Michael C. Hall knows curiosity is one of the reasons why people turn out to see him perform on the microphone with Princess Goes to the Butterfly Museum.

And the “Dexter” star is OK with that.

“I kind of like that people show up maybe having some sort of looky-loo curiosity or preconceived notions and then blowing their minds,” he says. “That’s fun.”

Still, that doesn’t mean that fans should walk up to the Princess vocalist after a show and say something along the lines of, “Wow! That Dexter sure can sing!”

“Well, if they say that, I’ll have to remind them that Dexter actually doesn’t really exist,” Hall says. “He’s just words on a page.”

Hall seems more concerned about the words that go into songs these days, as PGTTBM — the trio he fronts with keyboardist Matt Katz-Bohen and drummer Peter Yanowitz — is on its first North American tour, which kicked off Sunday.

Hall, of course, is the best-known member of the band, having starred not only in Showtime’s original “Dexter” and the recent reboot, but also in HBO’s acclaimed “Six Feet Under.” He also has an impressive Broadway pedigree, having appeared in “Cabaret,” “Chicago” and other hit musicals and plays.

Yet, the other two PGTTBM members also have impressive resumes and are highly regarded in their fields.

Yanowitz first came to fame as the original drummer for Jakob Dylan’s the Wallflowers. He then went on to set the beat on Natalie Merchant’s first three solo albums and work with such notables as Yoko Ono and even the legendary Beat Generation poet Allen Ginsberg.

Katz-Bohen has been performing and recording with Blondie for well over a decade, having first signed on with Debbie Harry and company for the Parallel Lines 30th Anniversary Tour in 2008. He’s also worked with Boy George and Cyndi Lauper.

The three artists first crossed paths when Hall was playing the lead in “Hedwig and the Angry Inch” on Broadway in 2014-15. Katz-Bohen and Yanowitz were also in the production, rocking out behind Hall as members of the fictional band.

“It’s a theater piece about a band playing live in concert,” Yanowitz says of the play. “It was just fun to be in a pretend band.”

Fiction would later turn to reality as Hall, Katz-Bohen and Yanowitz began gathering informally and making music as a trio.

“We just kept doing it and doing it without even really thinking about what it was,” Yanowitz says. “Pretty soon we had 10 songs and it felt like it had its own momentum. We were like, ‘I guess we should think of a name or maybe play a show.’ Everything just sort of occurred to us much later down the road, like, ‘Oh, I guess we’re a band.’”

And thus Princess Goes to the Butterfly Museum was born, delivering the dreamy synth-pop debut single “Ketamine” in 2019.

Of course, the group had a lot of different options when it came to deciding what type of music to make — especially given the varied backgrounds and expertise of the three participants. Yet, it seems the best decision was none at all.

“Luckily, we didn’t really have to decide. And we still don’t have to decide,” Katz-Bohen says. “It’s not like we are a genre band. We are not trying to make one type of music.”

The band members have never even talked about how they want to sound or even what kind of song they want to write, Hall says.

“We just let it happen,” he says. “Whatever alchemy exists when the three of our sensibilities intersect is what it is.”

Still, there is one condition that the music has to meet — and it ties into the band’s name.

“Whatever it is, it has to fit into our concept of a museum, which is sort of our aesthetic paradigm,” Katz-Bohen says. “So whenever we record something, we sort of are like, ‘Does that go in the museum?’ And we’re like, ‘Yeah, I think that fits into the museum.’”

The band members have come up with a couple of their own terms for describing the PGTTBM sound – “kaleidoscopic sound weather” and “gothadelic rocktronic” – which apparently are fun ways to say that the music draws from everything from classic glam rock and disco to modern electronic dance music and indie-pop.

The swirling mix is on full display on the group’s first full-length album, “Thanks for Coming,” which was released in 2021. The trio also has a self-titled six-song EP from 2020 as well as basically another full-length album in the can and waiting to be released at some future point.

Combine all of that and there’s more than enough to fill a concert set list.

“There’s no dearth of material,” Katz-Bohen says. “We have a lot to choose from. It’s kind of overwhelming.”

So don’t expect to hear band Katz-Bohen and Yanowitz borrow tunes from their other endeavors – notably, Blondie and Wallflowers — at a PGTTBM show.

And that’s also OK with Hall, who is certainly not itching to try out his best Debbie Harry impersonation on a cover of “Call Me.”

“Hell no!” he says. “I’ll leave that to the genuine article.”