It’s nice that the three members of Rush are still friends. Three and a half years after the prog band’s final show together, Geddy Lee, Alex Lifeson and Neil Peart haven’t gone their separate ways. “Alex and I just flew down to see Neil two weeks ago and hung for a couple of days,” Lee says, surrounded by the detritus of high tea in one of London’s grand but discreet hotels. “The first couple of months [after they disbanded], we were emotionally hungover. We didn’t know where the future was going to take us so we didn’t talk a ton then. And then we started to communicate again.”

Without Rush to sing and play bass for, Lee has kept himself busy compiling a coffee table book – Geddy Lee’s Big Beautiful Book of Bass – which sounds like one for a niche audience. Then again, that’s what people thought about Rush and they ended up filling arenas for 40 years and joined the Rock and Roll Hall of Fame, so who knows? The book goes to one side, though, as Lee surveys the career of the only prog band to have had a Hollywood bromance written around them.

The young Rush were rather out on their own, Canada not having many bands trying their hands at progressive hard rock. “We were playing a lot of high schools,” Lee says. “You’d pull up in Magnetawan, Ontario, set up your gear and start playing and the crowd would be looking at you to say: ‘What is this? I can’t dance to this!’ So we started by being unpopular on the high school circuit. And then, when we turned 18, we became unpopular on the bar circuit because we were too loud.”

They recorded their debut album, Rush, in late-night sessions – after playing five sets a night at a Toronto bar called the Gasworks – only to find the initial mixes were “wimpy and weak. When we heard it we were heartbroken.” Terry Brown was brought in to remix, and asked the band for more songs. “One of them was Finding My Way. We played him the song, and he loved it. ‘OK, let’s record that and one other song, and we’ll remix the others,’ he said. And that became the record. Finding My Way became a symbol to me of saving our first album.”

What transformed Rush from callow Led Zeppelin copyists into prog titans was the replacement of drummer John Rutsey with Neil Peart. “He was one of the goofiest looking guys I’d ever seen. He was very tall, lanky. He drove up in this little sports car, drums hanging out from every corner. He comes in, this big goofy guy with a small drumkit, and Alex and I thought he was a hick from the country. Then he sat down behind this kit and pummelled the drums – and us. As far as I was concerned he was hired from the minute he started playing.”

Peart suited Lee and Lifeson’s desire to make more complex music. Even better, he was happy to write the lyrics. The album 2112 was a last roll of the dice for Rush with their label, Mercury: “They really wanted us to be Bad Company 2 and we had loftier aspirations, so we stayed away from that and insisted on our own way of doing things. We were prepared to go down with the ship, and we almost did.”

Incredibly, 2112 was the making of the band, with fans flocking to its title track, a sidelong suite about (deep breath) how the priests of the Temples of Syrinx control life in the Solar Federation, and the struggle of the protagonist to express his individuality after discovering a guitar. It was inspired by Ayn Rand, which led to accusations that Rush were rightwing propagandists. “We were very, very surprised,” Lee says. “We were influenced by Ayn Rand, yes, but to me The Fountainhead was an artistic manifesto. The whole idea of that time in our lives was that we were trying to write original material and we didn’t want to compromise. The Fountainhead gave me comfort. When that whole thing came out in the press it seemed to take an odd political bent to it, which wasn’t really where we were at.”

‘Fans love it when we go into that crazy mode’ … Neil Peart, Alex Lifeson and Lee. Photograph: Fin Costello/Redferns

“That was a song where I would have to say our ideas exceeded our ability to play them,” Lee says of the nine-and-a-half-minute, 12-part instrumental suite that set new standards for bands hoping to go widdly woo on their instruments at great length. “We thought: ‘We’re going to write this long piece and then we’ll just record it live off the floor and boom!’ But it was really difficult. It was beyond us. I included it here because it surprised me how popular that song was among our fans. They just love it when we go into that crazy mode. Yes, it is an indulgence, but it seemed to be a pivotal moment for us in creating a fanbase that wanted us to be that way.”

Rush’s commitment to noodling made them the dream band for an audience that was overwhelmingly male. “There’s no getting around that,” Lee says. “We would joke about it backstage. ‘See any girls in the front row?’ ‘No. Some attractive boys. A lot of ugly boys.’ When things started changing – and they did – we noticed: ‘There’s girls in the front row’. Or there’d be a sign in the back: ‘Mythbusters: Girls who love Rush.’” Lee sighs, and laughs. “But we were too old to take advantage of it by that point.”

Lee didn’t want to include Tom Sawyer. “But how could I not? It changed our lives.” Tom Sawyer became a staple of classic rock radio, and regularly crops up in popular culture – Futurama, The Colbert Report, Freaks and Geeks, Family Guy, South Park. In the film I Love You, Man, it took a central role as the glue that binds Jason Segel and Paul Rudd (Rush themselves made an appearance, too). “When [the director] John Hamburg approached us about it, our instincts were to say no. But we were going through a phase where we decided to take the George Costanza approach to our career. We decided that anything we were going to say no to instinctively, we would now say yes to. It served us very well.”

Through all the years Peart was in Rush, Lee never wrote a lyric. He was always the vehicle for Peart’s words. “It has felt odd at times,” he says. “It has felt very comfortable at times, at times very uncomfortable. Being an interpreter for Neil has been a singular pleasure of mine and a really difficult job at the same time, because I’m not always on the same page as him. As we grew as a band, I became trusted by him to be his sounding board and his editor, and if I couldn’t get into a thing, he would leave it alone. That’s the beauty of a relationship that lasts.”

Through the 80s, Rush sidelined Alex Lifeson’s guitar and foregrounded synthesisers, but Roll the Bones saw a shift back. “Alex was driving at that point and he made it very clear we were drowning under a synthy noise and he wanted the guitar to return to its rightful position,” Lee says. “That’s fine. It unburdened me in a way. I accepted that maybe I had taken it a bridge too far.” Lee suggests he was always the least assertive member of the band, that at first Peart was the most strong-willed of the trio, and then “I think as our relationship evolved, Alex became more of the guy to convince, and Neil and I relaxed into our roles.” He smiles. “I just realised that.”

This is a rare Rush track that is, in Lee’s words, “rhythmically relaxed. It was really a way of calming down. Rush has a tendency to play very hyper, very fast. We were just not very good at playing in a relaxed state. Roll the Bones was our answer to that.”

Within 10 months between the summer of 1997 and summer 1998, Neil Peart lost his daughter (killed in a car accident) and his partner (to cancer). He retreated completely, and Lee assumed that was the end for Rush, and recorded a solo album. “After a couple of years I found solace in working and writing. I really did obsess over it and bury myself in it. Neil was so powerfully running away from all that pain that it was understandable to me if he didn’t want to return to the things that reminded him of the life that had been stolen from him. I didn’t think he would return, so this was a saviour for me.”

What Lee discovered, recording in Seattle with younger musicians including Matt Cameron of Soundgarden and Pearl Jam, was that he and Rush were loved. “I wasn’t used to that. All the local Seattle musicians of the time dropped by to say hi. They wanted to pay their respects, and I didn’t realise that I was iconic to them and the music they’d grown up with. Having grown up with no real music scene and no interaction with other musicians, to suddenly be in the middle of a thriving musical community was very good for me. I enjoyed the hell out of that experience.”

‘I wish I could do it all again’ … Rush backstage in Springfield, Massachusetts, during their All The World’s a Stage tour in 1976. Photograph: Fin Costello/Redferns

Rush did return, for three more albums, and they came full circle on their final record, another grand concept piece from Peart – Clockwork Angels – which also led to two novels and a series of comics. So, Geddy Lee, have you ever been able to make sense of any concept albums? “Not really.” Not even your own? “Barely. I grew up listening to Yes. I still can’t tell you what any of those records are about, honestly. I don’t think it matters, because the music and the lyrics create a sound, and that gives you a picture of a meaning. Sometimes that’s enough to make you love it.”

And have you ever thought, the one thing that would make this album better would be a novelisation? Lee laughs. “… or a graphic novel, and then the movie, and then the TV series, and then the cartoon. It was not my idea. I liked the concept of that record and I do believe it’s our best work. It was a tough record to make lyrically, getting it down to something that Alex and I could live with, that told enough of the story to satisfy Neil’s concept. There’s something about Headlong Flight that’s almost about the history of my band to me. It’s autobiographical in a way. Forty years into this career, and it goes by like that. The sentiment in that song is ‘I wish I could do it all again,’ and it’s true.”

‘I miss being on stage with those guys’ … Lifeson and Lee in 1983. Photograph: Rob Verhorst/Redferns

A live recording – from their final tour – of the song that was Rush’s breakthrough when it was picked up by WMMS in Cleveland in 1974. “It was my favourite song to play every night, and that’s why I wanted to include the live version.” It was also the final song Rush played at their last show, on 1 August 2015. Did Lee know at the time that was the end? “Not 100%. Neil was pretty adamant it was, and he played it like it was going to be the final show. And that’s why he actually left the drum throne and came out and gave us a hug on stage, which he swore he would never do. I guess I was a bit of an optimist. But nope. I think Alex accepted it more as the end. I thought we really killed it that night, but it was hard to tell because it got really emotional in the last 20 minutes. That’s the first time I ever got choked up at a microphone. So I guess a part of me knew.

“I miss playing with Rush. I don’t miss travelling with Rush. I miss being on stage with those guys because it was a singular honour to me. I’m sure I will play live again one day, but it will never replace that intensity of what a three-hour Rush show was like to perform: it challenged me to my max and that’s rare in this life.”

• Geddy Lee’s Big Beautiful Book of Bass is published by Harper Design

• Lee has curated a longer primer to Rush’s work, featuring the above tracks alongside other favourites from across their career; you can listen and subscribe to the playlist in Spotify below