# What can we deduce from measured resonance frequencies of trumpets concerning playing frequencies

René Caussé, Pauline Eveno, Joël Gilbert, Jean-François Petiot

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René Caussé, Pauline Eveno, Joël Gilbert, Jean-François Petiot. What can we deduce from measured resonance frequencies of trumpets concerning playing frequencies. ICA 2013, Jun 2013, Montréal, Canada. pp.2, 10.1121/1.4800050. hal-00951809

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# What can we deduce from measured resonance frequencies of trumpets concerning the playing frequencies?

René E. Caussé*, Pauline Eveno, Joël Gilbert and Jef Petiot

*Corresponding author's address: Ircam - UMR Cnrs 9912, 1 place Igor Stravinsky, Paris, 75004, France, France, Rene.Causse@ircam.fr

Lip type valve is responsible for sound production for brass instruments. The operation of the valve is controlled by feedback from a passive resonator. The purpose of this study is to compare experimentally how far the resonance frequencies of instrument, taken from their input impedance (which does not involve the intervention of the player's lips), are able to give information about the playing frequencies. A family of three trumpets made from a basic instrument for which the lead pipe will be slightly modified for each model, were considered for the experiment. Four musicians experts were asked to play the five playable notes, for four different fingerings and for three nuances. This exercise was repeated three times. All these notes allow to make a quantitative assessment of the relation between the resonance frequencies and the playing frequencies, using in particular statistical methods. Several results will be presented: the influence of the player on the overall intonation, the effect of nuances on the pitch and the relation between small changes of geometry and playing frequencies. Functions made from the input impedance, such as the « sum function » proposed by Wogram, do not bring more relevant information than the input impedance itself.
## Introduction - Generalities

As part of a larger project [1] aimed toward helping instrument makers to design and characterize their musical instruments, this work focuses on one variable related to quality of brass instrument, the intonation. It is part of the PhD thesis of Pauline Eveno [2].

How the bore resonances taken from the input impedance, can be related to the playing frequencies, without taking the musician's behavior into account, is one of the questions being asked here. This study examines also the extent to which functions derived from the input impedance as the “sum function” may be used as a predictor of this intonation.

To do this, we studied quantitatively and systematically the relationship between the frequencies of the resonances of passive resonators (trumpets) and the playing frequencies when the instrument is played by musicians. A somewhat similar study was carried out in the past by Pratt and Bowsher [3] on trombone qualities: timbre, ease of playing and intonation. However, the continuous nature of the slide action, allowing players to make corrections, has not allowed clear conclusions for the intonation.

The influence of the player on the sound production in brass instruments, trumpet in our case, is much stronger than in woodwind instrument, clarinet for example, since the player's lips play an integral part in the process of self-oscillation. When the player adjust, more or less easily depending on their degree of expertise, the lip(s) vibration frequency close to the frequency of one resonance of the resonator, a significant variation of pressure in the mouthpiece leads to an acoustical feedback from the resonator. Then, a self-sustained oscillation develops for which the best sound is emitted [4]. For this sound, the resonator imposes the timbre.

We will also try to determine in this work an order of magnitude of the musician's influence on the overall intonation of the instrument. The influence of the musician and dynamic level on the playing frequencies will also be discussed.

## Measurement of resonance frequencies

Three different trumpets have been used for the experiment, the first one with the normal leadpipe provided with the trumpet and the other two obtained by changing only the leadpipe of this trumpet. The three instruments are associated with the same mouthpiece. This leadpipe is made of four different interchangeable parts [5]. A letter has been given to each part of the leadpipe, corresponding to the dimensions of the radii. The trumpets that are thus assembled are called: NORM for normal, CHMQ and DKNR. The input impedance is then measured for four different fingerings 000, 100, 110 and 111. A pressed valve (or piston) will be noted 1 and a valve not pressed down will be noted 0.

The uncertainty of the resonance frequencies with the set-up used is about 0,5%, that is 5 cents [6].

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 shown in figure 1 for the three trumpets used in this study. The first resonances do not correspond to a normally playable note (regime of oscillation) on the trumpet and is not considered here.

These diagrams show that by increasing the length of the cylindrical part (growing from 000 to 111), the resonance frequencies become more distant from the frequencies of corresponding notes. Consequently, notes should be easier to play in tune with the 000 fingering than with the 111 fingering.

If in this study, the notes will 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 fingering 110 fingering, musicians play notes from the second regime to the fifth. The sixth is generally not used since E5 can be played with fifth regime of the 000 fingering.

For the 111 fingering, only regimes 2 and 3 are usually played. Regimes 4, 5 and 6 re an alternative way of playing the notes Gb4, Bb4 and Db5, for which musicians usually use the third regime of the 010 fingering (Gb4).
<figure><img src="image_2.png" /><figcaption>Figure 1. Intonation of trumpet (a) CHMQ, (b) DKNR and (c) NORM, realized 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 equally-tempered scale.</figcaption></figure>

Afterwards, it is interesting to focus on the harmonicity of the trumpet taking in figure 2 the example of the NORM trumpet. This diagram represents the difference in cents between the resonance frequency of the regime n and n.f<sub>4</sub>/4, where f<sub>4</sub> is the resonance frequency of the four regime, taken here as a reference since it is the regime most in tune for this trumpet. It shows how closely the peaks collectively match a harmonic series.

<figure><img src="image_3.png" /><figcaption>Figure 2. Inharmonicity of the NORM trumpet realized by calculating the difference in cents between f<sub>n</sub> (the resonance frequency of the regime n) and n.f<sub>4</sub>/4.</figcaption></figure>

The regimes of trumpet NORM are in harmonic relation ship, apart from the second regime of all the fingerings and surprisingly the six regime of the 000 fingering.

Measurement of playing frequencies and comparison playing frequencies vs resonance frequencies

Four musicians, one professor and three serious amateurs, were asked to play the three trumpets to record the sounds. The task for the musicians was different of what they normally played. Specifically, they were asked to play the note, after a short "warmed up", with the easiest emission and without trying to correct the intonation. Before playing a note, they have to say the name of the note in order to have a short rest between the notes and "forget" the pitch of the previous note. However, we were not able to verify, for example by measuring the control parameters of the musician (properties of the lips, pressure in the mouth, force exerted by the lips on the mouthpiece, tuning of the vocal tract) if this instruction was followed. For each trumpet, the four musicians played the first five regime, varied with four different fingerings, for three dynamics level and repeated three times.
The playing frequency has been analysed with the YIN software [7] during the stationary part of the note. As measures of the trumpets' input impedances and the recording of the notes were carried out at different temperatures, a correction of 16 cents, which is equivalent of 5.5°C, was introduced to resonance frequencies.

The primary objectives of these analyses, done using in particular statistical methods, is to estimate to what extent the resonance frequency can be used to predict the playing frequency.

They have showed the following main results:

- the playing frequencies are, on average, higher than the bore resonance frequencies, which is consistent with what we should expect from an outward striking reed.

- musicians have a limited influence on the overall intonation of the instrument. The pitch of the notes does not vary much with the dynamics levels. They play, surprisingly, at a higher pitch at the *piano* dynamic level than at *mezzo forte* or *forte*. Of course, for these experiments, the players are not allowed to use their finger-operated tuning slide.

- for relatively small variations in the geometry, the differences on the bore resonance frequencies can predict the differences produced in playing frequencies.

- the resonance frequencies is thus a good objective indicator for prediction the playing frequency as it does not take the influence of the musician into account. This is interesting for craftsman whose instrument need to be played by virtually any musicians.

- this study gives also an order of magnitude of the musicians' variability (around 8 cents).

## Beyond the input impedance

Wogram [8] introduced a “summenprinzip” (or “sum function”): the impedance values of an instrument at integral multiples of the fundamental frequencies combine at the player’s lips to determine the playing frequency. He argued that the playing frequency is directly related to the frequency at which maximum energy is transferred from the instrument to the surroundings, and this occurs at the maximum of the sum function. This function is therefore calculated by summing the real part of the impedance of an instrument at integral multiples of a fundamental frequency.

Figure 3 shows a comparison between the sum function and the impedance for the trumpet.

<figure><img src="image_4.png" /><figcaption>Figure 3. Comparison between measurements of the input impedance amplitude of the NORM trumpet with 000 fingering (in black) with the sum function calculated from this impedance (in grey).</figcaption></figure>

The sum function could be interesting for predicting the playing frequencies of the pedal note and of the second regime. Contrary to the trombone, the pedal is barely played with the trumpet. However, for the second regime it actually shifts these frequencies to the right direction but it over-corrects the discrepancy. The playing frequencies predicted by the sum function are further from the frequencies played by the musicians than the bore resonance frequencies. The sum function does not seem to give more information on the playing frequencies than the simple input impedance.
References

1, PAFI, Plateforme d'Aide à la Facture instrumentale, ANR Project, see http://www.pafi.fr/

2, P. Eveno, PhD thesis, L'impédance d'entrée pour l'aide à la facture des instruments de musique à vent : mesures, modèles et lien avec les fréquences de jeu, Université Pierre et Marie Curie, Paris, décembre 2012.

3, R.L. Pratt and J.M. Bowsher, The objective assessment of trombone quality, Journal of sound and Vibration 65(4), pages 521-547, 1979.

4, H. Bouasse, Instruments à vent, tome 1, page 309 and page 335, Delagrave publisher France, 1929.

5, E. Poirson, J.-F. Petiot et J. Gilbert. Integration of User Perceptions in the Design Process : Application to Musical Instrument Optimization. J. Mech. Des, vol. 129, no. 12, pages 1206-1214, 2007.

6, C. A. Macaluso et J.-P. Dalmont. Trumpet with near- perfect harmonicity : design and acoustic results. J. Acoust. Soc. Am., vol. 129, no. 1, pages 401-414, 2011.

7, A. de Cheveigné et H. Kawahara. YIN, a fundamental frequency estimator for speech and music. J. Acoust. Soc. Am, vol. 111, no. 4, pages 1917-1930, 2002.

8, K. Wogram. Ein Beitrag zur Ermittlung der Stimmung von Blechblasinstrumenten. Ph.D. thesis, Technische Universität Carolo Wilhelmina, Braunschweig, 1972.