The Music Thing (TMT) was an after-school Dick Clark dance/rock production. The show was a one-hour syndicated TV program which was apparently short-lived. It is only listed in LA Times TV schedule from 14 September 1975 through to 7 November 1975, and apparently was only shown on the US West Coast, on the (at the time) independent and trendy LA TV Channel 5. It does not appear either in Midwest or East Coast TV stations programming, so it was not a network TV program which may explain it not being listed in any of the TV reference books.

In his book ABBA by ABBA, Christer Borg writes that "in Los Angeles [in 1975] ABBA participated in ... American Bandstand and Music Thing, two separate pop programmes led by Dick Clark".

There's a reference in ABBA - The Book by Jean-Marie Potiez supposedly about "The Music Thing" on p.121 (top left): "During the course of this trip, 20th Century Fox invited the group to take part in a musical film alongside Barry Manilow and Phoebe Snow"

Apparently, ABBA appeared on one of the last The Music Thing shows, in LA in early November of 1975.

Information from Intermezzo Magazine. Thanks also to David Schrubba and Yuri.