The simulation results and sound examples in Section 7.3 were generated by a simple extension to the method described in Section 5.4. That section, with its side links, already explained in detail how to build a pluck simulation for a single string with known properties attached to an instrument body with a known bridge admittance. The method was based in the frequency domain, employing an inverse FFT to generate the final time waveform for each note.
The coupling of the string to the body was implemented via a standard “point coupling” result: if two subsystems are connected at a single point, the combined impedance at that point is simply the sum of the impedances of the separate subsystems. In terms of admittance, the inverse of the coupled admittance is the sum of the inverses of the two separate admittances. This result can be trivially extended to the case of three subsystems all connected at the same point: the combined impedance is the sum of three separate impedances. So to couple two strings to the body of an instrument, we simply have to make use of results we already have for the end impedance of each string, together with the measured bridge admittance of the body.
In the earlier work, the coupled impedance at the bridge was converted into a pluck response by making use of a reciprocal theorem. The pluck response we want is determined by the motion at the bridge in response to a step function of force applied at the plucking point on the string. By reciprocity, this is the same as the motion of the string at the plucking point in response to the same step function of force applied at the bridge. So for a “una corda” pluck, in which only one string of the pair is plucked, we can proceed exactly as before. Starting from the 3-way coupled impedance at the bridge, we apply the step function of force there and compute the motion of the relevant string at the chosen plucking point. Now to synthesise a “regular piano” pluck in which both strings are plucked simultaneously, we take advantage of the linearity of the whole system and use superposition: we can do the “una corda” calculation separately for each of the strings, then simply add the resulting bridge motions together to get the combined response to both plucks.
For the piano examples in Section 7.3, the assumed string properties were as follows. Both strings were identical, made of steel with a diameter of 1.0 mm and a length of 639 mm, as measured on the baby grand piano pictured in earlier sections. One string was tuned so that its fundamental was at the nominal equal-tempered frequency of $C_4$, while the other string was given a small additional tension to raise the frequency by the chosen mistuning ratio. For the lute examples, the properties of the 5th course on the chosen lute were used: a pair of identical fluorocarbon strings, of diameter 0.91 mm and length 625 mm for the open string. One open string was tuned to the nominal frequency $C_3$, while the other was tuned up slightly to achieve the chosen mistuning ratio. For the simulations of the higher note $G_3$, the length of both strings was reduced to correspond to stopping them at the 7th fret. For both instruments, the damping properties of the strings were taken from the earlier analysis in Section 7.2, using the relevant diameters and material properties.