Swinden's Unconformed Tonnetz, Chromatic Theory

Check this out.




This picture is a graph created by Kevin J. Swinden for his article in volume 27/2 of Music Theory Spectrum. Let me break it down for you:
- An upward pointing triangle is a major triad
- A downward pointing triangle is a minor triad
- Mix structures are denoted by the diagonals (Ab-C-E, A-C-Eb, Eb-G-B, E-G-Bb, etc.)
- Motions toward the Left/West denote moves towards the Subdominant (Tonic to Subdominant, Dominant to Tonic)
- Motions toward the Right/East denote moves towards the Dominant (Subdominant to Tonic, Tonic to Dominant)
- The main East-West Axis (in the center) contains the three primary functions (Dominant, Subdominant, and Tonic); Offset rows to the North and South denote "functional agents" (tendency tones)

I find that this is a fantastic way to visualize the relationships of triads, tendency tones (Harrison calls these "functional agents"), and how third relationships can move a piece towards or away from different tonal areas.

Here is the abstract to his article:

Daniel Harrison's 1994 study, Harmonic Function in Chromatic Music, questions the traditional mapping of chords onto function, and instead suggests that scale steps embody the source of harmonic function. His reformulation creates a new one-to-one mapping of scale steps onto harmonic function, which may be, at times, problematic. This article examines aspects of Harrison's theory and advances a different mechanism for the evaluation of harmonic function based on the Tonnetz. It examines a particular set of chromatic harmonies that display plural function, which may be organized according to a genus and species mode of classification.

For a more in-depth discussion of Swinden's analytical approach to chromatic music of the late 19th-Century refer to the articles listed below.

- Swinden, Kevin J, "When Functions Collide: Aspects of Plural Function in Chromatic Music," Music Theory Spectrum 27/2 (Fall 2005): 249-82.

- Harrison, Daniel, Harmonic Function in Chromatic Music: A Renewed Dualist Theory and an Account of its Precedents, Chicago and London: University of Chicago Press.
- Smith, Charles J, "The Functional Extravagance of Chromatic Chords," Music Theory Spectrum 8: 94-139.