"to tone down," 1915, figurative use from the noun (1856) in reference to the left foot-lever of a piano, which makes it quieter among other effects; from soft (adj.) + pedal (n.).
His advisers soft-pedaled the president's blunder
soften
softener
soft-hearted
softly
softness
soft-pedal
soft-shelled
soft-soap
soft-spoken
software
softy