There are only two ways of hunting in Germany I know of:
one is sitting on a "Hochsitz" (a wooden construction that has a ladder for climbing up and at the top a bench for sitting, often with a roof) and waiting for the deer and roe to appear.

The second involves lots of people with sticks going in a row through the woods beating the bushes, thus driving boars and other animals towards the hunters with the rifles.

I don't think there is a lot of (if any) horse-ridden fox-hunting.
At least in Bavaria we don't have the landscape for it - there is either forest (mostly fir-trees in mono-culture) or fields for grain, corn, ...
Most fields are very big (it was reorganized some decades ago from many small fields to fewer big fields). Hedgerows (or stone walls) like in Ireland or Great Britain to divide the fields are very rare (they might hinder the machinery ...) - therefore the number of small animals (hares, rabbits, pheasants, foxes, partridges ...) is MUCH smaller than in Ireland/UK ==> no need for/sense in hunting. Those few rabbits living here are mostly killed on the roads ...



As for hunting music:
As alanabit mentioned, there are some hunting songs. Searching for "Jäger" ("hunter") on www.ingeb.org lists 88 results - I did not check them ..

But I'd guess there is more "Jagdhorn-Blasen" (playing the "hunting horn"[???] - a simple brass instruments with no valves or holes) than singing


(disclaimer: this is mostly from what I see - as I am no hunter myself, I am not on the inside of hunting in Germany)

MudGuard (from Munich, Germany, thus not only a real German but even better: a real Bavarian)