(German: "tone-network", plural: Tonnetze)
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) \/ \/
/\ /\ /\ /\ /\ /\ /\ /\ /\ /\ /\ /\ /\
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
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.
Riemann, Hugo. [1877] 1971.
Riemann, Hugo. [1914-15] 1992.
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