The harmonic retrieval (HR) problem concerns the estimation of the frequencies in a sum of real or complex harmonics. Both correlation and cumulant-based approaches are used for this purpose. Cumulant-based HR algorithms use a single 1-D slice of the fourth-order cumulant that is estimated directly from the data. We present a new cumulant-based method for estimating a 1-D cumulant slice. It exploits an invariance property of the full fourth-order cumulant to increase the signal-to-noise ratio (SNR) of the harmonic signal. This procedure effectively suppresses both Gaussian and non-Gaussian noise. With simulations we illustrate that this new procedure enhances the performance of cumulant-based harmonic retrieval. It yields more accurate results than both the conventional cumulant- and correlation-based methods, especially when the signal is contaminated with coloured noise. It enables high-resolution harmonic retrieval using a limited number of available samples. It is therefore ideally suited for HR in short transients such as harmonic pulses, and is also relevant for direction-of-arrival (DOA) estimation.