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West and Wales - TWW didn't have a proper logo at all (just the initials in a box), HTV had three, 1968-1970, 1970-1992 and 1993-2002.
TWW's original logo was a monogram similar to ITN's - a circle with a large T in the centre and a smaller W on each side of it. The lettering inside a box emerged around 1962.
London Weekend - Four logos, the first one was the only one drastically different to the others.
Five, if you count the original white lettering on a black background that was used for about a year.
More than that. Version 1 was a large lion rampant with the word 'Scottish' above it and 'Television' below. Version 2 (from 1960) was the quadrant boxes with letters S, T and V in three boxes and the lion in the fourth. Version 3 - used briefly in 1964 - was the lion at the top, the words 'Scottish Television' in the centre and 'Channel Ten' at the bottom. This was replaced with version 4, the tumbling lion in his box (accompanied by a burst of 'a Hundred Pipers'). But the Lord Lyon King of Arms objected to this indignity and version 5 was hastily produced with the lion merely zooming into view. Version 6 was the tartan animation that revealed the monogram letters (accompanied by a few notes of 'Hey Jonnie Cope'), Version 7 was the one in which the monogram zoomed Lambie Nairn animated thistle (internally we called it 'Bertie Bassett') with specially composed music; Version 8 was the updated thistle with the blue thistle, Version 9 was the blue box, version 10 the big letter S, version 11 the current monogram!
Between the various versions of the original 'oscilloscope trace' (for that is what it was intended to represent) and the silver one (the 'golf club trophy' we called it) there was a brief period around 1964 when they used the word Ulster, with an underscore that formed the letters 'tv' to the right hand side. That was abandoned quite quickly. Oh, you missed Tyne Tees, who were happy with variants of their 'TTT' monogram; then the 'TTTV' monogram in various forms, interrupted only by the truly ghastly Channel 3 North East, which we won't talk about. A couple more observations: ABC had two logos - they started with the same logo as the ABC Cinemas (The letters ABC squashed into the triangle) and it was only around 1960 that they adopted the three segment triangle. And HTV started with its hallucinogenic animation before they sensibly dropped that and adopted the TV aerial motif which was, arguably, the cleverest logo of them all.
West and Wales - TWW didn't have a proper logo at all (just the initials in a box), HTV had three, 1968-1970, 1970-1992 and 1993-2002.
London Weekend - Four logos, the first one was the only one drastically different to the others.
Scottish - Seven logos, five completely different designs. The first was a lion, then colour saw the STV monogram introduced. CGI resulted in the thistle, which was later updated. Then they had the blue box, then the big S, and then the current stv with the tv in a triangle.
UTV - four logos. The second was an evolution of the first. The other two are just lettering in boxes.