A final note: the author's interest in gt theory stems from his desire to find a way of formalizing jazz harmony and voice leading. It is not intended to be an entire and comprehensive treatment of this area of study, but rather a foundation upon which the exploration of this complex subject matter can begin. For the author, the gt Tonnetz represents a way to stay connected to the source of the harmonic content that is applied to create jazz music. It also represents a way to transform the complex into the simple. Thus, much of this text is an attempt to explain the human tendency to simplify and to make sense of complex phenomena, among which is the harmonic organization and transformation of jazz.
As you can see, this method of analysis allows us to apply the same rules to a 5-dyad and get the same results, which is important when analyzing the structures of non-traditionally used chords. For more on this, see https://www.philosophyofmusic.net/music-theory-part-3-voice-leading/ .
The principles of voice leading in the Western system are of course more complex, and the above representations must be combined. At the very least, we require a half-step progression to form a 5-dyad, whereas a whole-step progression is not possible from a 5-dyad. A further complication is the existence of 7-dyads, for which the available voice-leading options are far more constrained. For a full discussion of the Western voice-leading system, consult Bach, Tom (2007). The Western tonal system. Advances in Research and Practice in Music Education (ARPME), 25(3), 391-423. .
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