The Ideal (Lossless) Stiff String

Simulation of a rigidly terminated, stiff string.

• All $N$ allpass filters $H_a(z)$ have had a minimum sample of delay extracted, corresponding to the delay at $f_s/2$
• All have been combined at a single point
• The delay-line length $N$ is the number of samples in $K$ periods of $f_K$ , where $K$ is the number of the highest partial (usually the last one before $f_s/2$ )
• In stiff strings, high frequencies travel faster $\Rightarrow$ allpass delay decreases with frequency
• An order 10 allpass gives high quality piano synthesis
• FFT convolution most efficient for maximum accuracy (FIR filter length $\approx N/2$ for note F1)