Neural networks have become the dominant technique for accurate pitch and periodicity estimation. Although a lot of research has gone into improving network architectures and training paradigms, most approaches operate directly on the raw audio waveform or on general-purpose time-frequency representations. We investigate the use of Sawtooth-Inspired Pitch Estimation (SWIPE) kernels as an audio frontend and find that these hand-crafted, task-specific features can make neural pitch estimators more accurate, robust to noise, and more parameter-efficient. We evaluate supervised and self-supervised state-of-the-art architectures on common datasets and show that the SWIPE audio frontend allows for reducing the network size by an order of magnitude without performance degradation. Additionally, we show that the SWIPE algorithm on its own is much more accurate than commonly reported, outperforming state-of-the-art self-supervised neural pitch estimators.