patterns are essentially the same, but with different intensities, because what
defines a pattern is therelativeintensities of its features. Patternr,incontrast,is
a distinct pattern because it contains the features in different proportions.
A vector space is an improvement over a Cartesian space for the depiction of
patterns because it makes explicit the equivalence of patterns that vary only in
intensity. If instead of points, we draw arrows from the origin to each point, we
get vectorsp,q,andr(depictedontherightinfigure19.2).Vectorsthatare
oriented in the same direction but have different lengths (e.g.,pandq)arecol-
linear and represent the same pattern with different intensities. Vectors that are
oriented in different directions (e.g.,pandr) represent different patterns. The
more divergent the directions in which two vectors point, the more dissimilar
are the patterns. Without the benefit of a visual depiction of a vector space (as
when the vector has more than three dimensions), one can tell if two vectors are
collinear by seeing if the multiplication of one vector by a scalar yields the
other (Appendix A).
In the perception of pitch and tonality, we are typically concerned with dif-
ferences between patterns rather than with differences in absolute intensity of
the same pattern. Neural nets are responsive to the differences between pat-
terns, and the absolute intensities play a minimal role. This is just one of several
reasons why neural net models hold promise for understanding pattern per-
ception, and why vector spaces are promising conceptualizations of these
models. In some models, the absolute intensities are ignored altogether in order
to simplify the computation. In Grossberg’s models, for example, all vectors
are normalized so that the sum of the squares of the intensities of all the fea-
tures equals one. This ensures that all vectors have unit length (i.e., all the vec-
tor arrowheads terminate at a unit circle or unit hypersphere centered at the
origin).
E. Composite Patterns
Theadditionof vectors yields aresultant vector(Appendix B) and is equivalent
to superimposing patterns. The resultant represents a pattern that is more sim-
ilar to each of the original patterns than they are to each other. Graphically, the
resultant vector is the diagonal of the parallelogram formed by the two sum-
med vectors, as shown in the left-hand panel of figure 19.3. It makes a smaller
Figure 19.2
Left: Patterns in Cartesian space. Right: The same patterns in vector space.
460 Jamshed J. Bharucha