model to the result of our correspondence analysis as displayed in upper Figure 6. (Cf. Appendix 6.2 for parameters.)

In order to facilitate comparison we choose a two-dimensional visualization of the three-dimensional model in Chew (2000). The projection of the model onto the X-Y-plane is circular. Therefore we can parameterize it as angle and length. We plot the vertical dimension (the elevation of the helix) versus the phase angle of the X-Y-plane (middle Figure 6). We interpret the phase angle of the X-Y-plane as the first angle of a torus, and the vertical height in the helix as the second angle of a torus. The helix is mapped on the surface of a torus by applying modulo to the height. Here is the distance on the vertical co-ordinate of two successive tones in the circle of fifths. We observe that upper and middle Figure 6 are very similar: The circles of fifths in Major and minor keys and in pitch classes curl around the torus three times. The only difference is that in the toroidal model derived from correspondence analysis Major keys and their relative minor keys are nearby, whereas in middle Figure 6 Major keys are closer to their parallel minor keys.

Consistency with a Cognitive Model (Purwins et al., 2000a). A very simple listener model comprises the following five stages:

1. Frequency analysis with uniform resolution on a logarithmic scale (constant Q transform of Brown (1991))
2. Compression into pitch class profiles by octave identification
3. Averaging of profiles across each piece
4. Generation of a reference set of profiles, one for each Major and minor key
5. Spatial arrangement of the reference set on a toroidal self-organizing feature map (Purwins et al., 2000b; Kohonen, 1982).

In this scheme, stage 1 can be considered a coarse model of auditory periphery. Stage 5 may be seen as a rough model of cortical feature maps (Obermayer et al., 1990). The constant Q transform is calculated from a digitized 1933/34 recording of Chopin’s Préludes Op. 28 performed by Alfred Cortot. The average cq-profiles for each single prelude are used as a training set for a toroidal self-organizing feature map (Purwins et al., 2000b; Kohonen, 1982). Again the resulting configuration (lower Figure 6) shows the circle of fifths and closely resembles the other configurations in Figure 6.