A kind of musical United Nations will gather in Trenton on Saturday.

That’s when the third annual Accordion Rally: The Republic of Squeeze will take place at the Trenton Grange Hall.

The first two rallies attracted more than 30 accordionists each year, according to Joe Niemczura, the event organizer and the master of Trenton Grange 550. Among the styles of accordion playing that have been represented are British, Bulgarian, Cajun, French, German, Polish, Slovenian and Yiddish. Songs played have ranged from the traditional polkas often associated with accordions to Jimi Hendrix’s “Purple Haze.”

“We never actually know who will show up and what they’ll do,” Niemczura said. “Everyone’s welcome. It’s just a really fun thing in that respect.”

The event got its start when Niemczura, who doesn’t play accordion, wanted to find a way to bring his accordionist friend, Nancy “3” Hoffman, and her Portland-area band, The Maine Squeeze, up to Trenton for the monthly contra dance the Grange holds on the third Saturday of every month.

“It just ballooned into inviting every accordion player we could think of,” Niemczura recalled.

Like the best of anything musical, the rally itself is rather free-form.

“We get together, drink some coffee, limber up the instruments and then people decide what they want to do,” Niemczura said. “There’s a real feeling of accordion brotherhood. That’s kind of the magic of it.”

The instruments used are primarily the traditional piano-style accordion or the button-oriented concertina, Niemczura explained, but there is a wide variety of more obscure styles, including a Slovenian two-row button accordion.

The talent level at the rally ranges from beginners playing borrowed instruments to Jeff Grosser, a Wiscasset dentist who once was a professional accordion player who appeared on the “Ed Sullivan Show” (Sullivan was also his agent).

The second part of the event is an evening recital, open to the public. The entire group will practice a couple of numbers during the day, with subgroups working on different kinds of songs.

Niemczura even tried his hand the first year … sort of. He bragged to other beginners that he would play that night, without practicing. Niemczura, who is the singer of the polka band King Pirogue, strapped on the accordion and took the stage. He folded his hands behind his back, and a hooded Tom Jamrog, King Pirogue’s accordionist, stuck his arms through Niemczura’s arms and played the accordion in his stead.

“We had all these accordion experts there, and they had never seen such a thing before,” he said with a laugh.

Through his association with the rally, Niemczura has learned quite a bit about accordionists. (He has even compiled a database of e-mail addresses for 57 accordionists in Maine.)

“People who play accordion have a personal relationship with their instrument,” he said. “Many have played it since childhood, or have a favorite accordion. Also, if someone finds out they’re accordionists, total strangers will give them accordions.”

Still, although he plays drums and trumpet, Niemczura hasn’t learned to play accordion yet.

“I’ve resisted the urge, because I’m a perfectionist,” he said. “It takes a lot to learn it, and I don’t have the time in the day.”

The Accordion Rally will be held at the Trenton Grange Hall, at the intersection of Routes 3 and 230 in Trenton, across from the Open Hearth Motel. The registration fee for the rally is $10, and the evening recital is $5 for audience members. For more information, call Joe Niemczura at 667-0260 or e-mail josephn@acadia.net.