Tonnetz Diagrams/Charts

Carson Farley

I have some simple questions about the Tonnetz charts and system:

1. Does the system [pre]determine the outcome of [tertian/tonal] harmonic results?

2. Is Tonnetz related to the harmonic series?

3. How do other types of triangle/stem triads work in Tonnetz charts (major, minor, diminished, augmented, suspension, M3b5) in asymmetrial combinations?  

4.  Are there other "templates" for Tonnetz that are irregular in symmetry?  

I'm curious for example how Tonnetz geometry might play out in a non symmetrical geodescic model (pentagon/hexagon combinations).  

Thank you.  

Nathan Baker

Hi Carson. In my master's thesis (written back in 2006), I discussed the use of Tonnetze whose axes feature interval classes other than 3, 4, and 5 (and 1, to name the lesser-examined diagonal) and analyzed some of Schoenberg's music on various Tonnetze. The literature review and the concluding chapter also point to other geometric configurations that might be fruitful for you to explore (although I'm sure there are more current explorations from the past 14 years as well). If you would like a copy, just shoot me an email: nbaker@caspercollege.edu

Cheers!

Thomas Noll

Hi Carson,

there is a quite systemmatic account in

Michael J. Catanzaro (2011) Generalized Tonnetze , Journal of Mathematics and Music, 5:2, 117-139,

DOI: 10.1080/17459737.2011.614448

all the best

Thomas

Jason Yust

Carson, 

Hi, good questions. 

(1) Do you mean, does it predict certain progressions over others? If that is the question, I think no. There might be some sense that using the Tonnetz implies that smaller moves should predominate over larger ones, but that is not a necessary assumption. If the question is, does it determine the effect of one progression over another, then the answer is yes, only insofar as it shows which progressions have more common tones and which less, and also aligns the directionality with possible axes which could have perceptual meaning (such as sharpness-flatness)

(2) Yes, in its original formulation by Euler, and also Riemann later. Tuning theorists still use it this way (usually as a square lattice rather than a tringluar one), but most modern usage has deemphasized this feature. See Rick Cohn's article in the Oxford Handbook of Neo-Riemannian Music Theories

(3) You lose me on "asymmetical combinations" here. Jay Hook has an article in the Journal of Mathematics and Music 7/2 that discusses representing seventh chords this way, and I believe also Cohn talks about this in his book. 

(4) Most generalized Tonnetze are symmetrical (not sure what it would mean to be "irregular in symmetry" though), although the method discussed by Bigo et al. in a couple articles, including one in the proceedings of the Mathematics and Computation in Music (MCM) conference in 2013, allow for assymetry by not including all transpositions of a given chord. In my recent paper in the Journal of Mathematics and Music (14/2, published online only right now) I state a transposability condition that shows why this would be the case.

About extensions of the Tonnetz to larger chords, there are a number of proposals. Dmitri Tymoczko (2012, JMT) and I extend the Tonnetz in different ways to larger simplexes. In other words, the change of cardinality corresponds to a change of dimensionality, and the chord remains a fully connected structure. For instance, a tetrachordal Tonnetz is a three-dimensional network of tetrahedrons (4-node simplexes). This is also true of Gollin's Tonnetz (1998, JMT), and a recent paper on a 5-dimensional Tonnetz of hexachords by Mohanty in the Journal of Mathematics and Music, and also the work of Bigo et al. I mentioned. You seem to be suggesting something else, that the Tonnetz remains two-dimensional (which implies limiting the within-chord connections). However, I think it still would be symmetrical in the same way as a regular Tonnetz, assuming you would include all transpositions of a given chord type. The best way to think of this is it is actually a trichordal Tonnetz (triangulation of a torus) and the larger chords are combinations of triangular regions. 

Nicolas Meeùs

In answer to Carson Farley's question

2. Is Tonnetz related to the harmonic series?

Jason Yust ansers:

(2) Yes, in its original formulation by Euler, and also Riemann later. Tuning theorists still use it this way (usually as a square lattice rather than a tringluar one), but most modern usage has deemphasized this feature. See Rick Cohn's article in the Oxford Handbook of Neo-Riemannian Music Theories

This is not exactly true. The fact is that most theorists in the 18th century and many in the 19th have been thinking in "just intonation". One way of justifying just intonation is by the harmonic series. However, just intonation was conceived in the 16th century before anyone was really conscious of harmonic partials and, so far as I remember, Euler does not mention them in relation with the Tonnetz – as a matter of fact, Euler does not even mention just intonation in that context, although he obviously is thinking in these terms.

Carson Farley

Thanks everyone for the responses and information.  I have ordered a Tonnetz chart that I can investigate more closely in a completed form.  It seems that the Tonnetz chart is related to "tonal" music specifically given the structure of triadic geometry.  It seems similar to the work of W. A. Mathieu's "Harmonic Experience" with similar lattice geometry and diagrams.  I can see how using the Tonnetz could be useful in exploring triadic harmony through common tone relationships or extended regions.  What bothers me intuitively about it however is the symmetrical design - nature does operate symmetrically and I think a more interesting lattice could be geodesic patterns (which are found in nature) and are asymmetrical.  I am not seeing the relationship to the harmonic series in Tonnetz yet.  Perhaps if someone can point that out.  I think a more asymmetrical geometry (geodesic) that incorporates more stem types (major, minor, diminished, augmented, sus, major/b5) could produce more contemporary/interesting results so that will be up to me to explore!  I would like to have posted a geodesic image of asymmetrical triangle patterns, but not sure how to do that on this site.   

Carson Farley

I might add that Mathieu's "Harmonic Experience" is specifically related to the harmonic series since harmonic overtones are specifically diagramed in the lattice charts he uses.    

Nicolas Meeùs

Carson Farley wrote:

I am not seeing the relationship to the harmonic series in Tonnetz yet.

There is no relationship to the harmonic series in the Tonnetz, Carson. Euler proposed it to demonstrate some properties of the musical system. He considered that the scale was formed of the consonances of octave (ratio 2:1), fifth (ratio 3:2) and major third (ratio 5:4). Like many theorists of his time, he considered that ratios using higher prime numbers were not used in music. (He later tried to build a system that used the ratios 7:6 and 8:7, but that was not really successful in his time).

Euler did not theorize any other construction of the musical system, he apparently was content to believe that consonances arose from simple ratios of integers. He mentioned neither the harmonic partials, nor the fact that the system he was describing was just intonation.

Horizontal lines in his Tonnetz (see the orignial here) show tones a 5th apart, vertical lines place them a major 3d apart:

F    C    G    D

A    E    B    F#

C#  G#  D#  A#

This is meant to illustrate, among others, that the 5ths D–A and F#–C#, which jump from one row to the next, are not true 5ths, they are a (syntonic) comma too narrow. For the same reason, the table is not cyclic: F is not a true 5th above A# (Bb), nor C# (Db) a true major 3d below F.

Harmonic partials are not involved in this unless you consider (which Euler did not) that just 5ths and just major 3ds are just because of a fusion of their partials.

In addition, it is usual today to consider the Tonnetz in equal temperament, or at least neglecting the enharmonic differences. The Tonnetz then becomes toroïdal – that is, it forms a circle both in the direction of the 5ths and in that of the major 3ds, as in the figure that can be seen here. There obviously is no question of harmonic partials in this case.

 

Jason Yust

Carson:

It is still unclear to me what you mean by "geodesic" and "asymmetrical": 

What bothers me intuitively about it however is the symmetrical design - nature does operate symmetrically and I think a more interesting lattice could be geodesic patterns (which are found in nature) and are asymmetrical.  

The word "geodesic" just refers to a generalization of a straight line to non-Cartesian spaces, which could be a torus or sphere. It is also used to refer to "geodesic domes" or "geodesic polyhedra," which are essentially triangulations of spheres. With that meaning of "geodesic" you are still talking about a symmetrical triangular lattice, with the difference being spherical vs. toroidal topology. But it is hard to arrange the pitch classes around a spherical surface in a meaningful way unless you (1) take a subset, or (2) use a higher-dimensional spherical surface (which is hard to visualize).