Originally the main criterion was that the writer was a Lowland Scots language poet, like MacDiarmid.
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The name is a tautology, since "burn" is a Lowland Scots/Northern English word referring to a small river or large brook.
Bray Hill (Lowland Scots: Brae a slope) formerly a country lane known as the Great Hill during the time of the ownership of the Duke of Atholl.