The semantics that underlies our handling of harmony operates in the 3D tonal space as used by Christopher Longuet-Higgins (and others before him).

For an excellent introduction to the tonal space theory, see the papers collected in Longuet-Higgins' book Mental Processes.

• Tonal Space Semantics: a description of the logical forms expressing the form of tonal space paths that our grammar produces.

Nearest Neighbour

Some of the rules and the logical form notation depend on a concept of the nearest possible choice out of an infinite set of points equated by equal temperament to a given starting point. For example, given the point C in the space, we might want to know what is the nearest possible point in the ET equivalence set of G (that including G-, G+, F##, F##-, F##+, etc). In this case it is indisputably the point that is one step to the right of C. In other cases, the choice is not so obvious.

• The document Equal Temperament Equivalences and Nearest Neighbours in the Tonal Space describes a policy for choosing the nearest neighbour.