A statement from the band said the Salford-raised musician died on Friday morning.
Alongside his frontman brother Shaun, Paul founded the Happy Mondays in 1980 and was credited with giving the band their signature rolling groove, present on hits such as Step On and Kinky Afro.
The group said on their official Facebook page said: “The Ryder family and Happy Mondays band members are deeply saddened and shocked to say that Paul Ryder passed away this morning.
“A true pioneer and legend. He will be forever missed.
“We thank you for respecting the privacy of all concerned at this time. Long live his funk.”
Mark Day, Paul Ryder (centre) and Shaun Ryder of The Happy Mondays Ian West
The original line-up also included drummer Gary Whelan and guitarist Mark Day, and they were later joined by maraca-wielding dancer Bez.
The band later achieved a successful crossover into the musical mainstream with hits from albums such as Pills ‘N’ Thrills And Bellyaches and also earned a reputation as hellraisers.
Synonymous with the “Madchester” music scene, their blend of psychedelia and alternative rock continues to influence other acts.
Mark Day and Paul Ryder (right) Ian West
Ryder remained an active member as the Happy Mondays broke up and reformed multiple times across the last 40 years.
Outside of the band, he also acted in several films including The Ghosts Of Oxford Street and Losing It, and also made a cameo as a gangster in the film 24 Hour Party People, about the Madchester music scene.
He also formed the band Big Arm and released a 2008 album titled Radiator, and played DJ sets around the world.
In January 2012, it was announced that the Happy Mondays were reforming with all of the original members, including Paul.
Oasis and Ride guitarist Andy Bell was among those paying tribute, tweeting: “Really sad news about Paul Ryder, RIP.”
The Stone Roses frontman Ian Brown added: “Rest in peace Ryder. A great friend, a great musician, a great fella, big love to Amelia, Jacob, Sonny, Chico and the family and band. Love ya longtime Pabs.”