He was 95.
Reportedly the last surviving member of the jazz band, Tanner had lived at Sunrise Assisted Living in La Costa.
CNN reported that Dick Darnall, Tanner's stepson, confirmed the death in a phone interview. The cause was pneumonia, Darnall told the network.
"Tanner was born in Skunk Hollow, Kentucky, in 1917, but soon left to tour the country with his five brothers and their father, all of them musicians, during the early 1930s," CNN reported, adding.
"Everybody was trying to survive during the Great Depression," Darnall said. After one gig, Miller, who scouted his own talent, came up from the audience and "asked Paul to become part of his band."
The trombonist went on to play with Miller, who also played trombone, from 1938 until 1942. But when the band leader joined the military and started a military band at military wages, the civilian band broke up. For the next few years, Tanner stayed in Hollywood, where he did studio work, Darnall said.
His second wife, Jeanette, was quoted by CNN as saying Tanner wrote a number of textbooks during his 23 years teaching at UCLA. His jazz history classes were among the most popular on campus.
"He used to crowd the auditorium area where he had his classes, and it got so full that the fire department closed the doors and wouldn't let any more in," she told CNN.
"Though he stopped teaching more than 30 years ago and moved to Carlsbad, "I still get letters from people that were his students," CNN quoted her as saying. "They said that he changed their life."
According to Wikipedia, "Tanner gained fame by playing trombone with Glenn Miller's band from 1938 until 1942, later working as a studio musician in Hollywood. He was a professor at UCLA and also authored or co-authored several academic and popular histories related to jazz."
Tanner also is credited with developing and played the electrotheremin, later known as the theremin, "an early electronic musical instrument controlled without discernible physical contact from the player."
Several songs by The Beach Boys used the theremin, with Tanner playing it himself—including on the iconic Good Vibrations.
A website devoted to Tanner said he became interested in the theremin in 1958, "when he played on Ray Heindorf's re-recording of the score to Hitchcock's film Spellbound. It continued:
Samuel Hoffmann, then the leading theremin player, performed on the soundtrack. ... Tanner saw the difficulties Hoffmann had with the instrument, which had no set positions from which Hoffman could take his cue. He kept relying on Tanner to provide a note to tune from, and this lead Tanner to consider if there was another way to produce the sounds of the theremin without having to master the subtleties of positioning hands around its two antennas.