Vogel's Tonnetz

Vogel's Tonnetz is a graphical and mathematical representation of the scale range of just intonation, introduced by German music theorist Martin Vogel 1976 in his book Die Lehre von den Tonbeziehungen (english: On the Relations of Tone, 1993). The graphical representation is based on Euler's Tonnetz, adding a third dimension for just sevenths to the two dimensions for just fifths and just thirds. It serves to illustrate and analyze chords and their relations. The four-dimensional mathematical representation including octaves allows the Evaluation of the congruency of harmonics of chords depending on the tonal material. It can thus also serve to determine the optimal tonal material for a certain chord.

The graphical representation of Vogel's Tonnetz is limited to the three dimensions for fifths, thirds, and seventh. In this representation tones separated by one or several octaves are depicted on the same nodes. The illustration shows the chord which is the most frequent 4-note chord in western music: the major seventh. In Euler's Tonnetz the B-flat is constructed from fifths and thirds. In Vogel's Tonnetz it is given as a just harmonic seventh.

Seventh chord

Representation in Euler's Tonnetz

Representation in Vogel's Tonnetz

The representation of this chord in Vogel's three-dimensional Tonnetz makes its statistical dominance much more plausible than its representation in Euler's two-dimensional Tonnetz: There is a distinct reference note (C), and all other notes are linked to this reference note via simple one-step intervals in this Tonnetz.

The mathematical representation of Vogel's Tonnetz is four-dimensional, considering also octaves. Each tone is represented by a quadruple of numbers specifying how many octaves, "fifths", "thirds", and "seventh" are needed to reach that tone in the Tonnetz (where the terms "fifths", "thirds", and "seventh" denote the prime numbers 3, 5, and 7, instead of the intervals 3/2, 5/4 und 7/4). The C-major seventh chord with the notes c', e', g', and b-flat' could (with reference to C)be represented by the numbers 4, 5, 6, and 7. This corresponds to the quadruple (2,0,0,0), (0,0,1,0), (1,1,0,0), and (0,0,0,1). The quadruple notations represents the prime decomposition of the numbers that are needed to describe the chord, limited to the first four prime numbers.

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