> Isn't most tolerable trade-off between multually-incompatible requirements another way of saying "best overall"?
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.
> 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.
This is totally true; tablature is better for beginning guitar players to learn to play specific songs on the guitar.
The only reason tablature doesn't supplant standard notation is that the metric under which it's superior is much narrower -- it's only for guitars, and only better than standard notation for beginners.
I don't think standard notation is necessarily the best overall, by I do think it happens to be the best overall, the best we've got today. And I'm not convinced it will ever become a choice, as opposed to standard notation evolving like it has in the past to incorporate new ideas.
Thanks for the explanation of the lattic layouts; I hadn't noticed the triangle orientation part, I only got as far as seeing that horizontal lines formed the circle of fifths. I can't tell what the plus and minus symbols mean, do you know? Usually those are used for diminished and augmented chords and not single notes, so is Bb- another name for A that is useful under the lattice system?
It's a way to identify distinct pitches that are usually treated as the same in equal temperament.
For instance, in just intonation 2 (the major second of the scale) has a frequency that makes a ratio of 9/8 relative to the tonic, but sometimes you might want a slightly flatter major second with a ratio of 10/9. So, that note is label 2- to distinguish it from the regular major second.
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.