This paper introduces TunesFormer, an efficient Transformer-based dual-decoder model specifically designed for the generation of melodies that adhere to user-defined musical forms. Trained on 214,122 Irish tunes, TunesFormer utilizes techniques including bar patching and control codes. Bar patching reduces sequence length and generation time, while control codes guide TunesFormer in producing melodies that conform to desired musical forms. Our evaluation demonstrates TunesFormer's superior efficiency, being 3.22 times faster than GPT-2 and 1.79 times faster than a model with linear complexity of equal scale while offering comparable performance in controllability and other metrics. TunesFormer provides a novel tool for musicians, composers, and music enthusiasts alike to explore the vast landscape of Irish music. Our model and code are available at https://github.com/sander-wood/tunesformer.