I always thought that the correct way to say this was "Violinenunterricht." However, it seems that "Violinunterricht" is more commonly used (there are also more hits on Google). Which one is correct and why? Is there a rule about this and is this comparable to the "Gedichtanalyse" vs. "Gedichtsanalyse" question? What is the formal term for this kind of construction?
This is a question about Fugenlaute (according joints) that are used as joiner between the parts of compound words.
Compound words are very often used in German Language, and so you very often have to decide which Fugenlaut to use. The problem ist, that the rules are very variable.
Very often the Fugenlaut is chosen that way, that the first part together with the Fugenlaut is equal to the plural or genitiv of the first part. In the case of »Violinenunterricht« you see such a plural-like construction, but it is not really a plural. It is just the first part »Violine« joint with a »n« that comes not from the plural, but just is a Fugenlaut.
Sometimes you omit any Fugenlaut like in Haustür, Tischdecke or Gasthaus. You can use this kind of joining also for Violine and Unterricht.
Here the e at the end of Violone has been omitted. This happens sometimes when a word that ends with an e is joint with a word that starts with a vowel, but is rarely done.
All three variations (Violinenunterricht, Violineunterricht and Violinunterricht) are correct German words.
some other facts about Fugenlaute
As said before, rules for Fugenlaute are not very clear, and so you often find more then just one solution for joining two words. Especially there are different rules in Germany and in Austria. I give you some examples:
Advent + Kalender
In Germany: Adventskalender (with an s between the parts)
In Austria: Adventkalender (no Fugenlaut)
Schwein + Braten
In Germany (without Bavaria): Schweinebraten (e)
In Austria and Bavaria: Schweinsbraten (s)
Schaden + Ersatz
In Germany: Schadensersatz (s)
In Austria: Schadenersatz (-)
Einkommen + Steuer
In Germany: Einkommensteuer (-)
In Austria: Einkommenssteuer (s)
You find more Information about Fugenlaute here:
Fugenkonsonanten bei zusammengesetzten Wörtern
Welche grammatische Form kann das erste Wort in zusammengesetzen Wörtern annehmen?
Is there any difference between "Vermögensteuer" and "Vermögenssteuer"?