The Tonnetz, or “network of tones”, is a theoretical model used in transformational music analysis to represent the harmonic relationships between pitches and chords in the equal-tempered system. It displays the interconnections between notes through the choice of two generating intervals, usually corresponding to the minor and major third.
In this specific Tonnetz, also indicated with (3,4,5), notes are arranged in a triangular grid where the diagonal axes represent minor and major thirds and the vertical axis corresponds to the perfect fifth. Triangles correspond to major and minor chords and three main elementary transformations enable to transform a given chord by keeping two notes and changing the third one by an interval of semitone or tone.
These transformations are called the Relative (R), the Parallel (P) and the Leading-Tone exchange (L). They transform for example a C major chord into its relative A minor chord (and vice-versa), a C major chord into its parallel C minor (and vice-versa) and, finally, a C major chord into the E minor chord (and vice-versa). The traditional (3,4,5) Tonnetz naturally extends to generic (a,b,c) Tonnetze where the numbers a and b correspond to the diagonal axes that generate the new harmonic grid.
In the case of the (3,4,5) Tonnetz, the two types of triangles correspond to minor chords – the left-pointing triangles having intervallic structure equal to (3,4,5) - and major chords – the right-pointing triangles having intervallic structure equal to (4,3,5).
In the generalized (a,b,c) Tonnetz, the left-pointing triangles will correspond to a chord whose intervallic structure is equal to (a,b,c) and the right-pointing will be their symmetrical, having intervallic structure equal to (b,a,c). For example, the (2,3,7) Tonnetz will have the diagonal axes generated respectively by the whole-tone and the minor third intervals. A left-pointing triangle of the grid will correspond to a chord containing the notes C, D, F.