Page  00000001 GROUP STRUCTURE AND EQUIVALENCE CLASSES IN EXTENDED TWELVE-TONE OPERATIONS Tuukka Ilomdki Sibelius Academy/Eastman School of Music ABSTRACT The classic set of pitch-class and row operations - transposition, inversion and retrograde - has been extended with new types of operations: first by Moperation and rotation, and more recently by a family of operations known as alpha, beta, gamma and delta. For a better understanding of these new operations, group theory enables us to find canonical forms that derive them, and to calculate the number of set-classes and row classes by tracking the invariances of the sets and rows under the operations. 1. INTRODUCTION In the formalization of a musical space we get a natural starting point by making a clear distinction between objects and operations [3]. These two concepts meet in the classification of the objects: typically two objects are considered equivalent if they are related by an operation. For example, the traditional classification of pitch-class sets (henceforth: pcsets) is based on transposition and inversion, and the classification of rows is based on transposition, inversion and retrograde. During the 20th century the set of available operations has grown considerably. The classic set of operations has been expanded, for example, by M-operation and rotation, and more recently by a family of operations known as alpha, beta, gamma and delta [2]. This paper presents formal methods drawn from group theory to analyze these new operations thus providing us with a better insight into their properties and their relation to the classic operations. 2. GROUPS AND EQUIVALENCE CLASSES Since Milton Babbitt introduced the idea of using formal methods in 1950s, group theory has been an essential tool in music theory. In fact, the properties of a group correspond nicely to the properties of pitch-class (henceforth: pc) operations. Formally, a group is defined as a non-empty set G with the following four properties: (i) G has a binary operation: a- b EG for all a, b E G. (ii) The binary operation is associative: a - (b - c) = (a - b) - c for all a, b, c E G. (iii) G contains an identity element e such that a - e = a - e- a for all a7& G. (iv) Every element a7-EG has an inverse element a1 such that a - a -e =a -a. As an example, let us consider transposition. (i) The composition of two transpositions is a transposition. For example T4T2 T6. (ii) The order in which transpositions are composed is insignificant. For example TI(T2T3) = T6= (TIT2)T3. (iii) Transposition at To is the identity element. (iv) Every transposition T,, has an inverse element T12-n. For example T4T8 = To. 2.1. Group action and equivalence A mathematical approach to operations and objects is to formalize the operations as a group acting on a set of objects. The usual pc and row operations satisfy the formal requirements of a group action: the identity operation leaves every object intact, and applying a composite operation F2F1 gives the same result as applying first F1 and then F2. Further, a group action induces permutations that form a group. Moreover, this permutation group induces an equivalence relation (a reflexive, symmetric and transitive relation) on the set of objects: x is related to y if there is a permutation that maps x to y. Reflexivity is due to the identity operation, symmetry is due to the inverse permutations, and transitivity is due to the binary operation defined in the permutation group. Hence, if the pc operations act on pcsets or the row operations act on rows, the resulting equivalence classes are the set-classes or row classes. 2.2. Extending groups An existing operation group can be extended by adding new operations. The new group generated by a given set of operations is produced by all finite combinations of the generating operations and their inverse operations. Determining the group structure requires tracking all possible combinations of generating operations that produce distinct operations. In general, the aim is to reduce the set of generating operations to "bare bones,"' from which the other operations can be deduced. For example, the group of transpositions can be extended by introducing inversion. This doubles the size of the group: each transposition Tn has a corresponding inversion TnI. In addition, transposition and inversion are closed under composition: any combination of them is a transposition or an inversion. Sometimes the addition of a new operation has an effect that can be difficult to predict. Adding an operation F to the group of transpositions might necessitate us to add new operations of forms TnF, FTnF, TnFTnF, etc. until no new combination produces a new operation. However, if F and G are subgroups (of a known group) whose intersection contains only the identity element and FG GF, then F and G generate a group of size |F| |G| and all group elements have a canonical representation fg where fE& F and gE& G. This result can be used to analyze combinations of pc

Page  00000002 operations since they all are permutations, and thus subgroups of the symmetric group S12. It behooves us to reduce the operations to a canonical form for easy reference. For instance, the transpositions and inversions can be written as TnI, and if the Moperation is added, the canonical form is Tn1MI. The 48 classic row operations can be written as RTnI. The canonical form "factorizes" the operations. Furthermore, the total number of different operations is the product of the orders of the subgroups generated by the factors. 2.3. Burnside's lemma Any group of operations divides a set of objects into equivalence classes. But simply dividing the number of objects by the number of operations does not necessarily result in the correct number of equivalence classes since not all equivalence classes have the same number of distinct elements. A key element here is the fixed points: objects that are kept the same in some operation(s). For example, applying transposition T4 to pcset {0,4,8} keeps the pcset fixed, but T4 to {0,2,4} yields {4,6,8}. A mathematical result known as Burnside's lemma provides a handy tool for computing the number of equivalence classes by using fixed points [1]. The reason for consulting fixed points is that they are easier to calculate than the equivalence classes. LEMMA (Burnside) The number of equivalence classes induced by a permutation group G = {t7, 2,..., 1n) is I. n7eG where yl(t) is the number of fixed points of tn. 3. FIXED POINT COUNTING STRATEGIES Counting the fixed points of an operation depends on whether it is applied to pcsets or rows (or something else). However, in both cases the cycle structure of the operation proves essential. Description of operations as products of cycles is well established [3]. For example, transposition T4 is a product of 4 cycles T4 = (0 4 8)(1 5 9)(2 6 10)(3 7 11). A pcset is a fixed point in an operation if and only if it is a union of complete cycles. For example, suppose that pcset S is a fixed point in T4. Now, if 0 is in S, then 4 must be also in S as 0 is transposed to 4 in T4. And if 4 is in S, then 8 must be also in S. Hence, if S contains a single pc of the cycle (0 4 8) it must contain all of them. Only the number of cycles affects the number of pcsets an operation holds fixed. The elements of each cycle either belong to a fixed point or they do not. Therefore, according to the "rule of product" an operation with n cycles has 2n fixed points. The calculation of fixed points of row operations is a bit more complicated. Row operations are combinations of pc operations (such as transposition and inversion) and order position operations (such as retrograde and rotation). No pc or order position operation (except To) is able to create a fixed point by itself. Pc operations by definition change pcs and order position operations change locations of pcs. In a fixed point, all pcs must remain in the same order positions. Hence, a fixed point is possible only in a combination of pc and order position operations that "cancel" each other in a row. Consider the row 967845BA2103 of Anton Webern's Symphony op. 21 and the operation RT6. The first pc 9 is transposed to 3 and then moved by retrograde to the last. Correspondingly, the last pc is transposed to 9 and moved to the beginning. Transposition T6 swaps all pcs pairwise and retrograde swaps them back; the row remains the same. In music theory literature this phenomenon is known as invariance. The key to counting the fixed points of a row operation lies in the cycle structures of the pc and order position operations. In the example above, both T6 and R have similar cycle structures: 6 cycles of length 2. A fixed point requires that the pc and order position operations have matching cycles. In each cycle, fixing the first pc fixes the others, too. For example, in RT6 if the first pc is n then the last pc must be n+6. For counting the total number of fixed points we need to calculate how many choices there are for the first pc of each cycle; the total number of fixed points is the product of all these choices. For example, in the case of RT6 there are 12 choices for the first cycle, but as the first cycle "uses" two pcs, there are only 10 choices for the second cycle and so on. 4. NEW OPERATIONS We will first examine rotations and M-operation as row operations, and then consider the family of operations introduced by Morris [2] applied to pitch-class sets. Morris' operations utilize the cyclic properties of the aggregate and belong to four families - alpha, beta, gamma and delta - that swap icI, ic2, ic3 and ic4 dyads, respectively. We also discuss a set of new operations swapping ic6 dyads. The rationale behind these operations is that they preserve dyads of certain intervalclasses while changing others and thus they "map sets with interval-class content similarity or identity across traditional set-group boundaries." While Morris' aim was to track the mapping of set-classes in these operations, our focus is their group structure. 4.1. Rotations Let us consider rows and row operations: transposition, inversion, retrograde, and rotation. This set of operations is symmetric as it has same operations for pcs and order positions: rotations correspond to transpositions, and retrograde corresponds to inversion. Two functions prove useful in the analysis of transposition and rotation. The greatest common denominator gcd(n,12) gives the number of cycles in a transposition Tn. As in transpositions all cycles are of equal length, the length of the cycles is 12/gcd(n,12). Euler's cp-function cp(n) denotes the number of positive numbers k < n such that gcd(k,n) = 1. The number of fixed points is given by the formula

Page  00000003 11 gcd(n,12) -1 p(12/gcd(n,12))- (12-(k-12/gcd(n,12))) = 479,057,472 n=0 k=0 where n runs through all transpositions, cp(12/gcd(n,12)) denotes the number of rotations with a similar cycle structure, and the product gives the number of possible choices in each of the gcd(n,12) cycles. Transposition T6 and odd inversions 12n+1 have 6 cycles of length 2. The corresponding order position operations are r6 and R combined with an even rotation r2n. Any combination of these has 12 - 10 - 8 ~ 6 ~ 4 - 2 = 46,080 fixed points. Hence, the six combinations of T6 with an even rotation of retrograde give 6 - 46,080 = 276,480 fixed points, the six combinations of an odd inversion with r6, give 6 - 46,080 = 276,480 fixed points, and the 6 - 6 = 36 combinations of an odd inversion with an even rotation of retrograde give 36 - 46,080 = 1,658,880 fixed points. Even inversions and a retrograde combined with an odd rotation have 5 cycles of length 2 and 2 cycles of length 1. The inversions keep two pcs fixed and the rotated retrogression keeps the locations of two pcs fixed. There are two ways to distribute the fixed pcs to the fixed locations, and for the cycles of length 2 we have 10, 8, 6, 4 and 2 choices. Thus, each combination has 2 - 10 - 8 - 6 - 4 - 2 = 7,680 fixed points, and in total we have 6 - 6 ~ 7,680 = 276,480 fixed points. Combining all fixed points and applying Burnside's lemma gives us that the number of row classes is 479,057,472+276,480+276,480+1,658,880+276,480 8, 576836,017. 576 4.2. M-operation The M-operation transforms the chromatic scale into a circle of fifths. Arithmetically, pcs are multiplied by 5 (mod 12). The cycles of the M-operation are M = (0)(1 5)(2 10)(3)(4 8)(6)(7 11)(9). Let us consider adding the M-operation to the previous set of row operations. Table 1 summarizes the Moperations involved in fixed points. The first and second columns line up the pc and order position operations with similar cycle structures. Any combination of these will result in the number of fixed points shown in the third column. The fourth column shows the total number of fixed points of the combinations of these operations. In sum, adding M-operation results in 576 - 2 = 1,152 operations with 483,163,776 fixed points. Applying Burnside's lemma gives us 419,413 row classes. pc operation op operation fixed points total T2M, T6M, TioM, r6, R, r2R, r4R, 12 ~ 10 ~ 8 ~ 6 ~ 4 ~ 2 5 ~ 7 46,080 T3MI, T9MI r6R, rsR, rioR = 46,080 = 1,612,800 T1M, T3M, T5M, r3,r9 12 8 4 = 384 6 - 2 - 384 T7M, T9M, T11M = 4608 T1MI, TsMI, r2, r10 12 - 6 = 72 4 2 72 T7MI, T11M = 576 Table 1. Fixed points in M-operation. M-operation could be applied also to order positions giving isomorphic sets of operations for both pcs and order positions: transposition, inversion, retrograde, rotation and M-operation for both. This results in 482 = 2,304 different operations. Using a similar argument to that given above, the number of row classes is 211,012. 4.3. Alpha The operations of the alpha family swap adjacent pcs. The two possible operations doing that are ai = (0 1)(2 3)(4 5)(6 7)(8 9)(10 11) a2 - (1 2)(3 4)(5 6)(7 8)(9 10)(11 0). Either one could be chosen as the basis of the operations; Morris chooses C3 = Tlal. However, for constructing a group the operation a = (03)2 = (0)(2)(4)(6)(8)(10)(1 9 5)(11 7 3) is more appropriate. While a does not commute with all of TnMI, the intersection of the subgroups generated by TnMI and a is {To} and TnMITk = ckTMI. Thus TnMITk affords a canonical unequivocal form for all group members (as opposed to Morris' oa, u2, and u3). As a has three distinct powers the group <T, M, I, a>1 has 3 - 48 = 144 operations. Given that the idea is to swap adjacent pcs, a may seem a surprising basis for the group. However, the "original" alpha operations can be expressed as ao = TIMIc2 and a2 = TIMIa. Table 2 shows the counts of cycles in the operations a, P, y, 6 and T. The number of fixed points in the alpha family is 24 - 21 + 28 - 22 + 20 23 + 8 - 24 + 4 - 25 +22 - 26 + 22 - 26 + 13 - 27 +2 - 29 + 1 - 212 =13,248. Using Burnside's lemma, dividing 13,248 by the number of operations 144 gives us the number of set-classes 92.2 S1 2 3 4 5 6 7 8 9 10 11 12 a 24 28 20 13 36 90 84 y 16 24 36 6 216 216 156 t 64 96 96 8 42 26 232 140 4 22 22 72 44 18 24 33 12 180 86 90 144 105 54 13 39 9 105 27 2 0 6 0 8 3 6 8 20 15 Table 2. Cycle counts in ca, 3, y, 6 and T. 4.4. Gamma The operations of the gamma family swap pcs ic3 apart. There are 8 possible operations doing that (23 = 8: each of the three ic3-cycles or diminished seventh chords can be split to ic3-dyads in two ways), for example 1 = (0 3)(6 9)(1 4)(7 10)(2 5)(8 11). Morris presents nine different gamma operations, but again we search for one that affords a canonic form for the group elements. Hence, we choose the operation y =(0 3 6 9)(1 2)(4 5)(7 8)(10 11) as our base. While y does not commute with all of TnMI, the intersection of the subgroups generated by TnMI and y is {To} and the sets TnMIy and ykTnMI are the same. Thus TnMIyk affords a canonical form for all group members (as opposed to Morris' operations). As y has 4 distinct powers the group <T, M, I, y> has 4 - 48 = 192 operations. Of the cycles of y only (0 3 6 9) swaps pcs 1 <T, M, I, a> is the group generated by the operations T, M, I, and a. 2 The list of set-classes in appendix of [2] lacks the alpha variant of the set-class 6-19. Consequently, table 10 lists an incorrect number of hexachords 17 (correct one is 18) in the alpha family.

Page  00000004 ic3 apart. However, all the "original" gamma operations can be expressed in terms of y. For example, y, = T6Iy. Consulting Table 2 and using Burnside's lemma results in a total of 98 set-classes.' In general more operations means less equivalence classes. The relation is not linear, however: y generates more operations than c, but the number of set-classes is larger in y. 4.5. Beta The beta family is based on swapping pcs ic2 apart. There are 4 possible operations doing that (22 4: both of the two ic2-cycles or whole-tone scales can be split to ic2-dyads in two ways), for example P I = (0 2)(4 6)(8 10)(1 3)(5 7)(9 11). However, this time no canonical form is available. Morris presents five variants of beta, and gives P5=(0)(3)(4)(7)(8)(11)(1 9 5)(2 10 6) a special status. Again, the generator somewhat hides relation between the generator and the original idea of swapping pcs ic2 apart. The group <T, M, I, P> contains 432 operations. All members can be written in the form TMIPTnMIPTn1MIl where P is any operation outside of the Tn1MI family. In this respect, P5 is not special. Alternatively, for 192 different betas (but none of Morris' betas), for example P = (0 8 4)(1 1 1)(2 6 10)(3 5)(7 9) also a simpler form T,, kTn1MI yields all group members. These forms are not unequivocal: many combinations yield the same operation. Consulting Table 2 and using Burnside's lemma results in a total of 61 set-classes. 4.6. Delta While the aggregate divides into 6 discrete pairs of dyads related by icl, ic2, or ic3, this is not possible with ic4, so Morris' deltas start with 3-note combinations (the four augmented triads), but uses 4 single pcs and four ic4 pairs for eight variants. For example, 61,= (0)(1)(2)(3)(4 8)(5 9)(6 10)(7 11) 62= (0)(1)(2)(7)(4 8)(5 9)(6 10)(3 11) 65= (0)(1)(3)(6)(4 8)(5 9)(2 10)(7 11). The equations P5=T,62T36 2and 62 =T6MP5TjP5T show that P5 and 62 can be expressed in terms of each other and thus generate the same groups.2 The equations 6, TM65T365 and 6s -6,To6,T,6,T, show that 6, and 6s can be expressed in terms of each other and thus generate the same groups. The operation 62 (and, consequently, also ps) can be expressed in terms of 6s (but not vice versa). Thus, the group <T, M, I, 62> is a subgroup of the group <T, M, I, 6s>. The group <T, M, I, 6,> is relatively large containing 1,296 operations altogether. All members can be written in the form Tn1MI6Tn1MI6TnMI, with 768 different deltas. 1 The list of set-classes in appendix of [2] lacks the gamma variant of the set-class 4-28. Consequently, table 10 lists an incorrect number of tetrachords 12 (correct one is 13) in the gamma family. 2 Morris' equation (T7182)2 1P5 shows that 1s can be expressed in terms of 82 but it does not show that 82 can be expressed in terms of 1P5 Alternatively, for 96 different deltas (all consisting of a single cycle of length 12), for example 6 = (0 5 6 11 8 9 2 3 4 1 10 7) also a simpler form T,,6kTn1MI yields all group members. The forms are not unequivocal. Consulting Table 2 and using Burnside's lemma results in 55 set-classes. 4.7. Tau The aggregate can be uniquely partitioned to six ic6 cycles. The combination of all six cycles is equal to T6 (which is why Morris discarded them). Nonetheless, we can define a family of operations - where one cycle is switched and the other pcs are kept fixed. For example, To= (0 6)(1)(2)(3)(4)(5)(7)(8)(9)(10)(1 1). The group <T, M, I, To> has 768 operations. No canonical form can be found. Somewhat surprisingly, all operations can be expressed as TiTn1MI-r where Ti and Tj both switch one ic6-pair. Consulting Table 2 and using Burnside's lemma results in a total of 92 set-classes. 4.8. Combinations The combination of alpha and beta operations result in the delta operations, since alpha and beta operations both are subgroups of delta and 61 IP5C3. Morris also suggests a new composite operation, labeled here as k: k =T379y5 =(0 3)(8 11)(1 6 5 4)(2 7 10 9). In practice, k gives the same result as taking both gamma and beta as generators, since y9 = T9MXT4XT8X and (5 =19kT8kT2kT2. However, the group <T, M, I, k> <T, M, I, y9, 5> contains no fewer than 1,036,800 operations. Groups of this size are very difficult to handle without a computer program. 5. CONCLUSIONS Group theory has proved to be an effective tool to analyze the new twelve-tone operations. The strategies for counting fixed points can easily be extended to other new operation types. The sizes of the new groups are relatively large. For alphas and gammas a canonical form can be found, not so with betas and deltas. The search for canonical forms sometimes obliges us to base the group extension on somewhat unintuitive operations. Combinations of the new operations result in up to 1,036,800 operations acting on 4,096 pcsets, which seems a dubious proportion at least. 6. REFERENCES [1] Liu, C. Introduction to Combinatorial Mathematics. McGraw-Hill, New York, 1968. [2] Morris, R. "Set Groups, Complementation, and Mappings Among Pitch-Class Sets." Journal of Music Theory 26(1), 10 1-144. [3] Starr, D. "Sets, Invariance, and Partitions." Journal of Music Theory 22(1), 1-42.