A few days after his arrival to Tredegarth, Erasmas is summoned along with other newcomers for Inbrase, and is expected to perform a chant.
I was expecting a follow-up on why the chant that he selected drew such a "rumble of astonishment" from the crowd, especially from the Thousanders. I searched around, but I couldn't find an explanation, and nothing followed further into the book.
I'm not sure my answer is complete, but here's my best shot. The chant that he chose was one that he had learned at Orithena, and Orithena is not a math, it is "the cloister of a lineage that was born a thousand years before Cartas and her Discipline," as Erasmas guesses correctly when he arrives there (p. 522). I think when he sang it (p. 601), the thousanders were surprised that Erasmas had had some kind of contact with thousander knowledge. Or they were surprised that he was confirming the existence of the Lineage. In an earlier conversation between Lio, Erasmas, and Criscan (p. 375), they discuss the rumours about a Lineage and why people are uncomfortable with the idea of the Lineage as a secret organization within the mathic world.
(Page numbers are from the hardcover first edition.)
Right before he does the chant, he explains where he got it from:
The Orithenans had used a system of computational chanting that, it was plain to see, was rooted in traditions that their founders had brought over from Edhar. To that point, it was clearly recognizable to any Edharian. It was a way of carrying out computations on patterns of information by permuting a given string of notes into new melodies. The permutation was done on the fly by following certain rules, defined using the formalism of cellular automata. After the Second Sack reforms, newly computerless avout had invented this kind of music. In some concents it had withered away, in others mutated into something else, but at Edhar it had always been practiced seriously. We'd all learned it as a sort of children's musical game. But at Orithena they had been doing new things with it, using it to solve problems. Or rather to solve a problem, the nature of which I didn't understand yet. Anyway, it sounded good—the results, for some reason, just tended to be more musical than the Edharian version, which was serviceable for computing things, but, as music, could be hard to take. I'd spent enough time among the Orithenans to hear some of it and to gain some familiarity with the system. I'd had one tune in particular stuck in my head during the flight to Tredegarh and my time in quarantine. Maybe if I sang it aloud, it would go away.
If the people in Orithena were using it to solve problems then the part of the chant he picked up was likely something the Thousanders found interesting. Erasmas doesn't understand the nature of the problem they're trying to solve, but the Thousanders likely do. In a way he's transmitting information from Orithena to the Thousanders, who are normally quarantined from such new information.
