Is it tonal like a human language, or double-tonal like IIRC some birds can make two tones at once?

Both. There are five tones, but some of the "vowels" are also polyphonic for certain tones--though the number of vowel-tone combinations exhibiting polyphony varies by dialect. See the descriptions of vowels and tones in the description in this linked post for more details.

As avian speech goes, Rŭ are somewhat conservative in our use of polyphony. Birds tend to use fancy wideband sounds more often in cluttered environments like forests, whereas Rŭ are enormous by songbird standards and spend more time out in the open. What you tend to see in birds adapted to such environments is that the favor narrowband sounds (think the squeaks of gulls and raptors.) So Yarŭ ends up sorta in between, but tilted toward the latter.