Differences between brass instruments arising from variations in brassiness due to nonlinear propagation Joël Gilbert, Murray Campbell, Arnold Myers, Bob Pyle

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Joël Gilbert, Murray Campbell, Arnold Myers, Bob Pyle. Differences between brass instruments aris- ing from variations in brassiness due to nonlinear propagation. International Symposium on Musical Acoustics (ISMA), Sep 2007, Barcelone, Spain. ￿hal-00475561￿

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DIFFERENCES BETW EEN BRASS INSTRUMENTS ARISING FROM VARIATIONS IN BRASSINESS DUE TO NONLINEAR PROPAGATION

J. Gilbert (1), D.M.Campbell (2), A.Myers (2), R.W .Pyle (3) (1) Laboratoire d‘Acoustique de l‘Université du Maine, CNRS, Le Mans, France, joel.gilbert@ univ-lemans.fr (2) University of Edinburgh, Edinburgh EH9 3JZ, UK, M.Campbell@ ed.ac.uk, A.Myers@ ed.ac.uk (3) 11 Holworthy Pl., Cambridge, MA 02138, USA, rpyle@ post.harvard.edu

Abstract The brightness of the sound generated by brass instruments at high dynamic level is mainly due to the essential nonlinearity of the wave propagation in the pipe. The bright instruments such as the trumpet and the trombone are different from more mellow brass instruments such as saxhorns or flugelhorns. The bright instruments have an almost cylindrical pipe segment just downstream of the mouthpiece. The conical bore of the saxhorns in this region implies a faster decay of the wave which reduces the nonlinear wave steepening. To investigate the differences between the brassiness of different instruments, crescendos were played using five different brass instruments representing the various brasswind families. A characteristic spectral enrichment parameter was extracted for each recorded sound. In parallel, a brassiness parameter was derived from weakly nonlinear 1D propagation theory. This parameter was calculated for each brass instrument from its bore geometry. The spectral enrichment parameter is found to be a monotonic increasing function of the brassiness parameter. This makes the brassiness parameter a useful tool to classify brass instruments from the point of view of their ability to play —cuivrés“ or brassy sounds.

INTRODUCTION

There is a large variety of shapes in brass instruments: some, such as the trumpet and trombone, have an almost cylindrical pipe segment downstream of the mouthpiece, while others, such as the saxhorns or the flugelhorn, have a steadily expanding bore [Herbert and Wallace, 1997]. Instruments in the latter category are known by brass players to be more mellow than those in the former category, which sound brighter. The brightness of the sound generated by brass instruments at high dynamic level is mainly due to the essential nonlinearity of wave propagation in the pipe, resulting in wave steepening and generation of shock waves [Hirschberg et al, 1996]. In expanding bores, a faster decay of the wave amplitude reduces the nonlinear steepening effect: this provides a hypothesis to explain the fact that —conical“ instruments are not as brassy as —cylindrical“ instruments. The aim of the present work is to test the above hypothesis by comparing the results of weakly nonlinear acoustic theory applied to non-uniform ducts with experimental measurements of brassiness obtained from a variety of brass instruments under playing conditions. A brassiness parameter B is calculated from the bore geometry of each brass instrument, making use of the coordinate stretching function ISMA 2007 Differences between brass instruments approach in weakly nonlinear propagation theory [Hamilton and Blackstock, 1998]. A spectral enrichment parameter A1 is estimated from measurements during playing [Gilbert, 2006]; this parameter is based on evaluation of the spectral centroid, which is strongly correlated with the perceived brightness of the sound. Finally, the rank orders of the different instruments with respect to the parameters B and A1 are compared to establish whether or not the calculated parameter B is indeed useful in predicting the ability of an instrument to generate —cuivrés“ or brassy sounds.

THEORETICAL BACKGROUND AND HYPOTHESES

W eakly nonlinear propagation in uniform ducts

Weakly nonlinear propagation in a cylindrical air-filled duct, assuming one- dimensional simple wave propagation, is described by the first-order nonlinear differential equation (1), which is approximated by the equation (2). These equations are limited to plane waves in a uniform lossless medium at rest. In tubes of finite length, waves propagate in both directions, independently in the linear limit, except for coupling at the ends. For weakly nonlinear waves it is also found that the simple waves propagating in each direction do not interact in the body of the fluid (see for example [Menguy and Gilbert, 2000]). That is described by the following nonlinear differential equation for the forward-travelling wave of pressure p and longitudinal velocity field u - propagation along the duct distance x: ∂ p » ≈ γ +1’ ÿ ∂ p + …co +∆ ÷uŸ = 0 , (1) ∂ t « 2 ◊ ⁄ ∂ x where co is the small signal sound speed and 9 the ratio of specific heats of the gas. Assuming (u/co)<<1 and locally p=ρoco.u, where ρo is the density of air, and using the x delayed time τ = t − in place of the time t, equation (1) becomes approximated by: co ≈ ’ ∂ p ∆ γ +1 ÷ ∂ p − ∆ 3 ÷ p = 0 . (2) ∂ x « 2 ρo co ◊ ∂ τ When the classical dependence of the viscothermal boundary-layer losses on the square root of the frequency is taken into account, the nonlinear differential equation (2) becomes a —generalized Burgers equation“ which is nonintegrable, and there is almost no chance of general analytical progress [Hamilton and Blackstock, 1998]. Then numerical methods are used in frequency-domain models to study permanent periodic regimes [Menguy and Gilbert, 2000], and can be applied to wind instruments: [Thompson and Strong, 2001], [Gilbert et al, 2005]. Using these frequency-domain models, the distortion of the wave along the propagation distance x can be illustrated. Figure 1 shows the evolution of the first five pressure harmonics Pn along x for a cylindrical tube (length 12 m, radius 6 mm) excited by a sine wave (frequency 350 Hz, amplitude 5000 Pa), the shock formation distance being estimated at 3.64 m. The spectral enrichment is estimated from the following dimensionless spectral centroid SC:

ƒn.Pn SC = n . (3) ƒ Pn n Figure 1 shows the rapidly increasing and then slowly decreasing evolution of SC, respectively before one and after three shock formation distance values. ISMA 2007 Differences between brass instruments

Figure 1: First five pressure harmonics Pn in [Pa] (above) and the spectral centroid SC (below) versus the propagation distance x in [m]. Simulations obtained for a cylindrical tube of 6 mm radius, excited by a sine wave (amplitude 5000 Pa, frequency 350 Hz) at x=0, shock formation distance equal to 3.64 m. Two calculations: with (solid lines) and without (dashed lines) wall viscothermal losses.

W eakly nonlinear propagation in nonuniform ducts

One-dimensional propagation also occurs in ducts of slowly varying cross section defined by their interior diameter D(x). It is assumed that the duct diameter D is small enough (kD<1, k being the wave number), and that the area varies sufficiently slowly on the scale of a wavelength, (1/kD).(dD/dx)<<1 , to justify a one-dimensional propagation model. Weakly nonlinear propagation can be described by another nonlinear differential equation (4) adapted for nonuniform ducts (forward-travelling wave, lossless approximation): ∂ p » ≈ γ +1’ ÿ ∂ p 1 d D + …co +∆ ÷uŸ + co p = 0 . (4) ∂ t « 2 ◊ ⁄ ∂ x D d x

Assuming again (u/co)<<1 and p=ρoco.u, and using the delayed time A, Equation (4) becomes: ≈ ’ ∂ p ∆ γ +1 ÷ ∂ p 1 d D − ∆ 3 ÷ p + p = 0 . (5) ∂ x « 2 ρo co ◊ ∂ τ D d x By introducing the coordinate stretching function z(x): x ≈ D ’ z(x) = ∆ o ÷.dy , (6) —0 « D(y)◊ and the spreading compensation function: w(x,τ ) = D(x).p(x,τ ) , (7) the nonlinear differential equation (5) for nonuniform ducts can be expressed in a similar form to that for uniform ducts Equation (2): ≈ ’ ∂ w ∆ γ +1 ÷ ∂ w − ∆ 3 ÷ w = 0 . (8) ∂ z « 2 ρo co ◊ ∂ τ ISMA 2007 Differences between brass instruments

Note that the linear limit of the nonlinear first-order differential equation (4) and its twin equation, the backward-travelling wave, lead to a set of the two following travelling wave equations and the wave equation (9): ≈ ’ 2 1 2 ∂ p ∆ ∂ p 1 d D ÷ ∂ ∂ ± co ∆ + p ÷ = 0 or 2 ( pD) − 2 2 ( pD) = 0 . (9) ∂ t « ∂ x D d x ◊ ∂ x co ∂ t These can be seen as a simplified version of the Webster-Lagrange horn equation (10): ∂ 2 1 ∂ 2 1 ∂ 2 D 2 ( pD) − 2 2 ( pD) − 2 ( pD) = 0 . (10) ∂ x co ∂ t D ∂ x Indeed, the dispersive effect due to the horn, last term of Equation (10), is not taken into account anymore in the simplified equations (4) and (9).

Brassiness parameter

[Pyle and Myers, 2006] suggested that a brassiness parameter might be based on the coordinate stretching function defined Equation (6) at the geometric length x=L. Indeed the stretched co-ordinate approach can be seen as defining a new equivalent length z(L) of a cylinder in which the nonlinear distortion of a sine wave propagating without losses develops to the same extent as it does in the actual instrument of length L. Thus, for example, if we have a cylinder of length Lcyl and a cone of length Lcone, the ratio z(Lcone)/z(Lcyl) lower than 1 defines in a sense the lower development of nonlinear distortion in cone compared to the cylinder. In other words, the cone of length Lcone is —equivalent“ to a cylinder of a shorter length z(Lcone). We wish to define a brassiness parameter B which will allow us to predict the ability of different instruments to obtain —cuivrés“ or brassy sounds. A simple approach is to define this parameter as proportional to z(L). To obtain a dimensionless parameter, the constant of proportionality could be chosen as the inverse of the geometric length L. However, this choice would not satisfy the desirable condition that for two instruments a and b, Ba/Bb = z(La)/z(Lb), since in general La≠Lb even for two instruments with the same nominal pitch. We therefore propose that the brassiness parameter is defined as: B z(L) = L(ecl) , (8) where L(ecl) is the equivalent cone length, L(ecl)=c/2f1, f1 being the fundamental frequency of the harmonic series best matching the playable notes. Indeed, we can consider z(L) as being a new equivalent length for nonlinear propagation: L(nlp)=z(L). Thus although L(ecl)a=L(ecl)b, since they play the same notes, and L(nlp)a > L(nlp)b for example. In fact, the increase is just proportional to the ratio of the brassiness parameters: L(nlp)a=(Ba/Bb)L(nlp)b. Moreover the brassiness parameter B defined in Equation (8) allows a comparison of brassiness between brasswinds of different pitch.

A set of —tenor“ brass instruments

Five brass instruments representing the variety of the family of tenor brasswinds have been chosen (see Figure 2): a baroque tenor trombone in Bb (Huschauer, Vienna, 1794 [euchmi 3205]), a modern bass trombone in Bb+F+Gb (Courtois, Paris, 2000), a bass saxhorn in Bb (A.Sax, Paris, 1867 [euchmi 4273]), a kaiserbaryton in Bb (Cerveny, Königgrätz, 1891-1901 [euchmi 3412]) and an ophicleïde in Bb, 9 keys (Gautrot ainé, Paris, 1860 [euchmi 3590]). The measured bore profiles and the calculated stretched ISMA 2007 Differences between brass instruments coordinates of the brass instruments are displayed in Figure 2. The bore profiles are characterised by the interior radius as a function of the axial distance, from the mouthpipe end. The stretched coordinate z is calculated from the interior radius of the bores using Equation (6). Knowing the equivalent cone lengths L(ecl), the Brassiness parameters B are calculated using Equation (8). The results reported in Table 1 show that the brass instruments can be ranked from the highest to the lowest B value as follows: tenor trombone, bass trombone, bass saxhorn, kaiserbaryton, ophicleïde.

Figure 2: The bore profiles (left, above) and the stretched coordinates (left, below) of the five brass instrument, tenor (orange) and bass trombone (red), bass saxhorn (blue), kaiserbaryton (green) and ophicleïde (black). The five brass instruments are tenor and bass trombone, bass saxhorn, ophicleïde and kaiserbaryton (from right to left on the photo).

L(ecl) in [m] z(L) in [m] B Tenor trombone in Bb 2.969 2.397 0.81 Bass trombone in Bb 2.969 2.205 0.74 Bass saxhorn in Bb 2.969 1.468 0.49 Kaiserbaryton in Bb 2.969 1.173 0.40 Ophicleïde giving A 3.145 0.969 0.31 (all holes closed) Table 1: Equivalent cone length L(ecl), maximum stretched coordinate value z(L) in [m] and dimensionless brassiness values of the brass instruments.

EXPERIMENTAL SET-UP, PROCEDURE AND DATA PROCESSING

Experimental set-up and procedure

Brass players were asked to perform crescendos typically 3 seconds long from pianissimo to fortissimo. They were asked to play the 6th regime three times for each of the five brass instruments played at its —natural state“ (first slide position, or no valves depressed, or all holes closed). The corresponding note was F4 for all the instruments except the ophicleide for which it was E4. All the performances were done using the same Courtois trombone mouthpiece. To measure the pressure in the mouthpiece (channel 1) a PCB microphone was inserted in the side of the mouthpiece back bore. The radiated sound (channel 2) was recorded at a distance of 20 cm in front of the bell, on its axis, using the same model of PCB microphone, the corresponding channel sensitivity being one hundred times higher thanks to the amplifier. The signals from the two microphones were then amplified and sampled at 44.1 kHz.

ISMA 2007 Differences between brass instruments

Preparing and processing the data

A time-frequency analysis is carried out on the signals. They are split into overlapping segments of 4096 samples each, windowed using a Hanning window, and analysed using the discrete Fourier transforms. The instantaneous fundamental frequency is estimated at each period by using the zero crossing method applied to the channel 1 signal. The mean value of the instantaneous fundamental frequency and the RMS value of the internal pressure (channel 1) P1rms are estimated for each segment of the time signals. Then the spectral centroids SC1 and SC2 associated with the channels 1 and 2, are calculated from the spectral data and the fundamental frequency for each segment using Equation 3. Because the spectral centroid results can be highly biased by the high frequency noise, all the spectrum amplitude values under a fixed threshold value have been set to zero. Spectral centroids SC1 and SC2 are increasing with the RMS pressure P1rms at the entrance of the instrument. They reveal the distortion of the signals during the crescendo. There are mainly two possible causes of the distortion process: (1) the localised nonlinear coupling at the entrance of the instrument between the vibrating lips and the acoustical resonator, (2) the cumulative nonlinear propagation phenomenon along the resonator itself. In a first approximation, the distortion mechanism (1) is revealed by the slight increase of SC1 (typically for the tenor trombone, from 1.2 to 1.6), and the distortion mechanism (2) is revealed by the large increase of SC2 (from 1.5 to 4.0). Nevertheless, a part of the distortion of the radiated pressure is due to the nonlinear coupling phenomenon. Because we are focussed on the nonlinear propagation supported by the resonator, we suggest compensating the nonconstant value of SC1 by analysing the ratio SC2/SC1. At the end of processing the data, we are looking for a single value characterising the spectral enrichment. This value, called A1, is extracted from a fit of the spectral centroid ratio SC2/SC1 function of P1rms. The fit is based on the empirically chosen function y=f(x) characterised by two coefficients A1 and A2 as follows:

y = f (x) = A2 [exp(A1 x)− (A1 x)] . (9)

RESULTS AND DISCUSSION

The experimental procedure described above has been carried out by three brass players. They have performed on the five brasswinds (see Figure 2), each one being played three times œ F4 played crescendo without any valves depressed, or with the slide in first position, except the ophicleide with played at E4. The spectral centroid ratios of one of the players are displayed on the left hand-side of Figure 3. The spectral enrichment œ i.e. the way the ratio SC2/SC1 is increasing œ seems to be smaller and smaller from the tenor trombone to the ophicleïde through the five brasswinds. This is confirmed by the corresponding values of A1, from the fit curves operated on the experimental results displayed on the right hand-side of Figure 3. All the data obtained from the three brass players playing three times each of the five brass instruments, have been processed. At the end, the values of A1 are displayed as a function of the brassiness parameter B in Figure 4. A1 is found to be a monotonic increasing function of the brassiness parameter B for each brass player. Thus the brassiness parameter B can be seen as a sensible tool to classify the brass instruments from the point of view of their ability to play brassy sounds. ISMA 2007 Differences between brass instruments

Figure 3: Experimental ratio SC2/SC1 (left), fit curves (right) as a function of P1rms - results corresponding to one player and the five brasswinds œ same colour convention as Figure 2.

Figure 4: The values of A1 as a function of the brassiness parameters B, results corresponding to three brass players.

The same kind of experiments could be carried out by using a loudspeaker, with a sinusoidal source playing near one of the acoustic resonance of the instrument, optimised to provide high level sounds in the input of the brass. Then a set-up could be defined to do systematic measurements on brass instruments avoiding the variability of real players.

CONCLUSION AND PERSPECTIVES

The brightness of the sound generated by brass instruments at high dynamic level, the brassy or —cuivrés“ sounds, is mainly due to the essential nonlinearity of the wave propagation in the pipe. Obviously, the brightness is depending on other parameters like the way the player is performing its instrument or the mouthpiece [Poirson et al, 2005]. A set of five brass instruments have been played by three players using the same mouthpiece, and their brassiness have been estimated, and ranked from the spectral enrichment parameter A1 during crescendos. The basic question of the present study was: is it possible to classify the brass instruments from the brassiness point of view without playing them? Knowing their bores and the acoustic pressure at the input of the instrument, it is possible to investigate that by doing simulations in ISMA 2007 Differences between brass instruments frequency domain ([Menguy and Gilbert, 2000], [Thompson and Strong, 2001], [Gilbert et al, 2005]), or in time domain ([Msallam et al, 2000], [Vergez and Rodet, 2000]). An alternative and attractive way is to define a brassiness parameter B directly from the bore. The preliminary results of the present study show that A1 is found to be a monotonic increasing function of the brassiness parameter B. Then the brassiness parameter B can be seen as a useful tool to classify the brass instruments from the point of view of their ability to play brassy sounds [Myers et al, 2007] without playing them. While the results obtained in the present paper are promising, it must be borne in mind that the brassiness parameter B presented here is derived from a simplified theory of weakly nonlinear propagation which considers only lossless propagation in the forward direction. Further study of the consequences of this simplification, and measurements on a wider range of players and instruments, will be necessary to establish the reliability of the parameter B as a predictor of the ease with which an instrument generates —cuivrés“ or brassy sounds.

Aknowledgments Emmanuel Brasseur, Michael Newton and Sam Stevenson are gratefully acknowledged for their help during the experiments. Marc Brossier, John Chick, Dominique Cormier and Jean-Loïc Le Carrou are thanked for supplying the preliminary sound recordings.

REFERENCES

Gilbert, J.; Dalmont, J-P.; Guimezanes, T. (2005) —Nonlinear propagation in woodwinds,“ Proceedings of Forum Acusticum, Budapest Gilbert, J. (2006) —Differences between cylindrical and conical brass instruments, the nonlinear propagation point of view from experiments and simulations,“ Joint Meeting between ASA and ASJ, Hawaii Hamilton, M.F.; Blackstock, D.T. (eds) (1998). Nonlinear Acousics, Academic Press Herbert, T.; Wallace, J. (eds) (1997). Brass Instruments, Cambridge university Press Hirschberg, A.; Gilbert, J.; Msallam, R.; Wijnands, A.P.J. (1996). —Shock waves in trombones,“ J. Acoust. Soc. Am. America 99, pp.1754-1758 Menguy, L.; Gilbert, J. (2000), —Weakly non-linear gas oscillations in air-filled tubes ; solutions and experiments“, Acta Acustica united with Acustica 86, pp. 798-810 Msallam, R.; Dequidt, S.; Caussé, R.; Tassart, S. (2000), —Physical model of the trombone including nonlinear effects, application to the sound synthesis of loud tones“, Acta Acustica united with Acustica 86, pp. 725-736 Myers, A.; Gilbert, J.; Pyle, R.W.; Campbell, D.M. (2007). —Non-linear propagation characteristics in the evolution of brass musical instrument design,“ Proceedings of International Congress on Acoustics, Madrid Poirson, E.; Petiot, J-F. ; Gilbert, J. (2005). —Study of the brightness of trumpet tones,“ J. Acoust. Soc. Am. America 118, pp.2656-2666 Pyle, R.W.; Myers, A. (2006) —Scaling of brasswind instruments,“ ASA Meeting, Providence Thompson, M.W.; Strong, W.J. (2001). —Inclusion of wave steepening in a frequency- domain model of trombone sound production,“ J. Acoust. Soc. Am. America 110, pp.556-562 Vergez, C.; Rodet, X. (2000). —New Algorithm for Nonlinear Propagation of a sound Wave, Aplication to a Physical Model of a Trumpet,“ Journal of Signal Processing 4(1)