Born as Clara Reisenberg in (present-day) Vilnius, Lithuania, Rockmore was a child prodigy on the violin and entered the Imperial conservatory of Saint Petersburg at the age of five. She studied violin under the virtuoso Leopold Auer, and remains to this day the youngest student ever to be admitted to the institution. Unfortunately, bone problems due to childhood malnutrition forced her to abandon violin performance past her teen years. That however led her to discover the newborn electronic instrument and become the most renowned player of the theremin.
Rockmore had several gifts that enabled her to play the theremin so well. Her classical training gave her an advantage over the many theremin performers who lacked this background. She possessed absolute pitch, helpful in playing an instrument that generates tones of any pitch throughout its range, not just those those defined by equal temperament. She had extremely precise, rapid control of her movements, important in playing an instrument that depends on the performer's motion and proximity rather than touch. She also had the advantage of working directly with Léon Theremin from the early days of the instrument's commercial development in the United States.
Rockmore, as the mature musician she was, saw the limitations of the original instrument and helped to develop the instrument to fulfill her needs, making several suggestions to improve the theremin as a performing instrument. Such suggestions, like a faster volume antenna, wider musical range, and control over the instrument's tone color were incorporated by the inventor in later versions. She had a special theremin tailored by Léon Theremin himself to meet her unique requirements.
She developed a whole technique for playing the instrument, including a fingering system, which allowed her to perform accurately fast passages and large note leaps without the much known portamento on theremin.
Her older sister was the concert pianist Nadia Reisenberg. Although Theremin proposed to her, Rockmore married attorney Robert Rockmore, and thereafter used his name professionally.She died in New York City on May 10, 1998, aged 87.