It’s been 35 years since the first time Ben Sidran didn’t meet Bob Dylan.

The Madison jazz pianist was playing a 1974 gig with Ry Cooder at Minneapolis’ Marigold Ballroom, and while on stage, Sidran heard that Dylan (who was in town re-recording some tracks for the “Blood on the Tracks” album) was sitting in the back of one of the booths, trying to remain incognito.

After the set, Sidran summoned up the courage to go over to Dylan’s table.

“I walked over to him and said, ‘Bob, I just want you to know, man, you’ve really influenced me a lot,’” Sidran recalled over coffee recently at Ancora Coffee on Monroe St., near his Near West Side home.

“And he was like this electrified wolf, huddled in the back of the booth, like sparks were coming off him. He was like, ‘Oh yeah?’ And I was like, ‘Yeah, man.’ That was it. It was a force field like I had never seen before. There was this yawning gap.”

The second time Sidran didn’t meet Bob Dylan came years later at New York’s Madison Square Garden, when Sidran was playing with Van Morrison. He was going up the stairs backstage to the dressing room, and Dylan was coming down the stairs, with a bodyguard in front of him clearing people out of the way.

“He walked right in front of me, and he had his hood up, and he looked absolutely like a ghost again,” Sidran said. “He looked like a ghost.”

Sidran, 66, has had a long and very busy career in jazz and rock, playing with all sorts of people, including Steve Miller, Diana Ross, Boz Scaggs and Mose Allison. He crossed paths a few more times with Dylan, never really meeting him. When Dylan was in Madison this past summer doing secret rehearsals at the Barrymore Theatre, Sidran imagined what it would be like to run into him on Monroe Street. But it never happened.

So instead, Sidran has come to know Bob Dylan in a different, and maybe more revealing way — through his songs. On Tuesday, Sidran’s new album “Dylan Different” will be released, featuring Sidran’s radical reinterpretations of a dozen Dylan songs, from “Tangled Up in Blue” to “Highway 61.”

Sidran had been playing Dylan songs as part of his repertoire, but knew that he had to bring something special to a full album of Dylan covers. Hundreds of artists have covered the legendary folk singer in the past; even Bob Dylan seems to be doing radically different covers of his own songs in concert now.

“You have to have a really good excuse to do Bob Dylan,” Sidran said. “There are hundreds of versions of everything. I had a sense of sorting through this pile of his tunes, stuff that I loved, that I felt I could open my mouth and the words could come through me and I wouldn’t feel ridiculous.”

The idea came to record the album in Europe with an entirely European cast of session musicians. In Europe, Sidran said, Dylan is seen as cutting a more romantic figure — “he’s a musician, sure, but he’s a poet, he’s a tortured human being, he’s a philosopher”— and Sidran hoped to capture some of that essence.

The album was recorded in a farmhouse in France, Sidran and his sidemen occasionally using instruments they found around the house on the tracks. In Sidran’s hands, “Highway 61” becomes a moody jazz tune, and the epic “Tangled Up in Blue” becomes a slinky little pop number. But Sidran credits his son Leo, a New York musician who produced the album, with creating one of its most starkly defining moments in “Blowin’ in the Wind.”

“I had re-harmonized it, put some hip changes behind it, figured out a way to do it,” he said. “And we just went the other way. Slowed it down to a funereal pace and just did this simple cowboy chorus. And there it was.”

Sidran said he’s especially proud of his vocals on the album, which are completely unadorned and naked.

“I loved being completely exposed vocally,” he said. “Bob Dylan was a messenger of authenticity — ‘Be Yourself.’ Of course, it’s ironic that he wasn’t himself when he said that, but he did such a convincing job of it.”

Right now, Sidran is busy on his usual fall tour of Europe, touring in support of “Dylan Different” and a couple of other projects in France and Spain. He doubts that he’ll do any touring in the United States to support the album, as his main focus now is on writing a book.

When Sidran was an artist-in-residence at the UW-Madison in 2003, he taught a course about Jewish-American musicians entitled “Jews, Music and the American Dream: From Irving Berlin to the Beastie Boys.” He’s now in the middle of writing a book on the subject.

“I love it and it’s the hardest thing you can do,” he said. “I’m totally captured by the ideas. I love the freedom to spend my time with ideas. It feels as good as playing piano in a club.”

Of course, Dylan himself will play a major role in the book. And, if Sidran’s personal brushes with the man won’t make it into the book, perhaps the insight he gained from living inside his songs for a while will have an influence.

“I think that’s a very powerful way to interact with Bob Dylan,” he said. “People I know who know him say the same thing. Day to day it can be very mundane, but the power of the guy is in the songs and in the music, and in that spectral presence. You can get the best of Bob Dylan for $15 on a CD. You don’t necessarily want to know your heroes.”