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.