This question is going to go a bit into the automotive industry, but I've always wondered how and with what methodology car manufacturers choose the specific gear ratios for their car transmissions. In order to try and gain some insight into the area, I decided to try my hand at figuring out how exactly a transmission's gear ratios might reasonably be decided by an experienced automotive engineer.
I started with first principles: what purpose exactly do discrete gears serve in a manual car? Primarily, they serve to extend the power band of the car's torque and horsepower curve so that it can accelerate optimally at a wider range of speeds than it otherwise could without a variable ratio transmission. To that end, then, the gear ratios should serve to keep the car in as optimal a power band as possible at all times, in order to optimize acceleration.
Going from there, in order to keep the car in it's optimal power band, which we assume is independent of actual car speed and only dependent on motor RPM, we should design gear ratios that allow one to shift between gears such that they always enter the beginning of the car's power band once shifting, after leaving it from the prior gear.
For example; lets say we have a simple car, which has an engine that revs up to a max of 7500 RPM. While I could draw a full power/torque curve here, let's just assume that the engine achieves maximum power between 5000-6500 RPM. For this car to accelerate as fast as possible at all speeds, it should try to stay between 5000-6500 RPM as often as possible. This means that the gearing should be such that, whenever we accelerate up to 6500 RPM, in order to keep the car at it's maximum power, the next gear up should immediately shift us back down to 5000 RPM at the start of the power curve, where we continue accelerating. This should be true for every gear in the car, meaning that no matter what gear you're in, if you shift up at 6500 RPM, you should go down to 5000 RPM.
If we assume this, then the ratio of every gear from the last should be fixed. Specifically here, the ratio of each gear should be $\frac{5000}{6500}\approx0.76923\%$ of the previous gear. If 1st gear has a ratio of 3:1, that makes our subsequent ratios ~2.3:1, 1.77:1, 1.365:1, 1.05:1, and 0.808:1 for gears 2-6. This exactly follows the exponential curve $3\cdot0.76923^{x-1}$, which describes the theoretical gear ratios of any number of gears as high as you might want to go.
My question is if this sort of gear organization allows for the best power transfer between gears and optimal car acceleration, why isn't this common in the automotive industry? For the most part, cars seem to have somewhat arbitrarily chosen gear ratios that maybe follow an exponential curve somewhat closely, but many do not, or seem to have large jumps between gears. Is there a factor I haven't considered?