cholidean-harmony-structure

Thanks for digging that up! That blog post covers the early "from scratch" version—essentially a mind experiment. Interestingly, Dmitri Tymoczko arrives at a similar prism structure (Figure 14b) in his paper "The Generalized Tonnetz" ( https://read.dukeupress.edu/journal-of-music-theory/article/... ). I reached a similar shape (Figure 11 in my pdf: https://jimishol.github.io/thoughts_on_harmony_en.pdf#page=2... ), but the specific, even arbitrary, twisting I used to realize the torus topology gives it a unique advantage: it immediately reveals the "hinge note" of a scale. I discuss that specific geometric comparison here: https://github.com/jimishol/cholidean-harmony-structure/disc... The new documentation in this repo ( https://github.com/jimishol/cholidean-harmony-structure ) represents the mature "Umbilic-Surface Grammar" that explains

Apologies for the delay! You don't need to run the code to see the theory in action. I have included a visualization GIF in the main README. While the demos in the installation section are short, they do contain audio-visual examples. Direct link to the README: https://github.com/jimishol/cholidean-harmony-structure/blob...

You would be better off posting the Github landing page with pictures and video links, since the number of music theorists on HN is probably in the single digits: https://github.com/jimishol/cholidean-harmony-structure/tree... Honestly, I don't think the observation of accidentals as a way of creating tension with an established harmony is especially novel, but I do like the 3d visualization despite its limitations.

OP here. Note on Terminology: This analysis uses specific geometric terms (like 'Station Shift' and 'P-Rotation') defined in the Grammar Specification. If the logic seems opaque, the definitions are here: 1. The Grammar Spec: https://github.com/jimishol/cholidean-harmony-structure/blob... 2. The Topological Basis: https://github.com/jimishol/cholidean-harmony-structure/blob...

OP here. This is a geometric decoding of Chopin's Prelude No. 4. I built a 3D music midi visualizer ( https://github.com/jimishol/cholidean-harmony-structure ) and realized that standard music theory couldn't explain the shapes I was seeing. So, I developed the Umbilic-Surface Grammar to map the topology of the harmony. This document demonstrates that the prelude's tension isn't random, but a rigorous conflict between 'Gravity' (Station Shifts) and 'Will' (Pivots). I am looking for feedback on the logic—specifically from anyone with a background in topology or music theory. Does this geometric proof hold up?

This is a real‑time, interactive model of 12‑tone equal temperament harmony, projected onto an umbilic torus. Harmonic ordering is encoded as a closed 3D curve tracing the circle of fourths; adjacency is defined along this spiral. Previous 3D projections of 12‑tone equal temperament — including Tonnetz‑on‑torus visualizations, chord‑space topologies, and other geometric or topos‑inspired models — often represent harmonic adjacency through triangular lattices or higher‑dimensional embeddings. While these preserve the underlying relationships, the added lattice can make the global structure harder to perceive as a single, continuous form. In this model, the umbilic torus is shown without such overlays, so the harmonic spiral is seen as one uninterrupted 3D path. The smooth curvature is used purely for visual continuity and aesthetic clarity