By applying simple operations such as transposition and inversion, she uncovered deep structures in the alien mathematics. Then she wrote codes that elevated these structures into music. With considerable effort she chose instruments and progression for the interweaving coherent lines, and the mathematics did the rest: tempo, cadence, details she did not fathom. After more hours of work she relaxed in her pod, letting the effects play over her. The equations led to cascading effects while still preserving the intervals between tones in a set. Her pod had descriptions of this. Notes in an equal-temperament octave form an Abelian group with twelve. Glissando moving upward, starting tones so each is the golden ratio between an equal-tempered minor and major sixth. Two opposing systems: those of the golden ratio and the acoustic scale below the previous tone. The proof for confocal hypergeometric functions imposes order on these antagonisms. Third movement occurs at the intervals 1:2:3:5:8:5:3:2:1…

All good enough, she thought, but the proof is in the song. Scientific proof was fickle. The next experiment could disprove a scientific idea, but a mathematical proof stood on logic and so, once found, could never be wrong. Unless logic somehow changed, but she could not imagine how that could occur even among alien minds. Pythagoras died knowing that his theorem about the relation between the sides of a right triangle would hold up for eternity. It was immortality of a sort, everywhere in the universe, given a Euclidean geometry.

But how to communicate proof into a living, singing, pattern-with-a-purpose—the sense of movement in the intricate strands of music? She felt herself getting closer.

Her work gnawed away through more days and then weeks.

Near the end, Cat asked her what a symphony had to be. Rachel, wrung out, all abuzz with her work, the bee swarm of music and math running hard in her, said, “A building made of math. I am writing in the pod, using my body to do the heavy lifting. If it’s an allegro that pursues me, my pulse keeps beating faster, I can get no sleep. If it’s an adagio, then I notice my pulse beating slowly. My imagination plays on me as if I were a clavier.”

“A…?”

“Stringed keyboard instrument. I feel like an instrument being played. Other times, just the strings.”

When she stopped in at her office between long sessions in the pod, she largely ignored the routine work. So she missed the etalk around the Library, ignored the voice sheets, and when she met with Catkejen for a drink and some crunchy mixed insects with veggies, news of the concert came as a shock.

“Prefect Masoul put it on the weekly program,” Catkejen said. “I thought you knew.”

“Knew?” Rachel blinked. “What’s the program?”

“The Sigma Structure Symphony, I think it’s called. Tomorrow.”

So Masoul had used her works in his Prefect way. His rank gave him that right. She allowed herself a small, thin smile.

She knew the labyrinths of the Library well by now and so had avoided the entrance. She did not want to see Masoul or anyone on his staff. Through a side door she eased into a seat near the front of the meeting space and stared at the assembled orchestra as it readied. There was no announcement; the conductor appeared, a slim woman in a stern white robe, and the piece began.