This paper identifies input strict-locality as a computational property of grammatical tone maps, a property shared with many tonal and segmental processes in phonology. Through a computational account of nominalizer tone circles in Cushillococha Ticuna (and an opaque interaction with a dissimilatory tonal operation), the current study shows that grammatical tone necessitates a framework that allows explicit reference to input structure. Crucially, the computational characterization pursued here defines both the tone circle and tonal dissimilation as a single map, meaning that they operate over the same input tonal and morphological structure in tandem. The input-orientedness of these definitions permits a straightforward account of the Ticuna data. It also identifies input strict-locality as a property of circular chain shifts in general; circular shifts like the Ticuna nominalizer tone circle are therefore rather simple from a computational perspective, despite the difficulties they pose to rule-based and constraint-based theories of phonology.