4.1 Tone-, Chord- and Keyspirals

The central idea in Chew’s approach is to attribute suitable points to all these objects within one and the same three-dimensional Euclidean ambient space in such a way that (1) the distances between these points are music-theoretically meaningful and that (2) transpositions are represented by Euclidean isometries (screw transformations). This is done by attributing suitable individual spirals to each of the five types, on which the corresponding discrete point sets are located. Chew defines five representation maps:

in the following way (Chew, 2000)[c.f.][p. 59]

1. tone representation where radius and height are free parameters.
2. major triad representation where is a weight vector of positive real numbers satisfying . The point is the ’center of effect’ of the three weighted points with respect to the weight vector . It this situated inside of the triangle spanned by the corresponding pitch points (c.f. Figure 4).
3. minor triad representation where is a weight vector of positive real numbers satisfying . The point is the ’center of effect’ of the three weighted points with respect to the weight vector .
4. major key representation where is a weight vector of positive real numbers satisfying . The point is the ’center of effect’ of the three weighted triad representations of the triad collection with respect to the weight vector . It this situated inside of the triangle spanned by the corresponding triad centers (c.f. Figure 4).