How It Works
Learn more about the theoretical foundations of the Music Harmony Analysis Tool.
Bachelor Thesis
My bachelor thesis, which explains the music theory this program is based on, can be found here:
Download PDFWhere do the 302 core modulations come from?
Each core modulation is a class of modulations. Every other modulation can be mapped to one (and occasionally more) of these fundamental types:
Modulations reducible by note-removal
Sometimes, when notes are removed from the pitch-class set, the ending keyset remains the same. We strip away tones until any further removal would change the ending keyset; the resulting minimal pc-set defines the core modulation.
(Certain modulations lead to more than one minimal form — in those cases they correspond to multiple core modulations.)
Modulations from keys other than C major or C minor
Transpose the entire progression so the starting key becomes C major or C minor. The harmonic "character" of a modulation is invariant under transposition — just as a major triad stays a major triad regardless of absolute pitch.
Modulations starting from a multi-key keyset
When the starting keyset has more than one possible key, we choose as the starting key the one that shares the largest tone overlap with a candidate ending key. The modulation resolves the harmonic ambiguity.
If several starting keys share the same minimal "distance" to an ending key, the modulation overlaps multiple core types.
Atonal modulations
As explained in the theory: remove tones from the atonal pc-set until it becomes tonal. Among the resulting tonal pc-sets, we pick the one where a possible ending key has the least distance from one of the possible starting keys. Then we use the method from point 3 for assigning to a core modulation.
If several tonal reductions satisfy this condition, the modulation might again map to multiple core types.
The hope is that this way, we group together modulations that have a very similar harmonic character, so that we can reason about modulations while only considering the core modulations. Else we would have millions of different modulations to consider - and that would be too much to handle.