tefilla - Why Don't we Say Tachanun on a Particular Day?

There are many days on which we don't say Tachanun, and much argument about whether we say it on any given day -- but is there any underlying reason why we wouldn't say it, a reason which might explain why we do or don't say it on a given day?

I looked in the Rambam and the M"B and all I saw was lists of when we do and don't say it -- no reasoning behind why. There were some explorations of why a specific day might be excluded, but no generalizable rule through which I could then predict whether on another day we would say Tachanun.

Is there a rule which determines saying or not saying Tachanun, or is the entire practice a series of individual cases with separate logic for each?

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_{A similar question, but about Tu Bishvat specifically.}