The Hayward Tuning Vine is a software interface for exploring the system of microtonal tuning known as Just Intonation. Based ultimately on prime number relationships, harmonic space in Just Intonation is inherently multidimensional, with each prime number tracing a unique path in space. Taking this multidimensionality as its point of departure, the Tuning Vine interface assigns a unique angle and colour to each prime number, along with aligning melodic pitch height to vertical height on the computer screen. These features allow direct and intuitive interaction with Just Intonation. The inclusion of a transposition function along each prime number axis also enables potentially unlimited exploration of harmonic space within prime limit 23. Currently available as desktop software, a prototype for a hardware version has also been constructed, and future tablet app and hardware versions of the Tuning Vine are planned that will allow tangible as well as audiovisual interaction with microtonal harmonic space.