Multiple pitch estimation consists of inferring the fundamental frequencies and the salience of the notes forming a music signal over short time frames. This mid-level representation can be exploited as a front-end for higher-level applications, such as music-to-score transcription or chord detection. One approach is to decompose the short-term magnitude spectrum of the signal into a sum of basis spectra representing individual pitches scaled by time-varying amplitudes, using algorithms such as nonnegative matrix factorization (NMF). Prior training of the basis spectra is often infeasible due to the wide range of possible instruments. Appropriate spectra must then be estimated from the observed data, which may result in limited performance due to inaccurately estimated spectra. In this article, we model each basis spectrum as a weighted sum of narrowband spectra representing a few adjacent harmonic partials, thus enforcing harmonicity and spectral smoothness while adapting the spectral envelope to each instrument. We derive a NMF-like algorithm to estimate the model parameters and evaluate it on a database of piano recordings, considering several choices for the narrowband spectra. Performance appears superior to unconstrained adaptive NMF and competitive with supervised NMF based on pre-trained piano spectra. We also apply our approach to woodwind data.