u/dooatito

Hi everyone,

I created this experimental website, which I called Z-Tonnetz, to test a new idea following my work on circleofthirds.com. This time I applied the same principle to the Neo-Riemann Tonnetz: I stripped the accidentals and made them change dynamically relative to context, leaving only 7 possible notes, each with their own letter name. This made new triad types appear that aren't on the regular Tonnetz: mainly diminished and augmented triads.

Working on this I realized that I was modeling a third dimension, several layers of the same Tonnetz but shifted so that each letter name were on the same column with a different accidental, like this: F𝄫, F♭, F, F♯, F𝄪...

This opened up a lot of possibilities. In this model, B and F aren't far apart anymore. They are neighbors, only on a different level. F is right below F#, and this made it possible to "close" the loop of tonal scales and have a visual model for these types of altered triads: they're represented by a tilted triangle.

This makes several musical concepts easy to grasp visually. The diminished fifth that is always present in the diatonic scale as the tritone can now be viewed as a geometric inevitability. Another example is the augmented 5th (C-G♯). This is a dissonant chord that happens to be enharmonic with the consonant minor sixth (C-A♭), and both intervals can be played using the same keys on a piano, which could make one wonder, how can the same interval be both consonant and dissonant? On the Z-Tonnetz, you can see that the C-G♯ augmented 5th crosses a layer to reach the note above the perfect fifth, and the C E G♯ triangle is tilted, while both C F A♭ (F minor) and C E♭ A♭ (A♭ major), which are perfect triads, are flat.

Functions of scale degrees can also be mapped out by the position of their triad relative to the "bend" of the tritone. The stable tonic, C E G (in C major) is the furthest away from it, and so is Am. In fact, all other triads in the diatonic scale touch the bend, which explains why in tonal music, degrees 1 and 6 are favored as stable centers.

Another geometric consequence of this model is the fully diminished 7th chord: it appears as two joined diminished triads, which create an oblique plane that cuts through two layers. If you toggle the "enharmonic" mode in the 3D view, you can see how it can reach several different layers, pointing to its versatile role in modulation. It also explains why we use a diminished 7th for this chord, rather than a sixth, which can be confusing to learners (why use B𝄫 when you can use A?...). This visualization provides a geometrical explanation to the rules of harmony and voice leading.

One application of this model is the enharmonic disambiguation algorithm I created: from a MIDI file that only encode 12 chromatic pitch classes, I worked out a way to score each possible spelling using their positions on the Tonnetz, and figure out which key a piece is in. As a proof of concept it actually works, though a lot more work is needed (for instance I don't differentiate yet between the actual key and a temporary tonicization). I added several demo files you can try out, or you can import your own. You can also test it out with the computer keys, or use a MIDI keyboard (you'll need to refresh the page after plugin it in).

I could say a lot more but this post is long enough. Not sure when I'll have more time to work on this, so I thought I'd share what I have now and hope that some people will find this niche project interesting. It has its limits, it doesn't explain everything (no theory or visualization ever could), but it's a novel way to look at some old concepts. I'll try to fix any bugs that remain in the near future, and hope to be able to work on this system some more when I can.

Link to the website: ztonnetz.com

Cheers,

Alex

acknowledgements:

Library (Meantonal) used for Pitch Class and Interval modeling by u/HexMusicTheory.

Theoretical foundation for this project by Andrew Milne can be found here: http://www.tonalcentre.org/