Consistency: Difference between revisions

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Formally, for some real ''d'' > 0, a chord C is consistent to span ''d'' in ''n'' ED''k'' if there exists an approximation C' of C in ''n'' ED''k'' such that:

* every instance of an interval in C is mapped to the same size in C', and

* all intervals in C' are off from their corresponding intervals in C by less than 1/(2''d'') ED''k''.

* all intervals in C' are off from their corresponding intervals in C by less than 1/(2''d'') steps of ''n'' ED''k''.

This more formal definition also provides an interesting generalisation of ''d'' from the naturals to the positive reals, as ''consistency to distance 1/2'' can be interpreted as meaning that all intervals in C are ''at worst'' represented using their second-best mapping, which can be tolerable for some purposes assuming sufficiently small steps. "Consistency to distance 1/2" can be nicknamed "semiconsistency", in which case ''C' '' is said to be a "semiconsistent" representation/approximation of C.

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 Formally, for some real ''d'' > 0, a chord C is consistent to span ''d'' in ''n'' ED''k'' if there exists an approximation C' of C in ''n'' ED''k'' such that:
 * every instance of an interval in C is mapped to the same size in C', and
-* all intervals in C' are off from their corresponding intervals in C by less than 1/(2''d'') ED''k''.
+* all intervals in C' are off from their corresponding intervals in C by less than 1/(2''d'') steps of ''n'' ED''k''.
 This more formal definition also provides an interesting generalisation of ''d'' from the naturals to the positive reals, as ''consistency to distance 1/2'' can be interpreted as meaning that all intervals in C are ''at worst'' represented using their second-best mapping, which can be tolerable for some purposes assuming sufficiently small steps. "Consistency to distance 1/2" can be nicknamed "semiconsistency", in which case ''C' '' is said to be a "semiconsistent" representation/approximation of C.