ï~~how many cells of a matrix of all possible intervals are used in the calculation of distance. A measure of
combinatoriality (C) of a metric might be given informally as C = I/Lm where I is the number of intervals used in the
calculation, and Lm is the number of possible intervals. For any given morphology M of length L, there are L-1
possible linear intervals, and Lm -= (L*(L+1)/2) - L possible (non-redundant) combinatorial intervals, constituting "half"
the matrix L x L (minus the diagonal).3 For example, for lengths 2-10 the number of intervals are 1, 3, 6, 10, 15, 21,
28, 36, 45. In general, Lm is the second binomial coefficient of L. Most linear metrics only consider intervals between
the cells of the matrix (el,e2), (e2,e3), (e3,e4) etc. As such, their measure of combinatoriality is (L-1)/Lm. By the above
suggested measure of a metric's combinatoriality (C), the measure of combinatoriality of strictly linear metrics of
morphologies of length 2-10 would be: 1, 1,.50,.40,.33,.29,.25,.22,.20. Obviously, the longer the morphology,
the less accurately a linear metric may measure its internal structure.
The simplest combinatorial metrics are very similar to the linear metrics in the examples above, but instead of
computing the means of successive linear intervals, or corresponding linear intervals, the means computed are from each
interval to all other intervals. Example 3 is an Unordered combinatorial magnitude (UCM) metric, based on Example
1:
L-1 L-j L-1 L-j
D(M1,M2) = I (. A(eli, el1i+j) - A(e2i, e2i+j) ))I
Lm Lm
This example takes the difference of the means of differences between all non-redundant cells of each morphology's
matrix. Note that in this example, the more general notation for interval distance function is used - which might
represent absolute value of arithmetic differences, ratio, or any other such function. 4
Example 4 is an Ordered combinatorial magnitude (OCM) metric. In this metric, the mean of the differences
between corresponding cells of the two matrices is considered:
L-1L-j
D(M1,M2) = ( I A( eli, eli+j) - A( e2i, e2i+j) I
Lm
The degree of order of a metric, and its linearity or combinatoriality, reflects the importance of sequence in the
judgement of morphological distance. In some cases order is not desired, as in, for example, purely statistical measures.
Similarly, there are situations in which a strictly linear metric can yield more useful results, as in most traditional
comparisons of melodic variation. The more combinatorial the metric, however, the more finely it measures the
differences in the internal structure of two morphologies.
2.2) Directional and Magnitudinal Metrics
Definition: A metric is directional if it considers the sign of an interval in a morphology, and magnitudinal if it
considers the value of that interval.
Comment: Directional metrics reduce a morphology to one of three values - "down, same, up"- usually
represented as -1, 0, and 1. Magnitudinal metrics measure actual parametric differences in morphologies. Directional
metrics measure contour, or direction of change. Magnitudinal metrics measure the amount of change. These two types
of metrics may easily be combined.
A simple and useful function for directional interval calculation is A(ei,ej) = sgn(ei,ej) where:
sgn(ei,e) = 1, ei > e; 0, ei = ej; -1, ei < e
Note that this interval distance function is not strictly a metric.
Example 5 is an Ordered linear directional (OLD) metric:
t
D(M1,M2) =. diff(sgn(eli,eli-1), sgn(e2i,e2il))
(L-l)
where diff(x,y) = 1, x y; 0, x = y -andsgn(ei,ej) = 1, ei > e; 0, ej = ej;-1, ei < ej; or sgn(ei,ej) = ( (ei- ej)/lei- e 1) / 1
In this example, only the sign of the interval is considered (directional), only intervallic direction between successive
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