Well depends if you already own a system?
The i7-6700k is a new chipset/socket than the i7-4790k and i7-5820k.
If you're just building now I would straight out go with the i7 6700k.
1. it's slightly faster at CPU intensive tasks than the i7-4790k (around 10-20% depending on workload)
2. it has newer features and newer motherboards.
3. Runs slightly cooler than the i7 4790k. It also overclocks slightly better if you're into that.
So personally I would go with the new i7-6700k skylake CPU, it is the newest CPU from Intel and will handle gaming easily, and other.
The i7 5820k does have 6 cores but each core is supposedly less powerful single cores, than to the other i7's. So in theory the 4790/6700 should perform better in less threaded games. The 6 cores however will be worth it in highly CPU intensive tasks with many threads.
The 4790 and 6700 are 4 cores with hyperthreading, with 8 threads on both.
It also depends on what music software you're using
I would still go the newer i7- 6700k for gaming and production. It cuts in the middle.
-good luck