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 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 dominant 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.
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 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 and 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. Vogel adopts the harmonic dualism of Arthur von Oettingen, with major and minor chords being mirror images of each other. This view is complemented by a quantitative computation of consonance (or rather dissonance) values.
For this purpose Vogel introduces virtual reference tones that are not necessarily part of the chord. These reference tones are chosen such that all chord tones have integer relations to these reference tones. For each chord there exists a lower and an upper reference tone, with all chord tones being integer multiples of the frequency of the lower reference tone and integer fractions of the frequency of the upper reference tone. In quadruple notation there are only positive (or zero) values if the chord is related to the lower reference tone, and only negative (or zero) values if the chord is related to the higher reference tone. To get a single numerical value describing the complexity of a chord, Vogel builds a weighted sum of the quadruples describing the notes of the chord. He suggests the weights 1, 3, 5, and 7 for the prime numbers 2, 3, 5, and 7. Vogel rejects the more obvious variant where prime number 2 is weighted with 2 because it leads to results that in his opinion do not comply with the perception of musically skilled listeners. Finally, the weighted sum is divided by the number of tones of the chord. This computation is done for both the higher and the lower reference tone. Depending on which of those two values is smaller, the chord is then labeled as "Oberklang" or "Unterklang" ("upper chord", if reference to the lower reference note, or "lower chord", if referenced to the upper reference note). The C major chord c’-e’-g’ could, for instance, be referenced to C. All three notes of the triad can be represented as integer multiples of the frequency of this reference tone (4, 5, and 6). The prime decomposition yields 2·2,5,2·3. Applying the weights suggested by Vogel one obtains a so-called consonance value of (1+1+5+1+3)/3 = 11/3 = 3.67. The same chord may also be referenced to b’’’’: this upper reference tone has 15 times the frequency of c’, 12 times the frequency of e’ and ten times the frequency of g’. The prime decomposition yields3·5,2·2·3,2·5. The consonance value computes to (3+5+1+1+3+1+5)/3 = 19/3 = 6,33. As the consonance value for the lower reference tone is better (smaller), the c major chord c’-e’-g’ is defined to be an upper chord referenced to C. The consonance value of the c minor chord c’-es’-g’ is identical. It is, however, reference to the upper reference tone of this chord, g’’’. In consequence, Vogel rejects the naming of this chord as C minor as its reference note is not C but G. He calls it "G lower chord". Vogel suggests a specific notation for upper and lower chords. The notation starts by denoting the reference tone in lower case. Upper chords are marked by an "O" (Oberklang) and are denoted from left to right, lower chords are marked by a "U" (Unterklang) and are denoted from right to left. The C major chord is denoted as cO, the C minor chord is denoted as Ug. Additional symbols for additional notes (7 for adding an upper or lower seventh) are added to the left or to the right, depending on whether it is an upper or a lower chord. The C7 chord depicted above would be denoted as cO7. In addition to the calculation of consonance values for single chords, Vogel suggests a computation of the consonance of chord transitions. When transiting from an n-note chord to an m-note chord all N·M note-to-note transitions are evaluated via prime decomposition and weighted sum, and a mean value for all these transitions is computed. Vogel also suggest to compute a consonance value for an entire piece of music, taking into account a central reference point similar to a final.
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