The Song of the Family means everything to Kino. His family are his whole world, his whole life. This is made quite clear in the very first chapter, when in the early morning he relaxes watching Juanita tending their baby Coyotito and singing a song which forms part of the Song of the Family.

Sometimes it rose to an aching chord that caught the throat, saying this is safety, this is warmth, this is the Whole. (chapter 1)

At this stage, the family are content and happy in their small home, happy just to be together, and so the song at this point represents security and comfort. This is just before the scorpion stings Coyotito, introducing a note of evil and discord into Kino's small world, threatening everything that he lives for and setting into motion a chain of events that ultimately results in catastrophe for the family and Coyotito's death. 

The Song of the Family changes as the family situation develops. To begin with, as seen in the quote above, it is gentle, lilting, soothing. However, by the end of the story when the family have lost everything and have been driven from their home, it becomes wild and desperate, 'fierce and sharp and feline as the snarl of a female puma' (chapter 6). This description conjures up an effective image of the family as a cornered animal fighting for its very survival.