Application to the Trumpet Using Empirically Derived Data

Acoustic pulse reflectometry techniques [6] were applied to obtain the impulse response of a trumpet (without mouthpiece). A piecewise cylindrical section model of the bore profile was reconstructed using an inverse-scattering method [1], taking into account the viscothermal losses (see Fig. 4). The piecewise cylindrical model corresponds well to the physical bore profile for non-flaring tube-segments, thus giving a good physical model up to the bell. The remaining cylindrical sections do not provide valid geometrical information, but they retain all relevant acoustical information of the bell reflectance, including the complex effects of higher transversal modes and radiation impedance.

Figure 3: Trumpet bore profile reconstruction. The valves and the final tubular bend show as `dents' in the profile. The main bore plus mouthpipe can be modeled with a cylindrical section preceded by a truncated cone (dashed lines).

The main bore of a trumpet is essentially cylindrical, with an initial taper widening (mouthpipe) (see Fig. 3). Thus, an accurate digital waveguide model of the trumpet can be derived by approximating the bore profile data with a cylindrical bore, plus a conical section to model the mouthpipe, and modeling the remaining part of the reconstruction as the isolated bell reflectance $H_{bell}(\omega)$. The complexity of the model can be further reduced by lumping the viscothermal losses of the main bore with the bell reflectance filter, yielding the ``round-trip filter'' $H_{rt}(\omega)$:

$$H_{rt}(\omega) = \frac{H_{bore}(\omega)}{H_{bore}^{'}(\omega)} \cdot H_{bell}(\omega),$$ (2)

where $H_{bore}(\omega)$ represents the response ``seen'' from the bell (see Fig. 3) while assuming an ideal closed end at the junction between the mouthpiepe and the main bore, and $H_{bore}^{'}(\omega)$ is the theoretical value of $H_{bore}(\omega)$ assuming no losses. The inverse Fourier transform $h_{rt}(t)$ differs from the theoretical Bessel horn response primarily in its two-stage build-up towards the primary reflection peak (see Fig. 4). This characteristic was observed for a variety of brass instruments. By adding another offset-exponential TIIR section (Eq. (1)) to the basic horn filter structure, the filter design methodology is sufficiently flexible to cover the two-stage build-up. The resulting impulse response and corresponding input impedance curve $Z_{in}(\omega)$ (``seen'' from the start of the main bore) are depicted in Fig. 4. The small amplitude deviations are mainly due to the fact that the TIIR approximation of the initial slow rise is insensitive to reflections caused by bore profile dents. Note that the resonance frequencies, controlled by the phase delay of $H_{rt}(\omega)$ are accurately modeled.

Figure 4: Round-trip filter (a) and ``main-bore'' input impedance (b) according to empirical data (dashed) compared to TIIR horn filter (solid). The vertical (dash-dot) lines in (a) indicate the response segmentation into 2 growing exponentials and a tail. The tail is modeled with a 4th-order IIR filter.