Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

thus defining a white or a black key) is omitted in the figure. The construction is analogous to the one of $X$ .

Of course the mappings of the given instrument parameters to their corresponding space coordinates in the symbolic gesture space are arbitrary in terms of absolute values and depend on the model being used (and finally being implemented) in the gesture space. In our case the measures of a real keyboard are used. The main reason for doing so is that this simplifies the transformation of the symbolic gesture to the physical one.

4.2 Curve Subdivision and Construction

Since onset time $E$ is also part of the curve, there is a lot of freedom in how to construct the curve. The following method was chosen in order to make a software implementation as simple as possible. First of all, all events are assigned to each symbolic finger according to the given fingering information. Each finger is handled separately. Then, the curve parameter $t$ is divided by the number of events for the current finger. In the case of figure 3, $t$ is divided into three intervals $[0,1/3[$ , $[1/3,2/3[$ and $[2/3,1]$ . Each of these intervals is again divided into three subintervals: the first is used for the transition before the event takes place, the second for the event itself (the vertical transparent bars in figure 3 highlight those intervals) and the last for the transition after the event.

For the construction of the curve, the event intervals are defined first: the values on the $E$ axis are set (in the case of figure 3 $e0$ and $e1$ for the first event), then pitch is mapped to $X$ and position $Y$ is set to a constant value, indicating the key is