The Relationship Between Bore Resonance Frequencies
and Playing Frequencies in Trumpets
Pauline Eveno, Jean François Petiot, Joël Gilbert, Benoît Kieffer, René Causse
To cite this version:
Pauline Eveno, Jean François Petiot, Joël Gilbert, Benoît Kieffer, René Causse. The Relationship
Between Bore Resonance Frequencies and Playing Frequencies in Trumpets. Acta Acustica united
with Acustica, Hirzel Verlag, 2014, 2 (100), pp.362-374. 10.3813/AAA.918715. hal-01106940
HAL Id: hal-01106940
https://hal.archives-ouvertes.fr/hal-01106940
Submitted on 24 Feb 2020
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The Relationship Between Bore Resonance
Frequencies and Playing Frequencies in
Trumpets
P. Eveno1)∗ , J.-F. Petiot2) , J. Gilbert3) , B. Kieffer1) , R. Caussé1)
1)
2)
3)
Institut de Recherche et Coordination Acoustique/Musique (UMR CNRS 9912), 1 place Igor Stravinsky,
75004 Paris, France. pauline.eveno@mail.mcgill.ca
Institut de Recherche en Communications et Cybernétique de Nantes (UMR CNRS 6597), 1 rue de la Noë,
BP 92101, 44321 Nantes Cedex 3, France.
Laboratoire d’Acoustique de l’Université du Maine (UMR CNRS 6613), Avenue Olivier Messiaen,
72085 Le Mans Cedex 9, France.
Summary
The aim of this work is to study experimentally the relationship between the resonance frequencies of the trumpet,
extracted from its input impedance, and the playing frequencies of notes, as played by musicians. Three different
trumpets have been used for the experiment, obtained by changing only the leadpipe of the same instrument.
After a measurement of the input impedance of these trumpets, four musicians were asked to play the first five
regimes of the instrument, for four different fingerings. This was done for three dynamic levels and repeated
three times. Statistical methods were implemented to assess the variability in the playing frequencies, and to
study quantitatively their relationships with the bore resonance frequencies. A limited influence of the musician
on the instrument overall intonation is observed, as well as a weak influence of the dynamic levels on the pitch of
the notes. The results show that for most of the regimes, variations of the resonance frequency lead to same order
variations of the playing frequency of the corresponding note. We noticed also that the sum function, derived
from the input impedance, does not give a better prediction of the playing frequency than the input impedance
itself.
1. Introduction
by Gilbert et al. [7] and Farner et al. [8] by using the harmonic balance technique adapted to self-sustained oscillations of wind instruments such as clarinets. The resonator
(i.e. the instrument body) is the linear part, treated in frequency domain, while the driving system (the reed) is the
nonlinear part, treated in time domain. The harmonic balance technique can also be used for brass instruments [9].
Three control parameters representing the “virtual” musician have to be defined: the pressure inside the mouth,
the resonance frequency of the lips, and the inverse of lips
mass density. Depending on the choice of these parameters, it is possible to obtain a series of playing frequencies,
such as those obtained by the musician. The coupling between the musician and the instrument can also be investigated using a simplified model in which a single mechanical lip mode is coupled to a single mode of the acoustical
resonator, as done by Cullen et al. [10] for the trombone. It
is also possible to predict the intonation of the instrument
by synthesizing the notes it can produce. Many studies are
carried out on physical modelling using temporal methods
[11, 12].
Measuring and computing wind musical instruments input
impedance is now well mastered [1, 2, 3, 4, 5, 6]. As part
of a larger project aimed toward helping instrument makers to design and characterise their musical instruments,
this work focuses on how the bore resonance frequencies,
taken from the input impedance, can be related to the playing frequencies. Indeed, instrument makers are primarily
interested in the overall intonation of their instruments in
playing situations, and therefore they need some predictive
indicators.
Some studies attempt to find a solution to this issue by
taking the coupling between the instrument and the musician into account. The case of reed instruments is treated
∗ New affiliation:
Computational Acoustic Modeling Laboratory, Schulich School of Music
of McGill University, 555 Sherbrooke Street West, Montréal, Canada.
pauline.eveno@mail.mcgill.ca
For the saxophone, for some advanced performance
techniques (bugling and altissimo playing), musicians can
1
use the resonance of their vocal tract to play a note close
to a weak bore resonance, or even decrease the sounding
pitch to several semitones below the standard pitch for the
same fingering [13, 14, 15]. It seems that this technique is
not used by trumpet players [16, 17].
The musician has a significant role in determining the
playing frequencies, this aspect being difficult to take into
account. Therefore, the first aim of this paper is to determine an order of magnitude of the brass player’s influence on the overall intonation of the instrument. Then,
it aims at finding some objective indicators from the input impedance which can predict the playing frequencies
without taking the musician’s behaviour into account, as
intended by Pratt and Bowsher [18] and previously by
Wogram [19]. This will be done by recording a large number of notes played by several musicians on three trumpets.
Section 2 presents some basic information about the
acoustics of the trumpet. Section 3 describes the recording
of notes played by the musicians on the different trumpets and the analysis of the data. From these measurements, an analysis of the musicians’ behaviour is presented in section 4. In section 5, the playing frequencies of
the recorded notes are compared to the bore resonance frequencies taken from the input impedance of the trumpets.
Section 6 presents the data first as a normal distribution
and then, in order to minimize the influence of the musician on the results, it focuses on frequency differences instead of the frequencies themselves. Finally, the relevance
of the sum function [19], a function made from the input
impedance to predict the intonation, is discussed.
Figure 1. Measurement of the input impedance amplitude (in dB)
and phase (in rad) of the trumpet called NORM with all the three
valves up, with the notes corresponding to each impedance peak
above (concert pitch of a B trumpet). The trumpet and the set-up
used for the measurement are presented in Section 3.1.
2. Trumpet resonances and playing frequencies: Preliminary discussion
Campbell and Greated [20] as well as Fletcher and Rossing [21] give a large overview on brass instruments. A
summary about trumpets is reported here as well as a discussion on the coupling with the musician.
The acoustic response of an instrument at different
frequencies can be characterised by its input impedance
(impedance computed or measured at the input of the entire instrument, that is to say at the input plane of the
mouthpiece). A typical input impedance of a brass instrument (see Figure 1) shows a large number of bore
resonance frequencies, where the impedance amplitude is
maximum and the phase is passing through zero. Some
of these resonance frequencies are associated with a note
(or oscillation regime) that the musician can play. In the
example of Figure 1, corresponding to the basic fingering
of a B trumpet where all the three valves are up, the resonances 2 to 6 correspond to the series of concert notes B 3,
F4, B 4, D5, F5 (harmonic series of B 2). The first resonance does not correspond to a normally playable note on
the trumpet. The three valves offer height combinations,
which allow the construction of the whole chromatic scale,
since the activation of a valve produces an elongation of
the air column which lower the resonance frequencies of
the instrument. The first valve brings down the frequency
of one tone, the second of a semitone, and the last one of
one and a half tones. In the rest of the article, a pressed
valve will be noted 1 and a valve up will be noted 0.
An initial estimation of the instrument intonation can
be carried out by comparing the bore resonance frequencies to their corresponding notes in the equally-tempered
scale, as it is shown in Figure 2 for the trumpets called
CHMQ, DKNR and NORM in this study. These instruments are presented in section 3. This representation is often used by instruments makers and is for example presented in the BIAS software1 . These graphs show that the
series of resonances of the trumpet can be considered as
harmonic with a [−20, +20] cents precision interval (the
resonance frequencies are almost aligned on an horizontal
line), apart from the resonances corresponding to the second regime of 100, 110 and 111 fingerings that are too low.
These diagrams also show that, by increasing the length of
the bore, the resonance frequencies become more distant
from the frequencies of their corresponding notes in the
1
BIAS is the Brass Instrument Analysis System developed at the Institute of Music Acoustics in Vienna: http://www.bias.at/?page_id=
5&sprache=2.
2
(a)
(c)
(b)
Figure 2. Intonation graph of trumpet (a) CHMQ, (b) DKNR and (c) NORM, obtained by calculating the difference in cents between
each resonance frequency of the input impedance of each trumpet for the four fingerings and its corresponding note in the equallytempered scale.
equally-tempered scale. Consequently, if we assume that
the resonance frequencies are representative of the playing frequencies, notes should be easier to play in tune with
the 000 fingering than with the 111 fingering.
However, this affirmation has to be considered cautiously because this graph forgets an important element of
the trumpet playing: the musician. Indeed, these diagrams
are only estimations of the intonation because the playing frequencies are not exactly equal to the bore resonance
frequencies. The differences between those frequencies result from a complex aeroelastic coupling between the lips
of the musician and the resonator. Thus, the intonation of
the instrument is not only controlled by the closest resonance frequency but possibly conditioned by upper resonance frequencies of the resonator [22].
Furthermore, a wind instrument is not an instrument
with a fixed sound, that is to say the musician can modify the pitch and the timbre of the played note by controlling his/her embouchure and “bending” the notes. The embouchure represents the capacity of the musician to control the mechanical parameters of his/her vibrating lips, by
modifying his/her facial musculature as well as the support
force of the lips on the mouthpiece. This also includes the
ability to control the air flow between the lips.
Figure 3. Parametrisation of the leadpipes used in this work
(from Petiot et al. [24]). Radii r1 , r2 , r3 and r4 are given in Table I.
leadpipes presented in the Table I and a leadpipe originally
provided with the trumpet, called NORM, as in “normal
leadpipe”. These three instruments are all playable and are,
at first sight, very similar from a musical point of view. For
all the tests, the position of the tuning slide was similar: it
was pulled out of a length of 1 cm for all the trumpets and
all the musicians.
The input impedances for these trumpets were then
measured for four different fingerings (000, 100, 110 and
111) using a set-up described by Macaluso and Dalmont
[4]. The first regime is not played with a trumpet. In this
study, the notes will thus be recorded for regimes 2 to 6
with these four fingerings. Nevertheless, some of these
notes do not correspond to the usual fingerings used by
the musician. Regimes 2 to 6 are normally played by musicians for fingerings 000 and 100. For the 110 fingering,
musicians play notes from the second regime to the fifth.
The sixth is generally not used since concert pitch D5 can
be played with the fifth regime of the 000 fingering. For
the 111 fingering, only regimes 2 and 3 are usually played.
Regimes 4, 5 and 6 are an alternative way of playing the
notes E4, G 4 and B4, for which musicians usually use the
third regime of the 010 fingering (E4), the fourth regime
of the 100 fingering (G 4) and the fifth regime of the 110
fingering (B4). These fingerings have been chosen in order to study the whole range of the trumpet frequencies,
from the lowest pitch to the highest. Furthermore, while
certain combinations of regimes and fingerings are almost
never used by musicians, it has been interesting to include
them in this study. Indeed, trumpet players are not used to
playing these notes so there is no “learning effect”, which
3. Set-Up and Data Analysis
3.1. Set-Up
A parametrised leadpipe, made of four different interchangeable parts, was designed [23, 24] as it is shown in
Figure 3. Several parts with various values for the radii r1 ,
r2 , r3 and r4 were manufactured with a numerically controlled turning machine. A letter has been given to each
part of the leadpipe, corresponding to the dimensions of
the radii. Thus, using the same B trumpet (Bach model
Vernon, bell 43) with the same mouthpiece (Bach 1 1/2
C) and the parametrised leadpipe, different instruments
with small different acoustical behaviours can be designed.
Three leadpipes were considered for the study: the two
3
Table I. Description of the two parametrised leadpipes used in this study (the radii are given in mm).
Part 1
CHMQ
DKNR
r1
4.64
4.64
Part 2
r2
5
5.45
r2
5
5.45
Part 3
r3
5.5
5.5
means that they are more likely to play without focusing
on the intonation.
Four musicians, one professor at a music school and
three experienced amateurs, were asked to play the three
trumpets to record the sounds. After a short warm-up, each
trumpet player had to play the five first playable notes
(regimes 2 to 6) by saying the name of the note before
playing, in order to have a short rest between the notes
and “forget” the pitch of the previous note. Indeed, trumpet players are interested in testing the flexibility of their
instrument and, if necessary, they bend the note in order
to correct the intonation defects. Nevertheless, the task for
the musician is different here since it consists of letting the
instrument guide him, even if it means playing out of tune.
The musicians were then asked to play the note with the
easiest emission, without trying to correct the intonation.
These recordings were made for three dynamic levels in
order to study their influence on the playing frequencies:
first mezzo forte, then piano, and finally forte. Afterwards,
each trumpet player had to move to the next fingering with
the same protocol and so on for the four fingerings and
the three trumpets. They had to repeat the whole process
three times in order to test their reproducibility. Finally, 4
trumpet players times 3 trumpets times 4 fingerings times
5 regimes times 3 dynamic levels times 3 repetitions give
2160 notes to analyse.
r3
5.5
5.5
Part 4
r4
5.7
5.825
r4
5.7
5.825
r5
5.825
5.825
Figure 4. Waveform (above) and evolution of the frequency of
the B 3 note played by a musician with the basic fingering (000)
at mezzo forte. Dashed lines separate, on the left the transient and
on the right the quiescent, from the rest of the quasi-stationary
signal. The dotted line represents the mean value of the frequency
during this quasi-stationary part. The error bar in the grey rectangle at the top left-hand corner represents twice the standard
deviation of the frequency during the quasi-stationary signal.
3.2. Data Analysis
standard deviation is then calculated in order to estimate
the ability of the trumpet player to play at a stable playing
frequency.
The measurements of the trumpets’ input impedances
and the recording of the musicians were carried out at different temperatures. The input impedance was measured
at 23◦ C whereas notes were played at a room temperature
of 25◦ C. According to Gilbert et al. [26] and Noreland
[27], we consider that the temperature of the air column
was around 28◦ C during playing. Consequently, for a reliable comparison, resonance frequencies need to be moved
forward from the equivalent temperature shift. Since resonance frequencies of both cones and cylinders are proportional tothe sound velocity, which can be written as
c = 331.45 T/T0 m/s with T the temperature in Kelvin
and T0 = 273.16 K, it can be considered that the resonance frequencies of a trumpet are proportional to the
square root of the temperature expressed in Kelvin. Consequently, the resonance frequencies from the measured
input impedances are increased by 14 cents which is the
equivalent of 5◦ C in order to be at the same level of the
playing frequencies’ temperature.
The playing frequency of the notes has been analysed with
the YIN [25] software2 , which is an estimator of the fundamental frequency specially calibrated for speech and music. Overlapping square windows of 68 ms length were
used. This is more than twice the largest expected period for all the measured notes. We noticed that musicians
were not able to play a perfectly steady note, and slight
oscillations around the playing frequency were observed.
Figure 4 shows an example of the frequency evolution of
one note, concert B 3, played by a musician with the basic fingering (000) at mezzo forte. At the beginning, the
frequency rapidly increases: this is a typical transient. The
same effect is happening during the quiescent. After removing the transient and the quiescent, a quasi-stationary
part stays, where the frequency is fluctuating a few hertz.
Therefore, from a measured signal like the one in Figure 4, what we call in this paper the playing frequency,
will be determined as the mean of the instantaneous frequency during the time t of the quasi-stationary part. The
2
It can be downloaded at http://audition.ens.fr/adc/
4
Finally, the frequency of the resonances is precisely determined with a peak fitting technique using a least square
method on the complex impedance [28]. This method represents the impedance in the Nyquist plot. In this plot, the
resonance is locally a circle that should go through the experimental points. Then, the resonance frequency is the
angle of the point, which is the furthest from the origin.
This method is the one used by Macaluso and Dalmont [4]
and leads to an estimation of the resonance frequency with
an uncertainty of about 5 cents. The resonance frequency
could also be determined with the phase zero crossing.
Nevertheless, as explained in [29, p. 149], the amplitude
of the impedance gives more information about the tuning and the ease of playing than the phase, that is why this
definition of the resonance frequency was chosen.
4. Musicians’ behaviours
4.1. Descriptive analysis of the playing frequencies
Figure 5. Boxplots representing the statistics of the playing frequencies of the notes played by each player for the three dynamic
levels (p, mf and f). Data are expressed in cents, as a difference
between each playing frequency and its corresponding resonance
frequency.
In order to study the behaviour of each musician, we represent by a boxplot the differences (in cents) between
each playing frequency and its respective resonance frequency for all the notes played. A boxplot is a convenient
way of graphically representing a distribution of numerical data through their five-number summaries: the smallest
observation (sample minimum), lower quartile (25th percentile, bottom of the box), median, upper quartile (75th
percentile, top of the box), and largest observation (sample
maximum).
One boxplot per dynamic level allows one to study the
influence of the dynamic level on the playing frequencies.
In each boxplot of Figure 5, there are 180 notes which correspond to 3 trumpets times 4 fingerings times 5 regimes
(2 to 6) times 3 attempts.
These boxplots show that the playing frequencies are,
on average, higher than the bore resonance frequencies
and that the four musicians play at a slight higher pitch
at the piano dynamic level (p) than at mezzo forte (mf) or
forte (f). The dynamic leads to less than a 10 cents difference on the playing frequency median. The four trumpet
players have a similar behaviour as they all play, on average, in the order of 8 to 20 cents above bore resonance
frequencies. These results could be consistent with a dominant outward striking regime of oscillations for the lips
that has been observed in previous studies [30, 31, 32].
Nevertheless, Figure 5 shows that musicians can also play
below the resonance frequencies for some notes. Players
can thus have different behaviours depending on the note
they play, on their embouchure, etc. and a single mechanical oscillator cannot model the complete behaviour of the
lip reed [33, 34, 35, 10].
[36, 37]. ANOVA is a collection of statistical models to
model a quantitative variable (the response) with qualitative variables (the factors). It belongs to the general frame
of the linear model, and proposes statistical tests to determine whether or not the means of different groups of data
are all equal, in the case of more than 2 groups (generalisation of the t-test).
In our application, the response is the playing frequency,
which is supposed to be modelled as the sum of different qualitative factors (independent variables). The general model in the case of a five-factors ANOVA is given
by
Fplayijklm = µ + αi + βj + γk + δl + ηm + εijklm
(1)
with
• µ a constant,
• αi the effect of the level i of the musician (i = 1 to 4
since there are four musicians),
• βj the effect of the level j of the dynamic level (j = 1
to 3 since there are three dynamics),
• γk the effect of the level k of the trumpet (k = 1 to 3
since there are three trumpets),
• δl the effect of the level l of the fingering (l = 1 to 4
since there are four fingerings),
• ηm the effect of the level m of the regime (m = 1 to 5
since there are five regimes),
• and εijklm the error term.
Each coefficient represents the influence of the level of the
factor on the response. From the measurements, a least
square procedure is used to estimate these coefficients
(minimization of the squared error between the measured
playing frequency and the playing frequency given by the
4.2. Modelling of the playing frequency with
ANOVA
In order to estimate the influence of each controlled factor of the experiment on the playing frequency, it can be
modelled using the analysis of variance method (ANOVA)
5
Table II. Results of the ANOVA model for all the data of the
study. Source means “the source of the variation in the data”, DF
means “the degrees of freedom in the source”, SS means “the
sum of squares due to the source”, MS means “the mean sum of
squares due to the source”, F means “the F-statistic” and P means
“the p-value”.
Source
DF
SS
MS
F
P
Musician
Dynamic
Trumpet
Fingering
Regime
3
2
2
3
4
1.7e3
2.0e3
1.8e1
4.9e6
4.6e7
5.8e2
1.0e3
8.9
1.6e6
1.1e7
1.6
2.8
0.02
4.5e3
3.1e4
0.184
0.063
0.976
<0.0001
<0.0001
model). A classical F-test is used to assess the significance
of the effect of the factors. The sources can be considered
to have a significant impact on the data if the probability
p is lower than 0.05 [36]. Table II gives the results of the
ANOVA model, the last column indicating the probabilityvalue p of the F-test (false rejection probability).
Only two factors have a significant effect on the playing frequency: the fingering and the regime (p < 0.0001).
Changing the fingering or the regime leads to important
modifications of the playing frequencies, which is obvious. The effects of the trumpet, the musician and the dynamic level are not significant at the 5% level. It means
that the influence of these factors on the playing frequency
is very weak. An analysis of the coefficients shows that the
piano dynamic level leads, on average, to a slightly higher
playing frequency than the mezzo forte and forte dynamic
levels. Moreover, it indicates that the first trumpet player
plays, on average, slightly lower than the other three. However these effects are negligible compared to those of the
fingering and the regime. Furthermore, an analysis of variance with interactions terms between each pair of factors
shows that interactions are not significant.
Figure 6. Playing frequencies (Fplay ) as functions of resonance
frequencies (Fres ), given in cents (taking the equally-tempered
scale as a reference) for all the 2160 recorded notes. There is
one figure for each regime, from regime 2 to regime 6, and a
different marker for each fingering: 000 in crosses, 100 in circles,
110 in diamonds and 111 in squares. The error bar on the left
represents twice the average standard deviation of a note. The
error bar on the right represents twice the average reproducibility
of the trumpet players.
5. Playing frequencies vs Resonance frequencies
thus calculated on these 9 notes and then the mean of these
standard deviations is calculated on all the notes played by
all the musicians (it is in fact a mean on 240 standard deviations). This error is equal to 8 cents, which is more important than the average standard deviation. Indeed, this
reproducibility is calculated by considering the three dynamic levels, which leads to more important variations
of playing frequency. It corresponds to an audible pitch
difference. Indeed, the just-noticeable difference (JND) is
about 3 Hz for sine waves and 1 Hz for complex tones below 500 Hz. Above 1000 Hz, the JND for sine waves is
about 10 cents [38, 39, 40]. Standard deviations σ1 and σ2
can also be calculated for each regime or for each trumpet player. The results are presented in Tables III and IV.
Table III shows that regime 2 induces more variation on
both the varying playing frequency of the note and the reproducibilty of the musicians. Table IV shows that players
have more or less the same reproducibility even if player
4 seems to play more straight notes and is more repeatable
than the others.
5.1. Visualisation of the raw data
Figure 6 presents the playing frequencies of notes (Fplay )
as functions of the bore resonance frequencies (Fres ) of
the corresponding regimes. Both of these frequencies are
expressed in cents taking the equally-tempered scale as a
reference. In every subfigure,there are two error bars at the
upper left-hand corner. The error bar on the left represents
twice the average standard deviation of a note, called σ1 .
Indeed, as explained in section 3.2, each played note is determined by an average frequency (which corresponds to
the playing frequency taken into account in the paper) and
a standard deviation σ. The error bar thus stands for the average σ over all the played notes, which is equal to 5 cents.
The error bar on the right represents twice the average
reproducibility of the trumpet players, called σ2 . Indeed,
each musician will repeat 9 times the same note (3 dynamic levels times 3 attempts). The standard deviation is
6
Table III. Average standard deviation σ1 and average reproducibility of players σ2 calculated for each regime (in cents).
Regime
2
3
4
5
6
All
σ1
σ2
8
11
5
7
4
7
4
8
3
8
5
8
where Fplayijklm is the value of the measured playing frequency for musician (i = 1 . . . I with I= 4, see Section 4.2
for more details on the variables), dynamics (j = 1 . . . J
with J= 3), trumpet (k = 1 . . . K with K= 3), fingering
(l = 1 . . . L with L= 4) and regime (m = 1 . . . M with
M= 5), Freskm is the value of the measured resonance frequency for trumpet k and regime m and εkm is the error
term.
In this case, the predicted value of the playing frequency, F̂playkm is given by
Table IV. Average standard deviation σ1 and average reproducibility of players σ2 calculated for each trumpet player (in
cents).
Trumpet player
1
2
3
4
All
σ1
σ2
5
10
5
8
6
8
3
7
5
8
F̂playkm = Freskm .
To estimate the quality of the model, two classical indicators can be computed [41],
• The mean square error MSE of the model. It quantifies
the difference between the observed value and the value
predicted by the model:
In Figure 6, for each regime, there are 12 columns of
points that represent all the combinations of the 4 fingerings for the 3 trumpets (a column is located at the value of
the resonance frequency of the regime). For each column,
there are 36 points that represent the notes played 3 times
by the 4 musicians for the 3 dynamic levels.
The results show first that for all the regimes, the range
of the data is important. Indeed, playing frequencies extend over 50 cents in average (and even more for the second regime). Secondly, for all the regimes, the playing frequency is higher than the resonance frequency (points are
almost all above the line of equation Fplay = Fres ). In particular, the playing frequencies of the second regime are
shifted up to the greatest extent with respect to the resonance frequencies. This observation can be related to the
inharmonicity of the resonances corresponding to the second regime, which were observed to be too low in Figure 2. For the 111 fingering in particular, where the inharmonicity is high, we notice that there is a “compensation
phenomenon” for the playing frequency, which is much
higher than the resonance. This may be due to the coupling musician/instrument, or just to the musician. Notes
played with this fingering are thus located in the three leftmost columns. On the other hand, for short tubes, as for
the 000 fingering, the playing frequencies are quite close
to the bore resonance frequencies. For regimes 3 to 5, playing frequencies are, in average, close to the resonance frequencies. Finally, for the sixth regime, playing frequencies
seem to be somewhat higher than resonance frequencies,
especially for the 000 fingering. This figure is interesting
to visualise the raw data, but we need to define a reference
for each player to draw more precise conclusions.
MSE =
1
(Fplayijklm − F̂playkm )2 .
I*J*K*L*M
(4)
i,j,k,l,m
• The MAPE (Mean Absolute Percentage Error). It is a
measure of accuracy of a method for constructing fitted
series values in statistics. It usually expresses accuracy
as a percentage:
100%
Fplayijklm − F̂playkm
MAPE =
. (5)
I*J*K*L*M
Fplayijklm
i,j,k,l,m
Several models can be fitted to the data, from the simplest
to the more complex, taken the different factors of the experiments into account. Four models are thus defined as
follows:
Model 1:
Model 2:
F̂playkm = Freskm ,
F̂playkm = aFreskm ,
(6)
(7)
Model 3:
F̂playkm = aFreskm + αi ,
(8)
Model 4:
F̂playkm = aFreskm + αi + βj ,
(9)
where a is the coefficient of the regression, αi represents
the effect of the musician and βj represents the effect of
the dynamics. A simple linear regression is used to estimate the coefficient a (Model 2), and analysis of covariance (ANCOVA) is used for Model 3 and 4 to estimate
conjointly the coefficient a and the parameters αi and βj .
Results in Table V indicate that, on average, the percentage of error of the four models is around 1%. Even for the
more complex model, Model 4, which takes all the experimental factors into account, the average error is around
1%. These results indicate that it is not possible to predict the playing frequency from the resonance frequency
with an average accuracy error lower than 1%, which is 16
cents. This is more than the noticeable difference in pitch.
The introduction of the dynamic level and the musician
in Model 4 does not give a significant improvement of the
model quality: the MSE decreases, which is normal since it
is a least square procedure, but the MAPE increases lightly
from Model 2 to Model 4.
5.2. Models of the playing frequency
The objective of this section is to estimate to what extend
the resonance frequency can be used to predict the playing
frequency. Different linear models can be proposed to predict the value of the playing frequency Fplay. The simplest
model than can be proposed is
Fplayijklm = Freskm + εkm ,
(3)
(2)
7
Table V. Estimation of the quality of the four models with the
MSE and MAPE.
Model 1
Model 2
Model 3
Model 4
MSE
MAPE
coeff a
101.03
87.28
86.69
86.23
1.18%
0.96%
0.979%
0.993%
1.0085
1.0097
1.0111
(a)
6. Quantification of the discrepancy between playing frequencies and resonance
frequencies
6.1. Histogram of the distribution of the playing
frequency
(b)
In order to have a global view on all the 2160 played
notes, it is possible to represent the data into a bar graph,
as shown on Figure 7(a). In that histogram, each playing frequency is given in cents, taking its corresponding
resonance frequency as a reference. The results seem to
be normally distributed somewhat around +20 cents, but
there are some abnormally high played notes around +100
cents. These notes in fact correspond to the second regime
since, as we saw in section 5.1, for fingerings involving
a long cylindrical part in the trumpet, playing frequencies
are much higher than resonance frequencies. By removing
all notes from regime 2, as shown in Figure 7(b), results
seem to better fit a normal distribution. The sampling distribution of mean µ is equal to 15.5 cents and the standard
deviation σ is equal to 12.7 cents. A 95% confidence interval can be determined with a normal distribution, given
the size of our samples (thousands of observations) and
the central limit theorem [42]. The 95% confidence for
√
√
the mean is thus [µ − 1.96σ/ n µ + 1.96σ/ n], with n
the number of samples. The mean µ therefore ranges from
14.3 to 16.7 cents. If we include the second regime in the
data, µ then ranges from 19.9 to 23.3 cents.
This representation of the results shows us that playing
frequencies are usually around 15 to 20 cents higher than
the resonance frequencies, taking the temperature into account. We must notice that the fact that the value of µ is of
the same order than the average accuracy error in estimating the playing frequency from the resonance frequency
with models from Section 5.2 is a coincidence. These
two quantities represent two different things: the MAPE
(around 1% - 16 cents), is the average prediction error of
the playing frequency modelled with ANCOVA and linear regression, whereas µ is the average deviation of the
playing frequency from the resonance frequency. For all
the regimes, the error margin of the 95% confidence interval is 1.7 cents. By removing the second regime this error
drops to 1.2 cents.
Nevertheless, these are absolute results whereas instrument makers are generally more interested in relative results. Indeed, a craftsman does not want his instrument
to play defined frequencies, especially since players can
tune their instruments in several ways. So, his interest is
Figure 7. (a) Histogram representing all the 2160 playing frequencies and (b) Histogram representing the same data but without the second regime. Data are expressed in cents, as a difference between each playing frequency and its corresponding resonance frequency.
to make an instrument that can play intervals in tune. Consequently, it is useful to study differences of playing frequencies instead of the frequencies themselves.
6.2. Differences of playing frequencies vs differences
of resonance frequencies
In order to study differences, references need to be chosen: one reference for the resonance frequencies (for each
trumpet), and one reference for the playing frequencies
(for each couple musician and trumpet). Concerning the
resonance frequencies, we propose to consider the fourth
regime of the fingering 000 (corresponding to the concert
note B 4) as the 0 cent reference. For the playing frequencies, we propose to consider the empirical mean of the frequency of the played note B 4 as the 0 cent reference. This
average is calculated on the 3 repetitions of the note B 4
played mezzo forte by a musician on each trumpet. There
are consequently 12 different references (3 trumpets and
4 musicians). This way of defining a reference is in fact
logical, because the note chosen to serve as a reference
8
Figure 8. dFplay as function of dFres (in crosses) for regimes 2 to
6 played by all the musicians on the trumpet CHMQ for the 000
fingering. Circles represent the average playing frequency and
the written numbers give the distance, in cents, from this mean
to the line representing dFplay =dFres .
Figure 9. dFplay as function of dFres (in crosses) for regimes 2 to
6 played by all the musicians on the trumpet CHMQ for the 111
fingering. Circles represent the average playing frequency and
the written numbers give the distance, in cents, from this mean
to the line representing dFplay =dFres .
corresponds to the tuning note generally used by trumpet
players to tune their instrument.
Figure 8 thus presents the differences of playing frequencies as function of the differences of resonance frequencies for the 000 fingering of CHMQ trumpet. These
differences are given in cents, taking the references defined above into account. The average of all the playing
frequencies is also given for each regime, represented with
a circle and its distance, in cents, to the line of equation
dFplay =dFres is indicated. First, it is important to notice
that, even if the fourth regime of that fingering is taken
as a reference, the average deviation is +3 cents and not
zero for that note. This is due to a discrepancy between the
frequencies of a same note played with the different dynamic levels (piano and forte). Then the deviation of other
regimes is +8 cents (regime 2), +1 cent (regime 3), +10
cents (regime 5) and +28 cents (regime 6). For the regimes
2 to 5, the deviation is very weak, in the same range as the
uncertainty in the determination of the playing frequency
and the repeatability of the musicians (see Section 5.1).
For these 4 regimes, it is thus possible to conclude that in
average, a variation of the resonance frequency leads to
a variation of the playing frequency in the same order of
magnitude. For the sixth regime, a variation of the resonance frequency leads to a much higher variation of playing frequency, which is a surprising unexpected result.
It has to be noticed that these conclusions represent only
an average behaviour of the instrument: by observing the
total variability of the playing frequency, we remark that
the data spreads out over about 30 cents for each regime.
This variability is inherent to the trumpet playing, where
several uncontrolled factors may modify the playing frequency of notes.
Figure 9 then shows the same kind of plot but for the
111 fingering of CHMQ trumpet. This time, regimes 3 to
6 are well centred on the line of equation dFplay =dFres ,
whereas regime 2 gives variations of playing frequency
much higher than variations of resonance frequency. Indeed, it was explained in section 5.1 that the longer the
cylindrical pipe, the more inharmonic the second resonance is. Moreover, for the second regime data are even
more spread out than for other regimes (about 70 cents).
9
Table VI. Deviation of the average of all the dFplay for each regime of each fingering on each trumpet to the line of equation dFplay =dFres
(in black) and the line of equation dFplay =dSF (in grey, these are results from section 6.3), given in cents with the fourth regime of 000
fingering taken as a reference.
Trumpet
000
CHMQ
DKNR
NORM
8
15
6
5
10
9
1
4
-2
13
11
7
3
3
4
3
3
4
10
6
7
12
7
10
28
33
24
34
39
29
100
CHMQ
DKNR
NORM
35
41
24
-7
-16
13
-3
2
-7
1
-5
-4
6
9
3
10
10
8
2
2
-1
2
2
0
11
11
7
13
13
9
CHMQ
DKNR
NORM
CHMQ
44
51
32
83
19
-27
-21
-35
-6
4
-7
-8
-7
-2
-6
-10
2
8
3
1
8
17
8
2
-1
1
-5
4
0
1
-3
6
3
6
1
-3
5
8
3
-3
DKNR
NORM
90
72
-38
-39
1
-14
-4
-12
6
-4
4
-5
6
-1
11
4
-3
-6
-4
-6
110
111
2
Regime
4
Fingering
3
5
6
cylindrical part of the trumpet gets longer. This result was
expected as it has already been observed in section 5.1.
The fact that playing frequencies for regime 6 are much
higher only for the 000 fingering is not an expected result.
We have seen that the variations of resonance frequencies are a good indicator of the playing frequencies variations but we also experienced some discrepancies for the
second and the sixth regime. In literature [19, 18], other
indicators like “sum functions”, have been defined in order to predict playing frequencies more accurately than the
resonance frequencies from the input impedance. A sum
function is thus evaluated by using our set of experimental
data in the following section.
6.3. The “sum function”: a supposed indicator of
the playing frequency
Figure 10. Comparison between the measurement of the input
impedance amplitude of the CHMQ trumpet with 111 fingering
(in black) with the sum function calculated from this impedance
with equation 10 (in grey).
Wogram [19] (who was quoted later by Pratt and Bowsher
[18]) introduced what he termed a “Summenprizinzip” (or
“sum function” in English): the impedance values of an
instrument at integral multiples of the fundamental frequency combine at the player’s lips to establish the playing frequency. Actually, the sum function is the sum of the
acoustic power entering the resonator for a forced oscillation with fixed flow rate amplitude and spectrum. In lip
reed instruments, the playing frequency strongly depends
also on the reed natural frequency which is not taken into
account in that function. One version of the sum function
can be calculated as
The results for all the fingerings on all the trumpet are
given in Table VI. The order of magnitude of the distances
are the same whatever the trumpet. This table shows that
it is possible to consider that a variation of resonance frequency leads in average to a variation of the playing frequency of the same order for regimes 3 to 6 of all fingerings, except regime 6 of 000 fingering. While regimes 3
to 5 have almost constant variations of playing frequency
over the different fingerings, regimes 2 and 6 have a completely different behaviour. For regime 6, a variation of resonance frequency first leads to a higher variation of playing frequency for fingering 000. Then, the deviation between dFplay and dFres decreases when the first two valves
are depressed. Finally, for the 111 fingering, the variation
of playing frequency becomes smaller than the variation
of resonance frequency. For the second regime it is the
contrary, dFplay differs more and more from dFres as the
S(f ) =
n
1
Re[Z(if )].
i
(10)
i=1
in which n is maximized such that nf < fmax , the highest
frequency for which Z is known.
An example of this sum function is given in Figure 10.
This function is thus supposed, as claimed by Wogram, to
predict the playing frequencies with a better accuracy than
the resonance frequencies from the input impedance.
10
Figure 11. dFplay as function of dFres (in black crosses) or dSF (in
grey crosses) for regimes 2 to 6 played by all the musicians on the
trumpet CHMQ for the 000 fingering. Dotted line represents the
average playing frequency and straight line is the line of equation
dFplay =dFres or dFplay =dSF.
Figure 12. dFplay as function of dFres (in black crosses) or dSF (in
grey crosses) for regimes 2 to 6 played by all the musicians on the
trumpet CHMQ for the 111 fingering.Dotted line represents the
average playing frequency and straight line is the line of equation
dFplay =dFres or dFplay =dSF.
frequencies to the right direction but it over-corrects the
discrepancy. Moreover, for the other regimes, the input
impedance allows predicting variations of resonance frequencies closer to those of playing frequencies than the
peak frequencies of the sum function.
Consequently, the sum function does not seem to give
more information on the playing frequencies than the simple input impedance.
In order to study if the sum function is able to predict the playing frequencies more accurately that the input impedance, we plot again dFplay as function of dFres
and as function of variations of the sum function peaks in
Figures 11 and 12. As previously done for the resonance
frequencies of the input impedance, a reference is taken:
the frequency of the sum function peak corresponding to
the regime 4. Then, each peak of the sum function is given
in cents, by calculating the difference with that reference,
and is written dSF.
Figures 11 and 12 show how variations of the resonance
frequencies taken from both the input impedance and the
sum function are able to predict the variations of the playing frequency. Deviations of the average dFplay from dFres
and dSF are summarised in Table VI. As pointed out in
the previous sections, there is a large discrepancy between
dFplay and dFres for the second regime of the 111 fingering. That is a reason why the sum function has been implemented and Figures 11 and 12 as well as Table VI show
that, for this regime, dSF is closer to dFplay than dFres . Nevertheless, the sum function actually shifts the resonance
7. Conclusion
This study proposed a quantitative assessment of the relations between the bore resonance frequencies and the
playing frequencies, based on experiments made on three
trumpets with four musicians for a large number of notes
(different regimes, fingerings and dynamic levels). Even if
it was already known that playing frequencies are close to
bore resonance frequencies, no detailed work had previously been carried out to quantify it.
First, this study shows that the dynamic level does not
have a strong influence on the playing frequencies and that
11
the four musicians have relatively the same “global” behaviour, as they all play on average in the order of 8 to 20
cents above the bore resonance frequencies.
Second, a closer analysis of the data shows that the average standard deviation of the playing frequency is about
5 cents, which means that a played note is stable with an
uncertainty of 5 cents. Furthermore, the average repeatability of a musician, calculated on his 9 repetitions of a
same note, is about 8 cents. Therefore, there is no need
to find a predictor of the playing frequency more accurate
than 8 cents.
Then, by representing the played notes as an histogram
it is possible to conclude that, from regime 3 to 6, playing
frequencies are in average 15 cents higher than the resonance frequencies. The error margin on the estimation of
that mean is 1.2 cents at a 95% confidence level.
Finally, by examining differences instead of just frequencies themselves, the impact of the musicians’ behaviour is diminished. Moreover, craftsmen often work by
making small changes in the geometry of their instruments
and studying the differences induced by the modification.
So, focusing on differences is a way to get closer to the
craftsman’s process.
Regime 4 played with the 000 fingering is thus taken
as a reference to calculate those differences, since it is the
note generally used to tune the instruments. Results show
that a variation of bore resonance frequency leads in average to a variation of playing frequency of the same order
for regime 3 to 6 (but surprisingly, except the sixth regime
for the 000 fingering). For regime 2, this rule is not satisfied because the notes are played at a higher frequency
than the bore resonance. The inharmonicity of the notes of
regime 2 could be a reason to explain this behaviour. These
results might show that the inharmonicity plays a role on
the control of the playing frequencies. It should indeed be
possible that, when the bore resonance frequency corresponding to the played note is in an harmonic relationship
with the other resonances, a variation of the resonance frequency leads to a variation of the playing frequency of the
same range. On the other hand, when the bore resonance
frequencies are inharmonic, that relation is not valid any
more. This is shown in the study of Dalmont et al. [43] for
one saxophone fingering. Nevertheless, further analysis is
required to support this explanation.
An attempt was made to model this effect with the sum
function. For the second regime, played frequencies are
actually closer to the sum function peaks frequencies than
to the bore resonances. Nevertheless, a discrepancy still
exists and the prediction of the sum function is less accurate for other regimes. In conclusion, the sum function does not seem to be more relevant than the input
impedance in order to predict playing frequencies. The
resonance frequency is thus a good objective indicator for
predicting the playing frequency, as it does not take the influence of the musician into account. This is interesting for
craftsmen whose instruments need to be played by virtual
musicians, and who often proceed by small adjustments on
their instruments.
Our results are obtained with three particular trumpets
that do not represent all the possible trumpets in the market. We must refrain any generalization of the results to the
trumpet in general, further studies are needed to prove the
robustness of the relationship playing frequency/resonance
frequency.
Also, for further work, it will then be interesting to compare these results with measurements using an artificial
mouth [44] and simulations.
Acknowledgement
This research was funded by the French National Research
Agency ANR within the PAFI project (Plateforme d’Aide
à la Facture Instrumentale in French). The authors would
like to thank all the trumpet players who participated in
this study as well as A. Burke, K. Cedergren, J.-P. Dalmont, D. Lopatin, A. Mamou-Mani and R. Piéchaud for
proofreading and valuable talks.
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