The Ivor Novello Award-winning singer last week pulled out of the Shrewsbury Folk Festival the day before she was due to appear.
She has now revealed to her 50,000 Twitter followers that the family bereavement was her father, David, a physics lecturer at St Andrews University.
The two shared a passion for astronomy. KT's 2004 debut album, which brought her to worldwide attention, was titled Eye to the Telescope.
KT was adopted at 18 days old and raised in St Andrews by her father and her teacher mother, Rosemary, who has two sons, Joe and Daniel.
The singer posted on Twitter: "A week ago I lost my brilliant, irreverent Dad. Mountaineer, marathon runner & great physicist. So loved & so brave."
She then added: "Thanks so much for all your kind tweets & very grateful for the understanding at my sadly pulling out of Shrewsbury Folk Fest. Much love x"
KT added a picture of herself with her father, writing: "Dad goes back to cosmic dust...I'll be following him one day...xxx"