Normally, the best interstation distance between subway or bus stops does not depend on population density. To resurrect past models, higher overall density means that there are more people near a potential transit stop, but also that there are more people on the train going through it, so overall it doesn’t influence the decision of whether the stop should be included or deleted. Relative density matters, i.e. there should be more stops in areas that along a line have higher density, for example city centers with high commercial density, but absolute density does not. However, there is one exception to the rule that absolute density does not matter, coming from line spacing and transfer placement. This can potentially help explain why Paris has such tight stop spacing on the Métro and why New York has such tight stop spacing on the local subway lines.
Stop spacing and line spacing
The spacing between transit stops interacts with that between transit lines. The reason is that public transportation works as a combined network, which requires every intersection between two lines to have a transfer. This isn’t always achieved in practice, though Paris has just one missed connection on the Métro (not the RER), M5/M14 near Bastille; New York has dozens, possibly as many as all other cities combined, but the lines built before 1930 only have one or two, the 3/L in East New York and maybe the 1/4-5 around South Ferry.
The upshot is that the optimal stop spacing depends on the line spacing. If the line spacing is tight – say this is Midtown Manhattan and there is a subway line underneath Lex/Park, Broadway, 6th, 7th, and 8th – then crossing lines have to have tight stop spacing in order to connect to all of these parallel lines. In the other direction, there were important streetcars on so many important cross-streets that it was desirable to intersect most or ideally all of them with transfers. With so many streetcar lines extending well past Midtown, it is not too surprising that there had to be frequent subway stops.
So why would denser cities have tighter line spacing?
Line spacing and density
The intuitive relationship between line spacing and density is that denser cities need more capacity, which requires them to build more rail lines.
To see this a bit more formally, think of an idealized city on a grid. Let’s say blocks are 100*100 meters, and the planners can figure out the target density in advance when designing the subway network. If the city is very compact, then the subway could even be a grid, at least locally. But now if we expect a low-density city, say 16 houses per block, then the subway grid spacing should be wide, since there isn’t going to be much traffic justifying many lines. As the city densifies, more subway is justifiable: go up to missing middle, which is around 30-40 apartments per block; then to the Old North of Tel Aviv, which would be around 80; then to a mid-rise euroblock, which is maybe 30-40 per floor and 150-200 per block; then finally a high-rise with maybe 500-1,000 apartments.
Each time we go up the density scale, we justify more subway. This isn’t linear – an area that fills 500 apartments per block, which is maybe 100,000 people per km^2, does not get 20 times the investment of an area on the dense side of single-family with 16 houses per block and 5,000 people per km^2. Higher density justifies intensification of service, with bigger and more frequent trains, as well as more crowding. With more subway lines, there are more opportunities for lines to intersect, leading to more frequent stop spacing.
Even if the first subway lines are not planned with big systems in mind, which New York’s wasn’t, the idea of connections to streetcar lines was historically important. A stop every 10 blocks, or 800 meters, was not considered on the local lines in New York early on; however, stops could be every 5 blocks or every 7, depending on the spacing of the major crosstown streets.
Dense blobs and linear density
Line spacing is important to stop spacing not on parallel lines, but crossing lines. If a bunch of lines go north-south close to one another, this by itself says little about the optimal spacing on north-south lines, but enforces tight spacing on east-west lines.
This means that high density encourages tight stop spacing when it is continuous in a two-dimensional area and not just a line. If large tracts of the city are very dense, then this provides justification for building a grid of subway, since the crosstown direction is likely to fill as well; in New York, 125th Street is a good candidate for continuing Second Avenue Subway Phase 2 as a crosstown line for this reason.
In contrast, if dense development follows a linear corridor, then there isn’t much justification for intense crosstown service. If there’s just one radial line, then the issue of line spacing is moot. Even if there are two closely parallel radial lines in the same area, a relatively linear development pattern means there’s no need for crosstown subways, since the two lines are within walking distance of each other. The radial urban and suburban rail networks of Tokyo and Seoul do not have narrow interstations, nor do they have much crosstown suburb-to-suburb service: density is high but follows linear corridors along rapid transit. Dense development in a finger plan does not justify much crosstown service, because there are big low-density gaps, and suburb-to-suburb traffic is usually served efficiently by trips on radial lines with a transfer in city center.