Jo roligere vuggevisen er, jo hurtigere falder barnet i søvn efter sin lur.

This is a very common Danish pattern used to express the ... the ... comparison:

• Jo roligere ..., jo hurtigere ...
• The calmer ..., the faster ...

So here:

• Jo roligere vuggevisen er = The calmer the lullaby is
• jo hurtigere falder barnet i søvn = the faster the child falls asleep

This use of jo is different from the other common Danish jo meaning something like after all / you know / indeed.

Roligere is the comparative form of rolig, which means calm or quiet.

• rolig = calm
• roligere = calmer
• roligst = calmest

So:

• rolig vuggevise = a calm lullaby
• roligere vuggevise = a calmer lullaby

In this sentence, roligere agrees with the comparison structure introduced by jo.

Hurtigere is the comparative of hurtig, meaning fast/quick.

• hurtig = fast
• hurtigere = faster
• hurtigst = fastest

In the sentence:

• jo hurtigere falder barnet i søvn = the faster the child falls asleep

So both halves of the sentence use comparatives:

• roligere = calmer
• hurtigere = faster

Vuggevisen is the definite singular form of vuggevise:

• en vuggevise = a lullaby
• vuggevisen = the lullaby

Danish often uses the definite form where English might also use the. In this sentence, it refers to the lullaby being sung/used in the situation.

So:

• Jo roligere vuggevisen er ... = The calmer the lullaby is ...

This is because Danish follows the verb-second (V2) rule in main clauses.

In the second clause, jo hurtigere comes first. When something other than the subject is placed first, the finite verb usually comes before the subject:

• Barnet falder hurtigere i søvn = The child falls asleep faster
• Jo hurtigere falder barnet i søvn = The faster the child falls asleep

So after the fronted element jo hurtigere, the verb falder comes before the subject barnet.

At falde i søvn is a fixed expression meaning to fall asleep.

Literally, it is:

• falder = falls
• i = into/in
• søvn = sleep

So it corresponds closely to English fall asleep, although Danish uses the preposition i.

You should learn falde i søvn as one chunk.

Sin is the reflexive possessive used when the possessor is the subject of the same clause.

Here, in the clause:

• barnet falder i søvn efter sin lur

the subject is barnet, and sin refers back to barnet. So sin lur means its own nap.

This is very natural in Danish.

Compare:

• Barnet savner sin mor. = The child misses its own mother.
• Barnet savner hendes mor. = The child misses her mother.
  (someone else’s mother)

So sin helps show that the nap belongs to the child itself.

Because lur is a common gender noun:

• en lur = a nap

For reflexive possessives, Danish uses:

• sin with en-words
• sit with et-words
• sine with plural nouns

Since lur is an en-word, the correct form is sin lur.

Examples:

• sin bog = his/her/its own book
• sit hus = his/her/its own house
• sine sko = his/her/its own shoes

En lur means a nap.

So:

• efter sin lur = after its nap

It is a very common everyday word in Danish, especially when talking about children.

Because the two clauses use different kinds of predicates.

First clause

• Jo roligere vuggevisen er
• Here er means is, linking the subject vuggevisen with the adjective roligere.

Second clause

• jo hurtigere falder barnet i søvn
• Here the main verb is falder = falls
• hurtigere modifies how quickly the child falls asleep

So the first clause is built around to be, while the second clause is built around to fall asleep.

Sometimes yes, depending on context.

Rolig can mean things like:

• calm
• quiet
• gentle

For a lullaby, English might naturally say:

• the calmer the lullaby is
• the quieter the lullaby is
• the more soothing the lullaby is

The exact best translation depends on the meaning already intended in context, but grammatically roligere is simply the comparative of rolig.

Yes. This jo ... jo ... pattern is very productive.

You can make many sentences with it:

• Jo mere du læser, jo bedre bliver du.
  = The more you read, the better you become.

• Jo længere vi venter, jo dyrere bliver det.
  = The longer we wait, the more expensive it becomes.

• Jo trættere han er, jo tidligere går han i seng.
  = The more tired he is, the earlier he goes to bed.

So this sentence is a great model for building your own comparative sentences.

Yes. It has two linked comparative clauses:

1. Jo roligere vuggevisen er
2. jo hurtigere falder barnet i søvn efter sin lur

Together they form one idea: a change in one thing is linked to a change in another.

A useful way to understand the structure is:

• Jo + comparative + clause, jo + comparative + clause

That is the core pattern.