Construction of a brass instrument with valves. When its played, you can see the pressure wave propagate through additional pieces of tubing at the valves and then rejoin the main air column.
It is not unexpected that brass instrument synthesis has been a major focus in the physical modeling world. Indeed, digital waveguide methods  are a good match, under linear conditions, if the tube is roughly conical or cylindrical, and waveguide brass was featured on the now-classic waveguide-based physical modeling synthesizer, Yamaha’s VL1 (which is coming up on 25 years old!) . Newer developments include the Arturia BRASS system .
We’re interested in some of the more subtle features of brass instruments. Three features specific to brass instruments lead to important perceptual effects which cannot be ignored (and lead to thorny technical considerations in simulation!) are:
- viscous boundary layer effects
- shock wave propagation
- variable instrument geometry
Boundary layer effects , especially in regions where the tube is thin, have a direct impact on the widths of resonances of the acoustic tube (the horn), and thus on playability; nonlinear behaviour (shock formation) leads to characteristic “brassiness” under loud playing conditions; and finally, variable geometry allows pitch changes (through slides and valves), and, in the modern brass repertoire, access to wide array of multiphonic timbres (in-between notes). The first  and second  features have been approached in simulation, for tubes of simplified geometry, but not for arbitrary instrument bores, usually by consolidating such effects in a terminating filter or memoryless nonlinearity, according to a digital waveguide framework. The third, perhaps the most important in the quest for a playable synthetic instrument, has never been approached for general instrument bores – some preliminary work on the valve appears in  and .
There are many simultaneous challenges that have been overcome in the production of a brass instrument environment: boundary layer modeling requires construction of effective filters (be they to approximate fractional derivatives  or constructed from circuit representations of the loss model [10,11]), variable geometry constitutes a moving boundary problem, and the user must be able to control the instrument in a way that doesn’t limit the possible sounds whilst still being useable. From this, we’ve created a synthesis instrument capable of emulating any existing instrument – and further, any imaginable design – see below for some sound examples using traditional instrument geometries along with a 10m long trumpet.
Moving valves on a trumpet:
Simulation of an acoustic tube coupled to an air box. Pressure in the tube (top left) is transferred to the room (right) and energy is conserved (bottom left).
Some preliminary work has been done on incorporating nonlinear propagation in the instrument, but a proof of stability still escapes us . We’ve also been experimenting with combining our 1D model with a 3D environment to allow for a more realistic playing environment. You can see some early work presented in  that shows the action of a cylindrical tube coupled to an air box – waves propagate through the tube into an acoustic space (and vice versa), with energy of the whole system being conserved.
Incidentally, it’s easy to code these instruments; learning to play them is a whole other matter. Try for yourself by downloading the environment on our software page. There’s some more information about the environment in  and .
 A. Chaigne and J. Kergomard. Acoustique des Instruments de Musique, Belin, 2008.
 J. O. Smith III. Acoustic Modeling Using Digital Waveguides, in C. Roads, S. Pope, A. Piccialli and G. DePoli (Eds.), Musical Signal Processing, 221—263, Swets and Zeitlinger, Lisse, The Netherlands,1997.
 M. Russ. Yamaha VL1 Review, Sound on Sound, July, 1994.
 C. Vergez and P. Tisserand. The BRASS Project, from Physical Models to Virtual Musical Instruments: Playability Issues, COMPUTER MUSIC MODELING AND RETRIEVAL
Lecture Notes in Computer Science, 2006, Volume 3902/2006, 24-33, DOI: 10.1007/11751069_2.
 R. Mignot, T. Hélie, and D. Matignon. Digital Waveguide Modeling for Wind Instruments: Building a State–Space Representation Based on the Webster–Lokshin Model, IEEE Transactions on Audio Speech and Language Processing, 18(4): 843—854.
 C. Vergez and X. Rodet. New Algorithm for Nonlinear Propagation of a Sound Wave Application to a Physical Model of a Trumpet, Journal of Signal Processing, 4(1): 79—87, 2000.
 S. Bilbao. Modeling of Brass Instrument Valves, Proceedings of the International Conference on Digital Audio Effects, Paris, September, 2011.
 R. Harrison and J. Chick. A Single Valve Brass Instrument Model using Finite-Difference Time-Domain Methods. Proceedings of the International Symposium on Musical Acoustics, LeMans, France, 2014.
 S. Bilbao and J. Chick, Finite difference time domain simulation for the brass instrument bore, Journal of the Acoustical Society of America, 134(5):3860-3871, 2015.
 S. Bilbao, R. Harrison, J. Kergomard, B. Lombard, and C. Vergez. Passive models of viscothermal wave propagation in acoustic tubes, Journal of the Acoustical Society of America, 138(2), 2015.
 S. Bilbao and R. Harrison. Passive time-domain numerical models of viscothermal wave propagation in acoustic tubes of variable cross section, Journal of the Acoustical Society of America, 140(1), 2016.
 R. Harrison and S. Bilbao. Comments on travelling wave solutions in nonlinear acoustic tubes: Application to musical acoustics, Proceedings of the 22nd International Conference on Acoustics, Buenos Aires, Argentina, 2016.
 R. Harrison and S. Bilbao. Coupling of a one-dimensional acoustic tube to a three-dimensional acoustic space using finite-difference tie-domain methods, Proceedings of the International Symposium on Musical and Room Acoustics, La Plata, Argentina, 2016.
 R. Harrison, S. Bilbao and J. Perry. An algorithm for a valved brass instrument synthesis environment using finite-difference time-domain methods with performance optimisation, Proceedings of the 18th International Conference on Digital Audio Effects, Trondheim, Norway, 2015.
 R. L. Harrison, S. Bilbao, J. Perry, and T. Wishart. An environment for physical modelling of articulated brass instruments, Computer Music Journal, 29(4):80-95, 2015.
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