Could tenor not be so dramatic? It's offensive! (8)
I believe the answer is:
tactless
'it's offensive' is the definition.
(thesaurus)
'could tenor not' is the wordplay.
'could tenor' becomes 'tact' (I can't explain this - if you can you should believe this answer much more).
'not' becomes 'less' (I've seen this in other clues).
'tact'+'less'='TACTLESS'
'be so dramatic?' is the link.
This may not be right. Some or all of it may be part of another bit of the clue.
Can you help me to learn more?
(Other definitions for tactless that I've seen before include "Lacking skill in handling delicate situations" , "Indiscreet" , "Lacking sensitivity toewards others, blundering" , "Gaffe-prone" , "Maladroit" .)