Tuesday, June 3, 2014

Geometric Representation of Whole Tone and Diminished Scales

Miles Okazaki shares a geometric representation of the two whole tone scales and three diminished scales in pitch-class space. 

Okazaki's hexagon with inscribed triangles is certainly more elegant than a traditional Tonnetz diagram of the same scales. 

Okazaki's diagram relates pitch classes to their proximate neighbors by step, allowing for the easy recognition of scales. The Tonnetz is intended to show tonal relations and is thus based on triads rather than steps. As a result, connecting pitch-classes in order to form scales is somewhat ungainly. 

On the other hand, it does allow for the visualization of the chords---augmented triads and diminished seventh chords---that form the whole tone and diminished scales. It also provides an easy way to visualize how the two whole tone scales and three diminished scales are related by transposition and how the diminished scales are related by inversion: by geometric "translation" and "reflection," respectively. 

Quite a bit more on this general topic here, and more about Miles

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