Why are music channels commercially viable with such a small audience?

They are commercially viable because having a small audience means advertisers can specifically target that audience and design an advert that will be attractive to the particular audience of that channel. An example of a company that has used this to great effect is Creative Zen Micro who advertised on only the channels owned by Emap.
Emap identified three key areas, Rock, hip hop and Pop. Creative also wanted to target a 16-34 age demographic, so, they designed nine adverts three for each one of the three areas (three pop, three rock and three hip hop) and each of these three adverts where specifically designed for the channel it was shown on e.g. a black man pulling along his whole record collection around for the hip hop channel
Over 6 weeks the advertising reached 76% of the core target audience and they had 46,991 entries into the competition normal entry into the same style of competition is around 1,000. Creative Zen Micro had sold out within weeks.
This proves that with the correct style of advertising you can make a profit from genre channels if you are targeting the correct audience e.g. a rock orientated advert might work on a rock channel but will not have the same effect on a hip hop audience.