“I heard back from the patent people. Surprisingly, they believe some of the Sigma music may be a success for us.” He allowed himself a thin smile like a line drawn on a wall.

“The Bach-like pieces? I studied them in linear processing mode. Great artful use of counterpoint, harmonic convergence, details of melodic lines. The side commentaries in other keys, once you separate them out and break them down into logic language, work like corollaries.”

He shrugged. “That could be a mere translation artifact. These AIs see language as a challenge. So they see what they can change messages into, in hopes of conveying meaning by other means.”

Rachel eyed him and ventured on. “I sense… something different. Each variation shows an incredible capacity to reach through the music into logical architectures. It’s as though the music is both mathematics and emotion, rendered in the texture. It’s… hard to describe,” she finished lamely.

“So you have been developing intricate relationships between music and linguistic mathematical text.” His flat expression gave her no sign how he felt. Maybe he didn’t.

She sat back and made herself say firmly, “I took some of the Sigma’s mathematics and transliterated it into musical terms. There is an intriguing octave leap in a bass line. Surprising. I had my pod make a cross-correlation analysis with all Earthly musical scores.”

He frowned. “That is an enormous processing cost. Why?”

“I… I felt something when I heard it in the pod.”

“And?”

“It’s uncanny. The mathematical logic flows through an array matrix and yields the repeated notes of the bass line in the opening movement of a Bach cantata. Its title is rendered in Anglish as ‘God’s time is the very best time.’ ”

“This is absurd.”

“The Sigma math hit upon the same complex notes as Bach! To them it was a theorem, and to us it is music. Maybe there’s no difference.”

“Coincidence.”

She said coolly, “I ran the stat measures. It’s quite unlikely to be coincidence, since the sequence is many thousands of bits long.”

He pursed his lips. “The Bach piece title seems odd.”

“That cantata ranks among his most important works. It’s inspired directly by its biblical text, which represents the relationship between heaven and earth. The notes depict the labored trudging of Jesus as he was forced to drag the cross to the crucifixion site.”

“Ajima was examining such portions of the Sigma Structures, as I recall. They had concentrated density and complexity?”

“Indeed, yes. But Ajima made a mistake. They’re not primarily pieces of music at all. They’re mathematical theorems. What we regard as sonic congruence and other instinctual responses to patterns, the Sigma Structure says are proofs of concepts dear to the hearts of its creators, which it calls the Overs.”

She had never seen a Prefect show surprise, but Masoul did with widened eyes and a pursed mouth. He sat still for a long moment. “The Bach cantata is a proof?”

“As the Sigma Structures see it.”