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demi-semitone

demi-ˈsemitone Music. rare.
  [demi- 9.]
  Half a semitone; a quarter-tone.

1866 Engel Nat. Mus. ii. 27 Councillor Tilesius informs us that the natives of Nukahiva..distinctly intone demi-semitones (quarter-tones) in their vocal performances.

Oxford English Dictionary

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