Greenfield died on 3 May after contracting Covid-19 during a prolonged stay in hospital for heart problems.
The Stranglers’ bass player Jean-Jacques “JJ” Burnel paid tribute to Greenfield, calling him a “musical genius”.
He said: “On the evening of Sunday May 3rd, my great friend and longstanding colleague of 45 years, the musical genius that was Dave Greenfield, passed away as one of the victims of the Great Pandemic of 2020.
“All of us in the worldwide Stranglers’ family grieve and send our sincerest condolences to Pam.”
Drummer Jet Black added: “We have just lost a dear friend and music genius, and so has the whole world.
“Dave was a complete natural in music. Together, we toured the globe endlessly and it was clear he was adored by millions. A huge talent, a great loss, he is dearly missed.”
The band's current vocalist and guitarist Baz Warne said: “We lost a true innovator, musical legend, and one of my dearest friends today.
“The word genius is bandied around far too easily in this day and age, but Dave Greenfield certainly was one.
“We stood together on the same side of the stage for 20 years, laughed, joked and shared our lives in the way that only band mates can. I’ll miss him forever.
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“Our thoughts and hearts are with his wife Pam, and to the millions of fans who worshipped at his altar, he’ll never be equalled.”
The band’s manager Sil Willcox said: “We are all in shock, Dave was a kind, generous soul who had time for anyone and everyone and it has been my privilege to have known him as both a close friend and his manager for over 40 years.
“Our thoughts are with his wife Pam at this sad time.”
Hugh Cornwell, the former guitarist and vocalist in The Stranglers, tweeted: "I am very sorry to hear of the passing of Dave Greenfield. He was the difference between The Stranglers and every other punk band. His musical skill and gentle nature gave an interesting twist to the band.
The Stranglers formed in 1974 in Guildford. The band had been due to perform a farewell tour this summer, but had to postpone it due to the coronavirus pandemic.
Greenfield, who was originally from Brighton, penned the band's biggest hit "Golden Brown". The song, which was about heroin, went to number two on the UK singles chart in 1982. It also won the group an Ivor Novello award.
Greenfield is survived by his wife Pam.
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