In Western music, there are $12$ notes up to transposition. In other words, music notes can be represented as integers modulo $12$ : $\big\{[0],[1],\dots,[11]\big\}$.
In Tonnetz grid, moving one step to the right gives the note "perfect fifth" above. That is, given a note $[a]$, the note on the right is $[a+7]$. Obviously, moving $12$ steps to the right gives the same note.
Moreover, given a note $[a]$, the note on the top-right is a "major third" above, that is $[a+4]$.
In the bottom right, the note "minor third" above lies, that is $[a+3]$.
Clearly, moving top-right $3$ times, or bottom-right $4$ times will give the same note.
Now, I am wondering, how this graph looks like if we glue the same notes together. It is stated that it is isomorphic to a torus, but I fail to see that result because there seems to be three independent cycles.
And if it is isomorphic to a torus, how should I place these $12$ points on the torus? I know, it does not really matter (because topologically they are the same) but I want to know if there is a "reasonable" way to place these notes on the torus?
For example, all of them on the top of the torus as a cycle. Or maybe $6$ of them on the top, $6$ of them on the bottom of the torus perhaps?
There are actually only two independent directions. If you can go $\nearrow$ and $\searrow$, then you create a step in the rightward direction. So there are two independent circles; the circle of three steps in the $\nearrow$ direction, and the circle of four steps in the $\searrow$ direction. This forms a torus: