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1 Chapter 1 a Brief History and Critique of Interval/Subset Class Vectors And 1 Chapter 1 A brief history and critique of interval/subset class vectors and similarity functions. 1.1 Basic Definitions There are a few terms, fundamental to musical atonal theory, that are used frequently in this study. For the reader who is not familiar with them, we present a quick primer. Pitch class (pc) is used to denote a pitch name without the octave designation. C4, for example, refers to a pitch in a particular octave; C is more generic, referring to any and/or all Cs in the sonic spectrum. We assume equal temperament and enharmonic equivalence, so the distance between two adjacent pcs is always the same and pc C º pc B . Pcs are also commonly labeled with an integer. In such cases, we will adopt the standard of 0 = C, 1 = C/D, … 9 = A, a = A /B , and b = B (‘a’ and ‘b’ are the duodecimal equivalents of decimal 10 and 11). Interval class (ic) represents the smallest possible pitch interval, counted in number of semitones, between the realization of any two pcs. For example, the interval class of C and G—ic(C, G)—is 5 because in their closest possible spacing, some C and G (C down to G or G up to C) would be separated by 5 semitones. There are only six interval classes (1 through 6) because intervals larger than the tritone (ic6) can be reduced to a smaller number (interval 7 = ic5, interval 8 = ic4, etc.). Chapter 1 2 A pcset is an unordered set of pitch classes that contains at most one of each pc. Using numeric notation, the set {C, D, E , F } would be represented as {0,2,3,6}. The aggregate (U) is the complete set of all twelve pcs. The interval-class vector (ICV) is an enumeration of the interval classes found between members of a pcset. Let X = pcset {0,2,3,6}. There are six distinct pairs of notes in any tetrachord; the ics within these pairs are as follows: {0,2} = ic2 {0,3} = ic3 {0,6} = ic6 {2,3} = ic1 {2,6} = ic4 {3,6} = ic3 We now place the numbers of each ic present into a six-argument vector, with each place in the vector representing the corresponding ic (the first argument in the vector represents the number of ic1s in the pcset, etc.). In other words, the number of ic ns is placed in argument n of the ICV; there are two ic3s in pcset X, so the third argument in ICV(X) is 2. Therefore, ICV({0,2,3,6}) = <112101>. From this, we can quickly see that there are no instances of ic5, two instances ic3, and one of each of the remaining ics. Even though there is one and only one interval present in a set of cardinality 2, a dyad (or dyad class) is not the same as an interval. An interval represents a distance between notes; a two-pc (cardinality 2, or #2) set is a group with two members. Despite this important distinction, it is possible to Chapter 1 3 think of the interval-class vector as a #2 subset-class vector—that is a listing of the number and type of #2 subsets embedded in a set.1 While such a vector would be identical in appearance to the ICV, conceptualizing the elements are subsets rather than intervals will allow us to generalize outwards and produce other subset-class vectors (e.g., a trichord-class vector, tetrachord-class vector, etc.). Before further discussing other subset-class vectors, it will be useful to define abstract pcset inclusion. Given two set classes (SCs) X (denoted /X/) and /Y/ where /X/ has the same number or fewer elements than /Y/ (#/X/ £ #/Y/),2 AS(/X/, /Y/) is a Boolean function (relation) which returns “true” if at least one X of /X/ is embedded in some Y of /Y/.3 AS(/X/, /Y/) returns “false” if no form of /X/ is embedded in some Y of /Y/. If AS(/X/, /Y/) = true then /X/ is said to be an abstract subset of /Y/.4 For example, let us consider three set classes: /A/ = [014], /B/ = [0145], /C/ = [0167]. AS(/A/, /B/) = true; AS(/A/, /C/) = false.5 David Lewin’s EMB(/X/, /Y/) function6 provides more specific information than does our AS relation. EMB(/X/, /Y/) returns the number of distinct forms of /X/ which are embedded in some Y of /Y/.7 If, for example, /X/ = SC(3-1) [012] and Y = {1,2,3,6,7,8,a}, EMB(/X/, /Y/) = 2; that is, both {1,2,3} 1There is a one-to-one relationship between the number of ic n in a pcset and the number of SC 2-n in the same pcset. 2#/X/ = cardinality of (number of elements in) pcset X. 3 We will, for the time, assume that forms of X are all possible transpositions (Tn(X)) and inversions (TnI(X)) of X. 4Because the use of set class is inherent to the notion of “abstract subset,” AS(/X/, /Y/) is functionally the same as AS(X, Y ). If X Ì Y, /X/ is an abstract subset of /Y /. 5As a numerical values, “true” and “false” are normally represented by the digits 1 and 0, respectively. 6Lewin 1977 and Lewin 1979-80a. In the latter article, Lewin uses the form EMB(/X/, Y), but comments that EMB(/X/, /Y/) provides the same information. (Lewin 1979-80a, 433.) Consistent with our definition of AS, we prefer the latter because it deals entirely with set classes. 7 We will, for now, use the common canonical operations transposition (Tn) and transposition and inversion (TnI) to calculate the distinct forms of /X/. Chapter 1 4 and {6,7,8} are forms of [012] and are included in {1,2,3,6,7,8,a}. Given that {1,2,3} (or any form of [012]) is an inversionally symmetrical set, it can map onto itself through two distinct operations: the identity operation (T0{1,2,3} = {1,2,3}); and under transposition and inversion (T3I{1,2,3} = {1,2,3}). These two transformations yield the same pcset and will be considered a single distinct form of 3-1. Abstract inclusion vectors, including n-class vectors (or nCVs8), such as the #2 subset-class vector (which, again, is equivalent in appearance to the ICV) can be derived by performing EMB(/X/, /Y/) for each distinct SC(X) where #X = n. When #X = 2, EMB(/X/, /Y/) returns an argument of the #2 subset-class vector of SC /Y/ (ICV(Y)). Formally, we can define each argument in the 2CV as follows: 2CV(X)i = EMB(i, /X/).9 1.2 Evolution of the interval class vector The ICV has played an integral part in the history of atonal theory, serving as the prototype for most other abstract subset inclusion vectors and, more specifically, to the new vectors and relations which will be introduced in chapter 2 of this study. In this section, we will chronicle its evolution and variations. 8This abbreviation is adopted from Castrén (Castrén 1995, 3) who, in turn, adopted it from Lewin (Lewin 1987, 106-7). 9We could also have defined the ICV using Lewin’s function COV(/X/, /Y/). This function, also presented in Lewin 1979-80, 434, returns the number of distinct forms of SC Y that cover—or include— a member of /X/. Where #X = 2 (that is, when X is a dyad), COV(/X/, /Y/) = EMB(/X/, /Y/). COV and EMB will also return the same value in all cases where pcset X is inversionally symmetrical. When pcset X is not inversionally symmetrical, these two functions return different values. Morris 1987, 90 provides an example where these two functions serve different purposes. Since we will be dealing with abstract inclusion of all pcset classes in the course of this study, EMB(/X/, /Y/) will serve as a more useful model than COV(/X/, /Y/). Chapter 1 5 1.2.1 Hanson, 1960 Howard Hanson’s chord or scale “interval analysis,” presented in his 1960 book Harmonic Materials of Modern Music: Resources of the Tempered Scale, is, to our knowledge, the first enumeration of the interval- class vector and thus its first use in comparing and relating different pcsets.10, The actual term “interval vector” was coined several years later by Allen Forte. Hanson represented each of the six interval classes with a different letter: p (“perfect”) represented ic5, m (“major third”) represented ic4, n (“minor third”) represented ic3, s (“major second”) represented ic2, d (“minor second”) represented ic1, and t (“tritone”) represented ic6. These letters were used to delineate SC and always fell in the order “pmnsdt.”11 Each letter was only used in a SC label if its respective ic was embedded in the set. E.g., SC(3-11) [037] = pmn in Hanson’s terminology. This is the same as saying that it has one ic5, ic4, and ic3, and none of the other ics. When there was more than one embedding of a particular ic in a SC, superscripts carrying the EMB(ic, /X/) number followed each letter. For example, SC(3-9) [027] = p2s. Hanson’s pmnsdt interval analysis was, in part, meant to show composers how they could project certain intervallic structures using particular SCs. His unique labeling scheme introduced a number of concepts that were reintroduced more formally by later theorists. For example, while Hanson labeled SCs using his mnemonic version of the ICV, he understood that ICV 10Even if that comparison is a bit less systematic than more recent attempts.
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