In 1920 and again in 1934 C. V. Raman showed that the right-hand head of the mridangam and the tabla, which carries a central loaded patch, produces overtones that are nearly harmonic, unlike an ordinary drum. He identified nine normal modes that group into the first five harmonics. This paper revisits Raman’s result with the finite-element method and asks what it implies for the standardised manufacture of drumheads. The head is modelled as an ideal circular membrane under uniform tension with a concentric patch of higher areal density. An axisymmetric solver using quadratic elements, applied to each angular order, and an independent two-dimensional solver using linear triangles are validated against the exact Bessel-function frequencies and agree with each other to within 0.13%. Optimising the patch radius and density ratio to minimise the root-mean-square (RMS) deviation of Raman’s nine modes from a harmonic series gives a radius ratio of 0.414 and a density ratio of 8.9, which reduces the inharmonicity from 216 cents for a uniform membrane to 37 cents. A two-zone patch, thicker at the centre, reduces it further to 15 cents. Sensitivity and Monte Carlo analyses show that harmonicity tolerates errors of ±5% in patch mass, is more sensitive to patch radius, and is almost unaffected by offsets of the patch up to 10% of the radius. These results translate into quantitative tolerances for a manufacturing specification.