Volume 19, Number 3, September 2013
Copyright © 2013 Society for Music Theory
Interval PermutationsDrew F. NobileKEYWORDS: Interval permutations, intervals, posttonal analysis, mathematical modeling, setclass, space, Schoenberg, The Book of the Hanging Gardens ABSTRACT: This paper presents a framework for analyzing the interval structure of pitchclass segments (ordered pitchclass sets). An “interval permutation” is a reordering of the intervals that arise between adjacent members of these pitchclass segments. Because pitchclass segments related by interval permutation are not necessarily members of the same setclass, this theory has the capability to demonstrate aurally significant relationships between sets that are not related by transposition or inversion.

Example 1. Schoenberg, Op. 15, No. 14, opening: rhythm and contour establish a relationship between the first two gestures in the piano and voice (click to enlarge and hear the audio) Example 2. All ordered pitchclass intervals are 6, 7, 10, or 11 in the opening two gestures of Song No. 14. The bracketed interval sequences, which represent instances of rhythmic motive x, are related by interval permutation (click to enlarge) [1] Example 1 shows the score to the opening measures of “Sprich nicht immer von dem Laub,” the fourteenth song from Schoenberg’s Book of the Hanging Gardens, op. 15. Upon first listen, one is drawn to the relationship between the piano’s gesture in the first measure and the vocal line in the second. The two most immediately perceptible domains—contour and rhythm—solidify this relationship: the two gestures have opposite contour and the voice presents the same two rhythmic motives as the piano in reverse order, as shown in the example. These surface relationships would lead any analyst to search for a deeper relationship between these two gestures, especially in the realm of pitch. While the first three notes of the voice’s gesture give hope that this might be a retrograde inversion of the piano’s, this turns out not to be the case. Setclass analysis also proves unsatisfying: one might point out that setclass (016) makes several appearances, but this does not seem to be a unifying principle, as it is difficult to account for every note without invoking questionable segmentations. A salient relationship emerges, however, when we consider the ordered pitchclass intervals between successive notes, as shown in Example 2. Specifically, we see that all of these intervals are either 6, 7, 10, or 11 semitones. If we focus on the intervals within each gesture’s rhythmic motive x, bracketed in Example 2, we see that both of these motives contain one instance of each of these intervals but in a different order: the piano’s interval sequence is 11, 10, 6, 7, while the voice’s is 11, 7, 10, 6. Despite the fact that the two pentachords involved in these interval sequences are not members of the same setclass, their shared interval content associates them in an aurally salient way. This type of association is the focus of this paper; I will call such a relationship an interval permutation. [2] I will return to this analysis in the final section of the paper, after providing a theoretical model of interval permutations and investigating their interactions with pitchclass set theory. The theory of interval permutations is based on two basic structures: the pitchclass segment and the interval series. A pitchclass segment is simply an ordered set of pitchclasses; these pitchclasses need not necessarily be distinct, though for practical reasons we will want to restrict ourselves to segments that contain at least two distinct pitchclasses. For example, the piano gesture in measure 1 of Example 1 can be represented as the segment (C, B, A, [3] An interval permutation is a reordering of an interval series. For example, [11, 6, 7, 10, 11], [10, 7, 6, 11, 11], and [6, 11, 10, 7, 11] are all interval permutations of the interval series [11, 10, 6, 7, 11] from the first measure of the Schoenberg example. An interval series of length n can be permuted in n! (“n factorial”) different ways if it does not have any duplicate intervals (the presence of duplicate intervals will reduce the number of distinct permutations). For example, a threemember series can be permuted in six different ways, as shown in Example 3 for the interval series [6, 1, 3]. In this example, the pitchclass segments are shown as vertical structures ordered from lowest to highest. The interval series can be alternatively written as a vertical stack, as shown between the staves in Example 3, which should be read from bottom to top. [4] Not all of the sonorities in Example 3 are members of the same setclass; in fact, three different setclasses are represented in this example, namely (0146), (0236), and (0136).^{(2)} In order to represent the relationship between pitchclass segments whose interval series are related by permutation I will introduce the terms series class and permutation class. A series class is a set of all interval series that are related to each other by permutation. A permutation class is a set of all pitchclass segments whose interval series are in the same series class. While these two equivalence classes are clearly closely related, they differ greatly in magnitude: there are six interval series in the same series class as [6, 1, 3] (these are precisely the six interval series shown below the staves in Example 3), while the permutation class related to this series class consists of 72 distinct pitchclass segments: the six shown in Example 3, plus those six transposed to begin on all twelve pitchclasses. A permutation class will always have twelve times as many members as its associated series class. [5] Permutation classes partition the set of all possible pitchclass segments such that all members of each permutation class can be said to be in some way similar to each other and dissimilar from every segment not in that permutation class. There are many definitions of the word “similar” and many musictheoretical concepts that purport to model similarity measures, and I make no claims that the current one is the similarity measure between pitchclass segments. I do, however, maintain that pitchclass segments that are related by interval permutation are similar in a directly audible way. This claim depends first on the assumption that the pitchclass segments reflect an ordering that is aurally apparent: we need to be able to perceive the pitchclasses in that specific order. This is not problematic when the segments reflect temporal ordering of melodic fragments (as in Examples 1 and 2) or vertical ordering of chords (as in Example 3), but other orderings might raise questions as to their aural salience.^{(3)} When we perceive a pitchclass segment in order, the intervals between adjacent elements are one of the most immediately perceived relationships. An aural comparison of two pitchclass segments in the same permutation class will reveal a strong similarity between them based on the fact that the intervals are the same—a similarity that will not arise between two segments of different permutation classes. Permutation groups and interval swaps [6] This section demonstrates the relationship between interval permutations and the “symmetric group of order n,” written S_{n}, which is the group of all permutations of the natural numbers from 1 to n. Specifically, the set of interval permutations on an interval series X of length n can be thought of as an action of S_{n} on the members of X’s series class.^{(4)} In order to show this, some formalism is required: we can formally define an interval series as a function INTVALUE from the natural numbers 1, . . . , n to the integers mod 12 (ℤ_{12}) such that the expression INTVALUE(s) = i means that the sth interval in the series has value i. For example, the interval series [6, 1, 3] is defined by the INTVALUE function such that INTVALUE(1) = 6, INTVALUE(2) = 1, and INTVALUE(3) = 3. If π is a permutation in S_{n}, then π sends every number from 1 to n to some (not necessarily distinct) number from 1 to n. The permutation π can therefore act on the INTVALUE function as such: π ◦ INTVALUE(s) = INTVALUE(π(s)) In other words, π sends the interval that originally occupied the sth position to the π(s)th position. Note that π does not act on the intervals themselves, but rather on the positions of the intervals. For example, the permutation that turns the interval series [6, 1, 3] into [1, 3, 6] can be defined such that π(1) = 3 (meaning that the first interval becomes the third interval), π(2) = 1 (the second interval becomes the first), and π(3) = 2 (the third interval becomes the second). With this action, any permutation in S_{n} will send an interval series to some other series within the same series class. Furthermore, for any two interval series X and Y that are in the same series class, there exists at least one permutation in S_{n} that will map X to Y.^{(5)} This group action is therefore equivalent to interval permutations on a given series class. Example 4. Any interval permutation can be written as the product of swaps or adjacent swaps (click to enlarge) [7] Demonstrating that interval permutations are equivalent to an action of S_{n} allows us to import results from group theory. Some of these results are obvious, such as transitivity (if A is a permutation of B and B is a permutation of C then A is a permutation of C) or the existence of an identity permutation (A is always a permutation of A), but some are more revealing: for example, any permutation can be written as a product of interval swaps, which are permutations that swap the positions of two intervals (Dummit and Foote 2003, 107).^{(6)} Example 4 illustrates this: the permutation in Example 4a sends the series [7, 4, 5, 9] to [9, 4, 7, 5], which Example 4b shows is equivalent to swapping the first and fourth intervals and then swapping the third and fourth intervals. Example 4c shows a different way of dividing this permutation into two swaps. Example 4d shows that this permutation can also be written as a product of adjacent interval swaps, i.e., swaps of intervals that occur next to each other in the interval series. Any interval permutation can be written as a product of adjacent swaps. Example 5. Bartók, String Quartet No. 3, measures 1–4 (reduction): the first violin’s melody moves against a held tetrachord, effecting several adjacent interval swaps (click to enlarge and hear the audio) [8] Adjacent interval swaps are interesting in a musical sense because they can result from motion of a single pitchclass from one segment to the next. In Example 4d, each successive chord retains four pitchclasses from the previous chord while a single pitchclass moves. Example 5 shows this process in the opening passage from Bartók’s String Quartet no. 3: in this passage, the cello, viola, and second violin hold a chromatic tetrachord throughout the first violin’s melody. The held tetrachord has interval series [1, 2, 11] (from lowest to highest notes), but the first violin’s melody lies in between the top two notes of this tetrachord, dividing the interval of 11 into two intervals.^{(7)} The first violin’s first two notes, Reversals, rotations, and circular interval series Example 6. A reversal and two rotations of a pitchclass segment. Reversals always preserve setclass membership, but rotations in general do not (click to enlarge) [9] In addition to swaps and adjacent swaps, there are two other types of permutation that are of interest: reversals and rotations. A reversal is a permutation that simply reverses the order of the intervals of a series. In the formal terms defined above, a permutation π is a reversal if π(s) = −s mod n for all s from 1 to n, where n is the length of the interval string. Example 6a shows the pitchclass segment (C, E, [10] A rotation preserves the ordering of an interval series but shifts the intervals cyclically. In other words, π is a rotation if π(s) = s + k mod n, where k is any integer. For example, there are four rotations of the interval series [4, 2, 3, 1]: itself, [2, 3, 1, 4], [3, 1, 4, 2], and [1, 4, 2, 3], the last two of which are shown in Examples 6c and 6d. Unlike reversals, rotations generally do not preserve setclass membership; the pitchclass segments in Examples 6a, 6c, and 6d are members of (02368), (01468), and (01368) respectively. There is a special type of interval series, however, for which rotations will always produce a member of the same set class: those whose intervals sum to zero mod 12, which I will call circular interval series. Circular interval series always describe pitchclass segments whose first and last notes are the same, and so rotating the series can produce the same pitchclass set in a different order, as shown in Example 7. In this example, the first interval series [9, 8, 5, 2] is circular because 9 + 8 + 5 + 2 = 0 mod 12. This interval series is rotated by one position to obtain the second series [8, 5, 2, 9], and as the pitch realization shows, the lowest two pitchclasses are thrown up to the top of the chord. The two pitchclass segments are different because the ordering has changed, but they both represent the same unordered pitchclass set.^{(9)} The same relationship exists between the second and third interval series in Example 7, and all three pitchclass sets are the same member of setclass (0237). Of course, the specific transposition level of the pitchclass segments is not given by the interval series, and so it is not always the case that rotations of a series will produce the same exact pitchclass set. However, rotations of a circular series will always produce a transposition of the original pitchclass set; in the specific case of the interval series [9, 8, 5, 2] shown in Example 7, all rotations will produce an Iform of setclass (0237) (i.e., a form related by inversion to the prime form of this setclass). Transposition and inversion Example 8. The inversion (b) of the interval series in (a) produces an inversion of the pitchclass segment (click to enlarge) [11] Transposition does not affect interval structure, as an interval series remains the same if its pitchclass segment is transposed. If the segment is inverted, however, the intervals will also become inverted. This applies no matter what the axis of (pitchclass) inversion is, just as long as the same axis is applied to all pitchclasses. Example 8a shows pitchclass segment (B, Relation to previous literature [12] The idea of describing an ordered set of pitchclasses by listing the intervals between adjacent members goes back at least to Richard Chrisman’s “successive intervalarrays” (1971), which were intended as a classification system for what we now call setclasses (Chrisman’s article predates Forte’s The Structure of Atonal Music [1973b]; he later relates his theory to Forte’s in Chrisman 1977). Successive interval arrays are essentially the same as interval series, but the pitchclass segments they describe are always in ascending order within the span of an octave (like a scale), and always begin and end on the same pitchclass. The successive intervalarrays therefore always represent circular interval series. The Romanian composer and theorist Anatol Vieru, who developed a theory of “modes” (pitchclass sets) independently of Forte and his North American colleagues, organizes them based on the interval successions that arise when the pitchclasses are arranged as a scale, resulting in a classification scheme very similar to Chrisman’s (Vieru 1993 and 1985; see also Nolan 2002, 294). While Chrisman does discuss “cyclical permutations” (= rotations) of the successive intervalarrays (which will always produce a transposition of the original pitchclass set), neither he nor Vieru discusses more general permutations that might produce members of different setclasses. [13] Roeder 1987 contains a more extensive study of the properties of interval series, with a focus on representing them geometrically. Roeder, who actually uses the term “interval series,” constructs “ordered interval spaces,” which are ndimensional toroidal graphs whose points represent interval series. (For example, the interval series [6, 1, 3], representing a fourelement pitchclass segment, is represented by the point (6, 1, 3) in a threedimensional torus.) These graphs reveal certain relationships among interval series, such as a shared interval in the same position or the presence of two adjacent intervals that sum to the same value. Roeder also constructs a metric for measuring “distances” between pitchclass segments based on their interval series, which provides an easy way to measure intervallic similarity (383–84).^{(11)} Certain permutations, such as reversals or rotations, are visible on Roeder’s graphs (through reflection about a linear axis), but more general permutations are not easily represented in Roeder’s geometrical spaces. Other authors, especially Morris (1995a) and Väisälä (1999), have discussed interval series as they relate to pitch (as opposed to pitchclass) structure. Morris’s PCINT is essentially an interval series, and he discusses the effects of certain pitch operations (such as Bernard’s “infoldings” and “unfoldings” [1987]) on PCINT and its pitchinterval counterpart INT.^{(12)} [14] With one exception, discussions of structures similar or identical to the notion of interval series do not consider the effects of permuting their component intervals. The lone study to deal in depth with the concept of interval permutations is Alan Chapman’s dissertation (1978), which introduces the concept of “VP [for Voice Pair] interval set”—his term for interval series. The VP interval set can be “prolonged” through expression in various permutations, such as “exchanges” (= adjacent swaps), “circular permutation” (= rotations) and “reversals.” Chapman deals exclusively with tetrachords presented as fourvoiced chords, and the VP interval sets always represent the three intervals in ascending order (bass/tenor, tenor/alto, alto/soprano). As a result, all of his permutations are rotations or reversals, or a combination of the two. Example 9. Based on Example 3.12 from Chapman 1978, 86. Permutations of a threemember interval series can express as many as three different setclasses. Interval series related by reversal are bracketed, and will always express the same setclass (click to enlarge) [15] One of the most interesting lines of inquiry taken in Chapman’s dissertation is an investigation of how different setclasses are related by interval permutation. Chapman notes that a threemember interval series can, when permuted, express as many as three different setclasses, the group of which he calls a “VP triple.” Example 9 adapts one of Chapman’s examples, which shows that the interval series [5, 4, 1] can be permuted to express setclasses (0237), (0137), and (0157). Three is the maximum number of different setclasses that a threemember series can express in permutation: of the six possible permutations of the series, three pairs will be related by reversal and therefore will express the same setclass, as previously demonstrated (these reversal pairs are bracketed in Example 9). Of course, many interval series cannot be permuted to express three different setclasses; in fact, as Chapman shows, only 34 of the 286 threemember interval series can be permuted to express three different tetrachordal setclasses (90). Chapman’s Chapter 3 categorizes these 286 series in terms of their various possibilities of setclass expression, demonstrating how the study of interval series relates to the study of pitchclass sets. Chapman’s categorization is somewhat unwieldy given the large number of possible interval series, and this is probably why this section was omitted from his Journal of Music Theory article based on his dissertation (Chapman 1981). It is also perhaps why the study of setclass relationships based on interval permutation did not take hold in the music theory community. Yet I believe that this aspect is one of the most promising in the study of interval permutations, and the following section represents an attempt to revitalize this idea by simplifying Chapman’s categorization and expanding on his preliminary investigation of the various ways in which different setclasses can be related by interval permutation. Interval Permutations and SetClass Relations [16] One source that was noticeably absent from my discussion above is Forte’s “basic interval patterns” (BIPs), defined and discussed in Forte 1973a. The reason for its omission is that the BIPs are fundamentally different from interval series: BIPs are unordered collections of interval classes. Nevertheless, one of Forte’s goals is to relate different setclasses based on their intervallic properties, which is also the goal of the current section. My approach will proceed in the opposite direction from Forte’s: Forte essentially asks, “given a setclass, what are the different configurations of intervals that can express this setclass,” while I will ask, “given an interval series, what are the different setclasses that are expressed by permutations of this interval series?” In the interest of space, the following discussion will be restricted to tetrachordal setclasses, with the understanding that the methodology employed may easily be extended to apply to setclasses of any cardinality. [17] To answer Forte’s question: For any tetrachordal setclass, there can be as many as 48 interval series of length three that express a member of that setclass.^{(13)} Example 10 shows the 48 different threemember interval series that express setclass (0236). The top line gives the 24 different orderings of the prime form {C, D, Example 11. a) Derivation of the four interval series [2, 1, 3], [1, 3, 6], [3, 6, 2], and [6, 2, 1] from the single circular interval series [2, 1, 3, 6]; b) The circular interval series [2, 1, 3, 6] plotted on a circle (click to enlarge) Example 12. The six circular interval series associated with setclass (0236) express six different orderings of the pitchclass set {C, D, (click to enlarge) [18] We can achieve a further reduction by dividing the 24 primeform interval series into six groups of four. Consider the four interval series [2, 1, 3], [1, 3, 6], [3, 6, 2], and [6, 2, 1], which correspond to the first chords of each “measure” in Example 10. These four series are almost rotations of one another, but not quite; in fact, they are all derived from the same interval series of length four. That series is the circular interval series [2, 1, 3, 6], and all four of the interval series mentioned above can be obtained by taking the first three intervals of some rotation of this circular interval series. Example 11 shows how this derivation works: Example 11a shows that if the intervals 2,1,3, and 6 are listed and repeated, then any three consecutive intervals will produce one of the four threeinterval series in question, while Example 11b plots this circular interval series on a circle such that any three consecutive clockwise intervals will produce one of these four threeinterval series. [19] All 24 of the primeform interval series of setclass (0236) shown in Example 10 can be derived in this way from one of the following six circular interval series: [2, 1, 3, 6], [2, 4, 9, 9], [3, 3, 8, 10], [3, 11, 4, 6], [6, 8, 1, 9], and [6, 9, 11, 10] Each of these circular interval series is associated with a specific ordering of the pitchclass set {C, D, Table 1. The 29 tetrachordal setclasses and their associated circular interval series (click to enlarge) [20] In each of the four “measures” of Example 10 (separated by double bars), there is exactly one series derived from each of these circular series. The total number of interval series for setclass (0236) has now been reduced from 48 to six such that each of the other 42 series is permutationally related to one of these six. Using this same method, we can find the circular interval series for every tetrachordal setclass. Table 1 shows the circular interval series associated with each of the 29 tetrachordal setclasses. To save space, the circular interval series in this table are written without brackets or commas and with the letters t and e standing for intervals 10 and 11. Each of these circular interval series can be rotated or reversed without changing its setclass association. Notice that the 15 symmetrical setclasses all have fewer than six circular interval series associated with them, while the 14 nonsymmetrical setclasses all have six. The relationship of interval series to symmetry will be explored in detail in a later section. Relationships Among SetClasses [21] Each fourinterval circular series can be permuted to express more than one setclass. While there are up to 24 distinct permutations of each fourinterval series, we know that reversals and rotations will not alter the setclass, so we will only want to look at permutations that are not equivalent to reversal and/or rotation. In fact, rotational and reversal equivalence partition the 24 permutations into only three equivalence classes—in other words, there exist three permutations that are not rotations or reversals of each other such that every other permutation is a rotation and/or reversal of one of these three.^{(14)} This means that every one of these circular interval series can be permuted to express only three (or fewer) setclasses. [22] Consider the circular interval series [2, 1, 3, 6]. We saw in the previous section that this series is associated with setclass (0236). Two permutations of this series are [1, 2, 3, 6] and [1, 3, 2, 6]—no two of these three series are related by rotation or reversal, and every other permutation of [2, 1, 3, 6] can be obtained by reversals and/or rotations of one of these three. Reading from Table 1, we see that the series [1, 2, 3, 6] expresses setclass (0136) and the series [1, 3, 2, 6] expresses (0146). In other words, setclasses (0236), (0136), and (0146) are related via permutations of the circular interval series [2, 1, 3, 6]. Example 13. Interval permutations on setclass (0236) produce members of setclasses (0136), (0146), and (0236) (click to enlarge) [23] This means that a pitchclass segment in setclass (0236) that instances the interval series [2, 1, 3, 6] can progress to a member of (0136) or (0146) via interval permutation. (By “instances the interval series [2, 1, 3, 6]” I mean that this segment’s interval series, which will have length three, is derived from the fourinterval circular series [2, 1, 3, 6] in the manner of Example 11.) Furthermore, this means that any interval permutation on this pitchclass segment will result in a member of (0136), (0236), or (0146). Example 13 gives some examples of these permutations: Example 13a shows an adjacent interval swap that takes a member of (0236) to a member of (0136); Example 13b shows an adjacent interval swap that takes a member of (0236) to a member of (0146); Example 13c shows a rotation that takes a member of (0236) to a member of (0136); and Example 13d shows a reversal that takes a member of (0236) to a different member of (0236). Table 2. The number of setclasses an interval series can express via permutation depends on how many distinct intervals it contains (click to enlarge) [24] Many fourinterval series express even fewer than three setclasses. The number of setclasses an interval series expresses depends on how many of its intervals are the same. If all four intervals are different (i.e., the series is of the form abcd), then it will have 24 distinct permutations, and so will express three different setclasses (as was the case with [2, 1, 3, 6] above). If two of the intervals are the same (aabc), there are only 12 distinct permutations, so this type of interval series will only express two different setclasses. For example, the series [3, 3, 8, 10] expresses (0236) in that order and expresses (0134) in the permutation [3, 8, 3, 10]. Every other permutation of this series is related to one of these two via rotation and/or reversal, so no other setclasses are obtainable by permutation. The same holds for series of the form aabb; for example, [1, 1, 5, 5] expresses (0127) and [1, 5, 1, 5] expresses (0167). Series of the form aaab or aaaa can only express one setclass because every permutation is equivalent to a rotation or a reversal of this series; an example of the former is [5, 5, 5, 9], which expresses setclass (0257), and an example of the latter is [3, 3, 3, 3], which expresses setclass (0369). These relationships are summarized in Table 2. [25] Since a setclass can have up to six fourinterval series associated with it, and each of those series can be permuted to express as many as two other setclasses, then it would seem that each setclass could be related via interval permutation to as many as 12 other setclasses. Actually, the maximum number of related setclasses is only six. As seen in the previous section, when all intervals in a series are inverted, the resultant pitchclass segment is in the same setclass as the original. Two interval series that are inversionally related will therefore express the same three (or fewer) setclasses in their various permutations. For example, we saw above that the interval series [2, 1, 3, 6] expresses setclasses (0236), (0136), and (0146) via permutation; the interval series [10, 6, 9, 11], which is a retrograde inversion (plus rotation) of this series, also express these same three setclasses (this can be verified in Table 1). The six fourinterval series associated with a particular setclass will group into three inversionallyrelated pairs, and each of these three pairs will be able to express up to two other setclasses. A setclass can therefore be related to up to six other setclasses by permutation of its interval series. Example 14. Setclass (0137) is related by permutation to six other setclasses (click to enlarge) Example 15. Setclass (0148) is related by permutation to three other setclasses (click to enlarge) Table 3. The 29 setclasses and their permutational relations (click to enlarge) [26] It is rarely the case that a setclass is in fact related to six other setclasses, however. For this to occur, all six of its associated fourinterval series would have to have four distinct intervals (see Table 2). In other words, this setclass must not have two instances of any single interval—this means that its interval vector must consist of all 1s and/or 0s. The only tetrachordal setclasses that have this property are (0137) and (0146), the two “allinterval tetrachords.” Example 14 graphically shows the allinterval tetrachord (0137) and its six related setclasses. The interval series that relate each group of three setclasses are shown within the triangles—these interval series are listed with their intervals in increasing order rather than in any specific order, since any ordering will result in a member of one of the three indicated setclasses. These interval series are grouped into inversionallyrelated pairs, as discussed above. Because (0137) is an allinterval setclass, all six of these interval series are of the form abcd. The setclass (0148), on the other hand, is associated with only three other setclasses, because all of its interval sets are of the form aabc, as shown graphically in Example 15. [27] Table 3 lists all of the permutation relationships among tetrachordal setclasses. The left column lists the 29 tetrachordal setclasses, and each of the three columns on the right lists setclasses that share a circular interval series with the setclass on the left, as well as the two inversionallyrelated interval series that relate that particular group of setclasses.^{(15)} For example, setclass (0146) appears in the left column of the second row; the second column shows that setclasses (0236) and (0136) are related to (0146) by permutation of the interval series [1, 3, 2, 6] and [6, 10, 9, 11] (which are abbreviated as 1326 and [28] Table 3 is organized based on how many other setclasses a given setclass is related to from most (six) to fewest (one). This organization divides the 29 tetrachordal setclasses into ten groups, as shown in the table. The first group contains those setclasses that are related to the maximum six other setclasses, which are precisely the two allinterval tetrachord classes (0137) and (0146), as discussed above. The second group contains setclasses that are related to five other setclasses; these are all nonsymmetrical setclasses that contain exactly one symmetrical trichord subset. For example, both (0125) and (0126) contain the symmetrical subset (012), (0135) contains (135), etc. Because of this symmetrical subset, these setclasses will contain two instances of one interval (and its complement); for example, (0125) and (0126) contain two instances of interval 1 (and 11), and (0135) contains two instances of interval 2 (and 10). Because this is the only symmetrical subset, no other interval will appear twice. Therefore, two of the six interval sets associated with each of these setclasses will be of the form aabc, and these two will be inversions of each other. This pair of interval sets will therefore relate to only one other setclass (see Table 2), while the other two pairs will relate to two. [29] There is an interesting relationship between the number of related setclasses and interval vectors. Of course, the allinterval tetrachords, which compose Group 1 in Table 3, both have interval vector <111111>. The setclasses in Group 2 do not all have the same interval vector, but their interval vectors are all very similar: they all have one 0, four 1s, and one 2. For example, (0125) has interval vector <211110> and (0136) has interval vector <112011>. All of the groups in Table 3 comprise setclasses whose interval vectors are related in this way. These interval vector entries are listed on the right side of the heading for each group in Table 3 in ascending order. The interval vector types are not necessarily unique to one specific group: for example, Groups 3, 4, and 5 all have the interval vectors with two 0s, two 1s, and two 2s. This is because this type of interval vector is achieved by three different types of setclasses: setclasses that have two different symmetrical trichord subsets, setclasses that have two pairs of trichord subsets of the same type, and setclasses that have two symmetrical subsets and two subsets of the same type. Example 16. Some interval permutations on tetrachords produce trichords with one note doubled (click to enlarge) [30] The setclasses in Groups 3 and 4 are all related to four other setclasses, but in Group 4, two of those four are trichords rather than tetrachords. A fourinterval circular series can produce a trichord when it is composed of two pairs of complementary intervals; i.e., if it is of the form [a, b, −a, −b]. Consider Example 16: the interval series [3, 11, 9] (derived from the circular series [1, 3, 11, 9]) expresses setclass (0134), on the left of the example, while its permutation [11, 3, 9] expresses the trichord setclass (013), in the middle of the example. This is because intervals 3 and 9 are complements (mod 12), so when they are adjacent they produce a doubled note. In this case, the A is doubled, and so we get an instance of the multisetclass (0013). The same logic applies to the permutation [3, 9, 11], on the right of the example, which expresses multisetclass (0114). Trichord multisetclasses are preceded by the symbol “#” in Table 3.^{(16)} [31] Groups 6, 7, and 8 contain setclasses in which one interval appears three times. The setclasses in Group 7 are generated by a single interval—for example, (0123) is generated by interval 1—and so they all contain two interval series of the form aaab, which will not express any other setclass (as demonstrated in Table 2). Groups 6 and 8, both of which consist of only one setclass, contain setclasses that have a (048) subset. These setclasses will contain three instances of interval class 4, but these three will never appear in the same interval series (since they sum to 0 mod 12, the fourth interval would have to be 0 to make it circular, which would express a trichord rather than a tetrachord). However, every one of their circular interval series will contain two instances of interval 4 (or 8), and hence will be of the form aabc, which limits the total number of related setclasses to three. Table 3 shows that Group 8’s setclass—(0248)—is related to only two other setclasses because it contains two (026) subsets. [32] Group 9 contains the two tetrachord classes that are transpositionally symmetrical, i.e., those that are made up of two disjoint tritones. Because of the two tritones, there will be two different interval series that express the same setclasses: for example, the interval series [1, 6, 6, 11] and [5, 6, 6, 7] both express (0167) and the multisetclass (0166), despite the fact that these two interval series are not related by inversion. Group 10, of course, contains only setclass (0369), which is the most symmetrical, and therefore least intervallically diverse, of the tetrachord classes. There is a loose relationship between these ten groups of setclasses and symmetry; for the most part, the higher the group number, the greater the level of symmetry. This is certainly true at the extremes: Groups 1 through 3 contain all nonsymmetrical setclasses, with the number of symmetrical subsets increasing with each subsequent group, and Groups 9 and 10 contain the most symmetrical setclasses. However, this is less true in the middle groups: Group 6 contains (0148) which is neither inversionally nor transpositionally symmetrical, while Groups 4 and 5 contain all inversionally symmetrical setclasses. It is always true, however, that members of the same group have the same level of symmetry. Paths Among SetClasses Example 17. A progression through different setclasses that alternates retaining interval content and pitchclass content (click to enlarge) [33] The relationships shown in Table 3 permit progressions like that shown in Example 17. This example shows a chord progression that alternately permutes the intervals and revoices the pitchclasses. The first chord is a (0137) tetrachord that instances the interval series [6, 8, 9], which progresses to an (0147) tetrachord via an adjacent interval swap. This (0147) tetrachord is shown as the pitchclass segment (F, B, [34] What if we were to continue the progression in Example 17 until we could not proceed to a setclass that we had not yet encountered? Would we be able to cycle through all 29 tetrachordal setclasses? The answer to that question is almost. In fact, we can construct a progression that hits all but four of the tetrachord classes. These four outliers are (0369), (0246), (0248), and (0268). Table 3 shows that (0369) is not related by permutation to any other tetrachord class, and that (0246), (0248), and (0268) are related only to each other.^{(17)} Therefore, it is impossible to progress to a member of (0369) via an interval permutation from any other setclass, and it is impossible to progress to (0246), (0248), or (0268) from any setclass that is not one of those three. The 25 other tetrachord classes can all proceed to each other via interval permutations and/or revoicings in the manner of Example 17. Example 18. Diagram showing tetrachordal setclasses that can be related by interval permutation (click to enlarge) [35] There can be many different paths between any two setclasses. For example, Example 17 shows that (0137) can progress to (0358) via (0147) and (0258). However, looking back at Table 3, we see that (0137) can bypass (0147) and progress directly to (0258) and then (0358). In fact there are many other paths between (0137) and (0358); for example, (0137) – (0136) – (0235) – (0135) – (0247) – (0358). Example 18 is a graphic version of Table 3 that visually represents these types of paths. This figure contains one node for each tetrachordal setclass, and setclasses that can be related by interval permutation are connected with double lines. Notice that (0369) is not connected to any other node, and (0246), (0248), and (0268) are separated from the rest of the nodes, as discussed above. The progression from Example 17 is traced in dark lines on this diagram: we begin with (0137) on the west (left) side, then proceed southeast to (0147), then north to (0258), then southeast to (0358) and southeast again to (0247). Interval Permutation Spaces Table 4. Distances between tetrachordal setclasses based on the space of Example 18. Undefined distances are marked with an asterisk (click to enlarge) Example 19. Straus’s “parsimonious voiceleading space for tetrachord classes.” (Reproduced from Straus 2005a, 56.) (click to enlarge) [36] Example 18 defines a compositional space on the tetrachordal setclasses.^{(18)} Setclasses that are connected in this space are related by interval permutation—meaning that there exists a pitchclass segment in one that is related by interval permutation to some pitchclass segment in the other. Based on this space, we can define a distance measure between two setclasses S and T as the smallest number of “moves” in the space that can take us from S to T. For example, the distance from (0137) to (0358) is 2, because (0137) is connected to (0258) which is connected to (0358), and there is no shorter path between these two setclasses. This distance function satisfies the criteria of a metric space: the distance between S and T is always nonnegative and is zero if and only if S = T, it is reflexive (DIST(S, T) = DIST(T, S)), and the triangle inequality is satisfied: DIST(S, T) + DIST(T, R) ≥ DIST(S, R).^{(19)} Table 4 gives the distances between any pair of tetrachordal setclasses based on the space in Example 18.^{(20)} These distances reflect how intervallically similar two setclasses are, with smaller distances corresponding to a greater level of intervallic similarity. [37] This space offers an alternative to the more typical voice leadingbased spaces, such as those presented in Straus 2005a and Callender, Quinn, and Tymoczko 2008 (hereafter CQT).^{(21)} Straus’s tetrachord space, reproduced as Example 19, relates tetrachordal classes that are almost transpositionally related—in Straus’s terms, that are related by “neartransposition.” Setclasses that have this relationship are related by parsimonious voice leading, i.e., there exists a parsimonious move (one voice moving a semitone while the others stay) that relates the two setclasses. CQT’s space is a continuous version of Straus’s space; while CQT derive their space by folding ℝ_{n} in various ways, the result is very similar to Straus’s.^{(22)} Although both of these spaces are based on voiceleading proximity, and hence are fundamentally different from the space in Example 18, it is useful to compare the different spaces to see what, if any, similarities do exist. One similarity that is immediately apparent is that the setclasses (0123), (0124), (0125). (0126), and (0127) form a line in both spaces. In addition, the two setclasses that only connect to one other setclass in Example 18—(0123) and (0167)—have this same property in Straus’s space. In general, however, the voiceleading spaces are quite different from the interval permutation space, and model different compositional practices. Analytical Examples: Schoenberg’s Op. 15 Song Cycle Song No. 1, measures 1–8 Example 20. Schoenberg, The Book of the Hanging Gardens, Op. 15, No. 1, measures 1–8, piano (click to enlarge and hear the audio) [38] This final section applies the theory of interval permutations in brief analyses of selections from Schoenberg’s Book of the Hanging Gardens song cycle, op. 15. These analyses are not intended to serve as validation of this theory, but rather they aim to demonstrate various ways in which interval permutations might be applied in conjunction with traditional setbased analysis to the posttonal repertoire. Example 20 shows the opening piano part of the first song of the cycle. This passage opens with the melodic pitchclass segment ( [39] Let us set aside the second phrase for now and focus on the other three. The first five notes form an important motive that recurs throughout the song in both the voice and piano parts. Allen Forte (1992, 288–90) points out that this motive’s setclass, (0124), dominates the entire phrase, as any four consecutive notes in the first two measures will constitute a form of this tetrachord class. (The second and third of these are formed by the interval series [11, 2, 2] and its reversal [2, 2, 11], shown under the staff in Example 20.) The third phrase, in contrast, expresses setclass (0135). Looking back at Example 18 or Table 3 will tell us that (0124) and (0135) are related by permutation of the circular interval series [2, 8, 3, 11], and Schoenberg exploits this relationship in the melodic intervals of these two phrases. Even though Forte considers these two phrases to be separate motives, we can see that they are connected in their interval structure, which unifies their sound despite the difference in setclass. After this third phrase, the held E is followed by a repeat of the opening motive, [40] The second phrase begins as a repeat of the first but gets interrupted after three notes by a high [41] Examining the interval structure of the opening measures to the op. 15 song cycle reveals connections beyond those of setclass analysis alone. We saw in Example 20 that the (0124) sets in the first and fourth phrases and the (0135) set in the third phrase are all related by interval permutation. The second phrase, on the other hand, is the anomaly with its unrelated interval series. This phrase dismantles the first phrase by abruptly interrupting what at first seems to be a repeat: the high Song No. 11, measures 1–4 Example 22. Song No. 11, measures 1–4 (click to enlarge and hear the audio) Example 23. Song No. 11, measures 22–24, voice (lower staff) and piano RH (click to enlarge) [42] Example 22 shows the first four measures of Song no. 11. The opening motive in the piano’s right hand is a linear statement of the pitchclass segment ( Example 24. Different interval series of Lewin’s X and N chords in measures 3–4 of Song No. 11 (click to enlarge) [43] David Lewin (1973) highlights two chords in measures 3–4, which are reproduced in Example 24a. Lewin calls these two tetrachords “X” and “N” respectively; these specific chords reoccur throughout the song, often with the same voicings. The X tetrachord, a member of (0125), has interval series [10, 5, 8] (reading upwards). This interval series can also express, via permutation, setclasses (0126) and (0146). The N tetrachord is a member of (0126), but its interval series [10, 6, 7] is not a permutation of the first chord’s. Lewin points out that both chords consist of a (012) trichord paired with pitchclass F; this (012) subset is transposed down a semitone from X to N, from {C, Song No. 14 [44] I will conclude this study as I began it, with some analytical observations on Song no. 14 from The Book of the Hanging Gardens. Example 25 shows the first piano gesture of this song, previously seen in Examples 1 and 2. This gesture has interval series [11, 10, 6, 7, 11], but as the example shows, the first four intervals of this series, namely [11, 10, 6, 7], form an important subseries. This shorter interval series is associated with what I called “rhythmic motive x” in Example 1, while the remaining interval 11 is associated with “rhythmic motive y.” Both of these rhythmic motives recur often throughout this short song, and rhythmic motive y—quarter note–eighth note—always pairs with the pitch motive of a descending semitone. In the case of this opening piano gesture, motive y’s descending semitone produces pitchclass A, which was already heard within rhythmic motive x earlier that measure. So the total pitch content of these six notes is a pentachord, specifically a member of setclass (01236). The normal form of this pentachord is shown on the lower staff of Example 25. The fourinterval series that expresses this pentachord in measure 1, namely [11, 10, 6, 7], has 24 distinct permutations, which can express twelve different setclasses (accounting for the fact that reversals express the same setclass). In fact, because the subset {11, 6, 7} sums to zero mod 12, any permutation in which these three intervals are concurrent (i.e., whenever interval 10 is first or last) will contain a duplicate pitchclass, and therefore will express a tetrachord rather than a pentachord. This applies to twelve of the 24 permutations of this interval series, and so six of the twelve permutationally related setclasses will be tetrachords. Specifically, this interval series can be permuted to express tetrachordal setclasses (0126), (0127), (0136), (0137), (0146), and (0157), and pentachordal setclasses (01236), (01237), (01257), (01268), (01368), and (02368). Example 26. Song No. 14, measures 2–3, voice (click to enlarge) Example 27. Song No. 14, measure 4, voice (click to enlarge) Example 28. Song No. 14, measure 11, piano (click to enlarge) [45] As we saw earlier, a permutation of this interval series is immediately presented in the vocal phrase in measure 2, shown in Example 26. This phrase contains rhythmic motives x and y in reverse order, beginning with y’s descending semitone and ending with x, the latter of which instances the interval series [11, 7, 10, 6], as shown bracketed in the example. This interval series is a permutation of the interval series in measure 1 and expresses setclass (01268). The “extra” two intervals that precede this series are 11 and 7—the same two that begin the series proper—and the interval 11 accompanies motive y. In contrast to measure 1, these extra intervals add one new pitchclass (D) to x’s pentachord, making the entire phrase a member of the hexachordal setclass (012678), as shown in the lower staff of Example 26. [46] Example 27 shows a different permutation of our interval series that occurs in the vocal line of measure 4. In this measure, the interval series expresses the tetrachordal setclass (0127) with the [47] The final measure of this song contains another permutation of this interval series, namely [11, 10, 7, 6], as shown in Example 28. This example, like Examples 25 and 26, expands this interval series by affixing repetitions of some of the component intervals after the interval series is complete. In this case, intervals 6 and 11 reappear at the end. This last interval 11 is created by a final iteration of motive y. The first half of this measure is related to the piano’s opening gesture (Example 25), which instanced rhythmic motive x with interval series [11, 10, 6, 7]. This series is related to the current series by adjacent interval swap, and the pitchclass segment (B, [48] The first five pitchclasses in measure 11 create a member of setclass (01237), which is a new setclass that is arrived at via interval permutation from our previous examples. The statement of motive y at the end adds the pitchclass D, which makes the total pitchclass content of this measure a member of setclass (012367). In these four examples from Song no. 14, we have seen members of setclasses (01236), (01268), (012678), (0127), (01237), and (012367) all created from various combinations of intervals 6, 7, 10, and 11. These intervals interact with rhythmic and pitch motives from the first notes of the song through the final measure. The analysis of this song through the lens of interval permutations demonstrates Schoenberg’s ability to present highly varied musical material under a single unifying principle. Conclusion [49] The concept of interval permutations is at its core a remarkably simple idea: rearrange the intervals of an ordered set of pitchclasses to get a different ordered set of pitchclasses. Perhaps this simplicity is to blame for why music theorists did not pick up on Alan Chapman’s exposition of the idea in 1978, despite the fact that the concept of interval series remained in common use through the work of Chrisman, Regener, Morris, Roeder, and others, as discussed earlier. Yet, as evidenced through the brief discussions above of a few Schoenberg songs, expanding our analytical vocabulary to include interval permutations reveals relationships that are often missed by setclassbased analysis alone. The framework’s simplicity is perhaps its most attractive feature, as it guarantees a direct connection with our perception: the basic elements are intervals and the basic operation is reordering, and so it is quite easy to perceive an aural similarity between two permutationally related segments. Furthermore, since interval permutations do not model traditional transposition and inversion operations, this similarity is fundamentally different from that between, say, two inversionally related members of (0146). This difference does not mean that the two analytical techniques are incompatible, though, and in fact interval permutations interact with setclass theory in interesting ways. [50] These implications were explored in the middle section of this article, specifically relating to tetrachordal setclasses. This led to a taxonomy of the 29 tetrachord classes based on their intervallic properties, as shown in Table 3, as well as a spatial representation of these setclasses in Example 18 and a metric for measuring distances between two setclasses in Table 4. While the discussion focused on tetrachords, the methodology is easily generalizable to larger setclasses. The complexity increases quickly as the sets get larger, however; each circular interval series for a pentachordal setclass can be permuted to express as many as twelve different setclasses (for tetrachords, this number was three), and for hexachords each series can theoretically express as many as 60 different setclasses, though in practice several of these will be duplicates. Nevertheless, the interval permutation framework has the capability of revealing salient relationships among sets of all sizes, and I am confident that further applications of this tool will be rewarding in both the analytical and theoretical realms.
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Väisälä, Olli. 1999. “Concepts of Harmony and Prolongation in Schoenberg’s Op. 19/2.” Music Theory Spectrum 21, no. 2: 230–59. Väisälä, Olli. 1999. “Concepts of Harmony and Prolongation in Schoenberg’s Op. 19/2.” Music Theory Spectrum 21, no. 2: 230–59. Vieru, Anatol. 1985. “Modalism—a ‘Third World.’” Perspectives of New Music 24, no. 1: 62–71. Vieru, Anatol. 1985. “Modalism—a ‘Third World.’” Perspectives of New Music 24, no. 1: 62–71. Vieru, Anatol. 1993. The Book of Modes, 2nd edition. Bucharest: Editura Musicala. —————. 1993. The Book of Modes, 2nd edition. Bucharest: Editura Musicala. Footnotes1. While pitchclass segments can theoretically be ordered in any way that the analyst deems salient, for the purposes of this paper I will stick to temporal ordering for melodic lines and registral ordering (lowest to highest) for simultaneities. 2. Setclasses will be identified by their prime forms following the conventions set out in Straus 2005b, with parentheses and no commas between successive elements. 3. The standard “normal” and “prime” orderings are convenient for identifying features of a pitchclass set but generally do not represent an immediately perceptible ordering. See Lewin 1977 and Benjamin 1979 for discussions of ordering of pitchclass sets. 4. Dummit and Foote’s Abstract Algebra textbook (2003) contains a basic discussion of symmetric groups in relation to group theory (29–33) as well as a detailed introduction to group actions (Chapter 4). 5. If all of the intervals in X and Y are distinct, then there will be exactly one permutation that maps X to Y, but if there are duplicate intervals there will be more than one. Furthermore, if there are duplicate intervals there exists at least one permutation besides the identity that maps an interval series to itself. The set of all permutations that map an interval series X to itself is called the “stabilizer of X,” and this set will be a subgroup of S_{n} (see Dummit and Foote 2003, 52–52). For example, the interval series [1, 6, 3, 3] maps to itself under the identity permutation as well as the permutation that swaps the third and fourth intervals. These two permutations together form a subgroup of S_{4}. 6. The standard mathematical term for these is “transpositions,” but to avoid confusion with pitchclass transposition, I will use “swaps” instead. 7. Chrisman (1971, 74–75) discusses such “partitionings” of intervals. 8. Straus (2005b, Chapter 5) discusses this relationship in reference to twelvetone series. 9. Since the doubled note is different, these represent different pitchclass multisets even when ordering is ignored. Multisets and multisetclasses will be discussed in the following section; see also Robinson 2009. 10. Contextual inversions are loosely defined as inversions whose axis is defined in terms of members of the set rather than a fixed point. The NeoRiemannian LPR group, for example, is a group of contextual transformations (see especially Cohn 1996 and 1998 and Douthett and Steinbach 1998). For investigations of the mathematical properties of contextual transformations, see Clough 1998, Kochavi 1998, and Fiore and Satyendra 2005, and for more general discussions of contextual transformations, see Lambert 2000 and Straus 2011. 11. Roeder’s metric is defined as “the absolute sum of the differences, considered as interval classes, of the intervals in each order position of X and Y respectively,” where X and Y are two interval series (1987, 383). So, for example, the difference between [6, 3, 1] and [5, 2, 4] would be 1 + 1 + 3 = 5. This idea foreshadows Straus’s “voiceleading offset” (2003), which measures how close two (unordered) pitchclass sets are to being transpositions of one another. 12. In earlier writings, Morris uses INT to denote an ordered set of intervals of either type. See Morris 1987 and 1991. 13. This is because there are 4! = 24 possible orderings of the four pitchclasses in both prime and inverted forms. 14. Since reversals have order two in S_{n}, in S_{4} there are 24/2 = 12 equivalence classes based on reversal equivalence; and since rotations have order four, these 12 will be divided into 12/4 = 3 equivalence classes based on both rotation and reversal equivalence. This means that any set of four or more permutations will necessarily include a pair of permutations that are related to each other by rotation and/or reversal. 15. The interval series are ordered based on the setclass in the left column, as they were in Table 1; for example, [1, 2, 4, 5] appears in the row of (0137) and its permutation [1, 4, 2, 5] appears in the row of (0157). 16. In general, an interval series of length n will express a setclass of size n + 1. However, if any subseries of that series sums to 0 (mod 12), that will produce a doubled note, reducing the size of the setclass by 1. Circular series have at least one such subseries (namely the entire series), so they will express setclasses of length n at the most. This is why the fourinterval series in Table 3 express tetrachords rather than pentachords. In general, a series of length n with k distinct (but not necessarily disjoint) subseries that sum to 0 (mod 12) will express a setclass of size (n − k) + 1. The multiset expressed by an interval series of length n will always contain n + 1 (not necessarily distinct) members (in other words, the pitchclass segment that this interval series expresses will have length n + 1, but some of its notes might be the same). 17. This is because all of these setclasses contain intervals that fall entirely within a subgroup of ℤ_{12}: (0369) contains only intervals that are multiples of 3, which are all members of the subgroup {0, 3, 6, 9}, or ℤ_{12}/ℤ_{3}, and (0246), (0248), and (0268) contain only even intervals, which are all members of the subgroup {0, 2, 4, 6, 8, 10}, or ℤ_{12}/ℤ_{2}. No combination of these intervals will sum to any interval outside the subgroup, and so permutations must remain in the subgroup. Every circular interval series of every other tetrachord class contains at least two odd intervals and at least two intervals that are not multiples of 3. 18. For more on the general concept of compositional spaces, see Morris 1987, 1995b, and 1998. 19. Because the space is not connected, however, this is not a welldefined metric space, since there exist S and T such that DIST(S, T) is undefined. The subspace of the 25 connected setclasses is a welldefined metric space. 20. Due to the constraints of two dimensions, the physical distances on Example 18 do not correspond at all to the distances shown in Table 4. 21. Other examples of voiceleading spaces can be found in Morris 1998, Cohn 2003, and Tymoczko 2011. 22. CQT’s space is difficult to visualize; an attempt is made in Figure S5, E through H, in the supporting online material to their paper. The relationship of Straus’s spaces to CQT’s is more easily seen in their respective trichord spaces; see Straus 2005a, 54, 54, and CQT 2008, 347. Tymoczko 2011, Chapter 3, expands on the spaces from CQT, but does not spend much time discussing setclass space. 23. See also the analysis in Straus 2005b, 47–52, which also focuses on statements of (0347) throughout the song. Straus admits that his analysis is “heavily indebted” to Lewin’s (78). While pitchclass segments can theoretically be ordered in any way that the analyst deems salient, for the purposes of this paper I will stick to temporal ordering for melodic lines and registral ordering (lowest to highest) for simultaneities. Setclasses will be identified by their prime forms following the conventions set out in Straus 2005b, with parentheses and no commas between successive elements. The standard “normal” and “prime” orderings are convenient for identifying features of a pitchclass set but generally do not represent an immediately perceptible ordering. See Lewin 1977 and Benjamin 1979 for discussions of ordering of pitchclass sets. Dummit and Foote’s Abstract Algebra textbook (2003) contains a basic discussion of symmetric groups in relation to group theory (29–33) as well as a detailed introduction to group actions (Chapter 4). If all of the intervals in X and Y are distinct, then there will be exactly one permutation that maps X to Y, but if there are duplicate intervals there will be more than one. Furthermore, if there are duplicate intervals there exists at least one permutation besides the identity that maps an interval series to itself. The set of all permutations that map an interval series X to itself is called the “stabilizer of X,” and this set will be a subgroup of S_{n} (see Dummit and Foote 2003, 52–52). For example, the interval series [1, 6, 3, 3] maps to itself under the identity permutation as well as the permutation that swaps the third and fourth intervals. These two permutations together form a subgroup of S_{4}. The standard mathematical term for these is “transpositions,” but to avoid confusion with pitchclass transposition, I will use “swaps” instead. Chrisman (1971, 74–75) discusses such “partitionings” of intervals. Straus (2005b, Chapter 5) discusses this relationship in reference to twelvetone series. Since the doubled note is different, these represent different pitchclass multisets even when ordering is ignored. Multisets and multisetclasses will be discussed in the following section; see also Robinson 2009. Contextual inversions are loosely defined as inversions whose axis is defined in terms of members of the set rather than a fixed point. The NeoRiemannian LPR group, for example, is a group of contextual transformations (see especially Cohn 1996 and 1998 and Douthett and Steinbach 1998). For investigations of the mathematical properties of contextual transformations, see Clough 1998, Kochavi 1998, and Fiore and Satyendra 2005, and for more general discussions of contextual transformations, see Lambert 2000 and Straus 2011. Roeder’s metric is defined as “the absolute sum of the differences, considered as interval classes, of the intervals in each order position of X and Y respectively,” where X and Y are two interval series (1987, 383). So, for example, the difference between [6, 3, 1] and [5, 2, 4] would be 1 + 1 + 3 = 5. This idea foreshadows Straus’s “voiceleading offset” (2003), which measures how close two (unordered) pitchclass sets are to being transpositions of one another. In earlier writings, Morris uses INT to denote an ordered set of intervals of either type. See Morris 1987 and 1991. This is because there are 4! = 24 possible orderings of the four pitchclasses in both prime and inverted forms. Since reversals have order two in S_{n}, in S_{4} there are 24/2 = 12 equivalence classes based on reversal equivalence; and since rotations have order four, these 12 will be divided into 12/4 = 3 equivalence classes based on both rotation and reversal equivalence. This means that any set of four or more permutations will necessarily include a pair of permutations that are related to each other by rotation and/or reversal. The interval series are ordered based on the setclass in the left column, as they were in Table 1; for example, [1, 2, 4, 5] appears in the row of (0137) and its permutation [1, 4, 2, 5] appears in the row of (0157). In general, an interval series of length n will express a setclass of size n + 1. However, if any subseries of that series sums to 0 (mod 12), that will produce a doubled note, reducing the size of the setclass by 1. Circular series have at least one such subseries (namely the entire series), so they will express setclasses of length n at the most. This is why the fourinterval series in Table 3 express tetrachords rather than pentachords. In general, a series of length n with k distinct (but not necessarily disjoint) subseries that sum to 0 (mod 12) will express a setclass of size (n − k) + 1. The multiset expressed by an interval series of length n will always contain n + 1 (not necessarily distinct) members (in other words, the pitchclass segment that this interval series expresses will have length n + 1, but some of its notes might be the same). This is because all of these setclasses contain intervals that fall entirely within a subgroup of ℤ_{12}: (0369) contains only intervals that are multiples of 3, which are all members of the subgroup {0, 3, 6, 9}, or ℤ_{12}/ℤ_{3}, and (0246), (0248), and (0268) contain only even intervals, which are all members of the subgroup {0, 2, 4, 6, 8, 10}, or ℤ_{12}/ℤ_{2}. No combination of these intervals will sum to any interval outside the subgroup, and so permutations must remain in the subgroup. Every circular interval series of every other tetrachord class contains at least two odd intervals and at least two intervals that are not multiples of 3. Because the space is not connected, however, this is not a welldefined metric space, since there exist S and T such that DIST(S, T) is undefined. The subspace of the 25 connected setclasses is a welldefined metric space. Due to the constraints of two dimensions, the physical distances on Example 18 do not correspond at all to the distances shown in Table 4. CQT’s space is difficult to visualize; an attempt is made in Figure S5, E through H, in the supporting online material to their paper. The relationship of Straus’s spaces to CQT’s is more easily seen in their respective trichord spaces; see Straus 2005a, 54, 54, and CQT 2008, 347. Tymoczko 2011, Chapter 3, expands on the spaces from CQT, but does not spend much time discussing setclass space. See also the analysis in Straus 2005b, 47–52, which also focuses on statements of (0347) throughout the song. Straus admits that his analysis is “heavily indebted” to Lewin’s (78).
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