  The term melodic template or skeleton refers to a basic melody which is
subject to variation during a music performance. In many oral music tradition,
these templates are implicitly passed throughout generations without ever being
formalized in a score. In this work, we introduce a new geometric optimization
problem, the spanning tube problem, to approximate a melodic template for a set
of labeled performance transcriptions corresponding to an specific style in
oral music traditions. Given a set of $n$ piecewise linear functions, we solve
the problem of finding a continuous function, $f^*$, and a minimum value,
$\varepsilon^*$, such that, the vertical segment of length $2\varepsilon^*$
centered at $(x,f^*(x))$ intersects at least $p$ functions ($p\leq n$). The
method explored here also provide a novel tool for quantitatively assess the
amount of melodic variation which occurs across performances.
