The Relationship Between Bore Resonance Frequencies and Playing Frequencies in Trumpets

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 di ﬀ erent 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 di ﬀ erent 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.


Introduction
Measuring and computing wind musical instruments input impedance is nowwell mastered [1,2,3,4,5,6]. As part of alarger project aimed toward helping instrument makers to design and characterise their musical instruments, this work focuses on howthe 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 theyneed some predictive indicators.
Some studies attempt to findasolution to this issue by taking the coupling between the instrument and the musician into account. The case of reed instruments is treated 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)i sthe linear part, treated in frequencydomain, while the driving system (the reed)isthe 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 frequencyofthe lips, and the inverse of lips mass density.D epending on the choice of these parameters, it is possible to obtain aseries of playing frequencies, such as those obtained by the musician. The coupling between the musician and the instrument can also be investigated using asimplified model in which asingle mechanical lip mode is coupled to asingle mode of the acoustical resonator,asdone 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. Manystudies are carried out on physical modelling using temporal methods [11,12].
Fort he saxophone, for some advanced performance techniques (bugling and altissimo playing), musicians can use the resonance of their vocal tract to play an ote close to aw eak bore resonance, or even decrease the sounding pitch to several semitones belowthe 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 as ignificant role in determining the playing frequencies, this aspect being difficult to takeinto account. Therefore, the first aim of this paper is to determine an order of magnitude of the brass player'si nfluence 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'sb ehaviour into account, as intended by Pratt and Bowsher [18] and previously by Wogram [19]. This will be done by recording alarge number of notes played by several musicians on three trumpets.
Section 2p resents some basic information about the acoustics of the trumpet. Section 3describes 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 6p resents the data first as an ormal distribution and then, in order to minimize the influence of the musician on the results, it focuses on frequencydifferences instead of the frequencies themselves. Finally,the relevance of the sum function [19], af unction made from the input impedance to predict the intonation, is discussed.

Trumpet resonances and playing frequencies: Preliminary discussion
Campbell and Greated [20] as well as Fletcher and Rossing [21] give al arge overviewo nb rass instruments. A summary about trumpets is reported here as well as adiscussion 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). At ypical input impedance of ab rass instrument (see Figure 1) shows al arge 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 an ote (oro scillation regime)t hat the musician can play.I nt he example of Figure 1, corresponding to the basic fingering of aB trumpet where all the three valves are up, the resonances 2to6correspond to the series of concert notes B 3, F4, B 4, D5, F5 (harmonic series of B 2).T he first resonance does not correspond to anormally playable note on the trumpet. The three valves offer height combinations, which allowthe construction of the whole chromatic scale, since the activation of av alvep roduces an elongation of the air column which lower the resonance frequencies of the instrument. The first valvebrings down the frequency Figure 1. Measurement of the input impedance amplitude (indB) and phase (inrad)ofthe trumpet called NORM with all the three valves up, with the notes corresponding to each impedance peak above (concert pitch of aB trumpet). The trumpet and the set-up used for the measurement are presented in Section 3.1. of one tone, the second of asemitone, and the last one of one and ah alf tones. In the rest of the article, ap ressed valvewill be noted 1and avalveupwill 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 2f or the trumpets called CHMQ, DKNR and NORM in this study.T hese instruments are presented in section 3. This representation is often used by instruments makers and is for example presented in the BIAS software 1 .These graphs showthat 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 showthat, by increasing the length of the bore, the resonance frequencies become more distant from the frequencies of their corresponding notes in the equally-tempered scale. Consequently,i fw ea ssume 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 acomplexaeroelastic coupling between the lips of the musician and the resonator.Thus, the intonation of the instrument is not only controlled by the closest resonance frequencyb ut possibly conditioned by upper resonance frequencies of the resonator [22].
Furthermore, aw ind instrument is not an instrument with afi xeds ound, 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 flowbetween the lips.

Set-Up
Ap arametrised leadpipe, made of four different interchangeable parts, wasd esigned [23,24] as it is shown in Figure 3. Several parts with various values for the radii r 1 , r 2 , r 3 and r 4 were manufactured with an umerically controlled turning machine. Al etter has been givent oe ach part of the leadpipe, corresponding to the dimensions of the radii. Thus, using the same B trumpet (Bach model Ve rnon, bell 43)w ith the same mouthpiece (Bach 11 /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 Figure 3. Parametrisation of the leadpipes used in this work (from Petiot et al. [24]). Radii r 1 , r 2 , r 3 and r 4 are giveni nT able I. leadpipes presented in the Table Iand aleadpipe 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 amusical point of view. For all the tests, the position of the tuning slide wassimilar: it waspulled out of alength of 1cmfor 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)u sing as et-up described by Macaluso and Dalmont [4]. The first regime is not played with at rumpet. In this study,t he notes will thus be recorded for regimes 2t o6 with these four fingerings. Nevertheless, some of these notes do not correspond to the usual fingerings used by the musician. Regimes 2to6are normally played by musicians for fingerings 000 and 100. Forthe 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 2and 3are usually played. Regimes 4, 5a nd 6a re an alternative wayo fp laying the notes E4, G 4and B4, for which musicians usually use the third regime of the 010 fingering (E4),t he fourth regime of the 100 fingering (G 4) and the fifth regime of the 110 fingering (B4).T hese 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 neverused 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 means that theya re more likely to play without focusing on the intonation. Four musicians, one professor at am usic school and three experienced amateurs, were asked to play the three trumpets to record the sounds. After ashort warm-up, each trumpet player had to play the fivefi rst playable notes (regimes 2t o6 )b ys aying the name of the note before playing, in order to have as hort 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,t heyb end 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. Theyh ad to repeat the whole process three times in order to test their reproducibility.Finally,4 trumpet players times 3trumpets times 4fingerings times 5regimes times 3dynamic levels times 3repetitions give 2160 notes to analyse.

Data Analysis
The playing frequencyofthe notes has been analysed with the YIN [25] software 2 ,which is an estimator of the fundamental frequencyspecially 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 ap erfectly steady note, and slight oscillations around the playing frequencyw ere observed. Figure 4shows an example of the frequencye volution of one note, concert B 3, played by am usician with the basic fingering (000)a tm ezzo forte. At the beginning, the frequencyrapidly increases: this is atypical transient. The same effect is happening during the quiescent. After removing the transient and the quiescent, aquasi-stationary part stays, where the frequencyisfluctuating afew hertz. Therefore, from am easured signal liket he one in Figure 4, what we call in this paper the playing frequency, will be determined as the mean of the instantaneous frequencyduring the time t of the quasi-stationary part. The standard deviation is then calculated in order to estimate the ability of the trumpet player to play at astable playing frequency.
The measurements of the trumpets' input impedances and the recording of the musicians were carried out at different temperatures. The input impedance wasm easured at 23 • Cwhereas notes were played at aroom temperature of 25 • C. According to Gilbert et al. [26] and Noreland [27], we consider that the temperature of the air column wasaround 28 • Cduring playing. Consequently,for areliable comparison, resonance frequencies need to be moved forward from the equivalent temperature shift. Since resonance frequencies of both cones and cylinders are proportional to the sound velocity,w hich can be written as c = 331.45 T/T 0 m/s with T the temperature in Kelvin and T 0 = 273.16 K, it can be considered that the resonance frequencies of at rumpet are proportional to the square root of the temperature expressed in Kelvin. Consequently,t he resonance frequencies from the measured input impedances are increased by 14 cents which is the equivalent of 5 • Ci no rder to be at the same levelo ft he playing frequencies' temperature.
Finally,the frequencyofthe resonances is precisely determined with apeak fitting technique using aleast square method on the compleximpedance [28]. This method represents the impedance in the Nyquist plot. In this plot, the resonance is locally acircle that should go through the experimental points. Then, the resonance frequencyi st he 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 frequencywith an uncertainty of about 5cents. The resonance frequency could also be determined with the phase zero crossing. Nevertheless, as explained in [29, p. 149], the amplitude of the impedance givesm ore information about the tuning and the ease of playing than the phase, that is whythis definition of the resonance frequencywas chosen.

Descriptive analysis of the playing frequencies
In order to study the behaviour of each musician, we represent by ab oxplot the differences (inc ents)b etween each playing frequencya nd its respective resonance frequencyfor all the notes played. Aboxplot is aconvenient wayo fg raphically representing ad istribution of numerical data through their five-number summaries: the smallest observation (sample minimum), lower quartile (25 th percentile, bottom of the box), median, upper quartile (75 th percentile, top of the box), and largest observation (sample maximum).
One boxplot per dynamic levelallows one to study the influence of the dynamic levelonthe playing frequencies.
These boxplots showt hat the playing frequencies are, on average, higher than the bore resonance frequencies and that the four musicians play at as light higher pitch at the piano dynamic level(p) than at mezzo forte (mf) or forte (f).The dynamic leads to less than a10cents difference on the playing frequencym edian. The four trumpet players have as imilar behaviour as theya ll play,o na verage, in the order of 8t o2 0c ents above bore resonance frequencies. These results could be consistent with adominant outward striking regime of oscillations for the lips that has been observed in previous studies [30,31,32]. Nevertheless, Figure 5shows that musicians can also play belowt he resonance frequencies for some notes. Players can thus have different behaviours depending on the note theyplay,ontheir embouchure, etc. and asingle mechanical oscillator cannot model the complete behaviour of the lip reed [33,34,35,10].

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 ) Figure 5. Boxplots representing the statistics of the playing frequencies of the notes played by each player for the three dynamic levels (p,mfand f).Data are expressed in cents, as adifference between each playing frequencyand its corresponding resonance frequency. [36,37].A NOVA is ac ollection of statistical models to model aquantitative variable (the response)w ith 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 2groups (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 afi ve-factors ANOVA is given by • α i the effect of the level i of the musician (i = 1t o4 since there are four musicians), • β j the effect of the level j of the dynamic level( j=1 to 3since there are three dynamics), • γ k the effect of the level k of the trumpet (k = 1t o3 since there are three trumpets), • δ l the effect of the level l of the fingering (l = 1t o4 since there are four fingerings), • η m the effect of the level m of the regime (m = 1t o5 since there are fiveregimes), • and ε ijklm the error term. Each coefficient represents the influence of the levelofthe factor on the response. From the measurements, al east square procedure is used to estimate these coefficients (minimization of the squared error between the measured playing frequencyand the playing frequencygiven by the 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", Fmeans "the F-statistic" and Pmeans "the p-value". model). Aclassical F-test is used to assess the significance of the effect of the factors. The sources can be considered to have as ignificant impact on the data if the probability p is lower than 0.05 [36]. Table II givesthe results of the ANOVA model, the last column indicating the probabilityvalue p of the F-test (false rejection probability).
Only twof actors have as ignificant 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 levela re 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 levelleads, on average, to aslightly higher playing frequencythan 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. Indeed, as explained in section 3.2, each played note is determined by an average frequency( which corresponds to the playing frequencytaken into account in the paper)and astandard deviation σ.The error bar thus stands for the average σ overall the played notes, which is equal to 5cents. The error bar on the right represents twice the average reproducibility of the trumpet players, called σ 2 .I ndeed, each musician will repeat 9t imes the same note (3 dynamic levels times 3a ttempts). The standard deviation is thus calculated on these 9notes and then the mean of these standard deviations is calculated on all the notes played by all the musicians (itisinfact amean on 240 standard deviations). This error is equal to 8cents, 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 3Hzfor sine wavesand 1Hzfor complextones be-low5 00 Hz. Above 1000 Hz, the JNDf or sine wavesi s about 10 cents [38,39,40]. Standard deviations σ 1 and σ 2 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 2i nduces more variation on both the varying playing frequencyofthe note and the reproducibilty of the musicians. Table IV shows that players have more or less the same reproducibility even if player 4seems to play more straight notes and is more repeatable than the others. In Figure 6, for each regime, there are 12 columns of points that represent all the combinations of the 4fi ngerings for the 3trumpets (a column is located at the value of the resonance frequencyofthe regime). Foreach column, there are 36 points that represent the notes played 3times by the 4musicians for the 3dynamic levels.

Playing frequencies vs
The results showfirst that for all the regimes, the range of the data is important. Indeed, playing frequencies extend over50cents in average (and even more for the second regime). Secondly,for all the regimes, the playing frequencyishigher than the resonance frequency(points are almost all above the line of equation F play = F res ). In particular,t he 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 lowi nF igure 2. Forthe 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. Forregimes 3to5,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 rawdata, butweneed to define areference for each player to drawmore precise conclusions.

Models of the playing frequency
The objective of this section is to estimate to what extend the resonance frequencycan be used to predict the playing frequency. Different linear models can be proposed to predict the value of the playing frequencyFplay.The simplest model than can be proposed is where Fplay ijklm is the value of the measured playing frequencyfor musician (i = 1 ...Iwith I= 4, see Section 4.2 for more details on the variables), dynamics (j = 1 ...J with J= 3),t rumpet (k = 1 ...Kw ith K= 3),fi ngering (l = 1 ...Lw ith L= 4) and regime (m = 1 ...Mw ith M= 5),Fres km is the value of the measured resonance frequencyf or trumpet k and regime m and ε km is the error term.
In this case, the predicted value of the playing frequency,Fplay km is givenbŷ To estimate the quality of the model, twoclassical indicators can be computed [41], 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:Fplay km = Fres km , Model 2:Fplay km = aFres km , Model 3:Fplay km = aFres km + α i , (8) Model 4:Fplay km = aFres km + α i + β j , (9) where a is the coefficient of the regression, α i represents the effect of the musician and β j represents the effect of the dynamics. As imple linear regression is used to estimate the coefficient a (Model 2),a nd analysis of covariance (ANCOVA )i su sed for Model 3a nd 4t oe stimate conjointly the coefficient a and the parameters α i and β j . Results in Table Vindicate that, on average, the percentage of error of the four models is around 1%. Even for the more complexmodel, 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 frequencyf rom the resonance frequency with an average accuracyerror lower than 1%, which is 16 cents. This is more than the noticeable difference in pitch.
The introduction of the dynamic leveland the musician in Model 4does not give asignificant improvement of the model quality: the MSE decreases, which is normal since it is aleast square procedure, butthe MAPE increases lightly from Model 2toModel 4. 6. Quantification of the discrepancy between playing frequencies and resonance frequencies 6

.1. Histogram of the distribution of the playing frequency
In order to have ag lobal viewo na ll the 2160 played notes, it is possible to represent the data into abar graph, as shown on Figure 7(a).I nt hat histogram, each playing frequencyi sg iven in cents, taking its corresponding resonance frequencya sar eference. 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 sawi ns ection 5.1, for fingerings involving along 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),r esults seem to better fit anormal distribution. The sampling distribution of mean µ is equal to 15.5 cents and the standard deviation σ is equal to 12.7 cents. A9 5% confidence interval can be determined with anormal distribution, given the size of our samples (thousands of observations)a nd the central limit theorem [42]. The 95% confidence for the mean is thus [µ − 1.96σ/ √ nµ + 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 accuracyerror in estimating the playing frequencyf rom the resonance frequency with models from Section 5.2 is ac oincidence. These twoq uantities represent twod i ff erent things: the MAPE (around 1% -16cents), is the average prediction error of the playing frequencym odelled with ANCOVA and linear regression, whereas µ is the average deviation of the playing frequencyf rom the resonance frequency. Fora ll the regimes, the error margin of the 95% confidence inter-valis1.7 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, ac raftsman does not want his instrument to play defined frequencies, especially since players can tune their instruments in several ways. So, his interest is to makeaninstrument that can play intervals in tune. Consequently,i ti su seful to study differences of playing frequencies instead of the frequencies themselves.

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 0cent reference. Forthe playing frequencies, we propose to consider the empirical mean of the frequencyofthe played note B 4asthe 0cent reference. This average is calculated on the 3r epetitions of the note B 4 played mezzo forte by amusician on each trumpet. There are consequently 12 different references (3 trumpets and 4m usicians). This wayo fd efining ar eference is in fact logical, because the note chosen to servea sar eference Figure 8. dF play as function of dF res (incrosses)for regimes 2to 6played by all the musicians on the trumpet CHMQ for the 000 fingering. Circles represent the average playing frequencya nd the written numbers give the distance, in cents, from this mean to the line representing dF play =dF res .
corresponds to the tuning note generally used by trumpet players to tune their instrument. Figure 8t hus presents the differences of playing frequencies as function of the differences of resonance frequencies for the 000 fingering of CHMQ trumpet. These differences are giveni nc ents, taking the references defined above into account. The average of all the playing frequencies is also givenfor each regime, represented with ac ircle and its distance, in cents, to the line of equation dF play =dF res is indicated. First, it is important to notice that, even if the fourth regime of that fingering is taken as ar eference, the average deviation is +3c ents and not zero for that note. This is due to adiscrepancybetween the frequencies of as ame note played with the different dynamic levels (piano and forte). Then the deviation of other regimes is +8c ents (regime 2), +1c ent (regime 3), +10 cents (regime 5) and +28 cents (regime 6).For the regimes 2to5,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). Forthese 4regimes, it is thus possible to conclude that in Figure 9. dF play as function of dF res (incrosses)for regimes 2to 6played by all the musicians on the trumpet CHMQ for the 111 fingering. Circles represent the average playing frequencya nd the written numbers give the distance, in cents, from this mean to the line representing dF play =dF res .
average, av ariation of the resonance frequencyl eads to av ariation of the playing frequencyi nt he same order of magnitude. Fort he sixth regime, av ariation of the resonance frequencyleads to amuch higher variation of playing frequency, which is asurprising 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 overabout 30 cents for each regime. This variability is inherent to the trumpet playing, where several uncontrolled factors may modify the playing frequencyofnotes. Figure 9t hen shows the same kind of plot butf or the 111 fingering of CHMQ trumpet. This time, regimes 3to 6a re well centred on the line of equation dF play =dF res , whereas regime 2g ives variations of playing frequency much higher than variations of resonance frequency. Indeed, it wase xplained 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).  The results for all the fingerings on all the trumpet are giveninT able VI. The order of magnitude of the distances are the same whatevert he trumpet. This table shows that it is possible to consider that av ariation of resonance frequencyl eads in average to av ariation of the playing frequencyofthe same order for regimes 3to6ofall fingerings, except regime 6o f0 00 fingering. While regimes 3 to 5havealmost constant variations of playing frequency overthe different fingerings, regimes 2and 6haveacompletely different behaviour.For regime 6, avariation of resonance frequencyfirst leads to ahigher variation of playing frequencyf or fingering 000. Then, the deviation between dF play and dF res decreases when the first twovalves are depressed. Finally,for the 111 fingering, the variation of playing frequencyb ecomes smaller than the variation of resonance frequency. Fort he second regime it is the contrary,d F play differs more and more from dF res as the 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 6a re much higher only for the 000 fingering is not an expected result.
We have seen that the variations of resonance frequencies are agood indicator of the playing frequencies variations butw ea lso experienced some discrepancies for the second and the sixth regime. In literature [19,18],o ther indicators like" sum functions", have been defined in order to predict playing frequencies more accurately than the resonance frequencies from the input impedance. As um function is thus evaluated by using our set of experimental data in the following section.

The "sum function": as upposed indicator of the playing frequency
Wogram [19] (who wasquoted 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 frequencyc ombine at the player'sl ips to establish the playing frequency. Actually,the sum function is the sum of the acoustic power entering the resonator for af orced oscillation with fixed flowrate amplitude and spectrum. In lip reed instruments, the playing frequencystrongly depends also on the reed natural frequencywhich is not taken into account in that function. One version of the sum function can be calculated as in which n is maximized such that nf <f max ,the highest frequencyfor which Z is known. An example of this sum function is giveninFigure 10. This function is thus supposed, as claimed by Wogram, to predict the playing frequencies with abetter accuracythan the resonance frequencies from the input impedance. Figure 11. dF play as function of dF res (inblack crosses)ordSF (in greycrosses)for regimes 2to6played by all the musicians on the trumpet CHMQ for the 000 fingering. Dotted line represents the average playing frequencyand straight line is the line of equation dF play =dF res or dF play =dSF.
In order to study if the sum function is able to predict the playing frequencies more accurately that the input impedance, we plot again dF play as function of dF res 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, ar eference is taken: the frequencyo ft he 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 showhow 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 dF play from dF res and dSF are summarised in Table VI. As pointed out in the previous sections, there is alarge discrepancybetween dF play and dF res for the second regime of the 111 fingering. That is ar eason whyt he sum function has been implemented and Figures 11 and 12 as well as Table VI show that, for this regime, dSF is closer to dF play than dF res .Nevertheless, the sum function actually shifts the resonance frequencies to the right direction buti to ver-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,t he sum function does not seem to give more information on the playing frequencies than the simple input impedance.

Conclusion
This study proposed aq uantitative assessment of the relations between the bore resonance frequencies and the playing frequencies, based on experiments made on three trumpets with four musicians for alarge number of notes (different regimes, fingerings and dynamic levels). Even if it wasalready 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 leveld oes not have astrong influence on the playing frequencies and that the four musicians have relatively the same "global" behaviour,astheyall play on average in the order of 8to20 cents above the bore resonance frequencies.
Second, acloser analysis of the data shows that the average standard deviation of the playing frequencyisabout 5c ents, which means that ap layed note is stable with an uncertainty of 5c ents. Furthermore, the average repeatability of am usician, calculated on his 9r epetitions of a same note, is about 8c ents. Therefore, there is no need to findapredictor of the playing frequencymore accurate than 8cents.
Then, by representing the played notes as an histogram it is possible to conclude that, from regime 3to6,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 a95% confidence level.
Finally,b ye xamining 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 aw ay to get closer to the craftsman'sprocess.
Regime 4p layed with the 000 fingering is thus taken as areference to calculate those differences, since it is the note generally used to tune the instruments. Results show that avariation of bore resonance frequencyleads in average to av ariation of playing frequencyo ft he same order for regime 3to6(but surprisingly,except the sixth regime for the 000 fingering). Forr egime 2, this rule is not satisfied because the notes are played at ah igher frequency than the bore resonance. The inharmonicity of the notes of regime 2could be areason to explain this behaviour.These results might showt hat the inharmonicity plays ar ole on the control of the playing frequencies. It should indeed be possible that, when the bore resonance frequencyc orresponding to the played note is in an harmonic relationship with the other resonances, avariation of the resonance frequencyleads to avariation of the playing frequencyofthe 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 wasmade to model this effect with the sum function. Fort he second regime, played frequencies are actually closer to the sum function peaks frequencies than to the bore resonances. Nevertheless, ad iscrepancys till 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 frequencyisthus agood objective indicator for predicting the playing frequency, as it does not takethe 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 anygeneralization 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.