# A Classi cation of Dodecaphonic Groups

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1 A Classi cation of Dodecaphonic Groups Franck Jedrzejewski CEA, Saclay (France) IRCAM, 15 janvier /25 Franck Jedrzejewski A Classi cation of Dodecaphonic Groups

2 Contents 1 Twelve-Tone Technique 2 Chord Diagrams 3 Transitive Groups 4 Serial Networks 5 The 26 Serial Groups 6 Primitive Groups 7 Block Systems 2/25 Franck Jedrzejewski A Classication of Dodecaphonic Groups

3 Twelve-Tone Technique Identify the 12 notes with Z n i.e. 0=do, 1= do, etc. Choose a 12-tone row. Example : Jean Barraqué...au-delà du hasard P = (0, 8, 7, 1, 4, 2, 10, 3, 11, 5, 6, 9) Apply the 4 basic following transformations Transpose up or down : T n (x) = x + n mod 12 Retrograde = reverse in time R(P) = 9,6,5,11, etc. Inversion = reverse in pitch I n (x) = x + n mod 12 Retrograde Inverse = composition RI (P, I, R, RI) = Klein group = dihedral group D 2 with presentation D 2 = a, b a 2 = b 2 = (ab) 2 = 1 with 1 = P, a = I, b = R Material for composition = at most 4 12 = 48 rows, called derivative forms 3/25 Franck Jedrzejewski A Classication of Dodecaphonic Groups

4 Enumeration of Tone Rows Action of the dihedral group D n on the permutation group S n Theorem In the n-tone equal temperament (n 3), under the equivalence of the derived forms (i.e. under the action of the dihedral group), there are { 1 ( n (n 1)! + 4 2(n 4)/ )! if n is even n 2 1 (n 1)! + ( ) 4 2(n 5)/2 n 1 2! if n is odd tone rows of n pitch classes. In particular, for n = 12, S = 9, 985, 920 rows. The problem of counting the tone rows of S is equivalent to count the number of orbits of the group G = D 12 Z 2 generated by the transformations T, I and R on the series P. D 12 denotes the dihedral group (of order 24). Burnside's lemma says that the number of orbits is the average number of xed points. 4/25 Franck Jedrzejewski A Classication of Dodecaphonic Groups

5 Tone Rows with Limited Derivative Forms Remark 12!/48 = 479, 001, 600/48 = 9, 979, 200 is not the number of 12-tone rows because some tone rows have less than 48 derivative series. Example 1 A = (0, 4, 8, 11, 3, 7, 1, 9, 5, 2, 10, 6) veries T 6 (A) = R(A) = (6, 10, 2, 5, 9, 1, 7, 3, 11, 8, 4, 0) Thus T n (A) = RT n+6(a) and I n (A) = RI n+6(a). The series A has only 24 derivative forms. Example 2 A = (0, 3, 11, 2, 10, 1, 6, 9, 5, 8, 4, 7) is equal to RI 7 (A). 5/25 Franck Jedrzejewski A Classication of Dodecaphonic Groups

6 Chord Diagrams How to build a Chord diagram? 1 Take a row of twelve pitch classes : 0,8,6,7,2,4,9,11,1,5,3,10 2 Place the numbers on a circle 3 Join the tritone pairs together 4 Just keep the structure This structure is called a chord diagram. 6/25 Franck Jedrzejewski A Classication of Dodecaphonic Groups

7 Chord Diagrams Proposition The 48 forms of the series are represented by the same chord diagram. i.e. Chord diagrams are invariant under the action of the dihedral group. Example : Retrogradation is obtained by mirror symmetry and rotation. 7/25 Franck Jedrzejewski A Classication of Dodecaphonic Groups

8 Enumeration of Chord Diagrams Theorem (A. Khruzin) Under the action of the dihedral group, the number of chord diagrams for the equal temperament with 2n degrees is d n = 1 2 (c n (κ n 1 + κ n )) c n = 1 n 2 n! ϕ(i)ν n (i), κ n = 2n k!(n 2k)! i 2n k=0 ϕ(i) is the Euler totient function and ν n is given for all divisors of 2n by i n/i (2n/i 1)!! if i is odd ν n (i) = n i ( ) 2n/i i k (2k 1)!! if i is even 2k k=0 8/25 Franck Jedrzejewski A Classication of Dodecaphonic Groups

9 Enumeration of Chord Diagrams For the rst 2n-equal temperaments, 2n c n d n Interest : about 10 millions rows are represented by only 554 diagrams. 9/25 Franck Jedrzejewski A Classication of Dodecaphonic Groups

10 Serial Groups Each chord diagram D n is associated with a permutation P n Luigi Nono Canto Sospeso D 358 is associated with (in cyclic notation) (0, 1)(2, 11)(3, 10)(4, 9)(5, 8)(6, 7) The generating power of this row is measured by the order of the Serial group dened by G n = P n, T, I T = permutation associated with transposition T = (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11) I = permutation associted with inversion I = (1, 11)(2, 10)(3, 9)(4, 8)(5, 7) 10/25 Franck Jedrzejewski A Classication of Dodecaphonic Groups

11 Serial Groups Luigi Nono, Canto Sospeso, S=(9,10,8,11,7,0,6,1,5,2,4,3) The permutation P n indicates the places where we can exchange the tritones P n = (0, 1)(2, 11)(3, 10)(4, 9)(5, 8)(6, 7) All combination of exchanges are obtained when we add T and I. G n = P n, T, I 11/25 Franck Jedrzejewski A Classication of Dodecaphonic Groups

12 Examples of Serial Groups A. Webern Concert for nine instruments, opus 24. S = 11, 10, 2, 3, 7, 6, 8, 4, 5, 0, 1, 9 a, b, c a2 = c G 549 = 2 = (b 1 c) 2 = (acb) 2 = (ab 2 c) 2 = (ab 1 ) 4 = (abab 1 ) 3 = 1 This group has 192 elements. Schoenberg's Ode to Napoleon, Kees van Baaren's Variations in Isometrical Series 5, 10, 6, 9, 7, 8, 2, 1, 3, 0, 4, 11 is associated with the diagram D 538. The associated group has 7680 elements. G 538 = P 538, T, I 12/25 Franck Jedrzejewski A Classication of Dodecaphonic Groups

13 Examples of Serial Groups Dallapiccola's Quaderno Musicale di Annalibera 10, 11, 3, 6, 8, 2, 7, 1, 5, 9, 0, 4 is associated with the diagram D 56. The group generated by the three operators G 56 = P 56, T, I has presentation a, b, c a2 = c G 56 = 2 = (b 1 c) 2 = (ac) 4 = acb 1 (abc) 2 = (acb 3 ) 2 = (ab 2 ab 2 ) 2 = b 12 = (acab 2 ) 4 It is a group of order Many diagrams have a maximum generating power S 12 = /25 Franck Jedrzejewski A Classication of Dodecaphonic Groups

14 The 26 Serial Groups Theorem The 554 chord diagrams generate only 26 serial groups, which form a subset of the 301 transitive permutation groups. Amongst the 26 serial groups, only two are primitive : PGL(2, 11) (r = 218) and S chord diagrams have a trivial serial group (S 12 ). Computations with GAP (http :// A system for Computational Discrete Algebra. r = index of the group in the table of the 301 transitive groups G = Order of the group (denote now G G ). Index of the chord diagrams in the table (Mathematical Theory of Music) Blocks = Block systems Type : B1,..., B9. 14/25 Franck Jedrzejewski A Classication of Dodecaphonic Groups

15 The 26 Serial Groups r G Chord diagrams Blocks , 554 B ,509 B B , 381, 553 B , 549 B , 359, 474 B B a 414 B b 383, 497, 507, 508 B c 136, 536 B B a 157, 552 B b 348, 530 B c 490 B , 516 B4 15/25 Franck Jedrzejewski A Classication of Dodecaphonic Groups

16 The 26 Serial Groups r G Chord diagrams Blocks , 161, 184, 241,395,417, 448, , 351, 440, 537 B B B , 282, 303 B ,61,111,347,355,382,412,441,504,538,551 B knots :24,51,56,150,163,183, ,371,376,419,420,439,481,514 B ,185,346,350,357,488,531,535,546,550 B ,174,285,289,313,321 B8 31 knots : 3,6,7,26,31,34,36,60,137,142, ,168,181,188,190,207,209,281,290,295 B7 305,334,352,362,418,434,482,494,511, The remainder : 431 chord diagrams 16/25 Franck Jedrzejewski A Classication of Dodecaphonic Groups

17 Example : The Serial Group G 24 of order 24 G 24 = a, b, c with : a=(0,1)(2,11)(3,10)(4,9)(5,8)(6,7) b=(0,1,2,3,4,5,6,7,8,9,10,11), c=(1,11)(2,10)(3,9)(4,8)(5,7) Action of G 24 : Let S be a 12-tone row of chord diagram D (358 or 554), then g G 24, g(s) D, (i.e. always have with the same diagram.) 17/25 Franck Jedrzejewski A Classication of Dodecaphonic Groups

18 Example : The Serial Group G 48 of order 48 G 48 = a, b, c with : a=(0,2)(1,7)(3,5)(4,10)(6,8)(9,11) b=(0,1,2,3,4,5,6,7,8,9,10,11), c=(1,11)(2,10)(3,9)(4,8)(5,7) Action of G 48 : Let S be a 12-tone row of chord diagram D (444 or 509), then g G 48, g(s) D, (i.e. always have with the same diagram.) 18/25 Franck Jedrzejewski A Classication of Dodecaphonic Groups

19 Properties of the Serial Groups Theorem Let S be a 12-tone row with chord diagram D, and G the serial group of D. The action of G on S is globally stable in the following sense : for all g G, the 12-tone row g(s) has a chord diagram associated with G or with one of its subgroups. Example 1 : The chord diagram of S={0,6,1,7,2,4,11,3,10,8,9,5} is D 50 of group G 216. For all g G 216 g(s) has chord 50 or 358. D 358 belongs to G 24 a subgroup of G 216. Example 2 : The chord diagram of S={0,8,6,7,3,11,9,2,4,1,10,5} is D 474 and its group G 216. For all g G 216 g(s) has chord 359 or 474 (The chord diagram 50 is not reached). 19/25 Franck Jedrzejewski A Classication of Dodecaphonic Groups

20 Lattice of the 26 Serial Groups Groups are represented by their order. Red cross = normal subgroup, g G, gh = Hg. 20/25 Franck Jedrzejewski A Classication of Dodecaphonic Groups

21 Stable Actions Theorem Only 9 chord diagrams have a stable action i.e. such that if S is a 12-tone row of chord diagram D with serial group G (depending on D), then for all g G, g(s) has the same chord diagram D, g G, g(s) D G CD , , b 497, b 530 The 12-tone rows created by the action of G on S have always the same chord diagram. 21/25 Franck Jedrzejewski A Classication of Dodecaphonic Groups

22 Primitive Groups A permutation group G acting on a set X is primitive if G preserves no nontrivial partition of X, otherwise, G is imprimitive. For X = {0, 1,..., 11}, there are only 6 primitive groups (R. Carmichael, 1937). Name G GAP index L(2,11) PGL(2,11) M M A S There are only 2 primitive serial groups (G 1320 and S 12 ). 22/25 Franck Jedrzejewski A Classication of Dodecaphonic Groups

23 Block Systems A subset B of X = {0, 1,..., 11} is a block of imprimitivity if g G, B g(b) = B or The trivial blocks, {x} and X are always blocks of imprimitivity. If the only blocks of imprimitivity are the trivial ones then the action of G on X is imprimitive. A partition of X composed of blocks of imprimitivity is a system of imprimitivity. 23/25 Franck Jedrzejewski A Classication of Dodecaphonic Groups

24 Block Systems Symbolic names of the Block Systems for Serial Groups (Table) B1 = [ 1, 3, 5, 7, 9, 11 ], [ 1, 4, 7, 10 ], [ 1, 5, 9 ], [ 1, 7 ] B2 = [ 1, 3, 5, 7, 9, 11 ], [ 1, 4, 7, 10 ], [ 1, 7 ] B3 = [ 1, 3, 5, 7, 9, 11 ], [ 1, 5, 9 ], [ 1, 7 ] B4 = [ 1, 3, 5, 7, 9, 11 ], [ 1, 5, 9 ] B5 = [ 1, 3, 5, 7, 9, 11 ], [ 1, 7 ] B6 = [ 1, 4, 7, 10 ], [ 1, 7 ] B7 = [ 1, 3, 5, 7, 9, 11 ] B8 = [ 1, 4, 7, 10 ] B9 = [ 1, 7 ] 24/25 Franck Jedrzejewski A Classication of Dodecaphonic Groups

25 Example : Imprimitivity of G 24 G 24 is generated by the permutations a=(0,1)(2,11)(3,10)(4,9)(5,8)(6,7) b=(0,1,2,3,4,5,6,7,8,9,10,11), c=(1,11)(2,10)(3,9)(4,8)(5,7) It has four block systems b1 = {{0, 2, 4, 6, 8, 10}, {1, 3, 5, 7, 9, 11}} b2 = {{0, 3, 6, 9}, {1, 4, 7, 10}, {2, 5, 8, 11}} b3 = {{0, 4, 8}, {1, 5, 9}, {2, 6, 10}, {3, 7, 11}} b4 = {{0, 6}, {1, 7}, {2, 8}, {3, 9}, {4, 10}, {5, 11}} The sub-blocks are limited transposition sets. Compositional motivation : We move from one block to the other or remain in the same block by the action of G. 25/25 Franck Jedrzejewski A Classication of Dodecaphonic Groups

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