Volume 14, Number 3, September 2008
Copyright © 2008 Society for Music Theory
Maximally AlphaLike OperationsGuy CapuzzoKEYWORDS: Maximally alphalike operations, alpha, Zrelation, Zpair, Mrelation, TTOs, mappings, pcsets, transformational network ABSTRACT: Any two Zrelated setclasses will map onto one another under 1) T_{n}M or T_{n}MI, or 2) T_{n}M or T_{n}MI in tandem with Morris’s alpha operations, or 3) maximally alphalike operations, the original contribution of the present paper. This brief “research notes” paper explores the theoretical formulation and analytical application of maximally alphalike operations.

Example 1. Dallapiccola, Quattro Liriche de Antonio Machado, III (1948), mm. 80–85 (click to enlarge) [1] Example 1 shows an excerpt from a Dallapiccola song.^{(1)} The excerpt contains four chords, labeled X, Y, T_{6}(Y), and T_{6}(X). The union of X and Y forms the pc aggregate, as does the union of T_{6}(Y) and T_{6}(X). The passage resists an overarching transformational network such as that at the bottom of Example 1 because there is no T_{n}, T_{n}I, T_{n}M, or T_{n}MI operation that will map the X and Y forms onto each other. The dashed arrows in the network represent this limitation. [2] The reason why X and Y cannot map onto one another is that they are Zrelated.^{(2)} However, not all Zpairs (two Zrelated scs) work this way. To explain, I shall divide the twentythree Zpairs (under the traditional equivalence operations T_{n} and T_{n}I) into three categories. Example 2 shows the first category, Zrelated/Mrelated. Here each sc maps under T_{n}M or T_{n}MI onto the other sc in the same Zpair; the two scs are thus Zrelated and Mrelated.^{(3)} Example 3 shows the second category, Zrelated/Mvariant. Here each sc maps under T_{n}M or T_{n}MI onto a sc in a different Zpair (thus the term “variant”). Example 4 shows the third category, Zrelated/Minvariant. Here each sc in the Zpair maps onto itself under T_{n}M or T_{n}MI (thus the term “invariant”). This is perhaps the most restrictive of the three categories, in that each sc can only map onto itself. The Zpair in Example 1, 6–Z28/6–Z49, belongs to this category.^{(4)}
[3] Robert Morris has noted that the Zrelation may appear or disappear depending on the canon of operations in use.^{(5)} This is evident in Example 2, where scs in Zpairs that do not relate by T_{n} or T_{n}I do relate by T_{n}M or T_{n}MI. To this end, Morris develops a number of operations designed to erase the Zrelation. The most often cited of these operations is alpha (α), whose mappings are
α1 = (01) (23) (45) (67) (89) (AB) For α1, Ian Quinn notes, “each pc in the even wholetone collection gets transposed up a semitone, and each pc in the odd wholetone collection down a semitone.”^{(7)} For α2, each pc in the even wholetone collection is transposed down a semitone, and each pc in the odd wholetone collection is transposed up a semitone. Applying α1 to a pcset X may yield quite different results than applying α2 to X. For instance, if X = {012478}, a member of 6–Z17[012478], applying α1 to X yields {013569}, a member of sc 6–Z28[013569]. However, applying α2 to X yields {12378B}, another member of 6–Z17. The fact that 6–Z17 and 6–Z28 belong to the same category of Zpairs (cf. Example 4) suggests that α may be of use in creating mappings for the Zpairs in Examples 3 and 4. Example 5. Adding α to the Zrelated/Mvariant scs (click to enlarge) Example 6. Adding α to the Zrelated/Mvariant scs (click to enlarge) Example 7. Two maximally α–like operations (click to enlarge) Example 8. Redo of the transformational network in Example 1 using 28 ↔ 49 (click to enlarge) [4] To test this hypothesis, Example 5 applies α to the scs in Example 3. The result is clear: α maps (the pcsets of) four of the eight Zpairs onto their Z partners, thus erasing the Zrelation for these scs (6–Z3/6–Z36, 6–Z25/6–Z47, 6–Z13/6–Z42, 6–Z50/6–Z29). The four Zpairs at the bottom of Example 5 do not map onto their Zpartners under α (6–Z4/6–Z37, 6–Z26/6–Z48, 6–Z24/6–Z46, 6–Z39/6–Z10). In like fashion, Example 6 applies α to the scs in Example 4. On the one hand, α resolves the Zrelations between 5–Z12/5–Z36, and between their abstract complements, 7–Z12/7–Z36. On the other hand, α turns the Zrelated/Minvariant hexachords into a new set of Zrelated/Mvariant hexachords (the set is new because the variances differ from those in Examples 3 and 5). The upshot is that the Zrelated/Minvariant hexachords are still unable to map onto their Zpartners. [5] The success of α in resolving every Zrelation save for four Zpairs in Example 5 and four Zpairs in Example 6 prompts me to create maximally αlike operations for those Zpairs.^{(8)} By “maximally αlike,” I am imagining operations whose cycles contain as many intervalclass 1s (ic 1s) as possible, since the cycles of α consist of six ic 1s. The ic 1 cycles result in a “small” voiceleading distance between two αrelated hexachords—no more than six ics of “work” are required to “move between” them.^{(9)} As a result, maximally αlike operations will come as close as possible to six ics of work in relating hexachords. Ideally, a maximally αlike operation will contain 5 ic 1s, but we shall see that certain cases permit only 4 or even 3 ic 1s. The following sections explore maximally αlike operations in detail. [6] Let us return to Example 1. There, X = {02458B} and Y = {13679A}. The maximally αlike operation 28 ↔ 49.1 = (01) (23) (47) (56) (89) (AB) maps X onto Y and vice versa. The label “28 ↔ 49.1” indicates that this operation maps the 6–Z28 member X onto the 6–Z49 member Y and vice versa. “.1” indicates that this is the first of two operations that will map X onto Y and vice versa. 28 ↔ 49.1 is maximally αlike because its cycles contain five ic 1s—(01), (23), (56), (89), (AB)—and one ic 3—(47). Underlines indicate the nonic 1 cycles. [7] Example 7 lists a second maximally αlike operation 28 ↔ 49.2 = (09) (12) (34) (56) (78) (AB) that also maps X onto Y and vice versa. 28 ↔ 49.2 also contains five ic 1s—(12), (34), (56), (78), (AB)—and one ic3—(09)—and is thus as αlike as 28 ↔ 49.1. In the abstract, the choice between 28 ↔ 49.1 and 28 ↔ 49.2 is essentially arbitrary, but in a specific musical context, factors such as instrumentation, register, and voicing may suggest one operation over another. [8] Example 8 renotates the transformational network of Example 1, using 28 ↔ 49. Because the registral spacing of the piano chords does not correspond to either of the 28 ↔ 49 operations, I use the generic label 28 ↔ 49 as opposed to the more specific 28 ↔ 49.1 or 49.2. The 28 ↔ 49 operation allows us to assert the relations that were not possible in Example 1’s network. By reading the network clockwise beginning from X, we follow the chronological procession of the hexachords in Example 1, <X, Y, T_{6}Y, T_{6}X>, and their respective transformations <28 ↔ 49, T_{6}, T_{6} 28 ↔ 49 T_{6}>. [9] A contextual factor in the definition of maximally αlike operations involves the two pcsets that will map onto one another. Up to this point, the 28 ↔ 49 operations have mapped X = {02458B} onto its literal complement, Y = {13679A}. However, to map X onto T1 of Y = {2478AB}, for example, it will not be possible to define a maximally αlike operation (1to1 and onto) since X and T1 of Y share common tones. A simple workaround involves retaining the alreadydefined 28 ↔ 49 operations, then transposing or inverting the resulting pcset. Because maximally αlike operations do not commute with T_{n} or T_{n}I, the initial choice of orthography must be adhered to. Throughout this paper, I use righttoleft orthography. For example, the compound operation T1 28 ↔ 49 maps X onto T1 of Y first through the application of 28 ↔ 49 to X (which maps X onto Y), and second through the application of T1 to Y. Example 9. Carter, Retrouvailles (2000), mm. 5–10 (click to enlarge) Example 10. Webern, Op. 7, No. 2 (1910) (click to enlarge) Example 11. Stravinsky, “Sacrificial Dance” from The Rite of Spring (1921 edition), R3 (click to enlarge) [10] Having defined maximally αlike operations for 6–Z28/6–Z49, I now proceed to the Zpair 6–Z17/6–Z43. Example 9 grounds the discussion with a passage from Carter’s Retrouvailles. Like the Dallapiccola excerpt in Example 1, Retrouvailles features an opening chord X with its literal complement Y, followed by transformations of X and Y that form a second aggregate. Here X = {03489A} and Y = {12567B}, and the lone maximally αlike operation that maps X onto Y (and vice versa) is 17 ↔ 43 = (01) (23) (45) (69) (78) (AB) (5 ic 1s, 1 ic 3) This operation permits the transformational network at the bottom of Example 9, which strongly recalls the network in Example 8. By reading the Example 9 network clockwise beginning from X, we follow the chronological procession of the hexachords, <X, Y, T_{B}I(X), T_{B}I(Y)>. [11] I now define the single maximally αlike operation for the Zpair 6–Z12/6–Z41. Example 10 provides a musical context for the discussion, reproducing a passage that Allen Forte discusses in detail.^{(10)} Forte observes two transformational relations among the chords in Example 10: first, that chord 3 is T_{9} of chord 1, and second, that chord 3 is T_{5}I of the literal complement of chord 2. The following operation formalizes Forte’s second observation: 12 ↔ 41 = (03) (12) (45) (67) (8B) (9A) (4 ic 1s, 2 ic 3s). Chord 2 is the 6–Z41 member {04567A} and chord 3 is the 6–Z12 member {234689}. 12 ↔ 41 maps {234689} onto its literal complement {0157AB} and vice versa. The arrows at the bottom of Example 10 indicate the T_{9} relation from chord 1 to chord 3, and the T_{5}I/12 ↔ 41 relations between chords 2 and 3.^{(11)} [12] Example 11 grounds the discussion of the final pair of Zrelated/Minvariant hexachords, 6–Z23/6–Z45, with a second passage discussed by Forte.^{(12)} The passage contains an opening chord X = {02359B} followed by T_{2} of X’s literal complement, {03689A}. Because the chords share pcs, a 1to1 operation from one to the other is not possible. For this reason, I shall list the two maximally αlike operations that map X = {02359B} onto its literal complement {14678A}:
23 ↔ 45.1 = (07) (12) (34) (56) (89) (AB) (5 ic 1s, 1 ic 5) Example 12 lists maximally αlike operations for the remaining hexachords in Example 5. [13] In this brief “research notes” paper, I have explored ways of mapping any Z sc onto its Z partner. For Zrelated/Mrelated scs (Example 2), this is accomplished by T_{n}M or T_{n}MI. For four of the eight Zrelated/Mvariant Zpairs (Examples 3 and 5) and two of the six Zrelated/Minvariant Zpairs (Examples 4 and 6), this is accomplished by a combination of α, T_{n}M, and/or T_{n}MI. Finally, for the remaining Zrelated/Mvariant hexachords (Example 5) and Zrelated/Minvariant hexachords (Example 6), this is accomplished by the primary contribution of this paper, maximally αlike operations.
[14] There exist a number of avenues for future work with maximally αlike operations. I begin with spaces other than pcspace. First, maximally αlike operations can be defined for pitches in pitchspace, or beats in beatclass (bc) space. Bcspace is particularly fertile ground for the development of new operations since, to date, theorists have defined bcsets primarily in terms of T_{n} and T_{n}I.^{(13)} Example 13 illustrates one such application, modeled on the 28 ↔ 49 operation (cf. §6 and Examples 7–8). The snare drum projects two mod12 bc aggregates. First, X = {02458B} precedes its 28 ↔ 49 transformation, Y = {13679A}. Second, T_{6} of Y = {790134} precedes T_{6} of X = {68AB25}. The network in Example 13 is isographic with that in Example 8, and the passage in Example 13 is isographic in bcspace to the passage in Example 1 in pcspace. [15] Returning to traditional pcspace, maximally αlike operations bear a number of similarities to models of fuzzy T_{n} and T_{n}I.^{(14)} For the latter models, the benchmarks are the traditional “crisp” T_{n} and T_{n}I operations, and offset (“degrees of divergence”) is measured from those cycles. In like fashion, maximally αlike operations measure offset from α by specifying the number and “size” of nonic 1 ics.^{(15)} Appendix: DefinitionsDEF 1: Zrelation: Two pcsets or scs are Zrelated if they share an ic vector but do not relate by T_{n} and/or T_{n}I. The standard gauge of T_{n}/T_{n}I equivalence is assumed. DEF 2: Zpair: Two Zrelated pcsets or scs (“Zpartners”). DEF 3: The two scs in a Zpair are one of the following: DEF 4: An operation is a mapping that is 1to1 and onto. DEF 5: Alpha (α) is an operation whose cycles are α1 = (01) (23) (45) (67) (89) (AB) or α2 = (B0) (12) (34) (56) (78) (9A) (Morris 1982). DEF 6: A maximally αlike operation is an operation whose cycles mimic those of α as closely as possible by containing the maximal number of ic 1 cycles. An example is (01) (23) (47) (56) (89) (AB). Underlines indicate nonic 1 cycles.
Guy Capuzzo Works CitedAlegant, Brian. 2001. “CrossPartitions as Harmony and Voice Leading in TwelveTone Music.” Music Theory Spectrum 23/1: 1–40. Alegant, Brian. 2001. “CrossPartitions as Harmony and Voice Leading in TwelveTone Music.” Music Theory Spectrum 23/1: 1–40. Babbitt, Milton. 1962. “TwelveTone Rhythmic Structure and the Electronic Medium.” Perspectives of New Music 1/1: 49–79. Babbitt, Milton. 1962. “TwelveTone Rhythmic Structure and the Electronic Medium.” Perspectives of New Music 1/1: 49–79. Buchler, Michael. 2000. “Broken and Unbroken Interval Cycles and Their Use In Determining PitchClass Set Resemblance.” Perspectives of New Music 38/2: 52–87. Buchler, Michael. 2000. “Broken and Unbroken Interval Cycles and Their Use In Determining PitchClass Set Resemblance.” Perspectives of New Music 38/2: 52–87. Cohn, Richard. 1992. “Transpositional Combination of BeatClass Sets in Steve Reich’s Phase Shifting Music.” Perspectives of New Music 30/2: 146–77. Cohn, Richard. 1992. “Transpositional Combination of BeatClass Sets in Steve Reich’s Phase Shifting Music.” Perspectives of New Music 30/2: 146–77. Forte, Allen. 1973. The Structure of Atonal Music. New Haven: Yale University Press. Forte, Allen. 1973. The Structure of Atonal Music. New Haven: Yale University Press. Forte, Allen. 1990. “A Major Webern Revision and Its Implications for Analysis.” Perspectives of New Music 28/1: 224–55. —————. 1990. “A Major Webern Revision and Its Implications for Analysis.” Perspectives of New Music 28/1: 224–55. Lewin, David. 1987. Generalized Musical Intervals and Transformations. New Haven: Yale University Press. Lewin, David. 1987. Generalized Musical Intervals and Transformations. New Haven: Yale University Press. Lewin, David. 1995. “Generalized Interval Systems for Babbitt’s Lists, and for Schoenberg’s String Trio.” Music Theory Spectrum 17/1: 81–118. —————. 1995. “Generalized Interval Systems for Babbitt’s Lists, and for Schoenberg’s String Trio.” Music Theory Spectrum 17/1: 81–118. Lewin, David. 1998. “Some Ideas about VoiceLeading Between Pcsets.” Journal of Music Theory 42/1: 15–72. —————. 1998. “Some Ideas about VoiceLeading Between Pcsets.” Journal of Music Theory 42/1: 15–72. London, Justin. 2002. “Some NonIsomorphisms Between Pitch and Time.” Journal of Music Theory 46/12: 12751. London, Justin. 2002. “Some NonIsomorphisms Between Pitch and Time.” Journal of Music Theory 46/12: 12751. Mead, Andrew. 1989. “Some Implications of the PitchClass/OrderNumber Isomorphism Inherent in the TwelveTone System: Part Two.” Perspectives of New Music 27/1: 180–233. Mead, Andrew. 1989. “Some Implications of the PitchClass/OrderNumber Isomorphism Inherent in the TwelveTone System: Part Two.” Perspectives of New Music 27/1: 180–233. Morris, Robert D. 1982. “Set Groups, Complementation, and Mappings Among PitchClass Sets.” Journal of Music Theory 26/1: 101–44. Morris, Robert D. 1982. “Set Groups, Complementation, and Mappings Among PitchClass Sets.” Journal of Music Theory 26/1: 101–44. Morris, Robert D. 1987. Composition with PitchClasses. New Haven: Yale University Press. —————. 1987. Composition with PitchClasses. New Haven: Yale University Press. Morris, Robert D. 1990. “PitchClass Complementation and its Generalizations.” Journal of Music Theory 34/2: 175–246. —————. 1990. “PitchClass Complementation and its Generalizations.” Journal of Music Theory 34/2: 175–246. Morris, Robert D. 1997. “K, Kh, and Beyond.” In Music Theory in Concept and Practice, ed. James Baker, David Beach, and Jonathan Bernard. Rochester: University of Rochester Press: 275–306. —————. 1997. “K, Kh, and Beyond.” In Music Theory in Concept and Practice, ed. James Baker, David Beach, and Jonathan Bernard. Rochester: University of Rochester Press: 275–306. Morris, Robert D. 2001. Class Notes for Advanced Atonal Music Theory. Lebanon, NH: Frog Peak Music. —————. 2001. Class Notes for Advanced Atonal Music Theory. Lebanon, NH: Frog Peak Music. Quinn, Ian. 2004. “A Unified Theory of Chord Quality in Equal Temperaments.” Ph.D. diss., University of Rochester. Quinn, Ian. 2004. “A Unified Theory of Chord Quality in Equal Temperaments.” Ph.D. diss., University of Rochester. Straus, Joseph N. 2005. “Voice Leading in SetClass Space.” Journal of Music Theory 49/1: 45–108. Straus, Joseph N. 2005. “Voice Leading in SetClass Space.” Journal of Music Theory 49/1: 45–108. Winham, Godfrey. 1970. “Composition with Arrays.” Perspectives of New Music 9/1: 43–67. Winham, Godfrey. 1970. “Composition with Arrays.” Perspectives of New Music 9/1: 43–67. Footnotes1. Buchler 2000, 52–3 discusses the chords in Example 1 in connection with interval cycles. 2. They are also ZCrelated (Morris 1982, 103), but the ZCrelation is not required for the present paper’s agenda. 3. The status of T_{n}M and T_{n}MI as equivalence operators on par with T_{n} and T_{n}I is controversial since T_{n}M or T_{n}MI exchanges ic 1 and ic 5 content (Winham 1970, 281–2, Morris 1987, 148, Morris 2001, 52). This can lead to drastically different “equivalent” pcsets, such as {012345} and {024579} (chromatic to diatonic). In the present paper, however, the sets under discussion are Zpairs, whose ic vectors are identical, thereby rendering this criticism moot. Winham 1970, 282, defends T_{n}M and T_{n}MI, stating, “it would not even be correct to say without qualification that I is a ‘closer’ relation than M5 or M7. For while M5 preserves the intervals 3 and 9 while complementing 2, 4, 8, and 10, and M7 does the opposite, I complements all of these and preserves none; so in that one sense I is the least ‘close’.” Nonetheless, while T_{n}M and T_{n}MI do not change the ic content of a Zpair, they do change the larger subsets embedded in each sc. 4. Morris 1982, 102–9, provides pertinent commentary. 5. Again see Morris 1982, 102–9. 6. Morris 1982, 115. Morris provides further applications of α in Morris 1990, 223–30 and Morris 1997, 304–6. Applications of α by other scholars include Lewin 1995, 103 ff., Mead 1989, 224 ff., and Quinn 2004, 36–8. 7. Quinn 2004, 36. 8. Morris 1982 develops operations other than α that change the mappings among Zpartners, but notes that the only way to address 6–Z17/6–Z43 and 6–Z28/6–Z49 is to create a system of equivalence in which the fifty T_{n}/T_{n}I hexachordal scs collapse into three scs (129–31). This system is not in widespread use. 9. The notion of “ics of work” comes from Lewin 1998 and Alegant 2001, 11. 10. Forte 1990, 247–9. 11. No maximally αlike operation whose cycles contain five ic 1s exists for 12 ↔ 41. 12. Forte 1973, 148. 13. Babbitt 1962, Lewin 1987, 23, Morris 1987, 299–305, Cohn 1992. Such “isomorphisms” between pitch and rhythm have long been controversial; a recent critique appears in London 2002. 14. See, most recently, Straus 2005, 45–50. 15. Thanks to Jonathan Salter for writing a computer program to calculate maximally αlike operations, Igor Erovenko (Department of Mathematics and Statistics, UNCG) for help with matters mathematical, and Clifton Callender, J. Daniel Jenkins, Evan Jones, Rachel Mitchell, Robert Morris, Robert Peck, Jonathan Pieslak, Adam Ricci, Caleb Smith, and
the anonymous MTO readers for their suggestions. Buchler 2000, 52–3 discusses the chords in Example 1 in connection with interval cycles. They are also ZCrelated (Morris 1982, 103), but the ZCrelation is not required for the present paper’s agenda. The status of T_{n}M and T_{n}MI as equivalence operators on par with T_{n} and T_{n}I is controversial since T_{n}M or T_{n}MI exchanges ic 1 and ic 5 content (Winham 1970, 281–2, Morris 1987, 148, Morris 2001, 52). This can lead to drastically different “equivalent” pcsets, such as {012345} and {024579} (chromatic to diatonic). In the present paper, however, the sets under discussion are Zpairs, whose ic vectors are identical, thereby rendering this criticism moot. Winham 1970, 282, defends T_{n}M and T_{n}MI, stating, “it would not even be correct to say without qualification that I is a ‘closer’ relation than M5 or M7. For while M5 preserves the intervals 3 and 9 while complementing 2, 4, 8, and 10, and M7 does the opposite, I complements all of these and preserves none; so in that one sense I is the least ‘close’.” Nonetheless, while T_{n}M and T_{n}MI do not change the ic content of a Zpair, they do change the larger subsets embedded in each sc. Morris 1982, 102–9, provides pertinent commentary. Again see Morris 1982, 102–9. Morris 1982, 115. Morris provides further applications of α in Morris 1990, 223–30 and Morris 1997, 304–6. Applications of α by other scholars include Lewin 1995, 103 ff., Mead 1989, 224 ff., and Quinn 2004, 36–8. Quinn 2004, 36. Morris 1982 develops operations other than α that change the mappings among Zpartners, but notes that the only way to address 6–Z17/6–Z43 and 6–Z28/6–Z49 is to create a system of equivalence in which the fifty T_{n}/T_{n}I hexachordal scs collapse into three scs (129–31). This system is not in widespread use. The notion of “ics of work” comes from Lewin 1998 and Alegant 2001, 11. Forte 1990, 247–9. No maximally αlike operation whose cycles contain five ic 1s exists for 12 ↔ 41. Forte 1973, 148. Babbitt 1962, Lewin 1987, 23, Morris 1987, 299–305, Cohn 1992. Such “isomorphisms” between pitch and rhythm have long been controversial; a recent critique appears in London 2002. See, most recently, Straus 2005, 45–50. Thanks to Jonathan Salter for writing a computer program to calculate maximally αlike operations, Igor Erovenko (Department of Mathematics and Statistics, UNCG) for help with matters mathematical, and Clifton Callender, J. Daniel Jenkins, Evan Jones, Rachel Mitchell, Robert Morris, Robert Peck, Jonathan Pieslak, Adam Ricci, Caleb Smith, and
the anonymous MTO readers for their suggestions.
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