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Simulation of Brass Instruments

Computer Model foundations

The computer model used in BIAS is founded upon electronic analogies found in acoustics research. An instrument is treated as a dissipative transmission-line. The sound pressure is analogous to the voltage and flow of the current. Thermoviscouse losses are accounted for by appropriate (complex) resistance. The circuit is terminated with a radiation impedance [Zt], that results in partially reflected waves, as in a real instrument.

The computation of the instrument over its length is done through a series of conical and cylindrical sections. The acoustical transmission properties of these single elements may be exactly determined through frequency-dependent matrices (Ai(f))- see [5] for exact matrices.

IWK Institut of Music Acoustics (Wiener klangstil)

Fig. 1: Sectioning of trumpet tube length (mensur) into smaller elements

IWK Institut of Music Acoustics (Wiener klangstil)

Pressure (p) and velocity (u) before and after the i-th element overlap.

IWK Institut of Music Acoustics (Wiener klangstil)

The impedance [Zi] at i results from

IWK Institut of Music Acoustics (Wiener klangstil)

The transmission characteristics of the complete instrument are achieved by calculating the product of the partial matrices

IWK Institut of Music Acoustics (Wiener klangstil)

The computation of the input impedance is completed in reverse order, beginning with the radiation impedance.

IWK Institut of Music Acoustics (Wiener klangstil)

Fig. 2: An impedance curve simulated using BIAS

Limitations of the model

Possible sources of error are:

In the case of a cylindrical tube with a known geometry, these limitations lead to a reliable simulation in the frequency range

IWK Institut of Music Acoustics (Wiener klangstil)

d is the maximum tube diameter [m] and c is the sound speed [m/s]. With complicated tube dimensions as they occur in real brass instruments, estimated accuracy of 60-80% is more realistic.

Diameter of the bell

ideal cut-off frequency

80% of the cut-off frequency

60% of the cut-off frequency

10,0 cm

2001,0 Hz

1600,8 Hz

1200,6 Hz

12,0 cm

1667,5 Hz

1334,0 Hz

1000,5 Hz

14,0 cm

1429,3 Hz

1143,4 Hz

857,6 Hz

16,0 cm

1250,6 Hz

1000,5 Hz

750,4 Hz

18,0 cm

1111,7 Hz

889,3 Hz

667,0 Hz

20,0 cm

1000,5 Hz

800,4 Hz

600,3 Hz

22,0 cm

909,5 Hz

727,6 Hz

545,7 Hz

24,0 cm

833,8 Hz

667,0 Hz

500,3 Hz

26,0 cm

769,6 Hz

615,7 Hz

461,8 Hz

28,0 cm

714,6 Hz

571,7 Hz

428,8 Hz

30,0 cm

667,0 Hz

533,6 Hz

400,2 Hz

Table 1: Validity range according to the formula for variable diameter with c=345m/s

Within the aforementioned limitations, the position of the resonance peaks may deviate on an average of less than 0.6% (10 cent) in calculations.

The height of the resonance peaks deviates with an error of 15-25%.

Comparison of measurements and computations

Comparisons of simulations with measurements lead to the following observations:


[1]  Caussé, René; Kergomard, J.; Lurton, X. (1984)
"Input Impedance of Brass Musical Instruments -- Comparison between Experiment and Numerical Models,"
Journal of the Acoustical Society of America (JASA) 75, 241-254.
[2] Keefe, Douglas H. (1984)
"Acoustical wave propagation in cylindrical ducts: Transmission line parameter approximations for isothermal and nonisothermal boundary conditions,"
Journal of the Acoustical Society of America (JASA) 75, 58-62.
[3] Lampton, M (1978)
"Transmission Matrices in Electroacoustics,"
Acustica 39, 239 ff.
[4] Levine, H.; Schwinger, J. (1948)
"On the radiation of sound from an unflanged circular pipe,"
Phys. Rev. 73, 383-406.
[5] Mapes-Riordan, Dan (1993)
"Horn Modeling with Conical and Cylindrical Transmission-Line Elements,"
Journal of the Audio Engineering Society 41, 471-483.


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