In many musical traditions, the melody line is of primary significance in a piece. Human listeners can readily distinguish melodies from accompaniment; however, making this distinction given only the written score -- i.e. without listening to the music performed -- can be a difficult task. Solving this task is of great importance for both Music Information Retrieval and musicological applications. In this paper, we propose an automated approach to identifying the most salient melody line in a symbolic score. The backbone of the method consists of a convolutional neural network (CNN) estimating the probability that each note in the score (more precisely: each pixel in a piano roll encoding of the score) belongs to the melody line. We train and evaluate the method on various datasets, using manual annotations where available and solo instrument parts where not. We also propose a method to inspect the CNN and to analyze the influence exerted by notes on the prediction of other notes; this method can be applied whenever the output of a neural network has the same size as the input.