Classification of Triadic Chord Inverstions Using Kohonen Self-Organizing Maps

1. Classification of Triadic Chord Inverstions Using Kohonen Self-Organizing Maps Luis Felipe de Oliviera, Luis Guilherme Pereira Lima, Andre Luiz Goncalves de Oliveria, Rael Bertarelli Gimenes Toffolo

2. Schönberg’s Hypothesis “The second inversion, as he argues, has an ambiguity constitution, being it related to its root position chord and to a chord a fifth above. This ambiguity has been lead to specific harmonic rules in the attempt to characterize the function of this chord.”

3. Inversion • A chord's inversion describes the relationship of its bass to the other tones in the chord. For instance, a C major triad contains the tones C, E and G; its inversion is determined by which of these tones is used as the bottom note in the chord.

4. Inversion cont’d • The term inversion is often used to categorically refer to the different possibilities, although it may also be restricted to only those chords where the bass note is not also the root of the chord (see root position below). In texts that make this restriction, the term position may be used instead to refer to all of the possibilities as a category. • http://en.wikipedia.org/wiki/Inversion_(music)

5. Harmonic Partials

6. Kohenen SOM Details • Input Pattern: 252 Chords (21 for each of the 12 tonalities • SOM Network 50 X 50 (2500 units) • Neighborhood Radius: starting with 30, ending at 1. • Learning rate of .1 and neighborhood learning rate of .037

7. Running 1500 iterations, error goes to nothing.

8. Topological Map

9. 3D topology

10. SOM Chord Topology