The Geometry of "Gravity": Why 3D Logic Beats Rote Memorization (A Retrospective)

[jimishol]

I am not a musician. I am a logician who looked at the structure of Western Harmony and saw a missing dimension.

Numerous times, I have tried to explain this structure to academic theorists. They see music as a continuous ocean of pitch where anything is possible. I see music as a system of Topological Gravity where certain paths are inevitable.

As I conclude my active development of the Cholidean Harmony Structure, I want to leave this community—specifically the Coders, Game Designers, and Music Students struggling with theory—with one final visualization that proves why this system works.

The "B D F" Proof: Gravity vs. The Void

Take the notes B, D, F.
In standard music theory, this is a "Diminished Triad." You are told to memorize that it resolves to C. You are given rules about leading tones and tritones. It is a list of instructions.

In the Umbilic-Surface Topology, this is not a rule. It is a geometric lock.

1. The Setup:
The key of C is defined by the tension between its boundaries: G' (Dominant) and Bb' (Supertonic).

2. The Shape (Visualized below):
I have attached two images to illustrate this.

• The Prism (Image 1): Existing theoretical models (like the Generalized Tonnetz) identify this structure as a Prism, establishing a toroidal topology by abstractly "gluing" the twisted opposite faces.
• The Umbilic Torus (Image 2): My work explicitly embeds this lattice onto the continuous surface of an Umbilic Torus. While the specific curvature might be arbitrary, the concrete 3D visualization reveals a distinct mechanical advantage.

In both views, but specifically on the Torus, look at the Supertonic (D).

• The G' Surface holds B (via diagonal DB).
• The Bb' Surface holds F (via diagonal DF).
• D is the Hinge.

The Prism (Image 1)	The Torus (Image 2)

Note: These images depict the full Dominant 7th (G-B-D-F), but the geometric locking mechanism via the Supertonic hinge (D) applies identically to the Diminished Triad (B-D-F).

3. The Gravity:
The images make the D-class note immediately visible as the organizing vertex. It physically connects the Dominant and Supertonic planes. When you activate these boundary surfaces, they pivot around D, creating a "vacuum" in the center.

The only natural extension of surfaces that can fill this vacuum without breaking the "walls" (F#' and Bb') are the surfaces that meet on C and E.

The Result:
The structure B D F does not "want" to go to C. It is geometrically falling into C because the Supertonic Hinge (D) aligns the planes to force that resolution. To make it go anywhere else (like Ab or F#) requires a "Kick"—an accidental that breaks the geometric lock.

The "Western Filter"

My grammar is a way to describe relative movements locally, always referencing the current cursor. With the exception that it uses m and M tokens for minor and major triads, the movement logic is neutral.

However, it is not a universal theory of all sound. It requires a Western Classical Filter applied to a neutral 3D lattice to be meaningful.

• The Lattice (The Umbilic Torus) is the hardware. It is neutral.
• The Grammar (The Rules of C|Am) is the software. It describes the specific aesthetic of Bach, Mozart, and Chopin.

Other cultures or styles (Jazz, Impressionism) might write different software for this same hardware. They might "kick" the rock differently. But for Western Tonal Music, this grammar is not just an option—it is a very elegant mnemonic, acting as its underlying source code.

To the Builders and Students

I have taken this project as far as I can. I have built the map.

• To the Students: Stop memorizing lists of rules. Look at the shape. If you can see the surface movements, you can predict the harmony progressions.
• To the Coders: You don't need to hard-code "music theory." You just need to code the geometry—or better, the topology—of the Torus. The harmony will emerge automatically.

The visuals are the true mnemonic. The grammar was just my way of translating what I saw into code. I leave this structure here for you to build upon, to visualize, and perhaps to find new grammars for new musics on the same beautiful surface.