The most successful system will likely combine several techniques. I doubt you'll find one technique that works well for melody, harmony, rhythm and bass sequence generation across all genres of music.
Markov chains, for instance, are well suited for melodic and harmonic sequence generation. This method requires analysis of existing songs to build the chain transition probabilities. The real beauty of Markov chains is that the states can be whatever you want.
- For melody generation, try key-relative note numbers (e.g. if the key is C minor, C would be 0, D would be 1, D# would be 2 and so on)
- For harmony generation, try a combination of key-relative note numbers for the root of the chord, the type of the chord (major, minor, diminished, augmented, etc.) and the inversion of the chord (root, first or second)
Neural networks are well suited to time series prediction (forecasting), which means they're equally suited to 'predicting' a musical sequence when trained against existing popular melodies/harmonies. The end result will be similar to that of the Markov chain approach. I can't think of any benefit over the Markov chain approach other than reducing the memory footprint.
In addition to pitch you will need duration to determine the rhythm of the generated notes or chords. You can choose to incorporate this information into the Markov chain states or neural network outputs, or you can generate it separately and combine the independent pitch and duration sequences.
Genetic algorithms can be used to evolve rhythm sections. A simple model could use a binary chromosome in which the first 32 bits represent the pattern of a kick drum, the second 32 bits a snare, the third 32 bits a closed hi hat and so on. The downside in this case is that they require continuous human feedback to assess the fitness of the newly evolved patterns.
An expert system can be used to verify sequences generated by the other techniques. The knowledge base for such a validation system can probably be lifted from any good music theory book or website. Try Ricci Adams' musictheory.net.