The Tonnetz – Euler's triangular lattice of sounds
Fifths to the right, thirds diagonally upwards: Leonhard Euler arranged the notes in a lattice whose triangles are the major and minor chords. It is the same triangular lattice the tetractys is laid out on.
With the numbers 1, 2, 3 and 4, the tetractys contains the three basic intervals of the Pythagoreans: the octave 1 : 2, the fifth 2 : 3 and the fourth 3 : 4 (see Number and sound). What it lacks is the third. The pure major third has the ratio 4 : 5, the minor third 5 : 6, and for these you need five and six.
From four to six
The music of the Middle Ages lived on fifths and fourths. Only with the polyphony of the Renaissance did thirds become consonances. The Venetian music theorist Gioseffo Zarlino drew the conclusion in 1558 in Le istitutioni harmoniche: he extended the tetractys by two numbers to the senario, the numbers 1 to 6. All consonances can be formed from them, and the major triad is nothing but 4 : 5 : 6.
Euler’s lattice
In 1739, in his music theory Tentamen novae theoriae musicae, Leonhard Euler arranged the notes in two directions: fifths in one, major thirds in the other. This creates a network in which every note has its pure neighbours. In the 19th century Arthur von Oettingen and Hugo Riemann gave the network its triangular form and the name Tonnetz, “tone network”.
To the right each step is a fifth further, diagonally up to the right a major third, diagonally up to the left a minor third down. Every triangle of three neighbouring notes is a triad. The upright triangles are major chords in the ratio 4 : 5 : 6, the inverted ones minor chords in the ratio 10 : 12 : 15.
Euler's Tonnetz in just intonation. Tap a triangle to play its triad, a circle plays the single note. Light triangles are major, bluish ones minor.
Neighbours share two notes
Two triangles sharing an edge share two notes. C major (C, E, G) borders on A minor (A, C, E), E minor (E, G, B) and C minor (C, E♭, G). To move from one chord to its neighbour, only a single voice moves, and only by a small step. These neighbourhoods explain much of what composers from Schubert and Wagner to film music do, and today’s music theory describes whole chord progressions as paths through the network.
The comma in the network
In the Tonnetz, E appears twice: once four fifths above C, once a major third above it. In just intonation these are not the same note. Four fifths give 81 : 64, the pure third 80 : 64. The difference, 81 : 80, is called the syntonic comma. In the panel you can listen to both Es one after the other.
That is why the pure Tonnetz is infinite. It never closes, every step leads to a new note. Only equal temperament, which spreads the comma over all intervals, bends the network into a ring and finally into a torus with just twelve notes. The purity of the ratios is traded for the freedom to play in every key.
The tetractys in the Tonnetz
The Tonnetz is a triangular lattice, exactly the lattice on which the Pythagoreans laid out their tetractys (see The triangle of dots). Lay the ten points of the tetractys on the network and it covers ten notes, six major triads and three minor triads. Start at C at the bottom left, and the bottom row holds the four notes of a chain of fifths, C, G, D and A, above them three, then two, and one at the top.
Where the tetractys contains the numbers 1 to 4 and with them the fifths, the Tonnetz unfolds the space that opens up with five. The old ratios remain one axis, the new number opens the second. The row becomes a surface, just as in the tetractys the line becomes the triangle.