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The musical notes produced by recorders and other flue instruments consist primarily of spectral components with frequencies given approximately by nf1, where n is an integer and f1 is the fundamental frequency. However, the real tones of these instruments contain other spectral components that have been observed and discussed by a number of authors. We report a study of spectral components in the tones produced by recorders that are odd half-integer multiples of f1, i.e., spectral components with frequencies n±12f1. Our results, obtained through a combination of experimental and simulation studies of soprano recorders, suggest that these components are associated with the air flow in the vicinity of the window region and the labium edge. We also show that these half-harmonics can be suppressed by modifications of the instrument that alter the pattern of air flow in those regions. Speculations concerning the importance of the half-harmonics and the degree to which they are perceptible by a listener are briefly discussed.