There have obviously been multiple transporter sounds seen in the various Trek series. I've been able to find out details about two of them, TOS and the Reboot films from a series of interviews with Ben Burtt, Senior Sound Designer for the new Trek films;
In the Bluray Extras for the 2009 Star Trek film, Ben Burtt – who
devised the film's sound effects – used the upper frequencies of a set
of studio chimes. "I was searching for a method by which they might
have created the materialization tones in the original transporter. I
wanted something like that," he related. "It was a magical sound but I
don't know how they did it. I experimented with a lot of different
things, and I found that if I started out with the very highest notes
[of the chimes] [...] and I just did a [steady finger] roll [...] you
got a really good approximation of something that sounded like
dematerialization or materialization." ("Ben Burtt and the Sounds of
Star Trek", Star Trek (Three disc Blu-ray) special features) Burtt
also used a recording of props from the film Frankenstein (1931) he
had acquired to go with the initial spark of electricity during a beam
He later confirms that he's identified the source of the transporter sounds for the Original Series;
Interviewer: What about the transporter?
Ben Burtt: There are several different elements to it in the original version, including, once again, a rising oscillator tone as
well as a “singing” ethereal tone. The transporter in the movie
looks and functions a little bit differently than the one in the
series, but I wanted to recreate the feeling of the original’s
shimmering, ringing tone. So I came up with something that was
reasonably close, using bar chimes and a lot of reverb.
And then promptly contradicts himself in another interview;
Case in point: the ever-present transporter, used to beam characters
up and down from the U.S.S. Enterprise. Burtt explained that the
current transporter sound has three elements. “There’s some chimes,
which are heavily echoed, and they’re in the same pitch and register
as what you might have heard in the original show,” he said.
And while researching the sounds from the classic series, Burtt
discovered that they were created with a Hammond chord organ.
“Going back and getting some organ recordings and playing with it, I
was able to fashion some things very similar to the transporter,
perhaps exactly the same way, so that’s in there,” he explained.