The systematic recording of thoughts on musical harmony led to the placement of the twelve musical tones on a constructible three-dimensional surface of an umbilic torus. To this visualized link of mathematics and music I gave the personal name Cholidean harmony structure.

The course of creation of the above link is recorded in pdf file. The broader background to its creation is set out at the beginning of this discussion.

Here are excerpts of the introduction and epilogue, the audio content, and the main images of the file.

The article is about Császár polyhedron.

(Its dual is the Szilassi polyhedron).

Tutorial

I was never able to construct Szilassi by myself, but there it follows how i constructed Császár polyhedron.

Algebraic start

The Euler characteristic X of a torus is zero.

So our torus has V-E+F=0, where V=vertices, E=edges and F=faces.

We can assume all faces are triangles. So, E=3F/2 (cause with 3F we count each edge twice).