## 3 Numerical Scheme
### Scheme Update
#### 3.6.3 [PERSON]'s Construction and Pitch Flattening  
Update Equation (75) is valid for the general case, for any friction curves obeying the condition \(v\varphi(v)\geqslant 0\). However, it is important to note that whenever the bow does not share any contact points with the fingers or the backboard (which is the case in most realistic playing scenarios), the last equation of System Equation (75) (corresponding to the bow friction force) is decoupled from the rest of the system. As a result, the bow friction nonlinearity, different in nature from that of the finger and fingerboard, can be treated separately, by the means of solving the nonlinear scalar equation:  
Given the known values \(b_{B}^{n}\) and \(\sigma^{n}\), Equation (77) is a scalar equation in terms of \(v_{\text{rel},B}^{n}\) only, and can be solved with the help of [PERSON]'s graphical construction [56]. Indeed, the solutions to Equation (77) are the abscissas of the intersections of the friction curve \(\varphi_{B}(v_{\text{rel},B}^{n})\) with the line defined by \(\eta_{B}\left(v_{\text{rel},B}^{n}\right)=-\frac{1}{\sigma^{n}}\left(v_{\text{ rel},B}^{n}+b_{B}^{n}\right)\), as shown in Figure 6, with \(\varphi_{B}\) in black and \(\eta_{B}\) in blue, for different values of \(b_{B}^{n}\). A solution on the vertical part of the curve corresponds to the sticking state (\(v_{\text{rel},B}^{n}=0\)); otherwise, the string is slipping against the bow (\(|v_{\text{rel},B}|>0\)). \[v_{\text{rel},B}^{n}+\sigma^{n}\varphi_{B}(v_{\text{rel},B}^{n})+b_{B}^{n}=0 \tag{77}\]  
where \(b_{B}^{n}\) is the last element of \(\mathbf{b}^{n}\), and \(\sigma^{n}\) is defined as:  
\[\sigma^{n}=\frac{hA}{2k}\left(\mathbf{j}_{B}^{n+1}\right)^{\mathsf{T}}\mu_{i}J_{ B}^{n}\left[f_{B}^{n}\right]_{+} \tag{78}\]As Figure 6 shows (case (c)), one must beware of the possibility of multiple solutions. With this force-velocity curve model, this possibility, although seemingly brought about by the particular discretisation chosen for the bowed string equations, does indeed resemble [PERSON]'s so-called _ambiguity_, giving rise to pitch flattening phenomena, as [PERSON] _et al._ have deduced [40]. They showed that the \"correct\" solution (Figure 6, i. and iii.) is the one that preserves the current oscillation phase (sticking (Figure 6, i.) or slipping (Figure 6, ii.)) for the longest time, and that the \"middle\" solution (Figure 6, iii.) is always physically unstable, and never seen in practice. This gives rise to hysteretic behaviour: the relative velocity solution jumps back and forth between the two branches of \(\varphi_{B}\) at different points depending on whether the transition is from stick to slip, or the other way around. This shows as a lengthening of the stick/slip cycles, and therefore a flattening of the pitch. This effect has later been found to be due to the naturally hysteretic thermal behaviour of the melting rosin, which is somewhat approximated by the hysteresis rule on the simpler friction curve model [42]. As can be seen in Figure 6, multiple solutions can only arise if the (negative) slope of \(\eta_{B}\) is small enough, provided by the condition:  
\[-\frac{1}{\sigma^{n}}\leqslant\min\left(\varphi_{B}^{\prime}\right) \tag{79}\]  
From Equation (78), it can be seen that for a given virtual string, as \(\mathbf{j}_{B}^{n}\) is normalised at all times, the variations of \(\sigma^{n}\) are directly proportional to those of the normal bow force: as \(\left[f_{B}^{n}\right]_{+}\) increases, \(\frac{-1}{\sigma^{n}}\) increases. The pitch flattening effect therefore is, for this scheme, a result of pushing down too strongly on the bow; this is indeed what can be observed in real bowed string playing.