fact that it does not satisfy the triangle inequality. It is actually a para-pseudo-distance according to our terminology in subsection 1.3. The following example schows a violation of the triangle inequality between the regional centers and . On the hand we have the On the other hand we have

How to music-theoretically interpret this violation? As long as the notion of distance serves just an illustrative metaphor there is not need to insist in the model of a metric space. In terms of the public transport metaphor one could easily switch from a shortest-path-principle to a cheapest-ticket-principle, where a violation of the triangle-inequality is typically compensated by a higher comfort of travel. Within Lerdahl’s theoretical framework, however, the violation has to be regarded as counterintuitive, because he simultaneously bilances the length (or cost) of pathways at several levels of reduction. It is as if a passenger in a local train - who manages to sleep with one eye closed at local stops - would be charged the high-comfort-price of an inter-regional train for the closed eye.

An alternative to the para-pseudo metrics is the pseudo-metrics , which minimizes the path lengths of all pathways between two loci in along the edges defined by all intra-regional, pivotal and inter-regional distances. But the Weber space is not isometrically embedded into .