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Measuring the resolution of Torus

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:

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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