Norma Waterson remembers Big Bill Broonzy scanning the menu in the little restaurant in Hull where she and her brother, Mike, and sister, Lal, had taken the blues legend for dinner.

Bill was not impressed. "They had haddock and chips, and patty and chips, but they didn't have what Bill wanted – whisky and chips," she says in her rich Yorkshire tones. "He liked his whisky, did Bill."

Broonzy was one of the many big names on the folk scene that the Watersons put up overnight when they came calling at the folk club the family ran in the Blue Bell pub during the 1960s. Doc Watson and Paul Simon were other visitors, but soon the hosts were attracting attention themselves for their potently harmonised ballads and, once they got invited down to London to sing, they soon became star attractions themselves. As the story goes, they left Hull with a gig to play at the Troubadour in Earls Court and returned with a recording contract.

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"They were exciting times," says Norma of the 1960s. "You'd get buses full of jazz, blues and folk musicians touring the country and stopping at every town, or so it seemed. You don't get that now – the business side of things has taken over – but back then it was all about the music and we heard all the greats, playing in a pub on the corner."

Norma and her siblings Lal and Mike, both now dead, grew up singing. It was, she says, just what they did. Their father died just after their mother, not long after they'd started school. He played guitar and banjo and was influenced by music American servicemen brought to East Yorkshire in the Second World War. Then skiffle arrived and the threesome picked up on that and went off in search of songs from closer to home.

"Nobody told us what to sing and we weren't taught," says Norma. "We'd just hear something we liked, learn it and sing it with our cousin Sheila at first. I can remember, later, when we got on stage, we'd experiment and sometimes it went okay and sometimes it didn't. But to begin with, Lal and I would sing into each other's mouths. I don't know why, we just liked the sound it made and something happened when we were standing right up close to each other. It gave me goosebumps."

These days Norma sings with her daughter, Eliza Carthy, who as the offspring of two of England's leading folk singers – her father is the great singer and guitar stylist Martin Carthy – has carried the family business into another generation. Mother and daughter had sung together lots of times over the years, notably in the Waterson:Carthy group, with Martin, but it took until 2010 for them to release their first album together as a duo, Gift.

Before they could tour and promote the album properly, Norma fell seriously ill after going into hospital with a knee infection and ending up in intensive care. She spent 12 weeks with a tube down her throat, and couldn't speak, let alone sing. It was deeply worrying for the family but now she's back on the road. The travelling, she says, is a necessary evil that lets her enjoy being onstage with her daughter.

"There's something special about singing with family," she says. "The same feeling I used to get singing with Lal especially is there when I sing with Eliza too. You instinctively know where one another's going and that makes learning songs together so much easier."

Their songs vary from traditional ballads such as Pretty Grey Hawk and folk songs including Poor Wayfaring Stranger, to jazz and old time-flavoured material. They even do the Amen Corner hit If Paradise Was Half As Nice, a throwback to Norma's time as a radio presenter in Montserrat in the 1960s.

"I look at the set list and think, 'this is my life'," she says. "I can remember my dad playing swing music and all the sing-songs we used to have at Christmas, New Year and especially Halloween – my gran, who looked after us after our parents died, was really big on Halloween. All these songs just go into your memory, and what goes in through your ears comes out through your mouth."

Songs generally have to have a good story – a good tune's a bonus, says Norma – and she and Eliza share pretty much the same taste in songs. "If I like a song and suggest it, Eliza usually likes it too. We don't have many arguments about music. But I don't tell her what to do – she's never done what's she's told anyway," she says with a chuckle. "Ever."

Norma Waterson and Eliza Carthy appear at the Tolbooth, Stirling, tomorrow.