# Tonnetz

(German: "tone-network", plural: Tonnetze)

[Joe Monzo]

A tonal lattice invented by Hugo Riemann as a model for Just Intonation. The Tonnetz has its roots in the theories of Leonhard Euler, and is the direct precursor of the lattice-diagrams used by modern tuning-theorists.

It is what tuning-theorists today call an "octave-invariant triangular lattice", with 3 axes forming the edges of a trangle which represents the basic concordanttriad of 5-limit just intonation, which has the ratios 4:5:6, and the tonal identities 1-3-5. Riemann did not actually draw horizontal lines, so his Tonnetz cells look like diamonds rather than triangles.

(Riemann actually used single / double / triple underlines where we use commas, and single / double / triple overlines where we use prime-marks.)

```   /    \    /    \    /    \    /    \    /    \    /    \    /    \    /    \    /    \    /    \    /    \    /    \
/      \  /      \  /      \  /      \  /      \  /      \  /      \  /      \  /      \  /      \  /      \  /      \
\/        \/        \/ (3. Oberterzen)  \/ dis,,, \/ ais,,, \/ eis,,, \/ his,,, \/fisis,,,\/cisis,,,\/        \/        \/
/\        /\        /\        /\        /\        /\        /\        /\        /\        /\        /\        /\        /\
\      /  \      /  \      /  \      /  \      /  \      /  \      /  \      /  \      /  \      /  \      /  \      /
\    /    \    /    \    /    \    /    \    /    \    /    \    /    \    /    \    /    \    /    \    /    \    /
\  /      \  /      \  /      \  /      \  /      \  /      \  /      \  /      \  /      \  /      \  /      \  /
\/        \/ (2. Oberterzen)  \/  h,,   \/  fis,, \/  cis,, \/  gis,, \/  dis,, \/  ais,, \/  eis,, \/        \/
/\        /\        /\        /\        /\        /\        /\        /\        /\        /\        /\        /\
/  \      /  \      /  \      /  \      //\\      //\\      //\\      //\\      /  \      /  \      /  \      /  \
/    \    /    \    /    \    /    \    //  \\    //  \\    //  \\    //  \\    /    \    /    \    /    \    /    \
/      \  /      \  /      \  /      \  //    \\  //    \\  //    \\  //    \\  /      \  /      \  /      \  /      \
\/        \/ (1. Oberterzen)  \/   g,   \//  d,  \\//  a,  \\//  e,  \\//  h,  \\/  fis,  \/  cis,  \/  gis,  \/        \/
/\        /\        /\        /\        /\\       /\        /\        /\        \\        /\        /\        /\        /\
\      /  \      /  \      /  \      /  \\     /  \      /  \      /  \      / \\      /  \      /  \      /  \      /
\    /    \    /    \    /    \    /    \\   /    \    /    \    /    \    /   \\    /    \    /    \    /    \    /
\  /      \  /      \  /      \  /      \\ /      \  /      \  /      \  /     \\  /      \  /      \  /      \  /
(Schlichte Quintenreihe) \/   es   \/   b    \\   f    \/   c    \/   g    \/   d   \\/   a    \/   e    \/        \/
/\        /\        /\        /\        /\\      //\\      //\\      //\\      //\        /\        /\        /\
/  \      /  \      /  \      /  \      /  \\    //  \\    //  \\    //  \\    //  \      /  \      /  \      /  \
/    \    /    \    /    \    /    \    /    \\  //    \\  //    \\  //    \\  //    \    /    \    /    \    /    \
/      \  /      \  /      \  /      \  /      \\//      \\//      \\//      \\//      \  /      \  /      \  /      \
\/        \/        \/  ces'  \/  ges'  \/  des'  \/  as'   \/  es'   \/   b'   \/   f'   \/   c'   \/ (1. Unterterzen) \/
/\        /\        /\        /\        /\        /\        /\        /\        /\        /\        /\        /\        /\
\      /  \      /  \      /  \      /  \      /  \      /  \      /  \      /  \      /  \      /  \      /  \      /
\    /    \    /    \    /    \    /    \    /    \    /    \    /    \    /    \    /    \    /    \    /    \    /
\  /      \  /      \  /      \  /      \  /      \  /      \  /      \  /      \  /      \  /      \  /      \  /
\/        \/ asas'' \/ eses'' \/ heses''\/ fes''  \/ ces''  \/ ges''  \/ des''  \/  as''  \/ (2. Unterterzen) \/
/\        /\        /\        /\        /\        /\        /\        /\        /\        /\        /\        /\
/  \      /  \      /  \      /  \      /  \      /  \      /  \      /  \      /  \      /  \      /  \      /  \
/    \    /    \    /    \    /    \    /    \    /    \    /    \    /    \    /    \    /    \    /    \    /    \
/      \  /      \  /      \  /      \  /      \  /      \  /      \  /      \  /      \  /      \  /      \  /      \
\/        \/feses'''\/ceses'''\/geses'''\/deses'''\/ asas'''\/ eses'''\/heses'''\/ fes''' \/ (3. Unterterzen) \/        \/
/\        /\        /\        /\        /\        /\        /\        /\        /\        /\        /\        /\        /\
\      /  \      /  \      /  \      /  \      /  \      /  \      /  \      /  \      /  \      /  \      /  \      /

```
• The horizontal axis represents the pythagorean perfect-5th of ratio 3:2,
• the southwest-to-northeast diagonal axis represents the just major-3rd of ratio 5:4,
• the northwest-to-southeast diagonal axis represents the just minor-3rd of ratio 6:5,

Part of Riemann's point in using the just intonation Tonnetz was to visually illustrate the enharmonic equivalence of what we would today call unison-vectors: in particular, the syntonic comma and the enharmonic diesis.

The Tonnetz fell out of use in mainstream music-theory in the early 1900s, concurrently with the universal adoption of 12-edo tuning, and particularly with the shift back towards a 1-dimensional (i.e., linear) pythagorean perspective but in its modern closed form in the "circle of 5ths". But the Tonnetz has been revived in academic music-theory in recent years (see issues of Journal of Music Theory from 1998-2003).

Tuning-theorists, by contrast, have been using lattice-diagrams related to the Tonnetz continuously ever since Riemann -- pre-internet examples include Shohe Tanaka and Max Meyer in the early 1900s, Adriaan Fokker, Erv Wilson, and Joel Mandelbaum in the 1950s-60s, and W. A. Mathieu and and Joe Monzo in the 1990s. Use of lattices is endemic on internet tuning lists.

###### REFERENCES

Riemann, Hugo. [1877] 1971.

Musikalische Sytnaxis: Grundriss einer harmonischen Satzbildungslehre.
[Leipzig: Breitkopf & Härtel.] Wiesbaden: Martin Sändig.

Riemann, Hugo. [1914-15] 1992.

"Ideas for a Study 'On the Imagination of Tone,'"
trans. Robert Wason and Elizabeth West Marvin.
Journal of Music Theory 36/1: 81-117.

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