Plucked/Struck String Pitch Estimation

• Take FFT of middle third of plucked string tone
• Detect spectral peaks
• Form histogram of peak spacing $\Delta f_i$
• Pitch estimate ${\hat f}_0\mathrel{\stackrel{\mathrm{\Delta}}{=}}$ most common spacing $\Delta f_i$
• Refine ${\hat f}_0$ with gradient search using harmonic comb:
  $$\begin{eqnarray*} {\hat f}_0 &\mathrel{\stackrel{\mathrm{\Delta}}{=}}& \arg\max_{{\hat f}_0} \sum_{k=1}^K \log\left\vert X(k{\hat f}_0)\right\vert\\ [10pt] &=& \arg\max_{{\hat f}_0} \prod_{k=1}^K \left\vert X(k{\hat f}_0)\right\vert \end{eqnarray*}$$
  

  where

  $$\begin{eqnarray*} K &=& \mbox{number of peaks, and}\\ k &=& \mbox{harmonic number of $k$th peak}\\ && \mbox{(valid method for non-stiff strings)} \end{eqnarray*}$$
  

Must skip over any missing harmonics,
i.e., omit $k$ whenever $\vert X(k{\hat f}_0)\vert\approx 0$ .

The text provides further details and pointers to recent papers on pitch estimation.