I think the answer is:
A tensile failure test
Help me, for I have become longer -- taller -- thinner
A tensile specimen fails by lengthening and becoming thinner
I am more ungenerous, more unwilling to give
It becomes harder (work hardens), which could be described as ungenerous (translating from the scientific term to the usual english usage), and gives with more difficulty (modulus increases).
Sharply, I will hurt you, wound you
Look, I don't really think this is the answer, so this gets more and more scretched (excuse the pun). Sharply could refer to a sharpie impact test, or the fact that a sharp edge makes a stress concentrator, which tend to be the break point, or the possible sharp edges left over after it snaps
A musician of the purest variety
Mmm. Well, I guess we could be performing the test on a musical string.
I burn, slightly and superficially
Often there is localized heating at the break point
A functional, right ratio
The ratio between the length and the cross sectional area (okay, okay, I know it's weak - I didn't see your answer written here!)
Here now abbreviated
Break = abbreviated
Inclusions can be the source of the weakness. Sigh It's just so weak. I feel broken. Torn into two. Rent asunder. Unable to hold it all together. Brittle. Someone give this dog a dog bone.