The Neo-Riemannian Tonnetz, a triangular net with keys as vertices and triads as triangles

The keys are denoted by their names, but also the key numbers from 0 to 11.
With enharmonic equivalence this triangular net forms a torus, and has 12 vertices and 24 triangles.
Unlike many tori, the left edge of the repeated parallelogram corresponds only to a part of the right edge (B♭-D-F♯).

Major triads are red, minor triads are blue.
Thin lines stand for minor thirds (major sixths), thicker lines for major thirds (minor sixths). The even thicker horizontal lines stand for fifths (fourths).

The relationship between triads whose triangles are horizontal neighbours is also seen in the circle diagram on the left:

• R (relatives), like a/C
  
• L (leading-tone exchanges), like F/a or C/e

By vertical neighbours the Tonnetz expresses another relationship:

• P (parallels), like a/A and c/C

Secondary operations can be constructed by combining these basic operations:

• RLP = N (nebenverwandt), like a/E or f/C
• LPR = S (slide), like A♭/a or C/c♯
• LPL = H (hexatonic pole), like D♭/a or C/g♯

Printable version: File:Neo-Riemannian Tonnetz.pdf

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