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Predicting the playing frequencies of brass
instruments
Michael J. Newton, Murray Campbell, John Chick
Acoustics & Audio Group, University of Edinburgh, UK.
Jonathan A. Kemp
School of Music, University of St Andrews, UK.
Summary
The frequency of a note played on a brass wind musical instrument is usually close to the frequency
of one of the peaks in the input impedance curve of the instrument. The exact playing frequency also
depends on factors, including lip tension and vocal tract shape, which allow an experienced player to
modify the pitch and timbre of a note without changing the physical shape of the instrument in any
way. This ability to ‘bend’ or ‘lip’ a note is useful in making subtle adjustments of intonation, and in
creating a musically expressive performance. This paper presents studies of the sounding frequencies
of notes played on brass instruments using both human players and an artificial mouth. The extent to
which the playing frequency can deviate from the acoustic resonance frequency is studied for different
playing regimes, and the results are compared with numerical predictions using a lip model with two
degrees of freedom.
PACS no. 43.75.Fg, 43.75.Yy
1. Introduction
Sound production in brass instruments results from
a nonlinear coupling between a mechanical oscillating
system, namely the lips of the player, and an air col-
umn, driven by an overpressure (energy source) sup-
plied by the lungs. To a first approximation, the be-
haviour of the air column is linear, and so may be
described in the frequency domain by an acoustical
impedance. A typical impedance curve for a trom-
bone is shown in Figure 1, which reveals a number
of resonances that lie somewhat close to a harmonic
series. The frequency of the played noted usually has
a fundamental component close to one of these acous-
tic resonances. However, the exact playing frequency
depends upon an interplay between the acoustical res-
onances of the instrument, the mechanical properties
of the lip-reed[1] and in some circumstances the reso-
nances of the vocal tract[8, 11].
Study of the playing frequencies of brass instru-
ments has seen significant work over the past decades,
and has involved a range of experimental and theoreti-
cal studies[9, 10, 11]. An important result of this is the
now widely accepted notion that the lip-reed exhibits
both inward and outward ‘striking’ behaviour[12]. It is
believed that an interplay between these two modes of
(c) European Acoustics Association
0 100 200 300 400 500 600 700 800
0
10
20
30
40
Frequency (Hz)
Impedance magnitude (MOhms)
Figure 1. An example impedance curve, for the trombone
used in this study in the fully retracted slide position.
operation, possibly corresponding to different degrees
of freedom in the lip motion, allows for the range of
‘lipping’ behaviours seen in real and artificial brass
players. Lipping is the process whereby the played
frequency may be shifted both above and below the
relevant acoustical frequency by adjustments to the
embouchure alone[18].
In this paper a review and update of some earlier
work[14, 15, 16] is presented, alongside some new com-
putational simulations. The objective is to further in-
vestigate the lip-reed - resonator interaction, for vari-
able embouchure and fixed resonator and vice versa,
with a view to exploring the interplay of factors that
determine the played frequency.
Newton et al: Predicting the playing frequencies of brass instrumentsFORUM ACUSTICUM 2014
7-12 September, Krakow
2. Background and Method
2.1. Brass instrument lip-reeds
A large body of literature exists on musical reed
physics (see for example [17]). Of particular interest
here are the definitions that describe so-called ‘in-
ward’ and ‘outward’ striking reeds.
An inward striking reed was originally defined by
Helmholtz[6] as one that undergoes a reduction in
aperture as an upstream overpressure is increased.
More recently, it has been shown[13] that for a purely
inward striking reed, a condition exists on the phase
relationship between the mouthpiece pressure fluctu-
ation pmp and the oscillating reed flow Ulr (assumed
to be in phase with the reed opening) such that en-
ergy can only be supplied to the air column for phase
difference magnitudes of π/2or less between pmp and
Ulr. Furthermore, a single degree of freedom inward
striking reed is expected to play only below its nat-
ural frequency, and below the relevant air column
resonance[17].
In contrast, an outward striking reed is supposed
to undergo an increase in aperture with increasing
overpressure, in a manner that might be expected for
a ‘swinging-door’ type reed. Such a reed can supply
energy to the air column when the phase difference
magnitude between reed opening (and flow Ulr) and
mouthpiece pressure again lies below π/2. Playing fre-
quencies are expected to lie above the reed’s natural
mode, for a single degree of freedom reed, and above
the relevant air column resonance[17].
In brass playing, the process of lipping allows a
player to play both above and below the relevant
acoustical resonance frequency, presumably by care-
ful adjustments to the embouchure. Such adjustments
can be made in a smooth manner, suggesting a con-
trolled transition between the inward and outward
striking operation modes. In the present work this
behaviour is investigated experimentally and numeri-
cally by studying the effect of maintaining a constant
embouchure whilst the acoustic resonances are altered
in frequency (section 3.1). A counter-example is then
studied (section 3.2) whereby the resonator is held
constant whilst the embouchure is adjusted to ‘lip’
the note both above and below the relevant acousti-
cal resonance.
2.2. Experimental setup
2.2.1. Artificial mouth for controlled constant em-
bouchure glissandos
Experimental investigations were carried out using an
artificial brass player[16]. This consisted of a pair of
water-filled rubber lips whose mechanical properties
could be varied via adjustments to the internal wa-
ter pressure and mouthpiece contact force. Once a
suitable embouchure was established, the mouthpiece
could be clamped securely in position to allow varia-
tions to the acoustical conditions, such as extending
the slide of a coupled trombone, without alteration to
the lip-reed mechanics. Such a procedure, described
here as a constant embouchure glissando, is rather
difficult to achieve in a repeatable manner using real
players.
2.2.2. Human playing measurements for controlled
lipping
In contrast to the case of a constant embouchure glis-
sando, it is difficult to use the artificial mouth to
produce smooth and continuously variable ‘lipping’
behaviour via embouchure adjustments alone. In or-
der to investigate this behaviour, measurements us-
ing a human player were carried out instead. A semi-
professional player was asked to smoothly transition
(‘slur’) from a playing frequency below to above a
given resonance, and then to continue lipping upwards
until a coupled regime with the next-highest acousti-
cal resonance was reached. The converse case, namely
downward lipping/slurring, was also explored.
2.3. Numerical setup: lip-reed and brass in-
strument model
A finite difference time domain (FDTD) model of a
trombone was constructed in Matlab as a numerical
counterpart to the experimental investigations. The
behaviour of the air column was based on the work of
Bilbao[2], and the lip-reed modelled using the single
mass with two degrees of freedom model of Adachi and
Sato[1]. Details concerning the combination of these
two models are described in Kemp et al.[3].
For this study the lip width was set to be b=10
mm, while the distance between the top of the lip
and the midline between the lips was given a value of
ξjointy
=6 mm. The mass of the lip was increased by
a factor of 1.5 in comparison with previous work[3] on
the trumpet, in accordance with the trombone mouth-
piece having a wider rim. The quality factor of the
lip was set to Q=1.5. It is noted that higher Q
factors give faster starting transients, and a Qvalue
of 5, as used in previous work[3], has been found to
give reasonable agreement with the starting transient
of notes higher up in the trumpet’s range. Higher Q
factors also increase the range of pitches that may
be sounded, giving the model increased ability to lip
notes up or down[7]. The lower value of Q=1.5 used
was found to be necessary in order to approximate
the amount of lipping seen in present experimental
results, using a fixed slide extension for the lower part
of the pitch range of the trombone. This result seems
to lie in broad agreement with previously measured
experimental data concerning lip-reed resonances of
real lips[12]. All other parameters (unless stated oth-
erwise) were identical to those in Kemp et al.[3].
Newton et al: Predicting the playing frequencies of brass instrumentsFORUM ACUSTICUM 2014
7-12 September, Krakow
3. Results and Discussion
3.1. Playing frequencies with variable acous-
tics and fixed embouchure
An artificial brass player was configured to produce
a steady note when coupled to the third impedance
peak (see Figure 1) of a trombone. The value of
the playing frequency was then measured as the res-
onator acoustics were continuously varied by exten-
sion of the trombone slide (i.e. variable acoustics and
fixed embouchure). Such a procedure might be called
‘anti-lipping’, in that the embouchure was deliber-
ately held constant during a period of several minutes,
which in impossible to achieve in real playing. Gradu-
ally extending the trombone slide in 5cm steps, from
a fully retracted position to an extension of 65cm,
had the effect of progressively lowering the acoustical
impedance frequencies. The total range of extension
corresponded approximately to the shift from first to
seventh position of the trombone slide, which has the
musical effect of lowering the pitch of each playable
note by six semitones. Note that mechanical response
measurements were carried out before the first (0cm
extension) and after the last (65cm extension) playing
frequency measurements, in order to ensure that the
mechanical lip-reed properties had remained constant.
Figure 2 shows experimental and numerically sim-
ulated measurements of the playing frequency of
the trombone system throughout the slide extension
range. (Note that the numerical simulations used a
slide extension step of 2.5cm.) The playing frequen-
cies were estimated by lowpass filtering (Hann filter,
cutoff 800 Hz) the mouthpiece pressure signal and cal-
culating the time between consecutive zero-crossings.
The two nearby experimentally measured lip-reed res-
onance frequencies, as well as the relevant nearby
acoustical resonances, are also shown. The lip model
used a constant lip frequency of 160Hz for both de-
grees of freedom, except that the vertical spring con-
stant and damping were increased during lip contact,
as described in Kemp et al.[3]. A runtime of 0.2s was
computed at each slide extension to allow extraction
of the local playing frequency.
The experimental lip-reed possessed at least two
relevant mechanical resonances, the lower of which
(125Hz) could be categorised as outward striking,
and the upper (180Hz) as inward striking (data not
shown here). Initially, at 0cm extension, the playing
frequency of 170Hz was almost the same as the fre-
quency of the third acoustic resonance, as expected.
This lay between the frequency of the two mechanical
resonances, which might suggest an outward striking
behaviour in a single degree of freedom interpretation
based upon the lower mechanical resonance (i.e. play-
ing frequency at or above both the mechanical and
acoustical frequencies).
As the slide was extended, the playing frequency
tracked downward at a slower rate than that of the
0 10 20 30 40 50 60
120
140
160
180
200
220
240
Slide extension length (cm)
Playing frequency (Hz)
Experimental playing frequency
Simulated playing frequency
Lip frequencies (experimental)
Lip frequencies (simulated)
Acoustical resonances
Figure 2. Playing frequency of the trombone system as a
function of slide extension, for fixed lip parameters. The
blue line is experimental data, the green is computed with
the model. The black lines show the frequencies of the
trombone’s nearest acoustic impedance peaks, and the red
lines the frequencies of the two relevant artificial lip eigen-
modes (experimental), held constant and measured sepa-
rately. The artificial mouth was initially configured to play
the third impedance peak (see Figure 1) in the fully re-
tracted position (roughly Bb3), associated with the lower
of the acoustical resonances shown. The lip model used a
single frequency (160Hz) for both degrees of freedom.
third acoustical resonance. After 30cm, the playing
frequency then jumped upwards to 176Hz, at which
point the acoustical coupling was primarily deter-
mined by the fourth resonance. However, this reso-
nance had a frequency of 187Hz, which meant that
the playing frequency was now below that of the dom-
inant acoustical resonance. Assuming for a moment
that only a single degree of freedom was relevant in
the lip-reed, presumably the resonance at 180Hz, this
would suggest an inward striking reed (i.e. playing fre-
quency at or below both the mechanical and acous-
tical frequencies). A further extension of the trom-
bone slide to 45cm and beyond then saw the playing
frequency smoothly cross over to lie above the local
acoustical resonance. A single degree of freedom in-
ward striking reed would not be expected to achieve
this behaviour (i.e. produce a playing frequency be-
low the mechanical frequency but above the acoustical
resonance frequency). Rather, such behaviour would
require at least two degrees of freedom, with each
resonance possessing a suitable phase characteristic.
These experimental results are broadly in agreement
with previous work[15, 16], as well as recent work from
Boutin et al.[18].
The close agreement of the simulated playing fre-
quency with this result (Figure 2), the underlying
model of which includes two mechanical degrees of
freedom, appears to confirm the requirement of the
lip-reed to include both inward and outward striking
characteristics, which may be smoothly combined to
produce controlled regime transitions. Interestingly,
despite the fact that both degrees of freedom were set
to the same frequency (160Hz), a very similar transi-
Newton et al: Predicting the playing frequencies of brass instrumentsFORUM ACUSTICUM 2014
7-12 September, Krakow
0 1 2 3 4 5 6
80
100
120
140
160
180
200
220
240
260
Time (s)
Frequency (Hz)
Frequency tracking based on mouthpiece signal: Upward lipping
Playing frequency
Acoustical resonances
Figure 3. Playing frequency of a trombone (fully retracted
position) and real player, detected from the mouthpiece
pressure (blue), for three different notes. The player was
asked to ‘lip’ the note around the relevant acoustic res-
onance, until the note ‘slurred’ upwards to the next
resonance. The black lines show the actual acoustical
impedance frequencies of the instrument.
tion behaviour is seen from the model. The implica-
tions are twofold. Firstly, to serve as a useful reminder
that the phase characteristic of the mechanical reso-
nance(s), more than the amplitude, is crucial in allow-
ing stable oscillations. Secondly, that a given playing
frequency may be achieved from more than one unique
embouchure characteristic. This point is perhaps not
surprising, given the range of lips that are able to
produce the same note on a given instrument. Fur-
ther work is needed to ascertain the true variability
of real players’ embouchures in brass playing.
3.2. Lip sluring: fixed acoustics and variable
embouchure
3.2.1. Experimental results
A semi-professional trombone player was asked to
slur between three notes on a trombone in the fully
retracted position (i.e. an experiment using fixed
acoustics and variable embouchure). The transitions
requested were between the second (111Hz), third
(169Hz) and fourth (227Hz) acoustical resonances, as
measured directly from the instrument (Figure 1).
The first exercise required the player to ‘lip’
notes from below to above the acoustic resonance as
smoothly as possible, and to continue lipping upwards
until the next resonance (i.e. the next highest ‘note’)
was reached, as shown in Figure 3. A counter-exercise
in downward lipping was similarly carried out, as
shown in Figure 4. The pressure signal in the instru-
ment mouthpiece was sampled during the exercises,
and frequency tracking analysis carried out.
3.2.2. Numerically simulated results
A comparative lipping/slurring simulation was car-
ried out using the numerical brass instrument model
described in section 2.3. Lip slurs were generated in
0 0.5 1 1.5 2 2.5 3
80
100
120
140
160
180
200
220
240
260
Time (s)
Frequency (Hz)
Frequency tracking based on mouthpiece signal: Downward lipping
Playing frequency
Acoustical resonances
Figure 4. As per Figure 3, but for downward lipping.
the model by linearly ramping the lip frequency be-
tween two nominated frequencies over a duration of 3
seconds.
To enable reasonable comparison with the experi-
ential results, an upward lip slur with a lip eigenmode
frequency ramping from 50 Hz to 260 Hz over 2 sec-
onds is shown in figure 5. A simulation of downward
lipping was also carried out, with the lip eigenmodes
ramped from 260 Hz to 50 Hz, as shown in Figure 6.
The trombone bore profile used in the model was
measured directly from the trombone used in the ex-
periments, for a fully retracted slide position. Thus
the acoustical impedance resonance frequencies would
be expected to be consistent between experiment and
simulation, provided other environmental parameters
were suitably matched.
3.2.3. Discussion
Consider first the human playing measurements
shown in Figures 3 and 4. For both upwards and
downwards lipping, the fundamental playing fre-
quency (as deduced from the mouthpiece pressure sig-
nal) is seen to traverse quite smoothly across each
of the relevant acoustical resonances. There does not
appear to be a strong tendency for the playing fre-
quency to persist either above or below the acoustic
resonance. However, note that the effect of lip protru-
sion on the tuning of the acoustical resonances was
not taken into account in the impedance measure-
ments, and so the ‘true’ relevant resonance frequency
should be treated as approximate. In reality is it likely
that lip protrusion into the mouthpiece, for both real
and artificial brass players, may mean that the acous-
tic resonances were tuned slightly higher than was
recorded by measurement at the mouthpiece entry
plane[4, 5]. Further work is ongoing to clarify the rel-
evance of this effect in brass playing.
For upwards lipping, the system frequency traverses
from below to above the acoustical resonance (e.g. be-
tween 2s and 3.5s in Figure 3). The converse is true in
downwards lipping, whereby the playing frequency be-
gins above the relevant resonance, and migrates down-
wards (e.g. between 0s and 1s in Figure 4). In all cases,
Newton et al: Predicting the playing frequencies of brass instrumentsFORUM ACUSTICUM 2014
7-12 September, Krakow
0 0.5 1 1.5 2 2.5 3
50
100
150
200
250
Time (s)
Frequency (Hz)
Frequency tracking based on mouthpiece signal: Upward lipping
Playing frequency
Lip frequency
Acoustical resonances
Figure 5. Playing frequency of the simulated trombone
system for a fully retracted slide extension, when the lip
eigenmode frequencies are linearly increased from 50Hz to
260Hz over an interval of 3s.
the observed behaviour shows that this human lip-
reed is able to smoothly transition from an outward
to an inward striking characteristic, notwithstanding
the absence of direct information about the lip eigen-
modes. It also demonstrates the remarkable control
that an experienced brass player can have over the
playing frequency of the instrument.
Consider now the simulated playing measurements
shown in Figures 5 and 6. These figures include addi-
tional information not easily obtained from the human
measurements, namely the actual lip frequencies as a
function of time. (Note that for the present simula-
tions both degrees of freedom in the lip model were
set to the same frequency.) The overall behaviour for
both upwards and downwards lipping is broadly com-
parable with the experimental measurements, in that
the playing frequency is able to take values both above
and below the relevant acoustical resonance. It is nev-
ertheless clear that the most stable regimes of the
model, particularly in the downward lipping exam-
ple, tend to lie below the acoustical resonance, which
taken on its own might suggest a primarily inward
striking behaviour. However, the playing frequency
tends to lie above the lip frequencies, suggesting an
outward striking behaviour. The overall lip-reed char-
acteristic must therefore include a mixture of these
operation regimes. Such a result might be expected
from an intuitive examination of the lip-reed model,
which allows motion in two degrees of freedom, thus
allowing differing phase characteristics between the
axial (presumably outward striking) and transverse
(presumably inward striking) motions. Further inves-
tigation is needed to ascertain whether alterations to
the lip eigenmode frequencies and damping along each
degree of freedom can allow for clearer separation be-
tween inward and outward striking behaviour, as ap-
pears to be the case from the human measurements.
Three further observations merit attention. Firstly,
the lip-reed model used here consistently produced
larger displacements along the axial degree of free-
0 0.5 1 1.5 2 2.5 3
50
100
150
200
250
Time (s)
Frequency (Hz)
Frequency tracking based on mouthpiece signal: Upward lipping
Playing frequency
Lip frequency
Acoustical resonances
Figure 6. As per Figure 5, but with lip eigenmode frequen-
cies linearly decreased from 225Hz to 100Hz.
dom than the transverse. This appears to be broadly
at odds with some experimental findings, and may in-
fluence the apparently larger degree of lipping achiev-
able in some simulated results.
Secondly, the initial 0.3s of the simulated upward
slur show that the playing frequency matches very
closely with the lip frequency, in the absence of any
strong nearby acoustical resonances. Self-oscillations
at such a minima in the acoustical impedance are only
possible with a two degree of freedom reed model, as
in the classic two-mass vocal fold model[19]. A similar
behaviour is also seen at the very end of the simulated
downwards slur.
Thirdly, the potential effect of temperature varia-
tion along the instrument during human playing was
considered. Measurements of the instrument’s acous-
tic impedance were taken just before and just after
one of the lipping runs, but the frequency deviations
were small enough (around 10 cents) to be considered
negligible in the present work.
4. Conclusions
The work in this paper provides a review and update
of some previous work into the playing frequencies
of brass instruments[15, 16]. Two experimental setups
were used, involving first an artificial brass player,
and then a real player. A numerical brass instrument
model, incorporating two degrees of mechanical free-
dom in the lip-reed, was used as a comparison.
The first investigation examined the variation in
playing frequency of a trombone as the embouchure
was held constant and the acoustical resonance fre-
quencies progressively lowered by extension of the
slide. Playing frequency variations during the experi-
ments were consistent with a lip-reed possessing both
inward and outward striking characteristics, and at
some points a combination of the two. Separate me-
chanical response measurements confirmed the pres-
ence of two suitable resonances at 125Hz and 180Hz.
Playing frequencies of the numerical model matched
Newton et al: Predicting the playing frequencies of brass instrumentsFORUM ACUSTICUM 2014
7-12 September, Krakow
very closely with the experiments when both lip eigen-
modes were set to 160Hz, suggesting that the phase
behaviour of the lip-reed is the really crucial feature
in ensuring stable system oscillations.
The second investigation provided an exact coun-
terexample to the first, whereby adjustments to the
embouchure were used on their own to alter the play-
ing frequency of the instrument. Such a procedure is
commonly used in real playing to ‘lip’ notes for tun-
ing or creative processes. Experimental results showed
that during extended upwards and downwards slurs
the playing frequency smoothly transitions from be-
low to above (and vice versa) the relevant acousti-
cal resonance. Only a lip-reed possessing both inward
and outward striking capabilities would be capable of
achieving this result. Numerical simulations using the
two degree of freedom lip-reed model were broadly
capable of reproducing this behaviour using a linear
lip frequency ramp. Further investigations are needed
on real players to determine the precise mechanical
configurations during playing that allow for this re-
markable flexibility.
Acknowledgement
The authors acknowledge the contribution of Adrien
Bitton and Amaya Lopez-Carromero in gathering
some of the experimental data used in this paper. One
of the authors (DMC) acknowledges financial support
by the Royal Society of Edinburgh
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