I've two related questions.
[1] Analogue record players rotate the disk at 16⅔, 33⅓, 45 or 78 RPM - apart from the first two values, there's no obvious relationship between all those figures. Why were those particular values chosen?
[2] Analogue tape speeds are 30, 15, 7½, 3¾, 1⅞ IPS (inches per second) and so on. Again, why those particular figures, and why the progressive halving? Why not a stepwise speed change instead, e.g. maybe 30, 25, 20, 15, 10, 5 IPS?
Emile Berliner created the first vinyl player with an electric motor, in 1888, which just happened to run at 3,600 rpm. That made the disc run close to 78rpm, allowing about 5 mins of play per record. It could have been anything - that's what it was. That was 1925, when it became an industry standard.
On a few years and new techniques developed, meaning smaller discs, due mainly to smaller grooves. But it took until after the War, 1948, years after it was launched, that Columbia took a market hold, as a concerto or such-like could be played on one side - no need to flip over. Hence 33rpm was the standard.
Faster revs meant better sound reproduction, and 45 was found to be good, with a reduction in size of the disc. Hence pop records of ~3-4 mins.
There's also the marketing wars similar to Beta/VHS which played their part.
According to a Wikipedia article that has no citation listed for this fact:
Tape recording first became common enough for the issue of compatibility between tape deck manufacturers to become an issue in the 1950s. At this time the most common speeds for professional recording were 30 ips and 15 ips, and some machines already supported both speeds. As the tape speed was determined by the speed of a synchronous motor driving a capstan, one way of achieving this was to switch the poles of the motor to a different configuration, halving or doubling the speed.
This system was extended to domestic tape decks, and so slower speeds as they were adopted tended to be exactly half the previous slowest speed
Despite there being no source for this, it is an extremely likely explanation, so I’m adding it as an answer while I look for support.
I’m starting to believe it’s just simple practicality. A synchronous motor with two poles in the circuit that is connected to North American 60 Hz power will rotate at 3600 RPM (60 revolutions per second). Adding two poles (which is electronically easy to do with a switch) cuts the motor speed in half. Building a more complicated circuit to divide speeds any other way is not worth the effort.
The only thing left unexplained is why 60 RPS was mechanically translated to 30 IPS. A likely explanation is that it is a good round number with a simple mathematical relationship to the motor speed.
At least for the second question, it makes way more sense to halve the speed because then you halve the resolution. This is the idea behind logarithmic scales: doing a normal (linear) step, say 5, would produce very little difference if your initial value was 100 (5%) but would produce a lot if your initial value was 10 (50%).
Doing it by halving (or any multiplier) gives you a something with respect to your initial value which makes way more sense
