Härpfer's Extended Indispensability Algorithm in Z

TL;DR

The paper addresses extending the $Indispensability\ Function$ concept to meters with irregular groupings by providing a $Z$-based specification and a generalized framework applicable to arbitrary $metric\ trees$. Using $GNSM$ and $MNSM$ formalisms, it presents a $Z$ specification of the $ExtendedIndispensabilityAlgorithm$ and a generalized variant that operates via a priority-based position sequence. It introduces a bottom-up construction and a recursive $\mathsf{descend}$ procedure to propagate indispensability values, and it clarifies how to switch to a priority-driven approach with $startPositions$. The work broadens the applicability of the indispensability concept to irregular meters and arbitrary metric trees, while acknowledging open questions about formal proof and the musical significance of the generalized approach.

Abstract

Since 1978, Clarence Barlow developed the ``Indispensability Function''. It operates on a metric tree that is bound to the same prime number of branches for all subtrees of each particular level. It assigns to all leaf postions of this tree a numeric value which indicates how important the acoustic presence of an event at this position is for the meter to be recognized as such. Bernd Härpfer extended this concept in 2015 to deal with meters which have arbitrary groupings into two or three at any position of the tree hierarchy. This is called ``Extended Indispensability Algorithm''. This article gives a specification of the Extended Algorithm in a slightly extended version of the Z specification language, and a possible generalization to arbitrary metric trees.