:
Being a Trumpet player since the age of 8, my inquisitive interest in the
instrument’s design and mechanic has been instilled ever since I played my first
notes. Among many other things, the main interest for me was the sheer power
and strength behind the instrument, be it within an orchestral context or lead
trumpet screamers of the big band era. The what appeared to be limitless range
of the instrument quickly became a personal highlight, of which, like many other
fellow Trumpet players, lead me to aspire to the Trumpet greats such as Maynard
Ferguson, Wayne Bergeron and Arturo Sandoval.
The relentless quest for greater range and endurance is now highly common to
nearly every Trumpet (and to some extent brass in general) player. With
impatience acting as a weight across every aspiring musicians shoulder, a quick
fix is often at the forefront of the mind. Therefore with various music instrument
companies now offering an almost “tailor-made” work ethic with the instruments
they offer, I naturally began to question what are they changing in order to satisfy
this degree of flexibiltiy. Are they designing by aesthetically driven trial and error or
is there an acoustical theorem defining every curve, radius and flare. Having
recently gone through the trial and error method when purchasing a new
instrument, this thesis aims to explore this hypothesis by collaborating with my
more recent endeavours within the audio industry, namely with the interest of
saving time and money in the future by making an educated first step when
attempting to find the “right” instrument.

4

Acknoweldgements:
Throughout writing this thesis, I have had the pleasure of collaborating with a
number of proffessionals and researchers alike. First and foremost my thanks and
gratitude go to Andy Farnell, with whom I attended an almost weekly tutorial slot
thoroughly discussing the research, development and resultant completion of this
research project as well as numerous emails and exchanges throughout.
Secondly, I greatly appreciate and am extremely thankful for the contribution and
personal guidance from various world leading researchers such as Julius O.
Smith, Maarten Van Walstijn, Richard A. Smith and Wilfried Kausel from the
“Institut für Wiener Klangstil”. Thirdly, my gratitude extends to the representatives
from the leading brass instrument manufacturers Smith Watkins, Yamaha and
Schagerl for there technical papers and further referrance as well as advice and
guidance from artists such as Onyx Ashanti and the world-reknown brass player
James Morrison.
I’d also like to personally thank Ian Cummings and David Runkel for the loan of
their instruments during testing, the teaching staff at SAE Institute for their
encouragement, the PureData community and finally to my close friends and
family for their much needed moral guidance and support throughout the entire
process.

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Contents:
1) Introduction pg4
2) Lead Pipe
2.1) Cavity Resonance pg9
2.2) Main Bore pg12
2.3) Mouth Pipe pg14
3) The Bell
3.1) Wave Impedance pg15
3.2) Horn Design pg18
3.3) Digital Filters pg19
3.4) Z-Transform pg21
3.5) Discrete Time modeling pg21
4) The Mouthpiece
4.1) Acoustical Influence pg22
4.2) Cup Volume pg24
4.3) Waveguide Implementation pg25
5) The Embouchure
5.1) Lip-Valve Mechanism pg27
5.2) Modeling Techniques pg29
5.3) Oral Cavity Resonance pg31
6) Final Waveguide pg32
7) Impulse Response
7.1) Linear Time-Invariance pg34
7.2) Methodology pg34
7.3) Results pg36
8) Conclusion pg39
9) Appendix pg45
10) Reference List pg60
11) Bibliography pg63

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Introduction:
Analysis of an instrument’s quality could be carried out within two discrete
categories. Firstly, the physical factors, such as the instruments internal design
and construction which are in turn measurable within an experimental context.
Secondly, the musical response and subjective analysis of the resultant effect of
the former. Smith (1986), states “attitudes towards a given instrument can vary
widely between players, audiences, recording engineers and so on”. Due to the
fickle and somewhat erratic nature of audience perception, it would therefore be
illogical to explore the instruments design through this category within a scholarly
context as it is likely the conclusions drawn would prove invalid. Therefore, in
order to establish what constitutes an instrument of “quality”, one must consider
the physical factors outlining the instrument’s performance.
Although an exploration of these factors through the method of physical
prototyping allows quantitative data to be collected and stepwise progression to
be adopted, it also brings about further complications. Namely, a lack of time
available to complete this thesis, as multiple versions of various prototypes would
need to be made and analysed. Additionally, by carrying out this development,
one includes the experimental variance of human error. Furthermore, the lack of
access to materials, funding and appropriate training needed to carry out these
prototypes to a comparable standard renders this option illogical. A highly more
cost efficient, time consuming and experimentally valid method would be to use
various forms of computational modelling.
The leading two instances of physical modelling consist of “Lumped Mass” and
“Waveguide” modelling. Both forms of physical modelling are based on numerical
integration of the “wave equation”, but both instances approach this mutual
objective via differing methodologies. Lumped Mass modelling essentially consists
of a physical comparison across mechanical interactions such as springs or
masses. With Newtonian law often defining said interaction, a scaled
mathematical representation can be drawn and is said in general, to be useful
when the dimensions in question are small relative to wavelength of vibration

7

(Smith, 1991), thus making themselves useful in situations such as the Trumpet
player’s lips. Due to the scope of this thesis, this approach will be considered but
not necessarily applied. Waveguide modelling on the other hand uses a simple bi-
directional delay line to model the interaction of transverse waves within a
predetermined acoustic system. Not only is waveguide modelling a
computationally inexpensive method of carrying out its mathematical equivalent, it
is also very simple to create and integrate within various forms of programs, such
as PureData.
Development of the patch within PureData poses many advantages with audible
feedback being the leading point of interest. Dimensions and properties within the
waveguide can be changed in real time, allowing the patch to not only be used as
an acoustical model, but also, through further development, as a software
instrument. Furthermore, the added interactivity via the user definable variables
within the waveguide expands the market into the creative composition
environment. Effects of non-realism and real time adjustment (within the
mathematical boundaries of the patch) such as the timbral effect of a considerably
extended or distorted flared bell can be investigated and the resultant synthesis
allows for exploration within an electroacoustic and algorythmic composition
context.
In order to create this interactivity between the user and the instrument, one must
approach the waveguide using Walstijn’s (2002) method by dividing the
instrument into its constituent parts, analysing each qualitatively, quantitatively and
finally integrating them within the waveguide. Initially, creating a fully functioning Bb
Trumpet waveguide was intended, but after researching the extended tolerance
and complications the valves (among many other variables) for instance bring
about, it was decided that the scope of the project should be reduced to a Natural
Trumpet, of which has no valves. The Natural Trumpet can therefore be broken
down into 4 parts.
Firstly, one must consider the most basic component, the lead pipe. The lead pipe
consists of 2 further parts, the mouth pipe and the main bore. Being essentially a

8

closed-open pipe configuration, this in itself dictates the pressure node positions
within the pipe. Splitting the main bore into two discrete waveguides, the main
bore can be accurately modelled as a basic tubular waveguide, whereas the
mouth pipe has to be dealt with separately as a truncated cone due to the
continuous change of wave impedance.
Secondly comes the introduction of the flaring bell, typical of all brass instruments.
The shape of the bell strongly influences the positioning of the resonances within
the tubing as well as how the instrument radiates the tones created. Benade
(1990) draws a useful analogy between the average bell shape and the
mathematical Bessel function, to which Smith + Walstijn (1998) later develops a
digital parallel in the form of a “Truncated Infinite Impulse Response” Filter (TIIR) to
model this effect within a waveguide context.
Thirdly, the introduction of the mouthpiece not only provides a comfortable
position for the player’s lips, but also further tailors the higher resonances within
the cavity towards a musically useful harmonic series. The complication within
these two systems arises from the deduction that the vibrations of the lips are
strongly influenced by the air column to which they are connected to (Benade,
1990) therefore dictating the order in which this waveguide should be developed.
With the acoustic cavity established, the next section would be to attach a
transducer to the end of the cavity, in this case a lip-valve mechanism to provide a
periodically varying flow of air. Unlike woodwind instruments, this periodic flow of
air is generated through the vibration of the player’s lips. This vibration then
becomes musically useful as the player controls the pitch of the vibration through
the tension and inertia of the lips. Another means of varying pitch is to increase the
velocity of the air traveling through the aperture. This is achieved through the
arching of the tongue, thus increasing air speed consequently reducing the oral
cavity. Drawing a comparison to that of speech phonetics, oral cavity resonance
therefore becomes integral within the transduction and will therefore be discussed
within the model.

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Finally, upon completion of the waveguide, in order to objectively assess the
validity of the model, impulse response testing shall be carried out. Results from
the waveguide and a selection of comparable instruments will be gathered and
analysed. Secondly, as a Trumpet player myself, I can accurately perform testing
on an aesthetic standpoint, analysing how the model behaves as an instrument
under certain typical playing conditions.
Due to the nature of the instrument, the scope of the waveguide model is
restricted within this thesis and therefore, the final version of the model will be
somewhat limited. Particular care has been taken to acknowledge the key
variables being omitted during development but in an effort to streamline the
scope of this thesis, not all possible variables will be considered. Furthermore,
comparing a software model to a real life tangible object brings about an added
level of inaccuracy, namely within the methodology as variables such as human
error and lack of access to materials/environments (for example, an anechoic
chamber) will continue to distort results. Despite this, intended further research will
be carried out during postgraduate study to amend these said variables.
Lead Pipe:
2.1) Cavity Resonance
In the interest of approaching each component of the instrument discretely, one
must first consider the lead pipe as a stand-alone acoustical body. In this case
generalising the lead pipe as a basic tubing of given length and diameter, which
therefore brings about certain acoustical properties.
Let us firstly consider two identical transverse waves travelling in opposite
directions within a tube of length 2x. When the two waves interact, a standing
wave is formed between them. If we consider a central point P being a distance of
x from either source, then the two waves interact in phase and therefore interfere
constructively forming an antinode. Contrastingly, if one was to move λ/4 closer to
either wave, the waves now posses a path difference of λ/2 and therefore a phase
relationship of π radians. The phase relationship during this interaction brings

10

about destructive inference thus forming a node. If we consider this tube to be
entirely lossless for the purpose of this analogy, the separation of adjacent nodes
will always remain λ/2 along length 2x.
Adams and Allday (2000) state that a standing wave will only form if the
wavelength is related to that of the length of tubing. Therefore, if one was to adjust
the frequencies of waves, the tube would resonate at certain frequencies with the
fundamental frequency being the lowest frequency resonance. The fundamental of
any tube length L therefore can be calculated assuming that the speed of sound
within length L remains constant.
!! =
!
2!
A tube closed at both ends has no musical importance since no sound can
radiate from it, therefore, one must consider variations upon this tube. If we
consider for example, a closed-open configuration, the analogy becomes more
useful within a musical instrument context. One can already draw from previous
conclusions that a boundary condition can be said to have maximum pressure
forming an antinode. If we consider the open end of a tube, the pressure outside
the tube is atmospheric and one can therefore assume pressure = 0, therefore
forming a node. Within these parameters, the fundamental of this tube is half that
of a tube of open-open configuration. Thus forming:
!! =
!
4!
Furthermore, since that one loop of the standing wave now occupies two-thirds
the length of the tube for the second harmonic, one can summarise that the
frequency of this mode is “3(c/4l)”. The modal frequencies for a closed open tube
are therefore odd integral multiples of the fundamental, contrasting to the even
multiples achieved from open-open configuration.
At the open end of an air column, air is moving in and out of the open end and its
motion extends beyond the limits of length l. Backus (1977) states that this
“makes the tube appear longer by an amount called the end correction” and goes

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on to state that for a cylindrical pipe of radius r, the end correction has been
calculated to be “0.61r”, therefore forming:
!! =
!
4! + 0.61!
The end correction varies with frequency but the effect is small enough to be
classed as negligible.
Now if one was to consider a pipe of the same length “l”, but with a conical cross
section, components within the standing waves begin to change. Initially, one
must first refer back to the open-open configuration. A travelling compression
impulse will propagate along the length l towards the open end, at which it is
reflected as a rarefaction (reasons for which will be discussed later in this thesis)
and the displacement impulse reflects unchanged. Now let us consider the same
compression impulse within a conical tube of the same length. The impulse still
travels with speed c but as it propagates through a decreasing area, the pressure
and displacement amplitudes increase with the displacement amplitude distorting
along length l. As the impulse reaches the open end the reflection is identical to
that of an open-open pipe, but as the impulse travels back down the pipe the
impulse reverts back to its original form. Due to the comparable reflection
behaviour, the conical shape makes no difference to the fundamental or the
multiples of that fundamental but it is worth noting that the length of the cone is
given by the internal length of the slant as apposed to along the axis.
(Wolfe, J n.d., Pipes and Harmonics [online], UNSW, Available from: http://www.phys.unsw.edu.au/jw/pipes.html
[Accessed: 28.8.2012])

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2.2) Main Bore
Having qualitatively discussed the effects of physical boundaries and shaped air
columns, one must now apply these theories within a waveguide context to then
apply these concepts within PureData.
By drawing an analogy with an ideal lossless string, Smith (1992) concludes that a
digital waveguide is based on spatial-temporal sampling of the traveling-wave
solutions of 1-D waveguides. One can then draw the conclusion that this can be
applied to that of an acoustic cavity by substituting string displacement for air
pressure fluctuation and transverse string velocity for longitudinal volume velocity
of air in the bore (see appendix E). Therefore, with this analogy one can derive
from the previous chapter that a standing wave within a cavity consists of two
oppositely propagating pressure waves interacting. Walstijn (2007) begins with the
wave equation within a lossless cylindrical duct showing pressure (p) as a function
of distance (x) and time (t):
!!
!
!"!
=
1
!!
!!
!
!"!
By referring to the previous chapter, one can conclude that the equation can
therefore be solved by the sum of the two oppositely travelling pressure waves
(pi
+
, pi
-
) travelling at speed c. Walstijn continues to say that given any point (x=xi) in
a duct of cross section S, the pressure (pi) and volume velocity (ui) can be given
by:
!! =
!!
!
− !!
!
(!"/!)
With ! denoting average air density we can therefore label “ !"/! ” as the
characteristic impedance (Z0).
Therefore, within the digital domain, assuming that sample period is T=1/fs,
propagation from left to right of cylinder length L can be modeled using a delay
line of N samples. Välimäki and Laakso (2000) outline a problem within this
system. Firstly, in order to achieve the realtime transition or more appropriately
within this context, being that in order for such musical instrument models to be

13

“in tune”, the delay lengths required are not multiples of the sampling frequency of
the system. Puckette (2007) also identifies that artifacts caused by varying time
delay become noticeable even at very small relative rates of change. Walstijn
(2007) accounts for this problem by using a fractional delay line (HFD) of non-
integer delay length D and states that “lower-order Thiran allpass filters and
Lagrange FIR interpolation filters are most commonly used”. Puckette (2007)
states that this problem can be solved by a 4-point interpolation scheme with fairly
reasonable computation efficiency by putting a cubic polynomial through the four
“known” points and then evaluating at then at point “D”. A disadvantage arises as
this scheme can be used for any delay of at least one sample due to the nature of
the interpolation. The system can be improved but Puckette outlines a trade-off
between quality and efficiency as well as the limitation introduced with higher
order interpolation increasing the minimum delay time.
A further point of interest arises when the fidelity of interpolating delay lines is
considered. Puckette states that variable delay lines introduce distortion to the
signals they operate on. Assuming the use of the aforementioned 4-point
interpolation, for sinusoids with periods longer than 32 samples, the distortion is
unnoticeable. Therefore at a 44.1khz sample rate, these periods would
correspond to frequencies up to about 1400hz, anything above this threshold
become noticeable non-linearity. If this is the case, then instead of increasing the
number of points of interpolation one can therefore increase the sample rate of the
system. But it could be argued that the distortion is merely being reduced as
apposed to being corrected.
Walstijn (2007) continues to improve the validity of the model by taking
viscothermal losses into account. Walstijn et al (1997) note that viscothermal
losses occur at the boundary layer of a tube and are dependent on the bore
radius and the temperature of air inside the tube. The major loss of energy due to
viscothermal effect is expressed in the transmission function of the main bore,
which can again be modeled through the use of a delay line. Abel et al (2003)
state that although the losses associated with viscous drag and thermal
conduction are distributed along the length of the tube, within a digital waveguide

14

it is more efficient to lump these losses by commuting a characteristic digital filter
(Hloss), such as a 4th
order IIR filter suggested by Walstijn (2007), to each end of the
waveguide.
(Walstijn, M 2007, Wave-Based Simulation of Wind Instrument Resonators, IEEE Signal Processing Magazine, March 2007)
A further variable not identified in Walstijn’s design is the effect of wall thickness.
Smith (1987) concluded that the thinner the material, the greater degree of
vibration and therefore harmonics being produced up to 2dB stronger greatly
affecting the resultant tone (see appendix G). One could assume that these
changes could be modelled by altering the lumped loss filter, but in order to
improve this generalisation further research would be required.
2.3) Mouthpipe
Considering the mouthpipe as a simple conical section of defined length, as with a
cylindrical waveguide, the wave equation can be solved through the sum of the
positive and negative pressure waves. Although despite this initial similarity, the
conical shape as explored in chapter 2.1, introduces amplitude discrepencies and
frequency dependent impedances.
Walstijn (2007) notes that now the pressure waves are scaled by the distance r
from the cone apex and goes on to state that this can still be modeled using the
same delay line, fractional delay line and loss filter explored in the previous chapter
but including a scaling factor to approximate the spread of the wavefront due to
inverse square law (see appendix F). The complication within this model arises
when you consider the impedance within the system now being frequency
dependent. If one was to consider this frequency dependent scattering within a
waveguide context, the losses may also be lumped into a scattering junction.

15

Walstijn et al (1997) concludes that the losses for a conical section can be
calculated as for a cylindrical tube with identical length and with a radius that
equals the mean radius of the cone. For example, considering the connection
between the mouthpipe and the main bore waveguide, one may consider the
conical profile of the mouthpipe to be a cylindrical pipe with a radius matching that
of the mean radius of the outward flaring cone, connected via a scattering junction
to a larger pipe with the radius matching that of the main bore.
Smith (2010) states that when a travelling wave encounters a change in wave
impedance, it will partially reflect the incoming signal, and the remainder will
transmit into the new impedance. Therefore, if we consider a pipe with impedance
R1 connected to another pipe of impedance R2 then the reflection coefficient (k1)
can be given by:
(Smith, J.O. 2010, Physical Audio Signal Processing, W3K Publishing, California.)
The Bell:
3.1) Wave Impedance
Benade (1990) poses an experiment within which a driving crank is loaded to a
pump cylinder providing sinusoidal variations of air pressure at the driving motor’s
frequency (see appendix D) using what is known as the “capillary excitation”
method. Let us now consider this arrangement with a pipe of a uniform cross
section with a microphone attached to the mouthpiece end. In order to avoid

16

disturbances travelling back from the end of the pipe we shall consider this pipe to
be of indefinite length. Benade notes that the pressure measured at the
microphone is independent of frequency, having a magnitude that is equal to the
product of the capillary driver’s source strength and the wave impedance of the
duct. For a pipe of cross-sectional area (A), the wave impedance is defined by:
!"#$ !"#$%&'($ = (
1
!
) !"
Therefore, assuming that A remains constant, wave impedance is the ratio of the
pressure to the volume flow injected into the duct.
Let us now consider a pipe with the same value of A, but having length
comparable to that of a Trumpet. The pressure wave created by the sinusoidal
disturbance propagates down the length of the pipe losing amplitude as it travels
due to viscothermal losses. At the pipe opening, the wave impedance changes
dramatically since the room can be imagined to be a second pipe of enormous
cross-sectional area. The pressure wave upon encountering this junction is almost
totally reflected back towards the mouthpiece end with the reflected wave being
inverted as a momentary rarefaction. This wave then interacts with newly injected
waves to produce the standing waves within the cavity, as described in chapter
2.1.
Just as the pressure at the driving end of a very long pipe is proportional to the
wave impedance, so also is it convenient to talk about a pipe of finite length
having an input impedance. Benade concludes that the measure of input
impedance is larger or smaller than the pipe’s wave impedance, depending on the
relationship of the excitation frequency to the natural frequencies of the duct.
Now considering this configuration from a musical standpoint, with the rapid
change in impedance at the open end, near to all of the frequencies arriving at the
end of the tube are reflected back into the tube. The standing waves formed thus
are frequencies far from those associated with the instrument in question and are
B = bulk modulus
d = air density

17

therefore musically redundant. Furthermore, Backus (1977) notes that the tones
produced from this configuration are “subdued, muffled and of poor quality”.
This observation is solved with the introduction of the flared bell at the end of the
tubing. Using the capillary excitation method mentioned earlier, Benade (1990)
compares the input impedances between a piece of cylindrical tubing and tubing
with a bell attached. Comparison between the curves shows that the addition of
the bell shifts the resonance peaks lower and with the lesser amplitude due to
anticipated viscothermal losses (see appendix B).
The resonances shift downwards because when a pressure disturbance
encounters a bell whose flare is rapidly increasing, the majority of musically useful
frequencies reflect a reasonable distance back from the bell. Only a small fraction
of the original disturbance can penetrate through the impedance barrier. Benade
hypothesises that the horn may be described metaphorically as a short pipe at
low frequencies and a long pipe at high frequencies. Parker (2009) thus concludes
“the node occurs at the point where the rate of flare is rapid compared to the
wavelength of the wave”, therefore, the lowest-numbered modes of a flaring horn
have higher frequencies than that of a cylindrical pipe of equal length, confirming
the observations made by Benade (1990) in the aforementioned study. The
amplitude difference is observed due to the higher number of reflections occurring
within the pipe due to the more severe change of impedance.
Considering this effect in the time domain, Fletcher and Rossing (1998) state that
when a note is played, it is not until the first reflection encounters the lip valve
does any form of interaction occur, therefore the period of the first mode can be
considered as the time taken during the preceding roundtrip. This concludes as to
why higher pitches on a Trumpet require much greater muscular input as the lips
carry out many oscillatory periods before they receive any acoustical support from
the horn.

The downward shift of the resonances tackles the previously encountered
problem of unstable notes. For example, if a player was to attempt to sound a

18

note based on the first-mode resonance, this note will be extremely difficult to
sound due to all the upper harmonics falling in resonant dips, therefore subjecting
the lip-valve to a great deal of disruptive influence. However, if the player was to
sound a note on the second resonance, Benade notes that peaks 2,4,6 and 8 are
harmonically related and so reinforce each other as a “regime” of oscillation
(further explored later within this thesis), making the note stable and easy to play.
3.2) Horn design
Bell designs can be generalised as an intermediate value between conical and
sharply flaring. Myers (1997) considers the resultant tonal effects between the two
concluding that if a section of a conical bell is removed, the pitch is raised without
affecting the tone. Whereas with a curtailed flaring horn the pitch is unaltered, but
the directional properties of the sound radiated changes. Olson (1967) reiterates
this and goes on to outline the main types of flare being, parabolic, conical,
exponential and hyperbolic. For the purpose of this waveguide, it is helpful to have
a mathematically controllable analogy of the horn shape. Myers (1997) outlines
such a parameter known as the Horn function (U) where r is the radius of the tube
and is defined by:
!! =
1
!

!!
!
!!!
As established in the previous chapter, low frequency waves are reflected in a
high value horn function, whereas high frequencies are hardly affected. Myers
notes that the peak function corresponds to the cut-off frequency of the horn can
therefore be given by:
!!"#$%% =
!
2!
!
In order to utilise the horn function and thus the cut-off frequency for real
instruments, a mathematical model was needed. Jansson and Benade (1974)
found that Bessel horns provide a useful application visually and mathematically.
For a Bessel-horn, the diameter (D) at any point is defined in terms of the distance
(y) from the large open end:
! =
!
(! + !!)!

19

Where B and y0 are chosen to give proper diameters at the small and large ends,
and m is the “flare parameter” which dominates the acoustical behavior of the air
column. Benade (1990) concludes that present trumpets and trombones
correspond closely to the shapes of Bessel horns having values of m lying
between the limits of 0.5 and 0.65. Characteristic frequencies of a closed Bessel
horn therefore can be given in terms of overall length (L), flare parameter m, and
the speed of sound (c).
!! =
!
4 ! + !!
2! − 1 +
2
! ! ! + 1
If we consider values of y0 comparable to that used in trumpets and trombones,
Benade assures that the formula functions within one percent difference between
those established through exact calculation. For intermediate values of m
(between 0-1), such as 0.5-0.65 mentioned earlier, the resonances calculated are
not arranged in a musically useful strategy and so as a standalone instrument,
would not prove viable in establishing useful oscillations. Despite this, one must
also consider the introduction of the leadpipe and mouthpiece onto the bell
function, which therefore accounts for the harmonic deviance.
3.3) Digital Filters
As previously established, digital waveguides simulate wave propagation by
solving the 1-D wave equation, therefore particularly suited to plane waves in
cylindrical bores and spherical waves in conical bores. Scavone (1997) states that
wave propagation through sections of non-cylindrical/conical nature become
multi-dimensional and are therefore no longer suited to digital waveguides. The
most common method of overcoming this problem is to model the bell as a
lumped impedance or reflectance.
Walstijn (2007) hypothesises that since the bell has a fixed reflectance it may be
modeled as a lumped-reactance filter that can be divided into two stages: the first
being a slow exponentially rising build up followed by an oscillatory decay. Both
instances of which can be implemented using digital filters.

20

Meddins (2000) outlines that there are two primary types of digital filter, a finite
response filter (FIR) and an infinite response filter (IIR). The key disadvantages of
IIR filters are the complexity of their design and the resultant instability making FIR
filters the favourable choice when linear phase is required throughout the system.
Output from a filter is made up from previous inputs and outputs using the
operation of convolution, or filter coefficients. A FIR filter achieves this through
summing delays and consequent multiplications directly to the input, dependent
on the pole of the filter. If such filter is fed an impulse, then once the impulse has
passed through the system, its consequent value must be 0 and is therefore finite
in duration.
(Smith, J.O. 2007, Introduction to Digital Filters with Audio Applications, W3K Publishing, California)
A FIR can be defined as:
! ! = !!! ! + !!! ! − 1 + !!! ! − 2 … !!!(! − !)
Where b is the filter coefficient and x being the previous input. An IIR filter
incorporates a recursive function in the form of a feedback loop within the filter
design and therefore must encompass the previous output within the equation
and thus involving a second, negative flowing filter coefficient:
! ! = !!! ! + !!! ! − !! − !!!(! − !!)
(Smith, J.O. 2007, Introduction to Digital Filters with Audio Applications, W3K Publishing, California)

21

The infinite nature of these filters arises from considering non-zero values for g
where the filter begins to oscillate positive and negative at each sample
corresponding to even and odd powers of g whereas Cook (2002) notes that if the
magnitude is greater than 1, the filter will grow without bound and therefore is
labelled as being “unstable”.
3.4) Z-Transform
In order for one to realise the equations shown above into the signal flow form, we
must consider the Z-transfer. Referring to the FIR signal flow diagram shown
above, we can understand that z-1
defines a unit delay within a system. To transfer
a filter into the z domain, Cook (2002) states that one simply capitalises all
variables of x and y and replace all time indicies (eg: n, n-1, n-2) with Z-n
.
Therefore, if one was to first consider filter equation:
y n = g x n + !!x n − 1 + !!x n − 2 + … !!x n − N −
(!!y n − 1 − !!y n − 2 − ⋯ − !!y n − M )

Through z-transform, the equation would take the form of:

! = g X + !!X!!!
+ !!!!!!
… !!!!!!
− (!!!!!!
− !!!!!!
… !!!!!!
)
And therefore becomes more applicable to signal flow application. Cook states
that a filter can be analysed by identifying its “transfer function” which is found by
solving the ratio of output (Y) to input (X) in z-transform. Smith (2007) describes
this process as the convolution theorem, in which for any signals x and y,
convolution in the time domain is multiplication in the z domain.
! ∗ ! ↔ ! ∙ !
3.5) Discrete time modelling
Reffering back to the conclusion outlined earlier by Walstijn (2007), the latter
oscillatory decay can therefore be approximated using an IIR filter, but an IIR
design collapses when trying to account for the initial exponential growth. It was

22

therefore evaluated that the simplest solution would be to model the initial growth
as an independent FIR filter. The completed filter design then consists of an N-tap
FIR in parallel to an IIR filter delayed by N samples, which Walstijn concludes as
still being considerably more efficient than a single FIR.
An alternate approach was later proposed by Smith and Walstijn (1998) who
developed a truncated infinite impulse response (TIIR) filter to replace the N-tap
FIR. Smith stated that the most efficient way to model the growing exponential
portion of the aforementioned design was by the means of an unstable one-pole
filter, but the TIIR filter provides a method of implementing this instability whilst
avoiding numerical problems. The concept is to synthesize a FIR as an IIR filter
minus an identical delayed tail cancelling IIR filter. In the stable case, this design
would function accordingly, but Smith states that once implemented in the
unstable case, the exponential growth of quantisation error eventually dominates,
resulting in a need for two independent instances of the tail cancelling filters. By
doing this, the non-active filter can be cleared in order to remove the accumulating
noise before re-activating and visa versa (see appendix H). Smith concludes that
whilst a 400 FIR filter (assuming impulse length of 10ms and 44.1khz sample rate)
functions faithfully as a Trumpet bell reflectance filter, the TIIR design greatly
reduces complexity whilst still maintaining accuracy.
Finally, in order to model the sound radiated by the instrument from the
perspective of a listener, Walstijn (2007) employs a transmittance filter of which the
power response is complementary to that of the reflectance filter, but notes that
accuracy of the high frequency component is compromised due to the
simplifications used in deriving the model.
Mouthpiece:
4.1) Acoustical Influence
By revisiting the results posed by Benade (1990) in chapter 3.1, one can conclude
that the resonant peaks disappear when the frequency of excitation is above
1500hz due to very little of the sound returning to set up standing waves. Looking

23

particularly at the heights (and therefore also depths) of the resonances, Benade
notes that the difference between the peaks/troughs and the wave impedance is
governed by a constant numerical factor, which we shall label Q0. Ie:
!"# !" !"#$ = !"#$ !"# × !!
One can already draw conclusions that the peak height is dependent on the
amplitude of the returning wave at the input end relative to the source and Benade
uses this theory to define Q0 in terms of the amplitude reduction (F) produced by a
single round trip.
!! =
(1 + !)
(1 − !)
Therefore, if one was to consider the mouthpiece as merely a cavity (cup) and
tapered a tube (backbore) attached to a tube of given length, then the first
conclusion to be drawn is that the tube is now effectively lengthened, therefore
lowering the fundamental. Backus (1977) fortifies this hypothesis by stating that
where the wavelength of a sound is longer to that of the mouthpiece, the amount
‘lengthened’ is analogous to tubing of comparable volume. Thus, in order to not
alter the fundamental of the leadpipe by adding a mouthpiece the equivalent
length with respect to the mouthpiece volume is removed. Backus continues on to
state that the higher resonances however, will not remain unchanged due to the
effect of the cup component.
Benade (1990) notes that with the introduction of the mouthpiece, the wave
impedance starts out equal to that of the pipe alone, rises at 850hz to a value five
times larger and decreases steadily, falling below the simple pipe value in the
region about 3500hz (see appendix C). It is worth noting that firstly, mode 1 still
remains in the same position as the pipe alone and secondly, the accentuated
peaks around 850hz, have the same Q0 value as that of the troughs. Therefore,
one can confirm that the effect is properly associated with the variation of wave
impedance.

24

This is due to the cup forming a cavity, which therefore has its own resonance, in
this case around 875hz. It is interesting to note that despite now becoming part of
the leadpipe, the mouthpiece retains its own resonant ‘popping’ frequency (Fp).
Backus (1977) therefore concludes that at this frequency, the mouthpiece
effectively adds its equivalent length to the tube and this effect increases with
respect to the frequency. As a consequence, the aforementioned shortened tube
will maintain the same lower resonances, but the higher modes are shifted down
from their original values, forming a musically useful harmonic sequence.
The lowering of these modes now further stabilises notes within this sequence by
forming appropriate regimes of oscillation. Take for example sounding the note G4,
the impedance maxima that influence this oscillation are peaks 3,6 and 9, the
dominant of which being peak 3 contributing primarily to the fundamental tone.
The amplitude of peak 3 is considerably higher than 2, therefore making it stable
when playing pianissimo (only exciting 1 mode) and as the tone gets louder, the
tallest peak, peak 6 enters the regime providing a great deal of energy, along with
to some extent the short 9th
peak (see appendix C). One can conclude that having
2 strong peaks forming the oscillatory regime, it proves to be often the strongest
and easiest note to play, made possible by the effect of the mouthpiece
resonance.
4.2) Cup Volume
Having already established through Backus’ (1977) work that a pipe of a
comparable volume to a mouthpiece can posses a mutual resonance, Benade
(1990) develops upon this hypothesis by further exploring the mouthpiece
equivalent length (Le) behavior across a frequency band. Benade concluded with
five discrete properties.
The first being for a cylindrical pipe, the Le of a mouthpiece at low frequencies is
equal to the length of cylindrical tube whose volume matches the total volume of
the mouthpiece, regardless of its shape.

25

Secondly, at Fp, Le is the length of cylindrical tube (open-closed) whose first mode
frequency equals FP, thus given by:
!! =
!
4!!
Thirdly, the total volume and Fp determines the variation of Le by anchoring it along
two points along the frequency scale. Subtle differences in Le at other frequencies
are caused by variations in the backbore/cup proportions.
Finally, Le increases steadily with frequency nearly to the top of the playing range
and mouthpieces with equal volume will only show greater change in Le if Fp is
made lower.
Let us consider a Trumpet playing a crescendo on G4 that plays flat as it gets
louder. One could therefore conclude that in order to correct this issue, the
frequency of peak 6 must be raised without moving the in tune fundamental.
Benade states that since changes in resonance frequency are correlated with
changes in the mouthpiece’s Le value, one must reduce the Le around peak 6
without affecting peak 3. The intuitive method of achieving this would be to raise
the popping frequency by reducing the cup volume, but this would be at the
expense of a reduction of total volume and therefore not ideal. Hence, the most
efficient option is to enlarge the back bore, which Benade states significantly
raises FP with only a negligible change in total mouthpiece volume.
4.3) Waveguide Implementation
Fletcher and Rossing (1998) state that in the low frequencies, the mouthpiece
cavity can be described as being analogous to an electrical shunt compliance
which can be defined as a function of cup volume (V), air density (ρ) and speed of
sound (c).
! =
!
!!!

26

In series with the cup, is the backbore component containing what can be
assumed as a conical cross section. Smyth and Scott (2011) continue the
electronic analogy by defining this as an inductance (inertance) given by:
! =
!!!
!!
Where lc is length, Sc is cross-sectional area of the constriction. Smith (1999)
asserts this analogy by describing the mouthpiece throat as a “resistance to air
flow giving the lips an “air spring” to vibrate against”. Also included in this model
are the viscothermal losses, therefore introducing a resistance.
Rabenstein and Petrausch (2005) continue this analogy by developing the design
through ‘block-based physical modeling”. With the mouthpiece acting as a
bandpass filter having predefined resonance, the mouthpiece is connected from
the non-linear (lip) element via a 3-port scatter junction in parallel with the
mouthpiece electrical equivalent, capacitor with the aforementioned inductance
and resistance in series.
(Petrausch, S + Rabenstein, R 2005, Application of Block Based Physical Modeling for Digital Sound Synthesis of Brass
Instruments, ForumAcusticum, FA 2005, Budapest Hungary, pp 703-708)
Walsitijn (2007) firstly hypothesises modeling the cup portion as a cylinder of equal
volume adhering to Benade’s conlcusions outlined in the previous chapters. This
model works theoretically within the low frequency domain, but in practice Walstijn
concludes that in order to model the high frequency component accurately, one
must approximate the cavity profile with a piece-wise series of very short tube
sections. However, in order to achieve this higher sample rates are needed thus a
resultant computational increase, hence making this option redundant.

27

A less computationally expensive method was outlined by modelling the cup as a
‘lumped’ component and the backbore as a simple conical waveguide with
appropriate scatter junctions between the two sections and the connecting
tubing. Walstijn notes that because the lips demonstrate an instantaneous
reflection, the cup volume unit will have to have a zero instantaneous reflection
towards the left which can be solved by selecting the appropriate port resistance
value within the scatter junction. Walstijn employs this through the use of the WD-l
design for the cavity (see below), which has a non-zero instantaneous reflection
towards the right. Thus, to avoid a delay free loop the left junction of the conical
section is modeled as a WD-l junction.
(Walstijn, M 2007, Wave-Based Simulation of Wind Instrument Resonators, IEEE Signal Processing Magazine, March 2007)
Walstijn concludes that this method proves to be successful in both the physical
domain when compared to impedance measurements of a comparable
instrument and thus, being particularly suited to sound synthesis.
The Embouchure:
5.1) The Lip-valve Mechanism
With the resonant cavity of the instrument outlined, the non-linear excitation
method, which Benade (1990) labels as a flow control mechanism, must be taken
into account.

28

A flow control mechanism converts steady air supply into oscillations of the air
column by causing the players lips to open and close rapidly in response to the
acoustical variations within mouthpiece. Olson (1967) considers the lips as being
closed at rest and the full pressure of the air being supplied from the lungs is
therefore acting solely on the lips. The lips are metaphorically held closed at rest
by muscular tension and the lips’ mass and resultant inertia. After a considerable
build up of pressure, the lips open and continue to open due to the inertial energy.
In accordance with Bernoulli’s theorem, the high velocity of air at this point causes
a reduction of air pressure upon the lips. Due to the lips being at maximum
opening, they can be generalised as being under maximum stress, and so the
restoring force of the lips is now greater than the pressure keeping them open,
thus restarting the cycle.
Benade (1990) draws a useful analogy here between the lip-valve mechanism and
a setup labeled as a Water Trumpet.
(Benade, A.H. 1990, Fundamentals of Musical Acoustics, 2nd, Dover, New York, pg 392)
Benade notes that water moving in a channel of varying cross section can be
defined by the same equations governing resonant air columns. The particular
point of interest here is the modeling of the input excitation. A water supply valve
opens progressively as the water level rises and consequently reduces flow as the
water level drops. Elementary physics state that to maintain an oscillation, the
excitation must be supplied at appropriately timed intervals and is thus modeled
through the valve mechanism.

29

The periodic movement of water creates fluid pressure deviation at the bottom of
the trough with respect to its average, or rest pressure. Therefore through this
analogy, one can conclude that the player’s lips are strongly influenced by
acoustic pressure variations that take place within the mouthpiece cup and that
the oscillation produced will favour the frequency closely matching one or another
of the air column’s natural frequencies.
Backus (1977) introduces the non-linear aspect of the lip-valve by comparing it to
a reed instrument. If one was to consider a Clarinet, the opening and closing of
the reed creates periodic flows of air with a considerable number of high
harmonics to excite resonances in the air column. Brass instruments however do
not abide by these mechanics. Through observation using transparent
mouthpieces, Backus concluded that the lips open and close almost sinusoidally
and that during most of the vibration cycle, the lips are open far enough so that
the pressure in the mouthpiece is equal to that inside the players mouth. However,
during a small portion of the cycle the lips are almost effectively closed causing
the mouthpiece pressure to drop considerably. This sharp drop results in a
number of harmonics being generated and are thus translated within the standing
modes of the instrument cavity.
5.2) Modeling Techniques
Berners (1999) notes that modelling nonlinearity within the waveguide is integral to
the quality of output spectrum, but the key defining point of the waveguide is the
mechanism that converts DC energy from the lungs into audio frequency. It is
therefore the spectral energy transfer by the lips that therefore must be modelled
either as time varying or a nonlinear component.
Berners splits said interaction into two categories, firstly lip dynamics, which
models the response of the lips to surrounding pressure fluctuations. This
collaboration can be modelled as a function of differential pressure and
instantaneous lip position. Secondly, lip acoustics are considered as the resultant
flow of air across the lips as a function of differential pressure and instantaneous

30

lip position determined as a “quasistatic approximation”, as velocity here is not
taken into account.
The model becomes extremely complex when one combines both acoustic and
dynamic models due to their shared non-linearity and therefore Berners concludes
that it is difficult to model within waveguide context. Various methods have been
hypothesised by numerous contributors. Noreland (2003) explores theories first
outlined by Helmholtz by identifying the lip-valve commonly being defined as
“outward striking”, where pressure increase in the oral cavity causes the lips to
deform outwards. It was later outlined that this is not always the case, particularly
in the high register where the motion becomes transversal due to the lips closing
solely under the influence of the Bernoulli force.
Adachi and Sato (1995) hypothesise that the lip motion can be modeled by the
first mode of flexaural vibration through “one-mass motion”. The lip is modelled as
a 2 dimensional harmonic oscillator having one mass, 2 springs and a damper
both parallel and perpendicular to the air flow. Adachi and Sato state that in an
effort to simplify the model, the upper and lower lips are assumed to have
symmetric motion by the axis of air flow and thus only one lip is considered.
Campbell (2003) on the other hand states that although one mass models are
widely adopted in physical modelling, there is much experimental evidence that
disproves the accuracy of the model.
(Kaburagi et al 2011, "A methodological and preliminary study on the acoustic effect of a trumpet player"s vocal tract",
Acous. Soc. of America, Japan, Vol. 130, Issue 1)

31

Cook (2002) develops upon Adachi and Sato’s (1995) hypothesis by determining
a model analogous to that of a clarinet reed. The bore pressure (pb) is calculated
through the waveguide and the mouth pressure (pm) is considered as an external
control factor representing the breath pressure inside the mouth of the player. The
net force on the reed can therefore be defined as:
!!"# = !(!! − !!)
where A is the area of the reed. Considering Hookes law, the displacement and
resultant reed movement can be determined from the spring constant of the reed.
From the reed opening, it is now made possible to calculate the pressure that
subsequently leaks into the model. If mouth pressure is higher than the bore, the
reed slams shut and visa versa, thus representing the desired asymmetric
nonlinearity in the system.
5.3) Oral Cavity resonance
Within a waveguide, one must consider both the positive travelling pressure waves
and the negative. If one was to consider the path of the negative pressure wave,
in order to accurately model the interaction between the instrument and the
player, the oral cavity must form a part of the negative pressure wave path. Most
models disregard this additional parameter as the assumption was that pressure
in the mouth remains constant. However, Elliot and Bowsher (1982) found that the
acoustic pressure in the mouth was 5-20% of that in the mouthpiece. Variation of
the wind impedance, for example, arching the tongue, could thus influence the lip
reed and so must be considered.
Kaburagi et al (2011) identifies that at frequencies near the main bore resonances,
the input impedance of the vocal tract is often less than that of the bore and can
therefore be regarded as negligable, but Kaburagi concluded through MRI scans
that the tongue rose toward the palate going from low to medium to high pitch.
This arching can create a resonance strong enough to restabilise the higher
frequency oscillations (see appendix L).

32

“The length of the vocal tract was estimated as 17.4 cm for the low, 16.6 cm for
the mid, and 17.0 cm for the high pitches. The larynx rose slightly for the mid
pitch, decreasing the vocal-tract length. The area and length of the glottis were
estimated as 0.14 cm2
and 0.12 cm, respectively, from MRI of the larynx.”
(Kaburagi et al, 2011)
Acous. Soc. of America, Japan, Vol. 130, Issue 1,Fig.1)
The Final Waveguide:
The completed design outlines the block diagram proposed by Walstijn (2007)
(see appendix K) which utilises piecewise modelling of the instrument
components. Certain piecewise waveguides were adapted from their original
design to suit the computational limits within PureData.
Instrument dimensions within the waveguide were derived from Walstijn’s (2002)
pulse reflectometry testing of a Boosey & Hawkes Trumpet (see appendix J). For
conical sections, as explored earlier, the implemented pipe radius is generalised
as an average across the cone and amplitude and frequency losses are lumped
between each waveguide.
Loss filters were initially approached as a low pass filter with variable cut off
frequencies, but a more advanced methods were developed in the form of a
biquad filter using the [biquad~] object in an attempt to model the observations
made by Walstijn.

33

For the non-linear excitation, the complexity of the previously outlined models
make them difficult to implement within PureData. The scope of this thesis
therefore required simplification and the resultant redesign consisting of
asymmettrical waveforms based on a “1/1+kcosx2
” pulse function and using a
“Chebyshev” polynomial to transform the pressure wave from square to almost
sinusoidal pressure wave analogous (seen below) to the observations made by
Yoshikawa (1994) for varying pitches (see appendix I).
(Chebyshev polynomial transforming the “1/1+kcosx2
” pulse function from square at low pitch to sinusoidal at higher pitch.)
The bell reflection filter was modelled in 3 ways, firstly being generalised as a basic
low pass filter with a centre frequency set to the cut-off identified in chapter 3.1.
Secondly, a more advanced low pass was designed using the [lp2_bess~] object
which performs as a low pass with bessel charecteristic and finally using
convolution. Initially, convolution was intended via the [partconv~] object but the
unsuccessful compiling of these externals rendered this option redundant.
The vocal tract was modelled, similiar to the mouthpiece, as a parallel compliance
and therefore an integral resonator within the system. Different tongue arch
shapes and cavity sizes were modelled through the use of a filter with a band pass
characteristic connected via a scatter junction. An analogy, drawing from Kaburagi
et al’s (2011) comparison with speech phonetics and formants, the band pass
filters were fed resonance frequencies corresponding to musically applicable
syllables from the International Phonetic Alphabet (IPA), in this case, “U”, “Uh”, “Er,
“Iy” and “I” for respective pitches. Various losses within the waveguide, for
example, negative flowing pressure waves not transmitted back into the
instrument were maintained within the scatter junctions to maintain accuracy of
the model.

34

Within the scatter junctions, the coefficients were inputted using number objects
within PureData allowing for realtime adjustment of resistances in both 2 port and
3 port designs. This same technique was used within the waveguide parameters
such as pipe length, radius, and loss filter allowing for the user to experiment
within the patch exploring the resultant audible changes without compromising the
patch’s computation.
Impulse Response:
7.1) Linear Time-Invariance
Considering the behaviour of non-linearity, it is a common assumption that the
input to a linear system, will emerge as a proportional equivalent of the former.
Considering this concept within audio, the output signal will not “gain” any new
information. Cook (2002) states that many systems of interest do not change their
behaviour quickly and can be treated as time-invariant over time intervals within
normal operating range. If a system is therefore linear and time-invariant (LTI), we
can characterise its behaviour by measuring its Impulse response. Defined
mathematically:
ℎ ! = ! ! , !"# ! ! = ! !
!ℎ!"! ! ! = 1, ! = 0, 0 , !"ℎ!"#$%!
If a system is said to be LTI, when we excite the system with a signal pulse of
sample 1 and 0 thereafter, the recorded output from the system will determine the
response of the system to an arbitrary input. Therefore, in order to test the
reliability of the waveguide, impulse testing was carried out on a variety of brass
instruments as well as the waveguide, and the results are therefore made
comparable.
7.2) Methodology
In order to use the same discrete impulse within both the waveguide and real
instruments, the pulse was generated within PureData using the [dirac~] object
with a reccuring period of 5 seconds. The output of the system was amplified and

35

fed to a crystal earpiece (piezo) which was then inserted into the leadpipe, the
amplitude of which was kept constant across every instrument. The mouthpiece
was omitted at this point during testing and in the waveguide signal flow due to
the lack of design continuity between the instruments being tested. Apart from the
Xeno, Custom Z and the LA trumpets, each instrument required their own
individual custom mouthpiece and so in the interest of maintaining controlled
variables, was removed during testing.
A controlled microphone was positioned at the bell end of the instrument at a
distance half the radius of the bell as Benade (1990) established this was the ideal
position for the most accurate tonal representation of the instrument. All
recordings were made within a recording studio that was adequately soundproof
from external interference.
A wide array of instruments were selected ranging from modern trumpets to
Baroque natural trumpets, spanning across multiple keys (Ab
,Bb
,C, D and Eb
),
materials and finishes. 3 impulses were recorded per instrument and averages
were taken to compensate for any anomalies in the signal although due care was
taken during recording to make sure there was no clipping within the analog,
digital or interpolated domains.
Equpiment: Rode K2 (cardiod), SE Reflexion Filter, Presonus HP4 amplifier, Soundcraft E12 mixer and M-Audio 1814 i/o
In order to test the accuracy of the mouthpiece component of the waveguide,
individual testing of the mouthpiece alone was carried out using the same method

36

as described above, using a variety of cup sizes, backbores and shapes. Scatter
junction values within the patch were adjusted to model the impedance change
from backbore to assumed atmospheric.
The final patch was fed the same dirac input as for the instruments and the
mouthpieces. The output was recorded via a [writesf~] object and both signals
were recorded at a 88.2khz sample rate to avoid aliasing and non-linear artifacts
outlined by Puckette (2007) in chapter 2.2. The impulse responses were finally
analysed through Sonic Visualiser in order to compare between patch and
instrument outputs.
7.3) Results
Due to the large number of uncontrollable variables within a real, tangible
acoustical system, it is understandable to predict that a digital representation will
not be able to accurately account for these, therefor bringing about experimental
discrepancy. As this is the case, trying to compare impulses between the model
and an instrument would be redundant as a complete match is extremely unlikely.
Therefor in order to critique the results as constructively as possible, the impulse
responses have been transformed so that the max peaks and troughs are
identified as a linear functioning of time, therefor producing discrete timing
positions and amplitudes of which can be more useful when trying to guage the
accuracy of the model. The graphical scale was kept constant throughout.
Initially, the completed patch was testing using the simplified filters, ie low pass
loss filter and reflection filter which displayed the following response.

37

Comparing to the Yamaha Custom Z response (left), one can firstly deduce the
issue with regards to impulse length. The extended resonance evident in the
waveguide response (right) was deduced to be due to the loss filters. With
viscothermal losses contributing to the dispersion and transference of energy
within the acoustical body, incorrectly configured filters would result in an
extended response and an overall lack of energy loss in the system. A secondary
comparison was made to the Yamaha LA trumpet which demonstrated a
considerably longer response due to the silver plating (see appendix), but the
correlation was too weak and so one can therefore conclude the extended
resonance is an inaccuracy within the model.
Secondly, the spacing between the peaks are wider in the model as well as a
noticeable interference creating three prominent low frequency oscillations not
evident in the instrument’s response. One could argue there are three resonances
present in the instrument shown by the annotated red lines, but their severity is
lesser than the models response. Thus, the conclusion was made that this was
due to the innacuracy of the reflection filter. Frequencies were not being redirected
back into the waveguide appropriately and so the standing waves generated,
differ between responses.
The filters were then changed to their more advanced counterparts and the
respective scatter junction values were adjusted in an effort to improve results.
The following response was then achieved.

38

The change in filter design and scatter junction values improved the accuracy of
the model (right) considerably. The response length was reduced and the natural
envelope became much more comparable to that of the instrument (left). The low
frequency oscilllation strength was reduced and the peaks were brought closer
together.
With the filter designs implemented, the mouthpiece was then tested discretely.
Comparing the patch output (right) to the A8 mouthpiece (left) of comparable
popping frequency to that in use within the patch.
The mouthpiece patch shows some signs of similarity within the envelope shape
but the observed spacing between peaks is wider and a few additional oscillations
are visible within the response. The extended response is evident as before but is
again on a similar scale to the deviations noticed within the instrument patch
which could therefore be a result of the loss filter inaccuracies identified.
Despite the waveguide response being comparable to that of the instrument
counterparts, the patch itself didn’t function as accurately on an aesthetical
standpoint. When running the patch using the lip valve as an input excitation, the
long notes perform as expected creating a stable tone. The analogy continues
when the input amplitude of the excitation is increased generating increased level
of harmonics, similar to that of the non-linear function present with the real
instrument. The discrepancy however arose when the frequency of the excitation

39

was changed. As the frequency changed, a new note formed but a ring or
overtone was audible as the input changed between frequencies. One could
therefor conclude that the model, despite the efforts within the scatter junctions,
the inherent connection and interaction between the lipvalve and the acoustic
cavity was missing. The component modelling this interaction in the waveguide is
the scatter junction between the lipvalve and the acoustical body. Therefore, the
first conclusion to be made was that the scatter junction values would need to be
adjusted, which would account for the apparent lack of interference between the
negative flowing pressure waves and the positive at the point of excitation.
Secondly, one could also conclude that the lip model, although modelled
mathematically, is considerably more complex and the generalisations made to
encompass the mechanic within PureData rendered the model interaction
inaccurate. Finally, the mouthpiece patch anomalies discovered during testing
would also account for this unexpected resonance.
An effort was made to improve the bell reflectance filter further using a convolution
filter, but due to computational restrictions within PureData, an external object
needed to be compiled within the system. Although the external appeared to be
compiled, the object did not function when included into the patch and was
therefor excluded from testing.
The overall amplitude difference was realised as an experimental discrepancy as
the pulse being fed into the patch is at full scale deflection (FSD) constituting of a
value of 0-1-0, whereas in the instrument testing, the output from the patch was
sent to an amplifier and then the crystal earpiece which both in turn posses
internal resistances contributing to signal loss.
Conclusion:
The impulse response testing of the patch highlighted particular areas of
weakness within the model’s design, the most integral of which being the bell
reflectance filter. An improperly configured reflectance filter would cause the
wrong set of frequencies to be reflected back down the negative component, at

40

incorrect times thus forming interference points at improper positions along the
length of the tube. This in turn, creates unknown and generally musically irrelevant
resonances within the waveguide. Secondly, the frequencies not reflected by the
filter contribute directly to the tone of the output signal, therefore further distorting
the response. By merely modelling the reflectance filter as a low pass, be it regular
or bessel characteristic, the bell section of the waveguide was being generalised
as a conical waveguide with the reflection filter being lumped at the end of the
waveguide. Walstijn (2002) explores the application of piecewise conical
approximations to a degree of success, but the main issue here is in harmony with
the point raised earlier by Benade (1990) in chapter 3.1 that musically useful
frequencies reflect a reasonable distance back from the horn. Referring back to
the conclusions made in chapter 2.1, moving the point of reflection increases the
distance between the two boundaries which therefore affects the manner in which
they interfere as a time discrepancy occurs due to the increased distance being
travelled. Despite this, the reflection filter functioned appropriately and showed
promising correlation during impulse testing, but in order to improve accuracy,
further experimentation within filter design would need to be carried out.
Despite not being able to successfully compile and test the convolution reflection
filter, Walstijn (2007) notes that when using the convolution method, any change in
parameter would require recalculations within the waveguide. Therefore, restricting
the user interactivity within the bell component and making the technique
unfavourable when considering the patch within a software instrument context.
Secondly, as discussed in the results section, the loss filter’s efficiency greatly
effects the resultant impulse. As it has been proved by Walstijn (2002) that the
positioning of the loss filters within the waveguide gives successful results, one
can conclude that the nature of the filter was not completely observed.
Improvements were made during re-design but the issue was not completely
solved. One could argue that this is not entirely a problem as viscothermal losses
mainly contribute to the overall length of the response generated. However, as
discovered through impulse testing and various studies carried out in this region,
namely Abel et al (2003) and Smith (1986), the instrument’s material composition

41

and thickness will greatly affect the viscothermal losses resulting in a change of
perceived tone or length of impulse. As mentioned in chapter 2.2, this effect
would be modelled within the loss filter, thus requiring further research as these
effects have not been considered by most contributors.
The final critical contributing factor is the resistance values within the scatter
junctions. Similar to the bell reflectance filter, the amount of the input signal
reflected back into the previous waveguide will greatly determine the standing
waves setup within the cavity. Benade (1990) identifies the importance of this
connection in forming regimes of oscillation and this is exemplified in an example
posed by Gilbert and Petiot (1997). In the case of a Trombonist for example, the
player is capable of forming oscillations over a large range of frequencies purely by
flexaural vibration. The same applies when connected to a mouthpiece, but
playing the instrument, the player is strongly guided by the instrument. Therefore,
one can conclude that to ensure accuracy within a performance standpoint, the
interaction between the excitation and the cavity is of crucial importance.
The accuracy of the mouthpiece model relies heavily on this factor, and could be
concluded as the main inaccuracy within its implementation. Rabenstein and
Petrausch (2005) have proven that this design as a reliable one, consequently
greater exploration within scatter junction values must be carried out in order to
improve performance. Explorations and adjustments were made to the scatter
junction values during production and testing, but as found with the impulse
response, the values would need to be explored further. However, the advantage
of the number objects within the waveguide design allows for easy adjustment of
said values in real time, allowing for exploration of the audible effects during this
process.
An additional variable however not accounted for in waveguide designs mentioned
thus far do not account for a component identified by Bowman (2003) as the
“venturi gap”. As the backbore and mouthpipe dimensions tend to vary between
manufacturers, the two often do not fully connect leaving a gap between the

42

backbore, the mouthpipe and the main bore which could be considered integral to
the interaction.
Despite these observations, the impulse response shown shows considerable
correlation to the instrument counterpart. As mentioned in chapter 7.3, modelling
a real-life acoustical system within the digital domain brings about many
generalisations and as a result the pursuit of accuracy is an ongoing task, and
hence a strong correlation was not to be expected, but positives can be drawn
from the similiarities outlined.
On an aesthetical standpoint, the instrument functioned well when producing and
supporting long notes, but deviated when changing the frequency of the input
excitation. As the accuracy of the lipvalve mechanism is justified mathematically in
chapter 6 and the waveguide having already been established as a comparable
representation by Walstijn (2002), the lack of interaction between it and the
instrument is therefore due to the scatter junction values and complexity of the
lipvalve. Oscillatory regimes formed between the lip and the cavity are still under
considerable research due to the intricacy of exploring such factors, for example,
human intuitive control of the lipvalve and result interpretation. Experimentation
using an artificial mouth, for example, studies by Bromage (2007) and Vergez and
Rodet (1997), allowed for more controlled research, but the exact nature of the
lipvalve is still under scrutiny, thus making modelling options very limited and often
very generalised. Studies from Noreland (2003) and Adachi and Sato (1995)
demonstrated how the mechanic of the lipvalve changes entirely as the pitch
increases. This variable was omitted in an effort to streamline the scope of this
thesis, but further study could be carried out within this area to improve
performance. Despite this, the unexpected resonance created when changing
frequencies reduced as the input excitation neared the cut-off frequency of the
reflection filter, thus adhering to the theories laid out by Fletcher and Rossing
(1998) in chapter 3.1 stating that note frequencies above the cut-off are
maintained purely by the muscular tension and receive no resonant support from
the instrument.

43

With regards to developing the patch as an instrument, much ground still needs to
be covered. Such topics as articulation and transients are still to be explored,
consequently expression within the patch as an instrument is particularly limited.
Also with the patch being a natural trumpet, only a restricted set of pitches are
available and therefore for a fully functioning, usable instrument to be developed,
further research within a valved trumpet needs to be carried out. Also more
attention would need to be paid with regards to the graphical user interface (GUI)
to make the patch more user friendly. Although an effort has been made to lay the
patch out in a logical manner, for users who are not familiar with waveguide
modelling or PureData, the GUI needs to be more intuitive and approachable for
the average consumer.
The development of the waveguide and the problems that occurred during its
development strongly outlined the importance of particular factors within brass
instrument design in general. Beginning with the mouthpiece and how the
relationship between cup size and backbore should be considered in order to
maintain tuning and intonation across all dynamics. This clearly identifies why
manufacturers such as Smith Watkins have started introducing adjustable
mouthpipe lengths in order to account for this problem. Arguably, the most
important factor discovered during research being the shape of the bell. Changing
the bell shape strongly affects the reflection filter characteristic and consequently
the radiation filter respectively, affecting both tone and instrument note stability.
Additional variables such as material composition arguably are becoming more
important in the design, greatly affecting vibration and the resultant energy loss
due to viscothermal effects. Not only do they therefore affect the length of the
impulse but also the harmonic content of the tone produced.
Performance points were also established by identifying the role of the vocal cavity
when producing tones. As already discussed, the instrument no longer supports
the sustained oscillation above that of the cut off frequency of the bell. An arching
of the tongue creates a resonance strong enough to stabilise the higher
oscillations not supported by the instrument and is thus a vital adjustment made
by the player in the upper range of the instrument.

44

One may argue that therefore the bell should be made larger, thus increasing the
cut off frequency, but a tradeoff must be considered when you refer back to the
conclusions made from both chapters 2 and 3. The player produces energy
through flexaural vibration that propagates through the bore and bell losing energy
due to viscothermal effects. By increasing the bell size then greater energy is lost
through transmission, whereas a smaller bell would be harder to play and less
sound would radiate. Therefore, depending on the mouthpiece, embouchure and
playing environment, the correct balance between bell and bore size is used to
provide the “right” instrument for the player.
It can thus be established that acoustically, every part of the instrument is equally
important within the tonal production, whereas, the performance aspect is entirely
governed by the connections between the components and the interactions there
after. The assumption within waveguide modelling is that the player selects a
vibration frequency that excites a harmonically related air column resonance,
whereas various studies identify that the synthesis is much more complicated.
Some success has been established already within this field, for example, BRASS
by IRCAM and Arturia, but an accurate model that performs both aesthetically and
mathematically requires further exploration. Once this balance has been
established, the model could pave the way for instrument design innovation and
create new boundaries within electroacoustic composition and advanced
synthesis.

45

Appendix:
Vincent Bach 1¼C Mouthpiece
Vincent Bach 2½C Mouthpiece

46

Warburton 4SV Cup 1 Backbore Mouthpiece
Warburton 4SV Cup, KT* Backbore Mouthpiece

47

Warburton 6SV Cup 1 Backbore Mouthpiece
Warburton 6SV Cup KT* Backbore Mouthpiece

48

A8 Mouthpiece
Yamaha Custom 16C4-GP

49

Yamaha Custom Z “Bobby Shew” Bb Trumpet, Brass finish
Yamaha Xeno Bb Trumpet, Matte finish

50

Natural Baroque Trumpet, Brass finish
Cavalry Trumpet/Horn, Brass finish

51

Yamaha LA “Wayne Bergeron” Bb Trumpet, Silver plated finish
Post Horn, Silver plated finish
Appendix A:
(Smith, R.A. 1978, Recent Developments in Trumpet Design, International Trumpet Guild, Vol. 3, Fig.2)

52

Appendix B:
(Benade, A.H. 1990, Fundamentals of Musical Acoustics, Dover, New York, Fig. 20.4 pg. 398)

53

Appendix C:
Appendix D:
excitation mechanism for study of pressure response of an air column

54

Appendix E:
(Smith, J.O. 1992, Physical Modeling using Digital Waveguides, Computer Mustic Journal, Physical Modeling of Musical
Instruments, Vol. 16, Issue 4, p. 74-91,Fig.1)
Appendix F:
(Berners, D.P. 1999, Acoustics and Signal Processing Techniques for Physical Modeling of Brass Instruments, PhD,
Stanford University,Fig 2.2-2.3, pg. 35)

55

Appendix G:

(Smith, R.A. 1986, "The effect of material in brass instruments; a review", Proceedings of Institute of Acoustics, Vol. 8, Part
1, pp. 91-96, Fig.3)
(Smith, R.A. 1987, "Holographs of bell vibrations", News and Views, Nature 329, 762, 29th October 1987.)
Measured vibrations of a trombone bell played at 1000hz (upper) and 630hz (lower).

56

(Smith, R.A. 1978, Recent Developments in Trumpet Design, International Trumpet Guild, Vol. 3, Fig.4)
Appendix H:
(van Walstijn, M. 2002, Discrete-Time Modelling of Brass and Reed Woodwind Instruments with Application to Musical
Sound Synthese, PhD, University of Edinburgh, Fig.7.12, pg. 163)

57

Appendix I:
(Yoshikawa, S. 1995, "Acoustical behaviour of brass player"s lips", Journal of Acoustic Society fo America, Vol. 97, Issue 3,
fig.8, pg. 1935)
Measured waveforms of mouthpiece pressure and lip strain for the notes F2, G3, F3, and A 3 played in mf level on the
French horn.

58

Appendix J:
Appendix K:

59

Appendix L:
Vocal-tract area function obtained from volumetric MRI. From top to bottom, the plots correspond to low, mid, and high
pitches

60

Reference List:
• Smith, R.A. 1986, "Ensuring high quality in the production of musical instruments",
Das Musikinstrument, April 1986, p. 131-132.
• Smith, J.O. 1991, "Wave Simulation of Non-cylindrical Acoustic Tubes",
International Computer Music Conference, Montreal, October 1991, p. 304-307.
• van Walstijn, M. 2002, Discrete-Time Modelling of Brass and Reed Woodwind
Instruments with Application to Musical Sound Synthese, PhD, University of
Edinburgh.
• Benade, A.H 1990, Fundamentals of Musical Acoustics, 2nd Edition, Dover
Publications, New York.
• Smith, J.O. + van Walstijn, M 1998, "Use of Truncated Infinite Impulse Response
(TIIR) Filters in Implementing Efficient Digital Waveguide Models of Flared Horns
and Piecewise Conical Bores with Unstable One-Pole Filter Elements", ISMA-98
Leavenworth, Washington, 28 June 1998, p. 309-314.
• Adams, S. + Allday, J. 2000, Advanced Physics, Oxford University Press, Oxford.
• Backus, J 1977, The Acoustical Foundations of Music, 2nd Edition, Norton.
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http://www.phys.unsw.edu.au/jw/pipes.html [Accessed: 28.8.2012])
• Smith, J.O. 1992, Physical Modeling using Digital Waveguides, Computer Mustic
Journal, Physical Modeling of Musical Instruments, Vol. 16, Issue 4, p. 74-91.
• van Walstijn, M 2007, "Wave-Based Simulation of Wind Instrument Resonators",
IEEE Signal Processing Magazine, March 2007, p. 21-31.
• Välimäki, V. + Laakso, T.I. 2000, Principles of Fractional Delay Filters, International
Conference on Acoustics, Speech, and Signal Processing, Instanbul, Turkey, 5-9.
• Puckette, M 2007, The Theory and Technique of Electronic Music, World
Scientific Publishing.
• van Walstijn, M. Cullen, J.S. Campbell, D.M. 1997, Modelling Viscothermal Wave
Propagation in Wind Instrument Air Columns, ISMA 97 Conference, Institute of
Acoustics, Volume 19: part 5, Book 2, pp. 251-583.
• Abel, J. Smyth, T. + Smith, J.O. 2003, "A Simple, Accurate Wall Loss Filter for
Acoustic Tubes", Proceedings of the 6th Int. Conference on Digitial Audio Effects,
8-11 September 2003.
• Smith, R.A. 1987, "Holographs of bell vibrations", News and Views, Nature 329,
762, 29th October 1987.
• Smith, J.O. 2010, Physical Audio Signal Processing, W3K Publishing, California.

61

• Parker, B. 2009, Good Vibrations: The Physics of Music, John Hopkins University
Press, Maryland.
• Fletcher, N.H. Rossing, T.D. 1998, The Physics of Musical Instruments, 2nd
Edition, Springer, New York.
• Myers, A. 1997, "The Horn Function and Brass Instrument Character",
Perspectives in Brass Scholarship: Proceedings of the International Historic, 1997,
p. 239-262.
• Olson, H.F. 1967, Music, Physics and Engineering, 2nd Edition, Dover
Publications, New York.
• Jansson, E.V. Benade, A.H. 1974, On Plane Spherical Waves in Horns with Non-
uniform flare, Acustica 31, Vol. 31, no.3, pp. 185-202.
• Scavone, G.P. 1997, An Acoustic Analysis of Single-Reed Woodwind Instruments
With and Emphasis on Design and Performance Issues and Digital Waveguide
Modeling Techniques, PhD, Stanford University.
• Meddins, R. 2000, Introduction to Digital Signal Processing, Newnes.
• Cook, P.R 2002, Real Sound Synthesis for Interactive Applications, A.K. Peters,
Massachusetts.
• Smith, J.O. 2007, Introduction to Digital Filters with Audio Applications, W3K
Publishing.
• Smyth, T. Scott, F.S. 2011, "Parametric Trombone Synthesis by Coupling
Dynamic Lip Valve and Instrument Models", Proceedings of the 8th Sound and
Music Computing Conference, 2011.
• Smith, R.A. 1999, "Exciting Your Instrument", Journal of the International Trumpet
Guild, May 1999, p. 44-45.
• Petrausch, S + Rabenstein, R 2005, Application of Block Based Physical
Modeling for Digital Sound Synthesis of Brass Instruments, ForumAcusticum, FA
2005, Budapest Hungary, pp 703-708.
• Berners, D.P. 1999, Acoustics and Signal Processing Techniques for Physical
Modeling of Brass Instruments, PhD, Stanford University.
• Noreland, D 2003, Numerical Techniques for Acoustic Modelling and Design of
Brass Wind Instruments, Acta Universitatis Upsaliensis Uppsala.
• Adachi, S. Sato, M. 1995, Brass Sound Simulation suing a 2D Lip Vibration
Model, Proceedings of the 15th International Congress of Acoustics, Trondheim,
Norway.

62

• Campbell, M. 2003, Brass Instruments As We Know Them Today, Proceedings of
Music Acoustics Conference, Stockhold, Sweden, Vol. 1, August 6-9, pp. 3850-
3861.
• Elliott, S.J. Bowsher, J.M. 1982, Regeneration in Brass Wind Instruments, J.
Sound Vibe., Vol. 83, pp. 181-217.
• Kaburagi, T. Yamada, N. Fukui, T. Minamiya, E. 2011, "A Methodological and
Preliminary Study on the Acoustic Effect of a Trumpet Player"s Vocal Tract",
Acoustical Society of America, Music and Musical Instruments, Vol. 130, Issue 1.
• Yoshikawa, S. 1995, "Acoustical behaviour of brass player"s lips", Journal of
Acoustic Society fo America, Vol. 97, Issue 3, pg. 1929-1939.
• Gilbert, J. Petiot, J. 1997, Brass Instruments, Some Theoretical and Experimental
Results, Proceedings of Institute of Acoustics, Vol. 19, Part 5, Book 2, pp. 251-
583.
• Bromage, S.R. 2007, Visualisation of the lip motion of brass instrument players,
and investigations of an artificial mouth as a tool for comparative studies of
instruments, Ph.D, The University of Edinburgh
• Bowman, J 2003, "Choosing a Trumpet Mouthpiece with Best Charecteristics",
Instrumentalist, May 2003.
• Vergez, C. Rodet, X. 1997, Model of the trumpet funcitoning: Real time simulation
and experiments with artificial mouth, Proceedings of International Symposium of
Musical Acoustics, pp. 425-432.

63

Bibliography:
• Abel, J. Smyth, T. + Smith, J.O. 2003, "A Simple, Accurate Wall Loss Filter for Acoustic
Tubes", Proceedings of the 6th Int. Conference on Digitial Audio Effects, 8-11 September
2003.
• Adachi, S. Sato, M. 1995, Brass Sound Simulation suing a 2D Lip Vibration Model,
Proceedings of the 15th International Congress of Acoustics, Trondheim, Norway.
• Adams, S. + Allday, J. 2000, Advanced Physics, Oxford University Press, Oxford.
• Backus, J 1977, The Acoustical Foundations of Music, 2nd Edition, Norton.
• Benade, A.H 1990, Fundamentals of Musical Acoustics, 2nd Edition, Dover Publications,
New York.
• Berners, D.P. 1999, Acoustics and Signal Processing Techniques for Physical Modeling of
Brass Instruments, PhD, Stanford University.
• Benson, D.J. 2007, Music: A Mathematical Offering, Cambridge University Press.
• Bromage, S.R. 2007, Visualisation of the lip motion of brass instrument players, and
investigations of an artificial mouth as a tool for comparative studies of instruments, Ph.D,
The University of Edinburgh
• Bowman, J 2003, "Choosing a Trumpet Mouthpiece with Best Charecteristics",
Instrumentalist, May 2003.
• Campbell, M. 2003, Brass Instruments As We Know Them Today, Proceedings of Music
Acoustics Conference, Stockhold, Sweden, Vol. 1, August 6-9, pp. 3850-3861.
• Cook, P.R. 1992, A Meta-wind instrument physical model, and a meta-controller for real
time performance control, International Computer Music Conference, Physical Modeling
and Signal Processing, California.
• Cook, P.R 2002, Real Sound Synthesis for Interactive Applications, A.K. Peters,
Massachusetts.
• Elliott, S.J. Bowsher, J.M. 1982, Regeneration in Brass Wind Instruments, J. Sound Vibe.,
Vol. 83, pp. 181-217.
• Fabre, B. Gilbert, J. Hirschberg, A. Pelorson, X. 2010, Aeroacoustics of Musical
Instruments, Annual Reviews.
• Fletcher, N.H. Rossing, T.D. 1998, The Physics of Musical Instruments, 2nd Edition,
Springer, New York.
• Gilbert, J. Petiot, J. 1997, Brass Instruments, Some Theoretical and Experimental Results,
Proceedings of Institute of Acoustics, Vol. 19, Part 5, Book 2, pp. 251-583.
• Jansson, E.V. Benade, A.H. 1974, On Plane Spherical Waves in Horns with Non-uniform
flare, Acustica 31, Vol. 31, no.3, pp. 185-202.
• Kaburagi, T. Yamada, N. Fukui, T. Minamiya, E. 2011, "A Methodological and Preliminary
Study on the Acoustic Effect of a Trumpet Player"s Vocal Tract", Acoustical Society of
America, Music and Musical Instruments, Vol. 130, Issue 1.
• Kreidler, J 2009, Loadbang: Programming Electronic Music in Pd, Wolke

64

• Lefebvre, A. 2010, Computational Acoustic Methods for the Design of Woodwind
Instruments, PhD, McGill University.
• Macaluso, C.A. Dalmont, J.P. 2011, Trumpet with near-perfect harmonicity: Design and
acoustic results, J. Acoust. Soc. Am., Vol. 129, Issue 1, pp. 404-414.
• Meddins, R. 2000, Introduction to Digital Signal Processing, Newnes.
• Myers, A. 1997, "The Horn Function and Brass Instrument Character", Perspectives in
Brass Scholarship: Proceedings of the International Historic, 1997, p. 239-262.
• Myers, A 1998, Characterization and Taxonomy of Historic Brass Musical Instruments
from an Acoustical Standpoint, Ph.D., The University of Edinburgh.
• Noreland, D 2003, Numerical Techniques for Acoustic Modelling and Design of Brass
Wind Instruments, Acta Universitatis Upsaliensis Uppsala.
• Olson, H.F. 1967, Music, Physics and Engineering, 2nd Edition, Dover Publications, New
York.
• Petrausch, S + Rabenstein, R 2005, Application of Block Based Physical Modeling for
Digital Sound Synthesis of Brass Instruments, ForumAcusticum, FA 2005, Budapest
Hungary, pp 703-708.
• Parker, B. 2009, Good Vibrations: The Physics of Music, John Hopkins University Press,
Maryland.
• Puckette, M 2007, The Theory and Technique of Electronic Music, World Scientific
Publishing.
• Rodet, X. 1995, One and Two Mass Model Oscillations for Voice and Instruments,
Proceedings of the International Computer Music Conference, Computer Music
Association, Banff, pp. 207-214.
• Scavone, G.P. 1997, An Acoustic Analysis of Single-Reed Woodwind Instruments With
and Emphasis on Design and Performance Issues and Digital Waveguide Modeling
Techniques, PhD, Stanford University.
• Shilke, R 2003, "Dimensional Charecterists of Brass Mouthpieces", Instrumentalist, May
2003.
• Smith, J.O. 1991, "Wave Simulation of Non-cylindrical Acoustic Tubes", International
Computer Music Conference, Montreal, October 1991, p. 304-307.
• Smith, J.O. 1992, Physical Modeling using Digital Waveguides, Computer Mustic Journal,
Physical Modeling of Musical Instruments, Vol. 16, Issue 4, p. 74-91.
• Smith, J.O. 2007, Introduction to Digital Filters with Audio Applications, W3K Publishing,
• Smith, J.O. 2010, Physical Audio Signal Processing, W3K Publishing, California.
• Smith, J.O. + van Walstijn, M 1998, "Use of Truncated Infinite Impulse Response (TIIR)
Filters in Implementing Efficient Digital Waveguide Models of Flared Horns and Piecewise
Conical Bores with Unstable One-Pole Filter Elements", ISMA-98 Leavenworth,
Washington, 28 June 1998, p. 309-314.
• Smith, R.A. 1978, Recent Developments in Trumpet Design, International Trumpet Guild,
Vol. 3.

65

• Smith, R.A. 1986, "Ensuring high quality in the production of musical instruments", Das
Musikinstrument, April 1986, p. 131-132.
• Smith, R.A. 1986, "The effect of material in brass instruments; a review", Proceedings of
Institute of Acoustics, Vol. 8, Part 1, pp. 91-96.
• Smith, R.A. 1987, "Holographs of bell vibrations", News and Views, Nature 329, 762, 29th
October 1987.
• Smith, R.A. 1999, "Exciting Your Instrument", Journal of the International Trumpet Guild,
May 1999, p. 44-45.
• Smyth, T. Scott, F.S. 2011, "Parametric Trombone Synthesis by Coupling Dynamic Lip
Valve and Instrument Models", Proceedings of the 8th Sound and Music Computing
Conference, 2011.
• van Walstijn, M. Cullen, J.S. Campbell, D.M. 1997, Modelling Viscothermal Wave
Propagation in Wind Instrument Air Columns, ISMA 97 Conference, Institute of Acoustics,
Volume 19: part 5, Book 2, pp. 251-583.
• van Walstijn, M. 2002, Discrete-Time Modelling of Brass and Reed Woodwind Instruments
with Application to Musical Sound Synthese, PhD, University of Edinburgh.
• van Walstijn, M 2007, "Wave-Based Simulation of Wind Instrument Resonators", IEEE
Signal Processing Magazine, March 2007, p. 21-31.
• Vergez, C. Rodet, X. 1997, Model of the trumpet funcitoning: Real time simulation and
experiments with artificial mouth, Proceedings of International Symposium of Musical
Acoustics, pp. 425-432.
• Välimäki, V. + Laakso, T.I. 2000, Principles of Fractional Delay Filters, International
Conference on Acoustics, Speech, and Signal Processing, Instanbul, Turkey, 5-9,
• Yoshikawa, S. 1995, "Acoustical behaviour of brass player"s lips", Journal of Acoustic
Society fo America, Vol. 97, Issue 3, pg. 1929-1939.

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