The answer - at least for 78.26 RPM - goes way back to around 1925, when it was standardized; primarily due to the advent of electrical recording technology by Bell Labs. Prior to that, individual record companies had their own speed standards, ranging from 78 to 82 RPM. Not a consumer problem, since playback machines had speed adjustments.
(RCA)Victor was the big dog then, they were the first customer for the new electrical equipment, and they said 78 RPM was it. Bell thusly declared it a standard. The reason was that a 3600 RPM synchronous motor w/ a 46 tooth worm gear drive resulted in a platter rotation of 78.26088 RPM.
At about the same time, a Bell subsidiary called Western Electric was active. Motion picture "talkies" using their Vitaphone System were developed. The earliest ones had the sound track on a 33.33 RPM disc that was synched with the projector. (Don't ask me how... or what speaker cables were used.) A single 35mm film reel ran for 11 minutes, the 10" 78 ran 3 minutes. ( I believe 12", 4.5 minute 78's came later.) To get the required 11 minutes play time, a speed of 1/2 of 78 was sought. The drive gear was increased to 54 tooth resulting in 33.33 RPM. Disc diameter was increased to 20". (11 minutes on a 20" disc probably had some *serious* levels and groove modulations. I can only imagine what was used for pickups in those times. Probably had a little mistracking here and there... These were still Direct-to-Disc recordings using wax masters.)
This system was soon replaced by the optical soundtrack placed directly on the film. The 33.33 RPM was maintained for recording radio broadcasts until tape recorders appeared in the early '50's (1947 by another account.). I think this is when the disc size went to 16" ("transcription" size).
The 12" 33.33 RPM came about in June 1948 by Columbia Records. With a longstanding rivalry with RCA, these companies had R&D departments in those days. This product was a breakthrough on two fronts. Shellac was replaced by Vinyl (read: unbreakable) and the Microgroove cutting process (read: Long Playing.)
As Jc2000 notes, RCA, clueless and footdragging, introduced in February 1949, the 45 RPM. (Note: 78-33= ...45) A year-long speed war resulted, with record stores facing major inventory problems with now 3 speeds available. RCA caved in early 1950 and bought into the LP model. Columbia, wisely, bought into the 45 RPM format for "singles." I'm guessing they also recognized what was happening with radio broadcasting by this time. The earliest inklings of a "popular" format; radios in automobiles becoming widespread (all-tubed, at that.) Shellac 78's hung on 'till the early-mid 1950's.
The first million-selling "Gold" record was
"Carry Me Back to Old Virginny", Alma Gluck; early 1900's.)
