Recent work in note onset detection has centered on deep learning models such as recurrent neural networks (RNN), convolutional neural networks (CNN) and more recently temporal convolutional networks (TCN), which achieve high evaluation accuracies for onsets characterized by clear, well-defined transients, as found in percussive instruments. However, onsets with less transient presence, as found in string instrument recordings, still pose a relatively difficult challenge for state-of-the-art algorithms. This challenge is further exacerbated by a paucity of string instrument data containing expert annotations. In this paper, we propose two new models for onset detection using bidirectional temporal and recurrent convolutional networks, which generalise to polyphonic signals and string instruments. We perform evaluations of the proposed methods alongside state-of-the-art algorithms for onset detection on a benchmark dataset from the MIR community, as well as on a test set from a newly proposed dataset of string instrument recordings with note onset annotations, comprising approximately 40 minutes and over 8,000 annotated onsets with varied expressive playing styles. The results demonstrate the effectiveness of both presented models, as they outperform the state-of-the-art algorithms on string recordings while maintaining comparative performance on other types of music.