How many shape characters does Torus need to identify tunes? Currently it uses an average of characters, although the number needed varies between and .

Looking in more detail, suppose we pick tunes in the database
uniformly at random and give Torus the first `k` characters of
their shape strings, for some fixed `k`. Then we can define two
possible measures:

- the
**uniqueness probability**= the probability that a tune is identified uniquely; - the
**resolution**= the reciprocal of the expected number of matches found.

We can express each of these measures as a number between 0 and 1:
the higher the value, the better Torus is at identifying tunes by
using `k` shape characters. We can then plot these measures
against `k` to get an idea of the number of shape characters we
might need to use:

The graph above suggests some puzzles: What shape should we expect the curves to be? Do they approach each other as the number of tunes becomes large?

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