“Halt and Catch Fire” is a platform for a fascinating, buried period of digital history. Illustration by Sachin Teng

In 1983, when I was a teen-ager, my brother told me that you could reach a BBS—an early online community—through our phone line, using a modem. I dialled in, and was met by that digital shrieking sound, the now nostalgic “handshake.” The sysop greeted me with a message in neon-green letters, something like “Hello, friend.” Unnerved, I signed off, and didn’t go online again until college.

“Halt and Catch Fire,” an AMC drama about the battle to create the first portable computer, takes place during that period, back when the pinging, whirring devices that have come to dominate our lives were still revolutionary and potentially frightening. The show is now completing its second season, which has been such a startling upgrade of the first that it begs for technological metaphors. Early on, the characters discuss the Doherty Threshold, a concept they describe as the speed at which a computer responds to the user’s fingertips: slower than four hundred milliseconds, and users will get frustrated and quit; faster, and they’ll be hooked. “Halt” ’s first season never quite hit the television equivalent of the Doherty Threshold: it was ambitious but jankily paced, decently cast but not quite good enough to recommend to strangers. Watching the show improve has provided its own drama, one that’s not uncommon in this new era of TV-making, as creators struggle to innovate, working from the older models of prestige cable.

From the start, “Halt and Catch Fire” was hurt by several factors, including that impossible-to-remember title. (It’s a computer command that causes a processor to overload, shutting down the machine.) The pilot felt like a wishful mashup, using elements scavenged from reruns. There was a Don Draper-ish salesman, Joe MacMillan (Lee Pace), who had a tormented family history, a closet full of Armani suits, and a gift for gassy monologues. There was a Walter White-like failed scientist, Gordon Clark (Scoot McNairy), emasculated by his sighing wife. There was a chick coder, Cameron Howe (Mackenzie Davis), a bleached-blond punkette straight out of “The Girl with the Dragon Tattoo” (or possibly Aimee Mann circa “Voices Carry”). In Dallas, these three rebels teamed up to build a new computer—or, actually, to improve on a prototype that Joe ripped off from his former employer, I.B.M. The entire enterprise felt surreally self-referential: a show about reverse-engineering a stolen computer was itself reverse-engineered from the stolen archetypes of better dramas.

And yet as “Halt and Catch Fire” proceeded it began to insert fixes, scene by scene, as if it were a product that had shipped too early, downloading updates to improve usability. It started to explore the rich, strange prehistory of today’s Silicon Valley gold rush, sketching out the culture clash among engineers and coders, silver-tongued marketers and crusty old-tech investors. The show looked lovely, washed out and green-gray, as if the nineteen-eighties took place after a rainstorm (even if the camera was often tilted, for no good reason, at a paranoiac forty-five-degree angle). For nostalgists, it offered the sensual pleasures of clunky beige consoles and pounding punk songs. There were moments of self-serious melodrama: Cameron brainstorming in lipstick on a mirror; Joe revealing his chest, covered in scars.

But there was also a sense of excitement as the team struggled to solve technical problems—to make the laptop light enough, to design a compressed motherboard, to code an operating system from scratch. And the writers gradually gave Donna (Kerry Bishé), Gordon’s nagging wife, a meaningful role. By the final episodes of Season 1, there was a legitimate thrill in seeing the team launch their compromised clone of a computer, the Cardiff Giant, at a Las Vegas expo. The episodes also featured an imaginative re-creation of a historic event, as tech geeks gathered in a hotel suite, lit by candles, to witness the début of the Apple Macintosh, gazing in awe as if gathered around the Nativity.

Season 2 jumps forward fourteen months, wrenching the ensemble into a new hierarchy. Joe is no longer the main character; he’s been sidelined and humbled, and the focus is now on Cameron and Donna, a former engineer herself, who have teamed up to run a small company called Mutiny, which starts out in gaming but evolves into a developer of chat rooms and discussion boards, with echoes of Compuserve and Prodigy. The gender switch isn’t entirely radical: this is still a show about a rule-breaker and a rule-follower debating digital philosophy. But the watchful, stubborn Donna, in her pastel corporate outfits, is a fresh type for TV, and the women’s chemistry is looser, releasing the show from the burdens of its gloomy forerunners. Meanwhile, Donna and Gordon’s marriage feels layered, no longer a cartoon of a genius husband and a henpecking supermom. The shambling house that serves as Mutiny’s office—all schlubby guy geeks, bickering and playing paintball—is a fun place to hang out. Even that tilted camera seems less a pretension and more an endearingly dorky trademark.

It’s not that the show is revelatory art—a plot about Gordon receiving a diagnosis of brain damage verges on bathos. But it doesn’t need to be. It’s effective doing what it does best, which is to be a platform for a fascinating, buried period of history. Along the way, we get oddly profound meditations on the nature of originality in the digital age, nested within relationship talk. In one great scene, Cameron and her new boyfriend argue over whether she should copy-protect their new game. Leaving code exposed is the only way to operate, he argues, amazed that she’d see it otherwise; they learned from stolen code—they even met through stolen code, when he hacked her game and improved it. That’s how anything creative gets stronger. But Cameron tells him that she isn’t willing to lose her money, or her credit, yet again.

It can’t be a coincidence that these are the very issues that haunt modern television. The medium is surging creatively, but it’s in a stage of economic chaos, reaching the viewer by many routes, through cables and antennae, over computers and tablets, sold via iTunes, remixed on YouTube, and traded, to the distress of producers, on BitTorrent. (To quote a retro P.S.A., “Don’t copy that floppy!”) I caught up on “Halt and Catch Fire” on my phone, streaming on Amazon, as I lay in bed with earbuds in, which would have been impossible only three years ago. There’s no way to talk about television as art without talking about television as technology: the kind of beauty we get is inevitably shaped by the way it’s delivered.

There’s a scene in the German spy series “Deutschland 83,” on Sundance, that encapsulates the year in which it is set. In West Germany, an East German named Martin stumbles upon a man selling black-market electronics. The man offers him options: Does he want a TV set? A Walkman, maybe? “What’s a Walkman?” Martin asks, baffled. Anyone who lived through this period will understand the look on his face as he hits Play: infatuation and awe, the kind of emotion that Cole Porter called “rapture serene.”

Like “Halt and Catch Fire,” “Deutschland 83” bears a resemblance to a more ambitious cable show: “The Americans,” FX’s series about Russian spies, starring Keri Russell and Matthew Rhys, which just finished a bleak and astonishing third season. But, if “Deutschland 83” doesn’t have quite that show’s depth, it has other charms. It’s a slinky thriller, well scored, well paced, cast with beautiful faces, and nearly as aesthetically aspirational as “Mad Men” ever was, if you’re in the mood to fantasize about being a chain-smoking German spy in green leather gloves. It’s gorgeous and it’s good enough.