Towards a Music Algebra: Fundamental Harmonic Substitutions in Jazz

In this paper the most common harmonic substitutions, at least as far as jazz music is concerned, are unconventionally addressed. The novelty consists in introducing a new method finalized to formally defining and logically applyingall the fundamental harmonic substitutions, by exploiting anunusually rigorous notation. After defining the substitutions and discussing their applicability, we resort to them in order to modify some simple harmonic progressions substantially based upon a banal major turnaround. As explicitly suggested by the title, the modifications are carried out by following an extremely formal line of reasoning: all the logic passages are accurately described by resorting to a notation so similar to the one commonly employed in mathematics and physics, that the harmonic analysis of a song turns out to be de facto comparable to the demonstration of a theorem.