Use the matrix multiplication method of Kalotas and Lee (K&L) to investigate the spectrum of nonuniform strings. First, reproduce Eqn (26) in Kalotas:

mat[ki_,wi_]:={{Cos[ki wi],-Sin[ki wi]/(ki)},{ki Sin[ki wi],Cos[ki wi]}};(mat[k1,w1].mat[k2,w2])[[1,2]]

Substitute specific values to get f[w] as in Fig 2 of K&L
w1 = w2 = 0.5 meter
mu1 = 0.01 kg/m
mu2 = 0.05 kg/m
t = 10 N

mu1=0.01;mu2=0.05;
w1=w2=0.5;t=10;
k1=2 Pi w Sqrt[mu1/t];k2=2 Pi w Sqrt[mu2/t];
f[w_]:=(mat[k1,w1].mat[k2,w2])[[1,2]];
Plot[f[w],{w,0,35},PlotRange->All]

The nonuniform string is 1 yard long (36”=0.9144 meters). Half of the string is wrapped and the other half isn’t. The heaver wrapped side (18”) weighs 12.6 grams (0.0275591 kg/m), and the lighter unwrapped side (18”) weighs 3.3 grams (0.00721785 kg/m).

mu1=0.0275591;mu2=0.00721785;
w1=w2=(0.9144/2);t=80.95*4.4484;
k1=2 Pi w Sqrt[mu1/t];k2=2 Pi w Sqrt[mu2/t];
f[w_]:=(mat[k1,w1].mat[k2,w2])[[1,2]];
Plot[f[w],{w,0,600},PlotRange->All]
sol=FindRoot[f[w]==0,{w,Range[75,500,25]}];
ansKal=Select[Union[sol[[1,2]]],#>0&]

Now import five seconds of one of the monochord plucks

data=First@Import[FileNameJoin[{NotebookDirectory[],"pluckLong.wav"}],"Data"];
wav=data[[1;;5*44100]];
spec=Abs[Fourier[wav,FourierParameters->{-1,1}]][[1;;10000]];
ListLinePlot[spec,PlotRange->All]

Zoom in to the low frequencies and measure them (about 0.2 Hz accuracy).

ListLinePlot[spec[[1;;5000]],PlotRange->All]
freqs=Select[FindPeaks[spec[[1;;5000]]],#[[2]]>0.001&]/5//N;ansString=First@Transpose[freqs]

The above plot has “samples” as its x-axis, to get Hz divide by 5. The actual frequencies of the peaks are: