## 2 Method
### Higher Frequency Region f2
#### 2.3.1 Far Field Recording  
In addition to the near field recordings, the radiated sound is also recorded with a microphone array \(\mathbf{X}_{\text{far},v,j}\) that samples the region in which the instrumentalist's head may be located during playing. We refer to this region as the _listening region_ and to the discrete sample points as _listening points_. The distance between equivalent sources on the sound board and the listening points lies in the order of decimeters to meters. For frequencies above 1.5 kHz, this means that the listening region lies in the far field.  
In the near field measurement, Section 2.2.1, one microphone array samples a planar region parallel to the sound board. In the far field measurement the microphone array samples a rectangular cuboid. The setup for the far field recordings is illustrated in Figures 4 and 11. As described in Sections 2.1 and 2.2.1, the sound board is excited with an exponential sweep. An array of \(J=128\) microphones samples the listening region. The microphones are arranged as a regular grid with a spacing of 4 cm. The array samples the complete sound field in the listening region for all wave lengths above 0.08 m, i.e., frequencies below 4 kHz. About \(V=11\) repeated measurements are carried out with a slightly shifted microphone array. Equations (1)-(5) describe how to excite the sound board and derive impulse responses for the \(A\times J\times V=87296\) different source-receiver constellations.  
These far field impulse responses provide a sample of the _desired sound field_\(Q_{\text{des}}(\mathbf{X},\boldsymbol{\kappa}_{a},\omega)\) in the region in which the instrumentalist is moving her head. In frequency domain, it can be described as the relationship between source signal \(S(\omega)\), complex transfer function \(G(r,\omega)\) and microphone array recordings \(P_{\text{des}}(\mathbf{X},\boldsymbol{\kappa}_{a},\omega)\)  
\[S(\omega)\times G(r,t)=P_{\text{des}}(\mathbf{X}_{v,j},\boldsymbol{\kappa}_{ a},\omega) \tag{14}\]  
where the recording of the sweep is aligned to receive the impulse response  
\[Q_{\text{des}}(\mathbf{X}_{v,j},\boldsymbol{\kappa}_{a},\omega)=P_{\text{des} }(\mathbf{X}_{v,j},\boldsymbol{\kappa}_{a},t)\times U(\omega). \tag{15}\]  
Figure 11: Procedure to derive the impulse response \(R^{{}^{\prime}}_{f2}(\boldsymbol{\Theta}_{l},\boldsymbol{\kappa}_{a},v,t)\) for each loudspeaker \(\boldsymbol{\Theta}_{l}\) and pressed key \(\boldsymbol{\kappa}_{a}\) in the radiation keyboard. The wooden plate represents the harpsichord sound board. The white plate represents the loudspeaker grid. The black dots represent microphones microphones in the listening region. At first, harpsichord and loudspeaker array create different sound fields in the listening region. Then, the loudspeaker signals are modified as to synthesize the harpsichord sound field at regular subset \(v\) of listening points that sample the listening region.  
The terms \(S(\omega)\) and \(G(r,t)\) in Equation (14) are known. So instead of microphone array measurements, \(P_{\text{des}}(\mathbf{X},\boldsymbol{\kappa}_{a},\omega)\) can be calculated by this forward-propagation formula. In [47] it was demonstrated that the forward-propagation equals the measurements.