We tackle the problem of multiscale regression for predictors that are spatially or temporally indexed with a Bayesian modular approach. The regression function at the finest scale is expressed as an additive expansion of coarse to fine step functions. Our Modular and Multiscale (M&M) methodology can be applied to very high-dimensional data that arise through very fine measurements. Unlike more complex methods for functional predictors, our approach provides easy interpretation of the results. Additionally, it provides a quantification of uncertainty on the data resolution, solving a common problem researchers encounter with simple models on down-sampled data. We show that our modular and multiscale posterior has an empirical Bayes interpretation, with a simple limiting distribution in large samples. An efficient sampling algorithm is developed for posterior computation, and the methods are illustrated through simulations studies and applications to time series classification and activity monitoring.