WHITE seem to be the starting point of song energy and the weakest, as it’s usually 'upgraded' to a different colour. It’s like a base that both Mylene and Basara start with when they start singing, and it's implied that it gets more powerful when coloured. I don’t think the colours have a ranking order, except one.
PINK is clearly love, and it was mostly utilised by Mylene, and it was always directed at people she loves. It's not strictly reserved for romantic love, as it occurred when Mylene was singing to her parents. It was also shown once in a Mylene-Basara duet but the situation was not romantic - since it was right after Basara regained his voice (ep27), it could shine for his love of singing. When Gigil was dying, Basara shone pink when he called out for him, but it was corrected to purple in the next episode recap.
PURPLE was most notably produced by Mylene and directed at Basara. Purple would be a mix of pink and blue - pink being love and blue being the following;
In ep46, when Basara and Mylene tried their hardest to free Gamlin, she was pink, while Basara’s energy was BLUE. That makes me assume that blue is friendship. This would make sense with the previous point, concerning Mylene’s feelings for Basara. As for why Basara didn't shine blue during Gigil's death, it could be because the feelings he had for Gigil were more complex - Gigil wasn't quite a friend, but he embodied something Basara had longed for - he listened to his song.
There isn’t a GREEN one, most likely because green is the light of spiritia and it would get confusing to see it used on both sides.
There’s also YELLOW, which is used a few times in very different situations. Once it was used by Mylene to bring back Basara’s voice, it occurred when Mylene fell out of her valkyrie, was caught by Gamlin, who wanted to take her to safety, but she told him to go back because she has to sing for Basara. This made me think that yellow might be something like loyalty. In episode 29 Basara dings to Sivil and is also glowing yellow, it could technically signify Basara's loyalty to the cause. But honestly, for now I have no idea what yellow might mean.
ORANGE might have been used once or twice by Basara, but I'm unsure if it was actually supposed to be orange light or it was Red and the lighting made it look orange. If I figure out what yellow means, I'll be able to conclude what orange might be.
And finally, RED seems to be most powerful and is, of course, Basara’s "main" colour. He always starts off white (they all do), but after he gets fired up and sings at full passion, his aura turns bright red, and I’m gonna say red is probably passion.