Would be nice to have a link to the music itself, both recording and sheet music.

This is the E minor prelude - I happened to recognize it by key but not by number.

See e.g. the Wikipedia article (https://en.wikipedia.org/wiki/Prelude,_Op._28,_No._4_(Chopin...) which has a recording embedded, although there are surely better ones.

Sheet music from IMSLP: https://s9.imslp.org/files/imglnks/usimg/3/3c/IMSLP319636-PM...

Thank you! That is a crucial omission.

I avoided including the MIDI/Score files in the repo to avoid licensing issues, but I have updated the analysis document immediately with the Wikipedia and IMSLP links.

The ideal experience (as shown in the README gif) is actually running the visualizer alongside a score editor like MuseScore via MIDI port sniffing, so you can see the geometric cursor sync with the sheet music cursor. But for reading the text, the recording is essential context. Thanks for the links.

Chopin's music is in the public domain and can be shared without licensing. Only particular recordings of the music are copyrighted/licensed by artists/labels.

There's two issues with that. Sometimes people publishing sheet music alter it and then they can claim copyright. Second is that that old original sheet music can be illegible to modern musicians – it has evolved over time.

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.

Yes. I have to admit I was expecting something a little more visual and leaning into the idea that sound is a 3d object changing shape through time. Thanks for the better link in that respect.

### HN reply

> Thanks — the main repo page includes demos, screenshots, and *installation instructions in `README.md`*.

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...

> The tension arises because the "Seventh" (A) is not a new addition, but a residue of the previous geometric state.

That residue in m.2 is common-tone voice leading. It's a technique that was used throughout the common-practice period to avoid tension, not introduce it. I'd bet the progression in m. 1-2 could be found in the figured bass at the beginning of a slow movement by Telemann or another Baroque composer.

Speaking of Baroque composers-- in the Coda of the 4th Ballade, Chopin has an exquisite passage of basso continuo plus accompaniment that would be right at home in a minor key aria by Handel. Except that:

1. There's no melody being accompanied.

2. It moves about 4x faster than it would have in the Baroque era.

I'd love to see a pianist play that passage by suddenly looking up and frantically nodding cues to an invisible, demonic singer.

You are absolutely right about the voice leading. Acoustically, it is the definition of smoothness.

When I wrote 'tension,' I was describing the Geometric Re-contextualization visible in the visualizer. The note (A) stays put, but the 'world' (the Surface) rotates underneath it.

To be clear: The Grammar is simply my attempt to describe the movements I observed in the 3D structure. I am not a theorist; I am a builder. I see the Structure itself (the lattice) as the real contribution.

My hope is that theorists will see this structure and develop their own grammars for it. I would love to see a 'Baroque Grammar' or a 'Jazz Grammar' that maps these geometric paths according to their specific stylistic rules. The geometry is neutral; the grammar is just the lens I used to read it.

I would love to see this but don’t want to run the code. Could you link a video? I understand if you have to omit or mangle sound to avoid strikes.

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...

Found his web page with some basic demos/vids. Had the same curiosity.

https://jimishol.github.io/post/tonality/

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 why those shapes happen.

This is very interesting! I can't wait to run it on some of my own compositions...

Please do! I would love to hear how the lattice interprets original compositions.

For the only piece I have ever composed ( https://jimishol.github.io/post/young_and_confused/ )—a harmony exercise written before I built this tool—I find the visualization acts primarily as a mnemonic device.

However, for pieces like Satie's Gnossienne No. 1, the geometry makes profound sense, while others like Flight of the Bumblebee are just beautiful to watch purely for the motion.

I actually just pushed a new update (v2.5.2-2) that attempts a topological estimation of keys based on the active surface. Give it a try and let me know what you find.