Generalized crochet Klein bottle instructions

Basic idea of how to crochet a Klein bottle. Klein bottles can have many different proportions, of course; you can mess around with the curvature.

Close-up of the Tonnetz

Let's take a closer look at the Tonnetz and its properties. First off, the triangles defined by three adjacent points represent major and minor chords. As you can see, isometries (geometric transformations) in the Tonnetz correspond to modulations (changes to a nearby key) in music.

Again, the Tonnetz can be tesselated/tiled indefinitely to fill a plane. Its sides connect with the sides of other Tonnetze. So picture the keys going on forever, as they do in music theory.

Baby's First Tonnetz 2.0

You know how I said you could model the Tonnetz (all major keys) on the surface of torus?

Each rectangual region is a Tonnetz, representing all twelve major keys separated by perfect fifths. There are twelve tesselations in the diagram. See how the top connects with the bottom, and the two sides connect? Voila, torus!

Additionally, I mapped the Circle of Fifths to the color wheel: keys that are closely related (close together on the Circle of Fifths) are closer on the color wheel.

I'm making a crochet model of this now. It will be soft and fuzzy and colorful, hence Baby's First Tonnetz.

By the time I finish this project, I will permanently associate the keys with twelve different colors. C will always be red. Give it to a baby and the kid will probably grow up that way too!