Suddenly hard to keep quiet (7)

I believe the answer is:

steeply

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'suddenly' is the definition.
Both the answer and definition are adverbs. Perhaps you can see a link between them that I can't see?

'hard to keep quiet' is the wordplay.
'hard' becomes 'steely' (similar in meaning).
'to keep' means one lot of letters goes inside another.
'quiet' becomes 'p' (music symbol for 'piano', ie soft or quiet).
'steely' going around 'p' is 'STEEPLY'.

Can you help me to learn more?

I've seen this clue in The Times.

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