Ther is a magnificent and thorough work: Percival Price, Bells and Man. He was a carillonneur and campanologist. He traveled and studied extensively. He inaugurated the Peace Tower (the bell tower of the Parliament of Canada) and was the first Dominion Carillonneur for 12 years. (You can hear a recording of the inauguration tunes.) It is a deep history of anything bell-like. Not just what you think of as bells as in Western churches, but Asian bells, crotals, sematrons, and even handbells. Not just the objects but their significance and use. Simple ringing, change ringing, Russian chiming, carillions.

He covers the topic on pp. 180-2, a subsection on clock chimes. Patterns played on the quarter-hour are called "quarters".

The formal name is "Cambridge Quarters" or "Westminster Quarters". There are four different tunes at each quarter, using some combination of 5 different measures.

1st quarter: measure 1 2nd quarter: measures 2, 3 3rd quarter: measures 4, 5, 1 4th quarter: measures 2, 3, 4, 5

(For example, the first two measures of the hourly signal are the half-hour signal, but they are no part of the 15- or 45-minute signals.)

"The Westminster Quarters were composed by William Crotch, an English musician, at about the age of seventeen, for a clock installed in 1793 at Great St Mary's Church, Cambridge, England, the University Church of Cambridge University. It is thought that he based it on four notes in the fifth measure of the aria, 'I know that my Redeemer liveth' from Handel's Messiah. ... [27] W. W. Starmer, Quarter Chimes and Chime Tunes (London, n. d.), pp. 6-8; Alan Phillips, The Story of Big Ben (London, 1959), p. 13". It was used in a few places but without much attention until it was put on the clock in what's now the Elizabeth Tower at the UK's Parliament in London in 1860. It went worldwide then. "By the late nineteenth century there was a reaction against the Westminster Quarters. An unnamed Englishman noted that they were played on seven public clocks within hearing distance of his residence. ... [30] Starmer, p. 8." "The only person who [composed quarter chimes that got any traction] was Richard Wagner, and he only provided the motif", the "bell motif" from Parsifal. It is played in the original at Riverside Church in New York City.

Are you referring to Westminster Chimes? To be precise, the second repetition isn't quite the same as the first, and often the full tune is only played on the hour.