We investigate in detail the spectrum of gravitational waves induced by a peaked primordial curvature power spectrum generated in single field inflationary models. We argue that the $f_{rm NL}$ parameter can be inferred by measuring the high frequency spectral tilt of the induced gravitational waves. We also show that the intrinsically non-Gaussian impact of $f_{rm NL}$ in $Omega_{rm GW}$ is to broaden its peak, although at a negligible level in order not to overproduce primordial black holes. We discuss possible degeneracies in the high frequency spectral tilt between $f_{rm NL}$ and a general equation of state of the universe $w$. Finally, we discuss the constraints on the amplitude, peak and slope (or equivalently, $f_{rm NL}$) of the primordial power spectrum by combining current and future gravitational wave experiments with limits on $mu$ distortions from the cosmic microwave background.
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