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COPYRIGHT 2000 University of Washington
THE THIRD SONG of Dallapiccola's Quattro Liriche di Antonio Machado ends with a series of six-note chords in the piano part (Example 1). (1) The first chord, repeated and held through measure 80, is a member of set class (SC) 6-Z28 [013569]; the second chord is its literal complement, a member of SC 6-Z49 [013479] (they form the two halves of a twelve-tone series). The third and fourth distinct chords (from the end of measure 81 to the end of the excerpt) reverse this pattern at a different transpositional level. While these chords types all have the same interval-class (ic) content (as is the nature of Z-related SCs), the subsets and intervals that they project differ greatly. The two realizations of 6-Z49 prominently feature two closely-spaced realizations of 3-5 [016]--one in each hand. In these chords, ics 1, 5, and 6 are most salient. By contrast, both realizations of 6-Z28 prominently feature a close-position augmented triad (3-12 [048]) in the right-hand part and an open-position diminished triad (3-10 [036]) in the left-hand part. Or, taking the lowest four notes of these 6-Z28 chords (including the lowest note in the right hand) yields a complete diminished seventh chord (4-28 [0369]), spaced as two tritones, nine semitones apart.
These differences between the 6-Z28 and 6-Z49 chords are not merely products of spacing. Even though both set types share the same intervallic profile, 6-Z49 embeds neither 3-12 nor 4-28. Because 6-Z28 does embed these two set types, both of which are complete interval cycles, I will argue that it has the potential of projecting ics 4 and 3 (their cyclic progenitors) more strongly than does its Z-equivalent, 6-Z49. This article will propose a series of pitch-class-based analytical tools (including a similarity index) that differentiate such set pairs, while still acknowledging their intervallic affinities.
Marcus Castren's recent work on measures of pitch-class set resemblance establishes a dichotomy between methods that compare only the sets' interval-class content and those that consider all subset classes. (2) Examples of interval-class-based resemblance measures include Morris's ASIM, Isaacson's IcVSIM and more recent ISIM, my own interval-class saturation similarity measure--or SATSIM(2),(3) and the new ANGLE measure by Damon Scott and Eric Isaacson. Examples of subset-based measures, which Castren calls "total" measures, include Rahn's ATMEMB, Castren's RECREL, and potentially Lewin's REL (depending on which subset classes are included in the TEST group). (4)
Castren, among others, objects to interval-class-based measures because they tend to produce a smaller number of distinct values than do total measures, and because they do not distinguish between Z-related set classes. (5) Total subset-based measures such as the ones mentioned above do distinguish Z-related set classes, and each of them produces a greater number of values than do any of the aforementioned interval-class based measures. However, I'm not convinced that there is any correlation between the number of distinct values produced and the quality--or effectiveness--of a particular measure. The measure that will be presented later in this article produces hundreds more values than any of these total measures, but I don't believe that this is necessarily an advantage.
Total measures, almost by definition, use different criteria in comparing sets that are not the same size. For example, if one wanted to compare two hexachords using a total measure, one would examine their mutual pentachord-, tetrachord-, trichord-, and dyad-class embeddings. If, however, one wanted to compare a hexachord to a trichord, one could only compare the mutual dyad-class (and perhaps trichord-class) content of the two sets. While each of these so-called total measures includes an algorithm to bring such unequal comparisons into a common range of values, they still create scenarios where different means are used to compare sets of unequal size.
Rather than judging resemblance by comparing interval classes or all available subset classes, I will propose a method that is based upon how each set is partitioned with respect to the six distinct interval-cycles. (Because interval 7- through 11-cycles may be understood as either retrogrades or inversions of interval 5- through 1-cycles, they will not be considered distinct.) This information serves as the basis for a new weighted six-argument vector that resembles the interval-class vector (ICV) in function (or at least in its function as data for similarity indices) but not in design. Each argument of the vector represents the degree to which instances of corresponding interval-class n are found in unbroken n-cycle segments. The assumption behind the weighting is that, for any set class X, the more that instances of interval-class n form a particular n-cycle, the more likely that X will project interval-class n. For example, one might reasonably claim that a four-note quartal (or quintal) chord projects i c5 more strongly than does a chord with three cyclically nonadjacent ic5s. Although I am addressing only pitch-class sets and not their particular orientations in pitch space, I believe that it is still legitimate to assert that many--or even most--realizations of set class [0257] will project ic5 to a great extent.
Before introducing the new vector types and similarity index, it will be useful to make a few comments on the cycles themselves and the ways in which they can be segmented and concatenated to form "cyclic sets." The group of cyclic sets has been discussed elsewhere in the theoretical literature, (6) but I will be undertaking an approach that is rather different in nature from these studies. My approach will lead toward a method for comparing two set classes based upon their shared and different cyclical construction.
Let us define an n-cycle (where n is a variable that represents any interval class in standard twelve-pc space) as a closed and finite ordered collection of pitch classes where one element maps onto the next (and the last onto the first) under transposition at a constant interval n. (7) The members of an n-cycle are defined as (x+n, x+[n.sup.2], x+[n.sup.3], ... x+[n.sup.p] = x) where p is the period of the n-cycle. For most values of n, there are several distinct n-cycles in the 12-pc aggregate. For example, there are four 4-cycles: (048), (159), (26a), and (37b). Because each n-cycle has p elements, there must be 12/p distinct cycles formed by interval n (we call this value m). The complete n-cycles are shown in Example 2.
Because all instances of ic n occur segmentally within the n-cycle(s), any pitch-class set that simply is a complete n-cycle naturally features the maximal amount of a given ic n for a set of its cardinality. {0, 2, 4, 6, 8, a}, for example, is maximally ic2 saturated; {0, 3, 6, 9} is maximally ic3 saturated, and so forth. The same is true of pcsets that are wholly the union of two n-cycles (for a single given n). Both (0, 1, 4, 5, 8, 9} and (0, 2, 4, 6, 8, a} can be formed by the union of two 4-cycles, (8) and consequently both hexachords maximally include ic4. For an interval n whose cycles have periodicity p, then, we know how to identify the "maximally n-saturated set types" whose cardinalities are p or integer multiples of p.
A pcset that is smaller than p will maximally saturate ic n if it is (again, wholly) a continuous n-cycle segment. For n = 2 (with p = 6), the two-through five-element set classes that maximally saturate ic2 are [02], [024], [0246], and [02468]. A pcset that is larger than p will maximally saturate ic n if it is the combination of however many complete n-cycles cardinallty permits (possibly just one) and an incomplete n-cycle of whatever length cardinality requires. For n = 4 (with p = 3), any combination of, for example, (0,4, 8} and some segment from one of the other three 4-cycles will produce sets that are maximally saturated with ic4 (e.g., (0, 1,4, 8}, {0, 1,4, 5, 8}, (0, 2,4, 8}, and {0, 2,4, 6, 81}).
We can condense the above conditions for maximal n-saturation into a single definition of what we shall call an n-set (for interval n). An n-set is comprised of some number of complete n-cycles (possibly none, one, or more than one) and, at most, one incomplete n-cycle segment. The complete list of all n-sets is the same as Tore Ericksson's maxpoint series. (9) All n-sets are maximally saturated with interval n and all pcsets that are maximally saturated with interval n are n-sets.
We will now return to the creation of several new vector types that reflect how the elements of a pcset are distributed with regard to the interval cycles. We will first examine such cyclic distribution, focusing on the number and position of any cyclic adjacencies. Next, we will create a version of the interval-class vector that distinguishes the size and quantity of all n-cycle segments. This amounts to a subdivided interval-class vector, the arguments of which will be weighted using a procedure that gives cyclic strings of intervals more prominence than equal numbers of the same intervals that are not all cyclically adjacent. My final construct will be derived by comparing these cyclically weighted interval-class vector arguments to what is possible given any set of the same size. This is what I have elsewhere called a measure of saturation. (10) These adjusted values will provide us with a relatively cardinality-neutral means of relating sets based upon their n-cycle subsets.
We will begin our transformation from an objective inventory of the intervals within a set to a weighted cyclic saturation vector by examining the manner in which elements of a pitch-class set are distributed among the n-cycles. Example 3 shows the cyclic distribution of set class 6-Z28 [013569] (interval-class vector: ). Each line of the example shows adjacencies within a particular n-cycle simply by ordering the elements of the "most normal" form of 6-Z28 along the cycle. Parentheses delineate the cycles, and adjacent pitch classes within the parentheses (including the wraparound) are n-cycle adjacencies, each producing a single embedded interval-class n. Dashes indicate vacant places in each n-cycle.
Example 3 illustrates how the pcset's elements are distributed among the cycles of any given interval. As a means of summarizing this data, we will create an array called [CycleSeg.sub.n](X). This construct lists the cardinalities of the n-cycle segments of X from longest to shortest. The sum of [CycleSeg.sub.n](X) numbers equals the cardinality of set X. Example 4 shows the cyclic segment lengths of our set, 6-Z28; compare these numbers with the patterns in Example 3. In Example 3, we can see that 6-Z28's elements fall into four disjunct segments of the 1-cycle, two of two elements and two of one; these are now represented by the array . Any realization of the set class (for example, {C, C[MUSICAL NOTES NOT REPRODUCIBLE IN ASCII], E[MUSICAL NOTES NOT REPRODUCIBLE IN ASCII], F, F[MUSICAL NOTES NOT REPRODUCIBLE IN ASCII],...
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