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Dmitri Tymoczko, Recasting K-nets


Figure 7. A graph of all the K-nets in 24-tone equal temperament that are positively isographic to {C, E} + {G}, and that represent set classes found in 12-tone equal temperament.
The graph is a 2-torus, whose top is glued to its bottom, and whose left edge is glued to its right. (Points A and B show how to attach the left and right edges.)
<Tx>, as I have defined it, moves 2x steps horizontally, while <Xx> moves 2x steps vertically. The two operations together define a coordinate system in the space,
represented here by the solid arrows. “Dual transformations” are an alternative coordinate system, represented here by dotted arrows.
Moving the major third up by semitone corresponds to diagonal SE motion; moving the singleton up by semitone corresponds to diagonal NE motion.