Since you're after artists, I'd recommend turning the query inside out and asking for artists, not genres. This will return artists names, IDs, and genres for all artists which include a genre of rock music or reggae:
[{
"id": null,
"name": null,
"type": "/music/artist",
"genre": [{
"name|=": [
"Rock music",
"Reggae"
]
}],
"g:genre": []
}]
It's be more robust to use the IDs for the genres rather than their names in case someone decides the genre should be called "Rock" instead of "Rock music."