Detecting the simultaneous activity of pitches in music audio recordings is a central task within music processing, commonly known as multi-pitch estimation or frame-wise polyphonic music transcription. Deep-learning approaches recently achieved major improvements for this task, but the lack of annotated, large-size datasets beyond the piano solo scenario is still a limitation for fully exploiting their potential. In this paper, we propose a strategy for training a CNN-based multi-pitch estimator on weakly aligned score--audio pairs of pieces in different instrumentations. To this end, we make use of a multi-label variant of the connectionist temporal classification loss (MCTC), recently proposed for image recognition tasks. We re-formalize the MCTC loss to be applicable for multi-pitch estimation and perform several systematic experiments to analyze its behavior and robustness to training conditions. Finally, we report on multi-pitch estimation results for common datasets using weakly aligned training with MCTC, which performs similar than systems trained on strongly aligned scores.