Deep Learning methods achieve state-of-the-art in many tasks, including vocal pitch extraction. However, these methods rely on the availability of pitch track annotations without errors, which are scarce and expensive to obtain for Carnatic Music. Here we identify the tradition-related challenges and propose tailored solutions to generate a novel, large, and open dataset, the Saraga-Carnatic-Melody-Synth (SCMS), comprising audio mixtures and time-aligned vocal pitch annotations. Through a cross-cultural evaluation leveraging this novel dataset, we show improvements in the performance of Deep Learning vocal pitch extraction methods on Indian Art Music recordings. Additional experiments show that the trained models outperform the currently used heuristic-based pitch extraction solutions for the computational melodic analysis of Carnatic Music and that this improvement leads to better results in the musicologically relevant task of repeated melodic pattern discovery when evaluated using expert annotations. The code and annotations are made available for reproducibility. The novel dataset and trained models are also integrated into the Python package compIAM1 which allows them to be used out-of-the-box.