Triangle Tonnetz vs. Hexagon Tonnetz

Question

I have just started to learn about Tonnetz and I've run across two very representations of the same information in different types of Tonnetz.

In the first one shown below the triangles represent a chord while the points represent a note:

In the second one shown below the hexagons represent a note while the points represent a chord:

It seems like they both state the same exact information, but in different ways. Is there a reason to use/prefer one over the other or is there a situation where one is more appropriate then the other?

Answer 1

These two diagrams form what mathematicians might call a dual graph, in that they are sort of mirror images of each other. I'll describe this briefly for the benefit of others.

In the first image, the nodes represent pitches (well, pitch classes), while in the second image, the nodes represent chords. To construct the second image from the first, place a point in the center of each triangle, label it with the name of the chord formed by the surrounding three pitches, and connect it to the centers of the three neighboring triangles. To construct the first image from the second, simply repeat this process: place a point in the center of each hexagon, label it with the name of the pitch that is common to the six surrounding chords, and connect it to the centers of the six neighboring hexagons. For example, the hexagon (c-C-a-F-f-A♭) represents the pitch C, and if you go one hexagon above it (g-G-e-C-c-E♭) this represents the pitch G. The line connecting these two hexagons is the same as the edge C-G in the first image.

Thus all of the intervalic relationships in the first diagram exist in the second diagram, but in a slightly less easy-to-see form (lines between centers of shapes rather than edges). Similarly, in the second diagram, the edges represent transitions between two chords that share two notes in common. But this same information can be seen in the first diagram by crossing an edge from one triangle to an adjacent one.

You might think that, since the second diagram focuses more on chords (by making them be the nodes), it would be better-suited for analyzing chord progressions. However, here we see the downside of the second diagram: not all chords are represented in the diagram, only major and minor triads. There are no diminished or augmented chords, much less suspensions, or sevenths, or other combinations. Furthermore, if you look at any pair of adjacent nodes as being a chord progression, they are always chords that differ by a single note.

On the other hand the first diagram is capable of displaying any possible combination of tones, regardless of whether it is a major or minor triad. However, chord progressions do become rather abstract, as one shape morphs into another.

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To summarize:

Triangles

• Nodes: All pitch classes
• Edges: Intervals, but only consonant ones (m3/M6, M3/m6, P4/P5)
• Shapes: Chords (any possible combination of nodes)

Hexagons

• Nodes: Chords, but only major/minor triads.
• Edges: Chord progressions, but only if they share two common tones.
• Shapes: Pitch classes (common tone of surrounding chords).

Given the clear limitations of the hexagon based diagram (which are easily overcome in the triangle diagram) I'm not sure why someone would choose it over the triangle diagram, given the choice (historically, I think it was developed first, though). Honestly, I'm having a hard time thinking of any cases where you can clearly and easily do something on the hex grid that you absolutely can't do on the triangle.

This agrees with the fact that it's more natural to think of a chord as a combination of multiple pitches, rather than to think of a pitch as the overlapping of multiple chords.

Answer 2

To expand on Caleb's and N1hk's excellent answers, I would just like to add my picture of the two tonnetze superimposed and rotated to show their relationship.

Interestingly, the hexagonal tonnetz as drawn in the question, has an enharmonic change of spelling which in my opinion further limits its usefulness since it obfuscates harmonic motion and chord progression analysis.

If a chord tonnetz is required I much prefer the 'chord lattice' of W.A. Mathieu

It shares some similarities with the hexagonal tonnetz while retaining the horizontal fifths/vertical thirds orientation of the triangular one, which in my opinion makes it easier to read.

I would also like to mention the 'five-limit lattice' which is more or less identical to the 'triangular' tonnetz but with a different orientation: drawn on a system of staves it has a slight slant to it but more importantly it is upside down.

My experience with lattices/tonnetze is mostly limited to simple harmonic analysis and singing practise over a drone. For these applications I've only ever used the five-limit lattice/triangular tonnetz.

(Image source: Harmonic Experience: Tonal Harmony from Its Natural Origins to Its Modern Expression, W.A. Mathieu)

Answer 3

I've never seen before the sort of Tonnetz you've provided with your second example! The first Tonnetz, however, is the kind I've encountered most often.

My immediate impression is that the first is much more navigable. The intervallic relationships are easily apparent: bottom-left to top-right are minor thirds, top-left to bottom-right are major thirds, and horizontally are the perfect intervals.

The second example obfuscates this intriguing logic with its edges: the intervals are either alternating between major and minor thirds or unisons. The consequence of these unisons is that there is a lot of redundancy, something the first example has much less of. Its enharmonicism also eliminates the potential for an infinite web of pitches with successively more accidentals, which the first example would do were it more expansive. With respect to the first example, you're right: there doesn't seem to be any reason to prefer the second one.

Slight digression: Hexagons are semantically relevant to study of transformational theory, which Tonnetze are a large part of. But the hexagons of the second example have little to do with the hexagonal "hexatonic systems" proposed by Richard Cohn in his landmark article "Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of Late-Romantic Triadic Progressions" (1996).