We propose modifications to the model structure and training procedure to a recently introduced Convolutional Neural Network for musical key classification. These modifications enable the network to learn a genre-independent model that performs better than models trained for specific music styles, which has not been the case in existing work. We analyse this generalisation capability on three datasets comprising distinct genres. We then evaluate the model on a number of unseen data sets, and show its superior performance compared to the state of the art. Finally, we investigate the model's performance on short excerpts of audio. From these experiments, we conclude that models need to consider the harmonic coherence of the whole piece when classifying the local key of short segments of audio.