The Tonnetz Diagram ( Music Theory )

[Jukka]

The Tonnetz diagram is a way to help us visualize various triadic ( three notes ) transformations in music theory. These sorts of transformation are often used in minimalist music, like from Philip Glass, John Adams, Alan Hovhaness, Hans Zimmer, and others. It also gets used often in a lot of film music. But it applies to all harmonic music as well.

Learning and using the Tonnetz diagram is one way to understand a portion of music theory. There are basic chord shapes ( images below ), and the six basic transformations, : P, L, R, N, F, S. These transformations all have a distinct chordal progression sound and are represented on the Tonnetz diagram as different directions of movement. ( Shown in the image above. )

Basic Chord Shapes

The light blue spots are root notes.

Here is a nice on-line tool that lets you experiment using the Tonnetz as a keyboard. It also allows you to record the results as standard MIDI files. Have fun !

Note : The online application is flipped from the diagram above. So the directions for the transforms are different. Sorry about that.

You will notice that harmonic music moves chordally in certain tight pattern across the Tonnetz.

The Tonnetz was first envisioned by the amazing mathematician Leonard Euler ( pronounced “oiler” ) in 1739 (!), it was largely forgotten for almost 120 years before being rediscovered. Based in the mathematical discipline called Graph Theory, the Tonnetz diagram can be described as a triangular grid, with the three “diagonals” being the Minor Third, the Major Third, and the Perfect Fifth/Fourth.

NOTE : Triangular grids are related to hexagonal grids, they are the “dual graphs” of each other.

In mathematical Graph Theory the Tonnetz can be considered a graph of the relationship of the twelve tones in the twelve tone system -OR- as the “dual graph” ( a concept from mathematical Graph Theory ) which is the graph of the twelve minor (blue) and twelve major (red) triads ( chords ) and the relationships between those chords. See the diagram at the top with the red and blue triangles.

In the online Tonnetz application above you can click on the vertex of a triangle ( the pointy spot ) and play a note, or you can point to the center of the triangle and play a triadic chord. These two alternate systems ( graphs ) are dual graphs of each other.

This form of the Tonnetz is for the equal temperament scale. There is also a Tonnetz that can be applied onto just intonation systems that are not repeating like the equal temperament Tonnetz, and stretches off to infinity ( at least theoretically ).

There are other diagrams ( triangular and otherwise ) possible based on other intervals, and potentially for other equal temperament divisions. While these may not have the fundamental musical interest that is represented in the Tonnetz Diagram, they could be used in alternate ( unusual sounnding ? ) musical systems.

You can play around with the numbers underneath the online Tonnetz application above. These numbers represent the tonal differrences between the notes along the three “dimensions”. You’ll likely find the 3,4,5 diagram the most interesting, it is the true Tonnetz graph diagram. Notice too that some of the other systems don’t span all twelve tones. ( Note : 3 = minor third, 4 = majir third, 5 = perfect fourth or inverted fifth. )

References :
Tonnetz Wikipedia Page

Daniel Lewis Music Theory Course Videos

Neo Triadic Transformations Intro
Parallel, Leading-Tone, Relative, and SLIDE Transformations
Using the Tonnetz

[Jukka]

There are two MIDI keyboards that i am aware of that can use the Tonnetz layout.
The Lumatone which you can define whatever key arrangement you’d like. ( thread )
The Akuto Studio AKT-0.1 which is currently being finish as a Kickstarter project. ( thread )

If you aware of others let us know.
Don’t post other arrangements or something-like-it posts here. Tonnetz only. There is a very nice Alternative Keyboard thread, that needs those posts.

[Jukka]

Just lookinng back at the basic chord shapes i am reminded of some music theory basics on the relationship of the minot 7th chord and the major 6th.

They’re exactly the same shape it’s just the roots are different.

Also you can see the minor triad in the major 7th, and the major triad in the minor 7th.

Also notice similarly the dominant 7th and the three diminished chords transition nicely. ( You’ll hear some of that in Thelonius Monk for instance. )

[psychetropic]

Thanks for posting this, it’s been a fascination of my own since I discovered it.

[handed]

Check out Navichord on iOS as well.

[psychetropic]

It’s also great for understanding modes: each tetrachord has a distinct shape, and you just line them up against each other the right way to make the permutations. Very helpful thinking in terms of modal modulation.

[Jukka]

Found this interesting video, from Will Daniels, that goes into detail of ways to use the Tonnetz to analyze the harmonic structure of music with more distant harmonic relationships.

First to set this up. His Tonnetz diagram is just like the one above, only turned 90 degrees to the right. Next you need to get an understanding of his idea of “luminosity” … brighter or darker sounds. He pretty quickly shows how those two work on the Tonnetz as a direction of flow.

He defines a Lum and a Flip direction of flow. The Lum direction along a channel is made of a combination of the L and the R transform above. The Flip direction across channels is made of a combinations of the P and the S transform.

Notice the facility he has going from the Tonnetz ways of describing things, to more conventional musical notation, and then to the hands on the keys. Working out the basic Tonnetz transforms ( as well as Lum and Flip ) on a conventional keyboard is a good skill to develop and practice to get them in your hands. Start with three note chords and work up from there.

He also goes through modal interchange, and shows an expanded circle of fifths with 24 stops going major to minor to major around the circle. ( You can also see that also in the combined circle of fifth diagram i put below that you may already know. )

This video takes some time to digest, it contains some denser music theory ideas, so be prepared to give it some time. From where i queued it there is 21:30 left.

How Kapustin Maximizes Harmonic Contrast: Brightness, Darkness, and Modal Interchange

Notice the compact shape that a major or minor scale makes on the Tonnetz. Interesting that he mentions Jacob Collier in reference to the “#15” interval.

Major and Minor Circle of Fifths