I'll approach the problem from several angles.
There's a library in Python for symbolic manipulation and feature extraction of sheet music, music21.
Generally discretely sampled time-series falls into a few domains. You can gain a broad overview by reading a book on signal processing, of which there are several accessible free resources. Such as Fourier and Wavelets, and the DSP Guide. A rigorous statistics approach is Brockwell and Davis. The book focuses on forecasting, but the mindset is invaluable.
If your samples were discretely-sampled continuously-valued sequences(like an audio recording), you'd reach for any of the commonly used shift-invariant feature extraction tools. Some popular ones are Mel Cepstrum, Stationary Wavelet Transforms and shift-invariant kernels. A useful intermediate step for Fourier based methods is Spectrograms, and for Wavelets there is the Continuous Wavelet Transform. Spectrograms have been segmented, clustered and histogrammed for cutting-edge bird classification performance.
Shift-invariance is notable here, as it is precisely the property that your features should possess. For example, a simple histogram of chords is shift-invariant.
There are undoubtedly many kernels that could be used for this problem, but you'll likely have to roll your own and verify both Mercer's conditions and shift-invariance. An oversimplified kernel that satisfies these conditions is Radial distance from a normalized histogram of chords. The simplest way to imbue temporal features would be n-gramming the chords before histogramming.
Finally, predicting music popularity is not sensible for many purposes(lastfm). A more frequent problem is building a Recommender system.
