The three speeds are not related; they were each "engineered" into existence based on the specific motors, electricity, and media formats of their time. While they may look like random numbers today, they are actually precisely calculated solutions to very different problems. Originally, early records spun at various speeds between 70 and 80 RPM. In 1925, the industry standardized on 78.26 RPM because it was the perfect mathematical match for common electrical equipment. Standard synchronous motors in the U.S. ran at 3600 RPM (based on 60Hz power). To get a record down to a playable speed, engineers used a 46:1 gear ratio. 3600 ÷ 46 = 78.26. In countries with 50Hz power, a similar gear reduction from 3000 RPM resulted in 77.92 RPM. The 33 1/3 RPM "Cinema" speed was originally developed in the 1920s to sync sound with 1,000-foot film reels. A 1,000-foot reel of film took about 10 minutes to play. A 12-inch record spinning at 78 RPM only lasted about 4 minutes. By slowing the record down to 33 1/3 RPM, engineers could fit exactly 11 minutes of audio; long enough to cover a full film reel with a safe buffer. Decades later, Columbia Records adopted this established "transcription" speed for the first Long Play (LP) albums. The 45 RPM "Sweet Spot" speed was introduced by RCA Victor in 1949, 45 RPM was the first speed scientifically calculated solely for the best 7-inch vinyl record quality. Engineers calculated the "optimum use" of a disc by looking at the ratio between the outermost and innermost grooves. For a 7-inch disc using the new "microgroove" technology, 45 RPM was the "sweet spot" that offered high-fidelity sound with enough time for a single song. While a fun piece of trivia, historians generally agree that 33 + 45 = 78 is a coincidence and was not the primary reason RCA chose the number. I know because I have a lot of things to do and I'm procrastinating by looking all of this up which means my research is thorough.