MONTREAL — “If the United Nations had a house band in 1962, hopefully Pink Martini would be it.” So says Thomas Lauderdale, who founded the remarkable miniature orchestra known as Pink Martini almost 30 years ago in Portland, Oregon. With all due respect, I think that’s being excessively complimentary to the UN.

Pink Martini — whose most prominent and consistent member, after the pianist-leader Mr. Lauderdale, is lead singer China Forbes — is to the United Nations what the 2020 miniseries “Hollywood” was to the actual Hollywood of the 1940s. Mr. Lauderdale and his troupe of a dozen or so traveling players and singers reimagine international popular music of the 1950s and ’60s.  

The major difference that they bring to the table might be described as either philosophical or political: Their version of cultural history functions as if the pop music industry had always been a true arena of equal opportunity, free from racial or gender bias, where artists of all nations and sexual orientations could express themselves on a level playing field. It’s as if the pan-global pop music of your grandparents has been given a diversity-driven facelift, and sounds all the more fresh and exciting because of it.

Mr. Lauderdale and company consistently prove that this music sounds even better now than it did then. In the era of Eisenhauer and Kennedy, people took it for granted that pop music would be rich in melody and harmony and have literate lyrics; in the 21st century, we can’t rely on any of those virtues. What record companies and TV variety shows would mass-produce in the 1950s sounds like something very rare and special today. 

In fact, it’s so rare and special that I was excited to fly to the Montreal Jazz Festival to hear Pink Martini, whose most recent album, “Je Dis Oui!,” was released in 2016 and who haven’t played New York in far too long. It’s because of this inherent underlying message of diversity that Pink Martini can deliver this music without guilt. They aren’t white imitators exploiting black innovators, North Americans culturally appropriating from Pan-Americans, or even contemporary musicians taking advantage of the past. Rather, they are everything at once: all nations, all musical styles, all time periods. It’s like comfort food without the fat or the sugar.

Their commitment to diversity also makes such notions as authenticity completely irrelevant — they might play a genuine Cuban classic, such as “Quizás, Quizás, Quizás,” or an ersatz Spanish song concocted by Hollywood, like “Que Sera, Sera,” and it couldn’t matter less because both sound great.

Their most consistent source of material comes from south of the border, both Central American boleros and rumbas in Spanish, and Brazilian sambas, choros, and bossa novas in Portuguese. Every show I have attended features Ms. Forbes singing “Amado Mio” (another Tinseltown bolero) and concludes with Ary Barroso’s “Aquarela do Brasil,” sung by Ms. Forbes in the famous English lyric by Bob Russell. 

Mr. Lauderdale’s well-crafted arrangements get a bigger sound than would be expected from 11 musicians: there’s only one trombone, one trumpet, one violin, but it sounds like much more. The one area where they go all out is percussion; at various points in the show there are as many as four drummers playing at once.

Apart from the en clave numbers, the focus tends to shift in accordance with the geography. There’s a wonderful YouTube video of a concert from Athens in 2013 where Storm Large, who like Ms. Forbes is reason enough to hear this band all by herself, treats us to “Children of Piraes” in the original Greek. The Montreal concert featured a disproportionate amount of French-language songs, not all of which I was able to recognize.

The band also features a rotating roster of guest singers. In Montreal we heard not only China Forbes but baritone Ari Shapiro, who sang one of several pop adaptations based on Franz Shubert; Jimmie Herrod, who crooned the Leon Gold movie theme “Exodus” in a high falsetto; flamenco-style singer Edna Vazquez doing “Quizás”; and percussionist-vocalist Timothy Nishimoto, who delivered “¿Donde estas, Yolanda?”

In a perfect world, Pink Martini would have a new album at least once a year, and play Town Hall on West 43rd Street — or anywhere other than the United Nations — more often than that.