First some history on how the various recording/playback speeds came to be.....
Though commonly referred to as 78's, ironically, none were designed to be played back at exactly 78 rpm. Even the term "78" is misleading, as the actual rpm later agreed on with the advent of the AC motor, is actually 78.26 rpm ! So how did the standard become 78.26 rpm instead of say a nice round 78 you're probably asking ? And for that matter, where did the original 78 rpm standard come from in the first place ?
Emil Berliner (the inventor of the gramophone) determined by subjective listening tests, the optimum speed for recording vintage records. The resulting optimum speed is dependent on a balance between the groove width, stylus and design of the cutter & reproducer. Best results were obtained at speeds ranging from a low of 70 to a high of 90 rpm, and Berliner and his British Gramophone Company, determined the best sound was obtained at 78 rpm. Since the Gramophone established the record format to begin with, other manufacturers followed Berliner's lead & 78 rpm became the defacto standard.
Later however, the standard was changed to 78.26 rpm - and is the standard that remains in effect to this day. The reason for the change is actually perfectly logical. The slight change came about with the introduction of the AC motor. All AC powered turntables in the US run off standard 60 Hz power, and use synchronous motors, which lock their speed to the line frequency. This yields a constant rpm, even though line voltage may fluctuate, thus minimizing any power induced fluctuations in the playback speed. Thus a standard 60 cycle synchronous motor will run at 3600 rpm (60 rev/sec = 3600 rpm). The closest integer gear ratio to reduce the 3600 rpm down to about 78 rpm is a ratio of 46:1. Dividing 3600/46 however, yields 78.26 rpm. Voila ! - - - - the new standard was set.....
The 33 1/3 speed came about with the advent of early sound films. In the early days of film, the audio was recorded on a separate record. 12 inch 78 rpm records using Gramophone groove widths, could hold about 5 minutes per side, yet a reel of film in those days, could run for about 11 minutes. Simply dropping the record recording speed down to 32 rpm would enable the full length film audio to be recorded non-stop on one side. Ultimately, 33 1/3 (3600 divided by a gear reduction ratio of 108:1) was agreed on as the final modern standard. Plus European power at 50 Hz yields a synchronous motor speed of 3000 rpm. - so an even gear reduction of 90:1 also yields 33 1/3. So 33 1/3 rpm was a "magic number" !
The 16 rpm speed came about as a necessity to maximize continuous recording times. 16 rpm was exactly 1/2 that of the 32 rpm used for recording early films. The low speed combined with massive 16 inch diameter disks (transcriptions) could yield up to about 30 minutes/side. Audio fidelity is rather poor due to the slow speed especially towards the hub(300 - 3000 Hz), but is perfectly acceptable for voice quality.
Up to this point, standardized speeds were determined by subjective listening, and "fine tuned" for compatibility with synchronous AC motors and gear reduction ratios.
The 45 rpm speed was the only speed to be determined by a more precise scientific approach conducted by RCA Victor in 1948. RCA showed that the optimum use of a disc record of constant rotational speed, occurs when the innermost track diameter is half that of the outermost recorded diameter. Given the adopted CBS vinyl groove dimensions (microgroove) and certain assumptions about the bandwidth and tolerable distortion, RCA plugged the numbers into their algorithm & a speed of 45 rpm was shown to be the optimum for this 7" record size format.