Learn Hardcore Greek: Στην καφετέρια η μουσική ακουγόταν πολύ δυνατά, αλλά η φωνή της σερβιτόρας ακουγόταν ήρεμη. - At the café the music was heard very loudly, but the waitress's voice was heard calmly.

You use σε (and therefore στην) with the accusative case in modern Greek, so καφετέρια here is in the accusative, even though it looks the same as the nominative.

What case is καφετέρια, and how can I tell?

Καφετέρια (café, coffee shop) is feminine singular and here it is in the accusative case.

You can tell from:

The article: στην = σε
- την (accusative feminine singular)
The preposition σε: in modern Greek, it is normally followed by the accusative

Feminine nouns in -α often have identical forms for nominative and accusative singular:

η καφετέρια (nominative – subject)
την καφετέρια (accusative – object / after a preposition)

So the article, not the noun ending, is your main clue.

What tense and voice is ακουγόταν, and what is a more literal translation?

Ακουγόταν is:

3rd person singular
Imperfect (past continuous)
Middle/passive voice of the verb ακούω / ακούγομαι

Active: ακούω = I hear
Middle/passive: ακούγομαι = I am heard / I sound

So:

η μουσική ακουγόταν
Literally: the music was being heard / was sounding
Natural English: the music was playing / the music sounded

The imperfect shows an ongoing state or action in the past – something that was happening over a period of time, not just once.

What is the difference between ακούω and ακούγομαι?

ακούω = I hear (active, someone hears something)
- Ακούω μουσική. = I hear music.
ακούγομαι = I am heard / I sound (middle/passive, something is heard)
- Η μουσική ακουγόταν δυνατά. = The music was heard loudly / the music sounded loud.

In this sentence, you want to describe how something sounds, not who hears it, so you use ακουγόταν (from ακούγομαι), not a form of ακούω.

Why is it πολύ δυνατά and not something like πολύ δυνατή?

Πολύ here is an adverb = very.
Δυνατά here is also used as an adverb = loudly.

Together:

πολύ δυνατά = very loudly

In Greek, many adjectives form adverbs by using the neuter plural form:

δυνατός, δυνατή, δυνατό → δυνατά (loudly)
καλός, καλή, καλό → καλά (well)

So δυνατά is not “neuter plural loud things” here; it functions as an adverb.
Δυνατή would be a feminine adjective (loud describing a feminine noun), e.g.:

δυνατή μουσική = loud music
Η μουσική ήταν δυνατή. = The music was loud.

Why is δυνατά an adverb here, but ήρεμη is an adjective?

Look at what each word is describing:

δυνατά describes the way the music was heard → it modifies the verb ακουγόταν, so it is an adverb (“loudly”).
ήρεμη describes the nature of the voice → it tells us what kind of voice it was, so it is an adjective agreeing with η φωνή (“calm”).

Grammatically:

η μουσική ακουγόταν πολύ δυνατά
= The music was being heard very loudly (how? loudly → adverb).
η φωνή της σερβιτόρας ακουγόταν ήρεμη
= The waitress’s voice sounded calm (what kind of voice? calm → adjective).

So:

Use an adverb (e.g. δυνατά) to describe how something happens.
Use an adjective (e.g. ήρεμη) to describe what something is like (a noun).

Why is it η φωνή της σερβιτόρας and not something like η σερβιτόρα είχε ήρεμη φωνή?

Both are possible, but they emphasize slightly different things.

η φωνή της σερβιτόρας = the voice of the waitress / the waitress’s voice
Here, φωνή is the main focus; της σερβιτόρας just specifies whose voice.
η σερβιτόρα είχε ήρεμη φωνή = the waitress had a calm voice
Here, η σερβιτόρα is more in focus as the subject.

In Greek, possession is very naturally expressed as:

[definite noun] + [genitive with article]
- η φωνή της σερβιτόρας
- το σπίτι του φίλου μου (my friend’s house)
- το αυτοκίνητο της μητέρας μου (my mother’s car)

So η φωνή της σερβιτόρας is exactly the normal Greek way to say the waitress’s voice.

Why does της σερβιτόρας end in -ας if it is feminine? I thought -ας endings were usually masculine.

Good observation: -ας is very common in masculine nouns (e.g. ο δάσκαλος, ο πατέρας), but here it marks genitive singular of a feminine noun in -α.

The noun is:

η σερβιτόρα = the waitress
- Nominative: η σερβιτόρα
- Genitive: της σερβιτόρας
- Accusative: τη σερβιτόρα

Many feminine nouns in -α form the genitive singular in -ας:

η γυναίκα → της γυναίκας (of the woman)
η δασκάλα → της δασκάλας (of the female teacher)
η σερβιτόρα → της σερβιτόρας (of the waitress)

So της σερβιτόρας is feminine genitive, not masculine.

Why is there a second η in αλλά η φωνή της σερβιτόρας? Could you leave it out?

In standard Greek you keep the article:

…αλλά η φωνή της σερβιτόρας…

Each noun phrase normally has its own article, even when you are contrasting two things:

Η μουσική ήταν δυνατή, αλλά η φωνή της σερβιτόρας ήταν ήρεμη.

Leaving out the second η (…αλλά φωνή της σερβιτόρας…) is not natural here and would sound ungrammatical or at best very marked.

Why is it ακουγόταν ήρεμη and not ακουγόταν ήρεμα?

Ήρεμη is a feminine adjective agreeing with η φωνή.
Ήρεμα would be an adverb (“calmly”).

The sentence says the voice itself was calm, not that it was speaking calmly as an action.

Compare:

Η φωνή της σερβιτόρας ακουγόταν ήρεμη.
→ Her voice sounded calm. (The quality of her voice.)
Η σερβιτόρα μιλούσε ήρεμα.
→ The waitress was speaking calmly. (The manner of speaking.)

So ήρεμη must agree in gender, number, and case with φωνή (fem. sing. nominative).

What does αλλά mean, and why is there a comma before it?

Αλλά means but, and it introduces a contrast.

The sentence has two independent clauses:

Στην καφετέρια η μουσική ακουγόταν πολύ δυνατά,
αλλά η φωνή της σερβιτόρας ακουγόταν ήρεμη.

In Greek, you normally put a comma before coordinating conjunctions like αλλά when they join full clauses, just like in English:

…πολύ δυνατά, αλλά η φωνή…
“…very loudly, but the voice…”

Is the word order fixed? Could I say Η μουσική στην καφετέρια ακουγόταν πολύ δυνατά instead?

The word order in Greek is fairly flexible, so several variations are possible and grammatical:

Στην καφετέρια η μουσική ακουγόταν πολύ δυνατά…
→ Sets the scene first: In the café, the music…
Η μουσική στην καφετέρια ακουγόταν πολύ δυνατά…
→ Focuses first on the music, then specifies in the café.
Η μουσική ακουγόταν πολύ δυνατά στην καφετέρια…
→ The location comes at the end; still natural.

All of these are understandable. The choice mainly affects emphasis and rhythm, not basic grammar.

Are both ακουγόταν forms in the same tense, and does that tell us anything about the time relationship?

Yes, both ακουγόταν forms are:

3rd person singular
Imperfect, middle/passive

This means:

The loud music and the calm voice were happening at the same time, as background situations in the past.
The imperfect tense gives a sense of ongoing or continuous action/state:
- The music was playing (ongoingly) loudly.
- The voice was sounding (ongoingly) calm.

So the sentence paints a simultaneous picture: loud background music, but a calm-sounding voice.