This work presents an interferometric study of the massive-binary fraction in the Orion Trapezium Cluster with the recently comissioned GRAVITY instrument. We observe a total of 16 stars of mainly OB spectral type. We find three previously unknown companions for $theta ^1$ Ori B, $theta ^2$ Ori B, and $theta ^2$ Ori C. We determine a separation for the previously suspected companion of NU Ori. We confirm four companions for $theta ^1$ Ori A, $theta ^1$ Ori C, $theta ^1$ Ori D, and $theta ^2$ Ori A, all with substantially improved astrometry and photometric mass estimates. We refine the orbit of the eccentric high-mass binary $theta ^1$ Ori C and we are able to derive a new orbit for $theta ^1$ Ori D. We find a system mass of 21.7 $M_{odot}$ and a period of $53$ days. Together with other previously detected companions seen in spectroscopy or direct imaging, eleven of the 16 high-mass stars are multiple systems. We obtain a total number of 22 companions with separations up to 600 AU. The companion fraction of the early B and O stars in our sample is about 2, significantly higher than in earlier studies of mostly OB associations. The separation distribution hints towards a bimodality. Such a bimodality has been previously found in A stars, but rarely in OB binaries, which up to this point have been assumed to be mostly compact with a tail of wider companions. We also do not find a substantial population of equal-mass binaries. The observed distribution of mass ratios declines steeply with mass, and like the direct star counts, indicates that our companions follow a standard power law initial mass function. Again, this is in contrast to earlier findings of flat mass ratio distributions in OB associations. We exclude collision as a dominant formation mechanism but find no clear preference for core accretion or competitive accretion.
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