Download or read book Generalized Transformational Voice-leading Systems written by David Ellis Orvek and published by . This book was released on 2019 with total page 147 pages. Available in PDF, EPUB and Kindle. Book excerpt: David Lewin writes that, “In conceptualizing a particular musical space, it often happens that we conceptualize along with it, as one of its characteristic textural features, a family of directed measurements, distances, or motions of some sort. Contemplating elements s and t of such a musical space, we are characteristically aware of the particular directed measurement, distance, or motion that proceeds `from s to t.’” This thesis is concerned with these measurements, distances, and motions as they relate to the voice leading between two pitch-class sets. We begin with Richard Cohn’s idea that we might understand the total voice-leading interval between two pitch-class sets as the mod-12 sum of the pitch-class intervals traversed by each voice. This “pairwise voice-leading sum” (PVLS) allows us to see that the total voice-leading interval is the same between several pitch-class sets within the same Tn/In set class and also that several pitch-class set transformations will produce the same voice-leading interval when applied to any one set. These sets that are equidistant from a given point are grouped into equivalence classes known as “SUM classes” (because all such sets will return the same value when their constituent pitch classes are summed together mod 12) and the transformations producing the same voice-leading intervals are grouped into equivalence classes known as “SUM-class transformations.” When the set class is not inversionally symmetrical, these transformations will be non-commutative, and we will be able to define a “dual” group of transformations for both the SUM classes and pitch-class sets that provide us with two different ways to navigate through the spaces. Together, the SUM classes/SUM-class transformations and pitch-class sets/pitch-class set transformations form two interrelated Generalized Interval Systems that allow us to conceptualize the “measurements, distances, and motions” of any of the Tn/In set classes and even, in a modified form, for all of the pitch-class sets of the same cardinality. What these constructions reveal, above all, is just how similar the set classes of the same cardinality really are as well as how many different ways there are to express the same background voice leading structures.