Jo roligere musikken er, jo bedre kan jeg slappe af i stuen.

It means the ..., the ... in English.

So:

• Jo roligere musikken er, jo bedre kan jeg slappe af
• The quieter the music is, the better I can relax

This is a very common Danish pattern for comparing two changing conditions:

• Jo mere jeg læser, jo mere lærer jeg = The more I read, the more I learn
• Jo ældre hun bliver, jo klogere bliver hun = The older she gets, the wiser she becomes

In this sentence, the idea is: if one thing increases or decreases, another thing changes too.

No. It is the same word in spelling, but not the same function.

In this sentence, jo is part of the fixed comparative pattern jo ... jo ....

So here it does not mean:

• yes
• after all
• as you know

It only helps create the meaning the ..., the ....

Because both are comparatives.

• rolig = calm / quiet

• roligere = calmer / quieter

• god = good
• bedre = better

In the jo ... jo ... structure, Danish usually uses the comparative form:

• jo større = the bigger
• jo mindre = the smaller
• jo hurtigere = the faster
• jo bedre = the better

So jo roligere musikken er literally means the quieter the music is.

Musikken is the definite form: the music.

• musik = music
• musikken = the music

Danish often uses the definite form when talking about a specific thing in the situation, such as the music currently playing.

So:

• Jo roligere musik er ... would sound wrong here
• Jo roligere musikken er ... = The quieter the music is ...

This is one of the big differences from English: Danish often puts definiteness at the end of the noun with a suffix:

• bog = book, bogen = the book
• stue = living room, stuen = the living room
• musik = music, musikken = the music

Because the two clauses behave differently.

First clause:

Jo roligere musikken er

Here, jo introduces a subordinate-type clause, so the subject normally comes before the verb:

• musikken er

Second clause:

jo bedre kan jeg slappe af i stuen

Here, the clause behaves like a main clause with inversion. Since jo bedre comes first, the finite verb comes next:

• kan jeg

This is very Danish: when something other than the subject comes first in a main clause, the finite verb usually comes second.

Compare:

• Jeg kan slappe af = I can relax
• Bedre kan jeg slappe af = Better can I relax

So in your sentence:

• jo bedre is fronted
• therefore kan comes before jeg

Normally, no.

In this jo ... jo ... construction, the first clause typically has subordinate-clause word order, so:

• Jo roligere musikken er ... is correct

Using er musikken there would sound wrong in standard Danish.

So this is a pattern worth learning as a chunk:

• Jo + comparative + subject + verb, jo + comparative + verb + subject ...

For example:

• Jo længere filmen er, jo trættere bliver jeg
• Jo dyrere det er, jo mindre køber folk

Because slappe af is a verb plus a particle, similar to English phrasal verbs like calm down or wake up.

The basic verb is:

• at slappe af = to relax

When used with another verb like kan, the particle often stays at the end:

• Jeg kan slappe af
• Hun vil slappe af
• Vi prøver at slappe af

So in your sentence:

• kan jeg slappe af i stuen

This is normal Danish word order.

Here it means in the living room.

• stue = living room
• stuen = the living room
• i stuen = in the living room

So stuen is definite because Danish usually says the living room, not just living room, in this kind of sentence.

Be aware that stue can sometimes mean other things in other contexts, but in everyday home-related Danish, stuen very often means the living room.

Because Danish often adds definiteness as an ending instead of using a separate word.

Examples:

• en stue = a living room

• stuen = the living room

• en bog = a book

• bogen = the book

• musik = music
• musikken = the music

So in this sentence:

• musikken already means the music
• stuen already means the living room

You do not need a separate word like English the.

Because the sentence has two linked clauses:

• Jo roligere musikken er
• jo bedre kan jeg slappe af i stuen

The comma separates the two parts of the comparison.

This is standard in Danish writing with this structure, and it also makes the sentence easier to read.

Yes, sometimes you may see jo ... desto ... instead of jo ... jo ....

For example:

• Jo roligere musikken er, desto bedre kan jeg slappe af i stuen

This means the same thing.

However, jo ... jo ... is very common and natural in modern Danish, so the sentence you have is completely normal.

The basic clause is:

• Jeg kan slappe af i stuen = I can relax in the living room

In your sentence, jo bedre is added to the front:

• jo bedre kan jeg slappe af i stuen

That fronting causes inversion:

• kan jeg instead of jeg kan

This is useful to notice because it helps you recognize the normal building blocks inside a more advanced sentence.