Tissot bumper made by omega or vice versa

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There are some of the old Tissot and Omega bumper movements, that are often described as equivalent or the same. What is actually the case? Has Omega developed the movements and Tissot build them under a different calibre name? Or has Tissot invented and Omega used them in their watches too? As an example: Tissot 28 and Omega 340.
 
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According to Estelle Fallet (TISSOT: 150 Years of History 1853-2003, pp. 90-92), Tissot started development of their bumper in 1942, and launched the new caliber in 1944-45. The system was supplied by Omega but Tissot designed and built its own caliber and were issued patents on various improvements.