In 1D, the densest packing of 0-sphere in a line is by apeirogon, placing their centre on the apeirogon's vertices.
In 2D, the densest packing of 1-sphere in a plane is by triangular tiling, which can be made by skew-stacking apeirogons.
In 3D, the densest packing of 2-sphere in the space is by tetrahedral-octahedral honeycomb, which can be made by skew-stacking triangular tilings.
Will this sequence continue analogously? What is the densest packing of 3-sphere in a 4D hyperplane? As @OscarLanzi says, the sequence doesn't continue to the densest packings. So I'm dropping the requirement of dense-ness.
To define "skew-stacking" more formally:
Let this sequence denoted by $a_n$, where $n$ is the dimension of the space. $a_{n+1}$ is formed by stacking $a_n$s so $n$-dimensional simplexes are formed, where $a_3$ is tetrahedral-octahedral honeycomb, not its gyrated version.
Note that $a_3$ doesn't consist only of tetrahedra.
I'm asking this to apply to musical tunings. Apeirogon represents 3-limit (Pythagorian) tuning, triangular tiling represents 5-limit (Just) tuning, tetrahedral-octahedral honeycomb represents 7-limit (septimal) tuning, and I-do-not-know-whether-exists $a_4$ would represent 11-limit (undecimal) tuning.
No, the later-by-layer model does not continue to work.
Your pattern (which I once believed existed until shown the light) would predict that the maximum number of $(n-1)$-spheres you can pack around a central sphere in $n$-dimensional space (the kissing number) is $n(n+1)$. But in all dimensionalities greater than or equal to four the kissing number takes off above $n(n+1)$ ($24$ rather than $20$ for $n=4$) and it grows exponentially from there.
Even though the kissing number matches the simple model in three dimensions, there are hints of the coming breakaway from the $n(n+1)$ shackles. If you pack the twelve spheres around the central one like they would be in a hexagonal or cubic close-packed crystal lattice, you discover that the polyhedron formed by the centers of the outer spheres has some square faces along with the triangular ones. This heralds the existence of gaps in the apparent close-packed structure even if you can't fit any more identically sized spheres in these gaps (in three dimensions). Alternatively, you can pack the outer spheres in an icosahedron as shown in this picture, but then the gaps appear as small alleys between adjacent outer spheres.
Either way, as you go to more dimensions the gaps grow and they get to the point where you can pack more spheres than would be suggested by the layer-by-layer process. Even today, we know the kissing number exactly only for certain numbers of dimensions and we have not yet nailed down the precise rate of the exponential growth.