5 ms·

I haven't thought about it in depth, but my guess is that it is not, because a lattice is not isomorphic with a cylinder or torus. Figuring out the relationshi

I haven't thought about it in depth, but my guess is that it is not, because a lattice is not isomorphic with a cylinder or torus. Figuring out the relationship in detail would be a good project.

stephencanon 9y ago
The Tonnetz is a torus, unwrapped onto the plane. On the first Wikipedia image (https://en.wikipedia.org/wiki/Tonnetz https://en.wikipedia.org/wiki/Tonnetz) you can see the circles of major- and minor-thirds on the 60˚ lines, whole-tone and semi-tone scales along 30˚ and vertical lines, and the circle of fifths on the horizontals. Different presentations skew it in different ways, to emphasize different features, but it's very hard to make a diagram like this that isn't topologically equivalent to the Tonnetz. If you find this sort of stuff interesting, you might also like to check out the writings of Fred Lerdahl and Dmitri Tymoczko, who come at these scale and chord topologies from a more academic perspective.
- lucasgonze 9y ago
  That sounds promising since it does disprove my earlier line of argument. That said, there is more work to be done to show equivalence.
- lucasgonze 9y ago
  I'd look to see how the spiral array model can map these together.
- elihu 9y ago
  It's equivalent to a torus in equal temperament, because the pattern repeats in a predictable way when the notes alias each other. In a just intonation 3-5 lattice, it extends infinitely without repetition.
  - stephencanon 9y ago
    Indeed--you can still map it onto the torus but the lattice doesn't line up neatly; you get irrational spirals that cover the surface. But, since this is in the context of Trane and Lateef, I think ET is a reasonable assumption.
- Maybestring 9y ago
  >it's very hard to make a diagram like this that isn't topologically equivalent to the Tonnetz. And more generally, it's just hard to have a good idea that Euler hasn't had first.

I haven't thought about it in depth, but my guess is that it is not, because a lattice is not isomorphic with a cylinder or torus. Figuring out the relationship in detail would be a good project.

stephencanon 9y ago

The Tonnetz is a torus, unwrapped onto the plane. On the first Wikipedia image (https://en.wikipedia.org/wiki/Tonnetz https://en.wikipedia.org/wiki/Tonnetz) you can see the circles of major- and minor-thirds on the 60˚ lines, whole-tone and semi-tone scales along 30˚ and vertical lines, and the circle of fifths on the horizontals. Different presentations skew it in different ways, to emphasize different features, but it's very hard to make a diagram like this that isn't topologically equivalent to the Tonnetz. If you find this sort of stuff interesting, you might also like to check out the writings of Fred Lerdahl and Dmitri Tymoczko, who come at these scale and chord topologies from a more academic perspective.

lucasgonze 9y ago
That sounds promising since it does disprove my earlier line of argument. That said, there is more work to be done to show equivalence.
lucasgonze 9y ago
I'd look to see how the spiral array model can map these together.
elihu 9y ago
It's equivalent to a torus in equal temperament, because the pattern repeats in a predictable way when the notes alias each other. In a just intonation 3-5 lattice, it extends infinitely without repetition.
- stephencanon 9y ago
  Indeed--you can still map it onto the torus but the lattice doesn't line up neatly; you get irrational spirals that cover the surface. But, since this is in the context of Trane and Lateef, I think ET is a reasonable assumption.
Maybestring 9y ago
>it's very hard to make a diagram like this that isn't topologically equivalent to the Tonnetz. And more generally, it's just hard to have a good idea that Euler hasn't had first.