The profile is described in terms of co-ordinates on the axes is the projection - in the -metric - of  profile onto the axis
Vice versa we have The profile is described in terms of co-ordinates on the axes

Each key is given by its pitch class profile In Figure 3 key is represented by its first two coordinates

Each pitch class is given by its key profile In Figures 3 and 7, pitch class is represented by its first two coordinates

6.2 Details of Chew’s Model with Choice of Parameters

We use a simplified instance of Chew’s more general model (Chew, 2000). It proposes a spatial arrangement such that tones, triads and keys are represented as vectors in three-dimensional space. For tones are denoted by , proceeding in steps of one fifth interval from index to index . We denote major and minor triads by , and Major and minor keys by , respectively.

The tones are arranged in a helix turning by and rising by a factor of per fifth:

Both Major and minor triads are represented as the weighted mean of their constituent tones: The keys are represented as weighted combinations of tonic, dominant, and subdominant, with the minor keys additionally incorporating some of the Major chords:

We choose the parameters so as to obtain a good fit with the data from Bach’s WTC I (Fugues), resulting in the following parameter settings: