What is it that is so appealing about accents? In America we have our southerners who sound so sweet and polite and ever-so-adorable. No disrespect to other sections of the US. I'm kinda partial to the sound of the south. Then you have the Brits, the French, the Italians...others too. It tickles the ear and soothes and excites and invites. I have no idea if our American accents thrill anyone. I mean west coast is not really definable. Many of us are transplants from somewhere else so how we sound is a pot pourri of everywhere. East coast can be rather more definite and sometimes is. North? Well the accent in the movie FARGO was awesome. Do people really talk that way up there or was it exaggerated?
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You had problems with understanding British accents in the Southern states of the USA? So do we English sometimes, when confronted with broad Geordie (from the Newcastle, NE England, area) and some other British dialects unfamiliar personally!
Many of the top-name British groups then were either Londoners or Northern English - especially Liverpool. The accents are very different, and not all tried to sing with an attempted American accent. I've wondered sometimes if those who did, sounded as fake to Americans, as Dick van Dyke's pseudo-Cockney in the children's-story film Mary Poppins was to British ears. (He played a chimney-sweep, a character not even in the original novel.)
The Beatles kept their Liverpudlian accents, as did singers like Cilla Black.
True Cockney is very specific to an area of East London; traditionally within earshot of Bow [Church] Bells. Accents do vary across London but you'd probably have to be a member of a London-native family to notice the more subtle differences. I couldn't, having always lived over 100 miles from London. There was a fad some years ago to adopt a fake Essex / East-End accent called "Estuary English", a coarse "Mockney" which is as false and pretentious as the worst "Received Pronunciation" version of the "posh" Home Counties voice from leafy-suburbia West of the city. The art critic Brian Sewell personifies this almost to self-parody. (Essex is the county to Greater London's immediate North-East, on the North bank of the Thames Estuary.)
My own accent puzzles people sometimes. I'm a Southerner, having always lived on the English Channel coast, but inherited from my parents Midland / Northern English basics which have strengthened somewhat through contact with friends "Oop North" - to whom I sound all Southern. I remember one day Our Mam (that's a dialect example!) and I being very amused by eavesdropping on my Southern-born younger sister and her equally-Southern friend when they were aged only about 5 or 6. They were playing in a sand-pit, and arguing over how to pronounce "sand castle".
"Carstle", [long, soft 'ah' sound] the friend insisted.
"No - it's 'Castle'", [short, hard 'a' as in 'lack'] according to our sister, with the same parental linguistic traits as mine.
One day I listened to two friends from NW England debating how to pronounce the name of a town spelt COLNE, in the county of Lancashire. "Colne", insisted one: hard 'o', 'l' sounded, 'n' noticeable, silent 'e' - . "No, it's Cone" replied t'other [as in ice-cream 'cone' but slightly sibilantly; and the true 'o' sound defies text-editors.] Eventually I said, "If you two Lancashire Lasses can't decide how to say your own place-names, what hope is there for a Southerner like me from 300 miles away to get it right?".
Wonderful things, accents and dialects, and it will be very sad if they all disappear through an increasingly homogenised, uniform, "international" society.