The widespread availability of music loops has revolutionized music production. However, combining loops requires a nuanced understanding of musical compatibility that can be difficult to learn and time-consuming. This study concentrates on the 'vertical problem' of music loop compatibility, which pertains to layering different loops to create a harmonious blend. The main limitation to applying deep learning in this domain is the absence of a large, high-quality, labeled dataset containing both positive and negative pairs. To address this, we synthesize high-quality audio from multi-track MIDI datasets containing independent instrument stems, and then extract loops to serve as positive pairs. This provides models with instrument-level information when learning compatibility. Moreover, we improve the generation of negative examples by matching the key and tempo of candidate loops, and then employing AutoMashUpper to identify incompatible loops. Creating a large dataset allows us to introduce and examine the application of Transformer architectures for addressing vertical loop compatibility. Experimental results show that our method outperforms the previous state-of-the-art, achieving an 18.6\% higher accuracy across multiple genres. Subjective assessments rate our model higher in seamlessly and creatively combining loops, underscoring our method's effectiveness. We name our approach the Deep Recombinant Transformer and provide audio samples available at: https://conference-demo-2024.github.io/demo/