Earlier quoted context omitted.
Another way to look at it is that the current notation system isn't the best overall, it's just the most tolerable trade-off between a bunch of mutually-incompatible requirements. Replacing standard notation for all uses may be doomed to failure, but replacing standard notation for some particular use case (especially new use cases that weren't anticipated when standard notation settled into its current form) may be…
> Another way to look at it is that the current notation system isn't the best overall, it's just the most tolerable trade-off between a bunch of mutually-incompatible requirements. Isn't most tolerable trade-off between multually-incompatible requirements another way of saying "best overall"? Totally agreed there are useful local overrides of standard notation. Tablature is one example, and there are others. I would…
I could have said that better. What I meant was that standard notation isn't better than every other system according to every metric we could use to compare such things.
Gary Garrett has more lattice demos on Youtube. Here's one that's an animation of an example in Harmonic Experience by W. A. Mathieu (which uses lattices extensively to explain harmony and is the best reference I know of for explaining how to understand them): https://www.youtube.com/watch?v=I49bj-X7fH0
A 3-5 lattice is a grid where one axis is fifths (powers of 3 in just intonation) and another axis is major thirds (powers of 5). Garrett implies a third axis for septimal flatted seventh (i.e. barbershop 7th) intervals. Since the grid is leaning to the right, the diagonals that lean the left are minor thirds. Powers of 2 (octaves) are usually ignored. Triangles that are flat on the bottom are major triads. Triangles that are flat on top are minor triads.
There isn't an obvious way to encode a whole song onto a single lattice diagram in a way that could be printed on a page and still be readable. They seem to work pretty well as animations or as static illustrations to explain chord transitions, though.