Pianos

Figure 11: Block diagram of piano synthesis using commuted waveguide synthesis.

Figure 11 shows a block diagram of a piano synthesis algorithm based on commuted waveguide synthesis [166,162,182].¹⁷Commuted synthesis introduces an enormous computational simplification based on the commutativity of linear, time-invariant systems [161]. When cascaded systems are commutative, their order can be interchanged. This is why the soundboard model appears before the string model in Fig. 11. The ``trigger'' signal occurs when a piano key is pressed, and it starts a playback of the soundboard impulse response from a table. The tapped delay line feeds three lowpass filters which depend on the key velocity $v_c$. The filter outputs are summed and fed to a digital waveguide string model (including coupled-string effects [13,196,162]). The sum of the filter impulse responses models the interaction-force between the string and the piano hammer, as shown in Fig. 12. Sound examples from the commuted piano synthesis model can be heard at the on-line version of [162].¹⁸

Figure 12: Illustration of the building of a hammer-string interaction force as the superposition of three filter impulse response.