Morphisms in a Musical Analysis of Schoenberg, String Quartet No. 1, Opus 7

Transcription

1 Morphisms in a Musical Analysis of Schoenberg, String Quartet No., Opus Thomas M. Fiore joint work with Thomas Noll and Ramon Satyendra,

2 Transformational Analysis of Schoenberg, String Quartet No., Opus

3 Schoenberg, String Quartet No., Opus Opening Theme Materials, Measures -3. &b. # œ œ œ. œ œ œ œ œ œ œ œ mf shark &b œ # œ œ œ œ œ œ œ œ œ jet triad strain octatonic subset Octatonic set {, 2, 4, 5,, 8, 0, } with subsets Jet {2,, 5} Major {0, 2, 5} Strain {0, 2, 4} Shark {, 5, 4} 9 5 : Z 2 Z 2 denotes the affine function x 5x + 9

4 Schoenberg, String Quartet No., Opus End of First Section, Measures 88-92, Major/Minor RP-Cycle { &b 88 & b œ J nœ œ nœ œ Œ? 90 & Œ œ œ nœ œ # œ # œ œ Gb Ebmi Eb Cmi C Ami A F#mi bn # œ nn Octatonic set {0,, 3, 4, 6,, 9, 0} contains major/minor chords 6t R 6t3 P t3 R 30 P 04 R 049 P 49 R 96 ##œ Consecutive chords have maximal overlap (well known neo-riemannian operations) Complete cycle These major/minor chords are a maximal cover of the octatonic.

5 Certain Affine Image of Major/Minor RP-Cycle is Jet/Shark RP-Cycle : Z 2 Z 2 denotes the affine map x x +. This maps the octatonic of mm to the octatonic of the opening theme, and a RP-cycle to a RP-cycle : & & triadic series in mm nœ œ nœ 6 t œ # œ œ œ # œ nœ e t 2 5 transformation of the triadic series by nœ # œ # œ œ # œ œ 254 is in opening theme: Jet 25 and Shark 54. Note: Affine image of maximally overlapping chords also consists of maximally overlapping chords, cycle cycle, maximal cover maximal cover

6 Affine Image is Piece-wide Narrative Constructed from Opening Cell in Measures and 2 m. & # œ œ. œ œ mf ff 2 4 m. C30 m. 85 J œ œ œ & œ # œ œ œ œ œ œ 3 p œ œ œ nœ 5 6 œ nœ m œ œ b œ?? n # œ œ# œœ œ œ œ # œ m. O3 œ nœ f # œ 8 œ # œ and 2: Opening violin motive 3 and 4: Fortissimo return of main theme, doubled in all four instruments 5,6, and : Just before major/minor RP-cycle, violin imitation and 8: Return of opening theme in parallel major, with forms and 8 as registral extremes

7 Further Instance of RP-Pattern within Third Octatonic, Measures 8-0 > > œ œ > œ > { &b # œ b œ œ nœ b # œœ nœ nn œ # œ n # b œ œ œ n œ ? b œ œ nœ œ # œ Third Octatonic {2, 3, 5, 6, 8, 9,, 0} contains major/minor chords Consecutive chords have maximal overlap (well known neo-riemannian operations) However, pattern breaks off before attaining maximal cover.

8 Desiderata of a Mathematical Theory for this Analysis To be of use in this analyis, a mathematical theory should: Associate a neo-riemannian-type group to 3-tuples (x, x 2, x 3 ) of pitch-classes, e.g. for (2,,5) in opening theme 2 Have attendant theorems about duality with transposition, inversion, and affine maps, e.g. commutation with 3 Show how to move between the associated neo-riemannian type groups associated to two such 3-tuples (x, x 2, x 3 ) and (y, y 2, y 3 ) when related by an affine map, e.g. the major (, 6, 0) and the jet (2,, 5) related by 4 Show how to obtain substructures of neo-riemannian type groups, e.g. for octatonic and its maximal major/minor cover or jet/shark cover 5 Show how to move between these substructures, e.g. between the three octatonics 6 Determine which subsets of the octatonic set generate maximal covers with simply transitive action under a neo-riemannian type group action

9 The Neo-Riemannian PLR-Group

10 The Neo-Riemannian Transformation P We consider three functions P, L, R : S S. Let P(x) be that triad of opposite type as x with the first and third notes switched. For example P 0, 4, = P(C-major) = The set S of consonant triads Major Triads Minor Triads C = 0, 4, 0, 8, 5 = f C = D =, 5, 8, 9, 6 = f = g D = 2, 6, 9 2, 0, = g D = E = 3,, 0 3,, 8 = g = a E = 4, 8, 4, 0, 9 = a F = 5, 9, 0 5,, 0 = a = b F = G = 6, 0, 6, 2, = b G =,, 2, 3, 0 = c G = A = 8, 0, 3 8, 4, = c = d A = 9,, 4 9, 5, 2 = d A = B = 0, 2, 5 0, 6, 3 = d = e B =, 3, 6,, 4 = e

11 The Neo-Riemannian Transformation P We consider three functions P, L, R : S S. Let P(x) be that triad of opposite type as x with the first and third notes switched. For example P 0, 4, =, 3, 0 P(C-major) = c-minor The set S of consonant triads Major Triads Minor Triads C = 0, 4, 0, 8, 5 = f C = D =, 5, 8, 9, 6 = f = g D = 2, 6, 9 2, 0, = g D = E = 3,, 0 3,, 8 = g = a E = 4, 8, 4, 0, 9 = a F = 5, 9, 0 5,, 0 = a = b F = G = 6, 0, 6, 2, = b G =,, 2, 3, 0 = c G = A = 8, 0, 3 8, 4, = c = d A = 9,, 4 9, 5, 2 = d A = B = 0, 2, 5 0, 6, 3 = d = e B =, 3, 6,, 4 = e

12 The Neo-Riemannian Transformations L and R Let L(x) be that triad of opposite type as x with the second and third notes switched. For example L 0, 4, =,, 4 L(C-major) = e-minor. Let R(x) be that triad of opposite type as x with the first and second notes switched. For example R 0, 4, = 4, 0, 9 R(C-major) = a-minor.

13 Example: Oh! Darling from the Beatles f L D R b P R L E E+ A E Oh Darling please believe me f D I ll never do you no harm b E Be-lieve me when I tell you b E A I ll never do you no harm

14 The Neo-Riemannian PLR-Group and Duality Definition The neo-riemannian PLR-group is the subgroup of permutations of S generated by P,L, and R. Theorem (Lewin 80 s, Hook 2002,...) The PLR group is dihedral of order 24 and is generated by L and R. Theorem (Lewin 80 s, Hook 2002,...) The PLR group is dual to the T /I group in the sense that each is the centralizer of the other in the symmetric group on the set S of major and minor triads. Moreover, both groups act simply transitively on S.

15 Extension of Neo-Riemannian Theory Theorem (Fiore Satyendra, 2005) Let x,..., x n Z m and suppose that there exist x q, x r in the list such that 2(x q x r ) 0. Let S be the family of 2m pitch-class segments that are obtained by transposing and inverting the pitch-class segment X = x,..., x n. Then: The transposition-inversion group acts simply transitively on S and has order 2m. 2 Its centralizer is also dihedral of order 2m, and is generated by any PLR-type operation and { T Y if Y is a transposed form of X Q (Y ) := T Y if Y is an inverted form of X.

16 Constructing Sub Dual Groups

17 Dual Groups Definition (Dual groups in the sense of Lewin) Let Sym(S) be the symmetric group on the set S. Two subgroups G and H of the symmetric group Sym(S) are called dual if their natural actions on S are simply transitive and each is the centralizer of the other, that is, C Sym(S) (G) = H and C Sym(S) (H) = G. Example The T /I -group and the PLR-group are dual in the symmetric group on consonant triads. Example (Cayley) All dual groups arise as the left and right multiplication of a group on itself.

18 Constructing Sub Dual Groups Theorem (Fiore Noll, 20) Given a pair of dual groups, G and H acting on S, we may easily construct a subdual pair, G 0 and H 0 acting on S 0, as follows. Pick G 0 G and s 0 S. 2 Let S 0 := G 0 s 0 and H 0 := {h H hs 0 S 0 }. 3 Then G 0 and H 0 are dual groups acting on S 0. 4 If k H, and we transform S 0 to ks 0, then H 0 transforms to kh 0 k.

19 Example: {C, c} Example G = PLR-group H = T /I -group S = consonant triads. Pick G 0 = {Id, P} and s 0 = C. 2 Then S 0 = G 0 s 0 = {C, c} H 0 consists of solutions to T i C = C and I j C = c, H 0 = {Id, I }. 3 Then {Id, P} and {Id, I } are dual groups on {C, c}. 4 For any T l, {Id, P} and {Id, T l I T l } are dual groups on {T l C, T l c}.

20 Example: Hexatonic (Cohn 996) and (Clampitt 998) Example G = PLR-group, H = T /I -group, S = consonant triads. Pick G 0 = P, L and s 0 = E. 2 Then S 0 = G 0 s 0 = {E, e, B, b, G, g} H 0 = {Id, T 4, T 8, I, I 5, I 9 } consists of solutions to T i E = E T j E = G I l E = e I m E = g T k E = B I n E = b. 3 Then P, L and {Id, T 4, T 8, I, I 5, I 9 } are dual groups on {E, e, B, b, G, g}. 4 The transforms are the hyperhexatonic. Underlying set is {2, 3, 6,, 0, }.

21 Example: Octatonic Example G = PLR-group, H = T /I -group, S = consonant triads. Pick G 0 = R, P and s 0 = G. 2 Then S 0 = G 0 s 0 = {G, e, E, c, C, a, A, f } (measures of Schoenberg!) H 0 = {Id, T 3, T 6, T 9, I, I 4, I, I 0 } by solving equations 3 Then R, P and H 0 are dual groups on {G, e, E, c, C, a, A, f }. 4 The transforms are the hyperoctatonic. Underlying set is octatonic {0,, 3, 4, 6,, 9, 0}.

22 Other Octatonic Covers Proposition Any 3-element subset X of the octatonic {0,, 3, 4, 6,, 9, 0} Z 2 generates a simply transitive cover with respect to the set-wise stabilizer {T 0, T 3, T 6, T 9, I, I 4, I, I 0 }. {0, 4, } = {0, 4, } = major type {0, 4, } 2 = {0, 8, 2} = strain type {0, 4, } 5 = {0, 8, } = shark type {0, 4, } = {0, 4, } = jet type {0, 4, } 0 = {0, 4, 0} = stride type {0, 4, } = {0, 8, 5} = minor type.

23 Simply Transitive Groups for Other Octatonic Covers Only multiples of major chord that have 3 notes: {0, 4, } = {0, 4, } = major type {0, 4, } 2 = {0, 8, 2} = strain type {0, 4, } 5 = {0, 8, } = shark type {0, 4, } = {0, 4, } = jet type {0, 4, } 0 = {0, 4, 0} = stride type {0, 4, } = {0, 8, 5} = minor type. All of these types occur in Schoenberg, String Quartet, Opus. Fiore Satyendra 2005 Each multiple has a neo-riemannian type group associated to it. Fiore Noll 20 Each multiple generates a simply transitive octatonic cover (take dual to set wise octatonic T /I stabilizer).

24 Morphisms of Simply Transitive Group Actions

25 Morphisms of Group Actions Definition (Morphism of Group Actions) Suppose (G, S ) and (G 2, S 2 ) are group actions. A morphism of group actions (f, ϕ): (G, S ) (G 2, S 2 ) consists of a function f : S S 2 and a group homomorphism ϕ: G G 2 such that commutes for all g G. g S f S 2 S f S 2 ϕ(g)

26 Affine Maps are Morphisms for Groups Generated by P, L, and R-Analogues Definition (Affine Map) A function f : Z 2 Z 2 is called affine if there exist m and b such that f (x) = mx + b for all x Z 2. Proposition Suppose f : Z 2 Z 2 is affine. Then f commutes with P, L, and R-analogues, that is Pf = fp Lf = fl Rf = fr. Example (2,, 5) R (, 5, 4) (, 6, 0) R (6,, 3)

27 Return to Musical Analysis

28 The Affine Morphism Indicates a Parallel Organization Between the RP Major/Minor Cycle and the Jet-Shark Piece-wide Narrative. &b 88 {& b œ J? 90 nœ œ nœ œ Œ & Œ œ œ nœ œ # œ # œ œ Gb Ebmi Eb Cmi C Ami A F#mi bn # œ nn ##œ m. mf & & œ & # œ œ. œ œ triadic series in mm nœ œ nœ 6 t # œ œ œ # œ nœ e t 2 5 ff 2 4 m. C30 m. 85 J œ œ œ & œ # œ œ œ œ œ œ 3 transformation of the triadic series by p œ œ œ nœ 5 6 œ nœ m nœ # œ # œ œ # œ œ b œ œ œ n # œ œ# œœ œ œ œ# œ m. O3?? œ nœ # œ f 8 œ # œ

29 Total Network

30 Total Network Schoenberg, String Quartet No., op. piece-wide, trichordal cycle triadic melody, mm RICH RICH RICH RICH RICH RICH e et et2 RICH RICH RICH RICH RICH RICH 6t 6t3 t cello melody, mm. 8-0 triadic melody, m RICH RICH RICH RICH RICH RICH RICH RICH t inner voices, m. 9 0 RICH 0 40t inner voices, m. 90 RICH 3 39