For Faster Real-Time Computation

Pre-solve the graphical intersection and store the result in a look-up table

Let $h$ denote half-pressure $p/2$. Then

$$p_b^{-}= \hm - \rho(p_{\Delta})\cdot h_{\Delta}^{+}$$

Subtracting both sides from $p_b^{+}$ and solving for $\rho$ gives

$$\rho(p_{\Delta}) = \frac{p_{\Delta}}{h_{\Delta}^{+}}-1$$

Now, for each $h_{\Delta}^{+}= \hm-p_b^{+}$, find $p_{\Delta}$ graphically, and store the resulting reflection coefficient $\rho(p_{\Delta})$ as a function of $h_{\Delta}^{+}$:

$$\begin{eqnarray*} \hat\rho (h_{\Delta}^{+}) = \rho(p_{\Delta}(h_{\Delta}^{+})) \end{eqnarray*}$$

Then the real-time reed computation reduces simply to

$$p_b^{-}= \hm - \hat\rho (h_{\Delta}^{+})\cdot h_{\Delta}^{+}$$

This is the form chosen for implementation above

Table-Reduced Reed Reflection Coefficient

• Control variable = mouth half-pressure $\hm$
• $h_{\Delta}^{+}= \hm-p_b^{+}$ computed from incoming bore pressure by a subtraction
• Table is indexed by $h_{\Delta}^{+}$
• Result of lookup is multiplied by $h_{\Delta}^{+}$
• Result of multiplication is subtracted from $\hm$
• Total reed cost = two subtractions, one multiply, and one table lookup per sample

Simple Piecewise-Linear Reed Table

$$\hat\rho (h_{\Delta}^{+}) = \left\{ \begin{array}{ll} 1-m(h_{\Del... ...c-\hdp)^{-1} 1, & h_{\Delta}^c\leq h_{\Delta}^{+}\leq 1 \\ \end{array}\right.$$

• Corner point $h_{\Delta}^c$ = smallest pressure difference giving reed closure
• In fixed-point, $h_{\Delta}^{+}\mathrel{\stackrel{\mathrm{\Delta}}{=}}p_m/2 - p_b^{+}$ is confined to $[-1,1)$
• Embouchure and reed-stiffness set by $h_{\Delta}^c$ and slope $m$ [ $m=1/(h_{\Delta}^c+1)$ in the figure]
• Zero at maximum negative pressure $h_{\Delta}^{+}= -1$ is not physical but is practical for inhibiting overflow
• A brighter tone is obtained by increasing the curvature as the reed begins to open [E.g., $\hat\rho ^k(h_{\Delta}^{+})$, for $k\gg 1$]