Learn Hardcore Finnish: Tämän kappaleen rytmi on rauhallinen.

Tämän – demonstrative pronoun (this), in the genitive case, working as a determiner.
kappaleen – noun kappale (piece, track) in the genitive singular (kappale → kappaleen). Together tämän kappaleen means of this piece.
rytmi – noun in the nominative singular (rhythm). This is the subject of the sentence.
on – 3rd person singular of the verb olla (to be). This is the main verb / copula.
rauhallinen – adjective (calm, peaceful) in the nominative singular. This is the predicative complement, describing the subject (rytmi).

Structure in English terms:
[Of this piece] [the rhythm] [is] [calm].

Why are tämän and kappaleen both in the genitive case? Why not just one of them?

Finnish often makes the possessor phrase genitive all the way through:

tämä kappale = this piece (basic noun phrase, nominative)
tämän kappaleen rytmi = this piece’s rhythm / the rhythm of this piece

Here:

kappaleen is genitive because it’s the possessor of rytmi.
tämän must agree in case with kappale (its head noun), so it is also genitive: tämä → tämän.

So you get a little “genitive chain”:

tämän kappaleen rytmi
literally: this.GEN piece.GEN rhythm.NOM
= the rhythm of this piece.

You cannot say *tämä kappaleen rytmi; the demonstrative has to match the case of kappale.

Why is kappale in the form kappaleen here, and how does this noun decline?

kappale is an -e-ending noun. Its genitive singular is formed by adding -en:

Nominative: kappale (piece, track)
Genitive: kappaleen (of the piece)
Partitive: kappaletta (some of a piece / a piece, in partitive use)

So in the sentence, kappaleen is used because we need the genitive for a possessor:

kappaleen rytmi = the piece’s rhythm / the rhythm of the piece.

Other forms you might see:

kappaleessa – in the piece
kappaleesta – from the piece
kappaleeseen – into the piece

Why is rytmi in the basic form (nominative) and not something like rytmin?

rytmi is the subject of the sentence, so it appears in the nominative singular, the default subject form:

Rytmi on rauhallinen. – The rhythm is calm.

The possessor is in the genitive (tämän kappaleen = of this piece), but the thing possessed (the subject) remains nominative:

Tämän kappaleen rytmi on rauhallinen.
this.GEN piece.GEN rhythm.NOM is calm.NOM

Putting rytmi in genitive (rytmin) would make it another possessor and break the structure.

Why is it rauhallinen, not rauhallista or something else?

In Finnish, when you say X is adjective, the adjective (the predicative) normally agrees with the subject in case and number:

Rytmi (nominative singular) on rauhallinen (nominative singular).
Rytmit (nominative plural) ovat rauhalliset (nominative plural).

So here:

Subject: rytmi – nominative singular
Predicative: rauhallinen – nominative singular to match

The partitive form *rauhallista would be wrong in this neutral sentence, because rytmi is a regular, fully defined subject in the nominative. As a beginner rule: in simple “X is Y (adjective)” sentences, keep the adjective in the nominative to match the subject.

What is the relationship between rauha and rauhallinen? What does the ending -llinen / -linen do?

rauhallinen is derived from the noun rauha (peace) plus the adjectival ending -llinen:

rauha (peace) → rauhallinen (peaceful, calm)

The ending -llinen / -linen is very common in Finnish and usually means “having X, characterized by X, full of X”.

Some examples:

onni (luck) → onnellinen (happy, fortunate)
väri (colour) → värillinen (coloured, having colour)
huoli (worry) → huolestunut (different pattern) but also huolestunut ihminen vs huoleton ihminen (care-free person)

So rauhallinen rytmi is literally a rhythm characterized by peace / calmness → a calm rhythm.

Could I change the word order, like Rauhallinen on tämän kappaleen rytmi, or is the given order fixed?

The given order is the neutral, most natural one:

Tämän kappaleen rytmi on rauhallinen.

Some changes are possible:

Tämän kappaleen rytmi on hyvin rauhallinen. – adding an adverb
Kappaleen rytmi on rauhallinen. – dropping tämän

Marked but grammatical variants:

Rauhallinen on tämän kappaleen rytmi. – puts strong emphasis on rauhallinen; stylistic, a bit poetic or contrastive.

However, you cannot break up the possessor phrase incorrectly:

*Rytmi tämän kappaleen on rauhallinen. – ungrammatical
The genitive possessor (tämän kappaleen) must come right before the noun it possesses (rytmi).

Do I really need tämän? How does the meaning change if I just say Kappaleen rytmi on rauhallinen?

You don’t have to use tämän. Both sentences are correct, but there’s a nuance:

Kappaleen rytmi on rauhallinen.
= The rhythm of the piece is calm. (which piece is understood from context, or you are talking about “the piece” in general)
Tämän kappaleen rytmi on rauhallinen.
= The rhythm of this piece is calm. (you specifically point to or distinguish this one from others)

So tämän adds the meaning of “this (one here)” and contrasts it with other possible pieces.

What is the difference between kappale, laulu, and biisi in musical contexts?

All three can appear in discussions about music, but they’re not identical:

kappale – very general, neutral word: a piece, track, or item.
- musiikkikappale – musical piece
- Works well in more neutral or formal speech and writing.
laulu – literally song, something that is sung, has lyrics.
biisi – colloquial / informal, borrowed from English “piece” (via Swedish, etc.), now often used like “song / track” in everyday speech:
- Tää biisi on tosi hyvä. – This track is really good.

In your sentence, kappale is a safe, neutral choice: you are talking about the rhythm of a musical piece/track.

Is rytmi a Finnish word or a loanword, and how do you pronounce it?

rytmi is an international loanword (related to English rhythm, etc.), adapted to Finnish spelling and sounds.

Pronunciation:

r – tapped / rolled r
y – front rounded vowel /y/, like French u in lune, or German ü in für (not like English “y”)
t – unaspirated t
m – as in English
i – short “ee” sound, like the i in ski

Syllables: RYT-mi (stress on the first syllable, as always in Finnish).

How is on formed, and do you always need a form of olla (“to be”) in sentences like this?

on is the 3rd person singular present of olla (to be):

minä olen – I am
sinä olet – you are (singular)
hän on – he/she is
se on – it is
me olemme – we are
te olette – you are (plural / polite)
he ovat – they are
ne ovat – they are (non-human / informal they)

In Finnish, in sentences of the type X is Y (adjective/noun), you do use the verb olla explicitly:

Rytmi on rauhallinen. – The rhythm is calm.
Kappale on kaunis. – The piece is beautiful.

You cannot normally omit on in the present tense the way you can sometimes omit “is” in Russian or Arabic.

How would I make this sentence negative in Finnish?

To negate a sentence with olla, you use the negative verb ei plus the connegative form of olla (ole), and keep the predicative as it is:

Tämän kappaleen rytmi ei ole rauhallinen.
= The rhythm of this piece is not calm.

Breakdown:

ei – negative verb (3rd person here: form doesn’t change in the 3rd person)
ole – connegative form of olla
rauhallinen – stays nominative singular, still agreeing with rytmi

So the only change from the positive sentence is on → ei ole.