Ring was originally a game reward or prize related to riding a. Most carousels did not have 'jumpers' - animals which move up and down as the carousel spins - The outer row animals were still/fixed. Many riders of carousels tended to ignore the outer rows in favor of the jumping animals. In order to encourage more riders to sit on the outer rows, someone devised the ring game and it became a common fixture of the carousel. The object of the game was simple.
A rider sits on the outer row and when the carousel spins around, the rider reaches out and grabs one of the metal rings from the dispenser. Since not all the rings were brass - typically one brass ring per ride among many non-brass rings - getting ring was a special treat. Ring usually entitled the holder to a free ride when they returned the brass ring to the attendant. Or could be kept as a good luck charm, etc. It's important to note that in their heyday, most carousels were pay-to-ride amusements.
Back when a couple pennies, was still significant change, a free ride on the carousel would seem much more relevant. Therefore, to get the elusive brass ring was to win a notable prize equated with fun, gratification, fulfillment, etc.
New York City-based, Herb Alpert-esque outfit from the 1960s, led by reed player.The group specialized in a short-lived musical style called 'the Now Sound,' which was largely instrumental. The Brass Ring had several minor hits in the 1960s, their best-known being 'The Love Theme from the Flight of the Phoenix,' while another of their songs, 'The Dis-Advantages of You,' was used in a series of TV commercials for Benson and Hedges cigarettes. The Brass Ring issued several albums during their brief career, including such titles as Love Theme from the Flight of the Phoenix, Lara's Theme, The Dis-Advantages of You, The Now Sound, Gazpacho, and Only Love, in addition to several best-of compilations.