:		This extended patch is a self-retrograde.
:		This subset of the former patch is a self-
		retrograde as well as as a self-inversion;
		the four tones have equal pitch.
		This patch is transformed into itself by
		a retrograde followed by an inversion.
:		These tones form a self-retrograde.
		All three tones have equal pitch, henceÊ
		the set is a self-inversion.

Comparing these sets of symmetry patches immediately reveals the differences, both in the occurring and the numbers of corresponding symmetry patches. It is left to the reader to draw conclusions concerning the question mentioned in the Introduction. Instead, we want to emphasize three important phenomena that have to be taken into account in the general situation. First, reduction by removing non-maximal subsets may lead to a loss in information: The set is contained in , but has a bigger isotropy group, i.e., there are more transformations mapping it into itself than in the case of the larger set. Second, there are “non-trivial” orbits consisting of a single patch, in the sense that the associated isotropy group is non-trivial. Third, in general there will be “degenerate” symmetry patches like those formed by tones of equal pitch. They seem to be rather irrelevant for the musical analysis but nevertheless carry the information about pitches occurring more than twice in the melody together with their frequencies.

All that is to follow will start from , discuss some of its features and try to grasp more pieces of the encoded information by means of several numerical characteristics.

3 Numerical Characteristics

In this section we define, for a given melody and a chosen paradigmatic group, two kinds of rates, the first expressing the proportion of the number of symmetry patches and the second that of the number of associated orbits to the number of all subsets with the same cardinality. Apart from one general fact concerning upper bounds for the symmetry rate, the focus will be on examples.