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WORDS 2007

WORDS 2007. Thomas Noll David Clampitt Manuel Dominguez Barcelona New Haven Madrid. What Sturmian Morphisms reveal about Musical Scales and Tonality. A Circular Definition of the “generic” Diatonic Scale:. Each Fifth is four Steps.

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WORDS 2007

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  1. WORDS 2007 Thomas Noll David Clampitt Manuel DominguezBarcelona New Haven Madrid What Sturmian Morphisms reveal about Musical Scales and Tonality

  2. A Circular Definition of the “generic” Diatonic Scale: Each Fifth is four Steps One Step is two Fifths (mod Octave). Each Step is two Fifths (mod Octave).

  3. A Circular Definition of the “generic” Diatonic Scale: Fifth = four Steps One Step is two Fifths (mod Octave).

  4. A Circular Definition of the “generic” Diatonic Scale: 4 * 2 = 1 mod 7 Fifth = four Steps Step = two Fifths (mod Octave).

  5. Haendel-Passacaille Circular Definition of the Diatonic: A Fifth is four Steps. A Step is two Fifths (mod Octave). D C Bb A (G) G C F Bb Eb A D G G F Eb D C F Eb D C Bb Eb D C Bb A D C Bb A G G C F Bb Eb A D G 12

  6. b b b b a b a a a a a a b b b a a a b a a a a b b a b b b Norman Carey & David Clampitt’s Theory of Wellformed Scales The well-formed generatedness of the diatonicis equivalent to the fact that each generic intervalsecond, third, fourth, fifth, sixth, seventh comes in exactly two different sizes: m2, M2, m3, M3, P4, A4, D5, P5, m6, M6, m7, M7

  7. ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 2 3 3 4 1 0 1 0 1 5 3 0 1 5 1 1 7 2 1 2 2 b b b b a b a a a a a a b b b a a a b a a a a b b a b b b L R L Well-formed Scales are associatedwith Semiconvergentsof Log2(3/2). L R R ( ) 3 7 5 12

  8. Pythagorean Lattice pitch height vector (log2(3/2),1) zero pitch level octave axis fifth axis zero pitch level

  9. Corresponding upper and lower mechanical words Tonality = A discrete isotonic line ?

  10. Basic Motivation of this Talk: 4 * 2 = 1 mod 7 b a a a a a b a a a b a a b Lydian

  11. Basic Motivation of this Talk: 4 * 2 = 1 mod 7 b a a a a a b a a b a a b a Mixolydian

  12. Basic Motivation of this Talk: 4 * 2 = 1 mod 7 b a a a a a b a b a a b a a Aeolian

  13. Basic Motivation of this Talk: 4 * 2 = 1 mod 7 b a a a a a b b a a b a a a Locrian

  14. Basic Motivation of this Talk: 4 * 2 = 1 mod 7 b a a a a a b a a b a a a b Ionian

  15. Basic Motivation of this Talk: 4 * 2 = 1 mod 7 b a a a a a b a b a a a b a Dorian

  16. Basic Motivation of this Talk: 4 * 2 = 1 mod 7 b a a a a a b b a a a b a a Phrygian

  17. a a b a | a a b Ionian a b a a | a b a Dorian b a a a | b a a Phrygian a a a b | a a b Lydian a a b a | a b a Mixolydian a b a a | b a a Aeolian b a a b | a a a Locrian

  18. How to cut this equation: 4 * 2 = 1 mod 7 ?

  19. Involution Involution w f w* f* = = Christoffel Words xyxyxyy aaabaab Special SturmianMorphisms = = D~GG GGD f*(xy) = f(xy)* f*(xy) = Conjugate(f(xy)*) Can one fine-tune the involution on words (or on morphisms)such that f*(xy) = f(xy)* holds?

  20. Well-formed Words Remark: A word is well-formed iff it is conjugate to a Christoffel word

  21. w*=a a b a a a b substitute a with |w*|b and b with -|w*|a 2 2 -5 2 2 2 -5 calculate partial sums 0 2 4 -1 1 3 5 (0) create word as single loop starting from the minimal value.(go left = y, go right = x) 0 2 4 -1 1 3 5 y x y x y x y Plain Adjoint w*of a Well-formed Word w Idea (by Example): w= y x y x y x y Balance of w: substitute x with |w|y and y with -|w|x -3 4 -3 4 -3 4 -3 Accumulation of w: calculate partial sums 0 -3 1 -2 2 -1 3 (0) create word as single loop starting from the minimal value.(go left = b, go right = a) 0 -3 1 -2 2 -1 3 a a b a a a b

  22. Plain Adjoint w*of a Well-formed Word w w= y x y x y x y w*=a a b a a a b Remarks: (1) The plain adjoints extend involution of Christoffel words(2) Standard words are mappped to standard words(3) f(xy)* = Reverse[f](xy) for f in <G,D> and f in <G, D~> (4) The plain adjoints of amorphous words ( bad conjugates) are generally not amorphous(5) Standardicity is equivalent to divider incidenceThis property is music-theoretically particularly illuminating. It distinguishes the ionian mode (Major mode) from the seven modes.

  23. -b a F y -3 4 -3 4 -3 4 -3 x 0 G C 0 -3 1 -2 2 -1 3 (0) y x -3 1 A D y x -2 2 E B y Octaves -1 3 F# Fifths (0) yx|yxyxy DGG(x|y)

  24. a b a a b a a F C’ Renaming: a for major step bfor minor step 0 G C 1 -3 A D 2 -2 E B 3 -1 aaba|aab GGD(a|b) -y x

  25. a b a a b a a F C’ (0) Renaming: a for major step bfor minor step 0, -1 G C 1, 1 -3, 0 A D 2, 3 -2, 2 E B 3, 5 -1, 4 aaba|aab GGD(a|b) -y x

  26. -b a y x y x y x y DGG(x|y) GGD(a|b) a a b a a a b -y x

  27. DGG(a,b) = DG(a,ab) = D(a,aab) = (aaba, aab) GGD(a,b) = GG(ba, b) = G(aba, ab) = (aaba, aab) GGD(x,y) = GG(yx,y) = G(yx, yxy)= (yx, yxyxy) DGG(x,y) = DG(x, xy) = D(x, xxy)= (yx, yxyxy) Ionian

  28. DGG~(a,b) = DG(a,ba) = D(a,aba) = (abaa,aba) G~GD (a,b) = G~G(ba,b) = G~(aba,ab) = (abaa,aba) G~G~D~(x,y) = G~G~(xy,y) = G~(xy, yxy)= (xy, yxyxy) D~G~G~(x,y) = D~G~(x,yx) = D~(x, yxx)= (xy, yxyxy) Dorian

  29. DG~G~(a,b) = DG~(a,ba) = D(a,baa) = (baaa,baa) G~G~D(a,b) = G~ G~(ba,b) = G~(baa,ba) = (baaa,baa) GG~D~(x,y) = GG~(xy,y) = G(xy, yxy)= (xy, xyyxy) D~G~G(x,y) = D~G~(x, xy) = D~(x, xyx)= (xy, xyyxy) Phrygian

  30. D~GG(a,b) = D~G(a, ab) = D~(a, aab) = (aaab,aab) GGD~ (a,b) = GG(ab, b) = G(aab, ab) = (aaab,aab) GGD~(x,y) = GG(xy,y) = G(xy, xyy)= (xy, xyxyy) D~GG(x,y) = D~G(x, xy) = D~(x, xxy)= (xy, xyxyy) Lydian

  31. D~GG~(a,b) = D~G(a, ba) = D~(a, aba) = (aaba,aba) G~GD~(a,b) = G~G(ab, b) = G~(aab,ab) = (aaba,aba) (yy, xyxyx) No morphism Mixolydian

  32. D~G~G~(a,b) = D~G~(a,ba) = D~(a,baa) = (abaa, baa) G~G~ D~(a,b) = G~G~(ab, b) = G~(aba, ba) = (abaa, baa) G~G~D(x,y) = G~G~(yx,y) = G~(yx, yyx)= (yx, yyxyx) DG~G~(x,y) = DG~(x, yx) = D(x, yxx)= (yx, yyxyx) Aeolian

  33. (baab,aaa) No morphism GG~D(x,y) = GG~(yx,y) = G(yx, yyx)= (yx, yxyyx) DG~G(x,y) = DG~(x, xy) = D(x, xyx)= (yx, yxyyx) Locrian

  34. Sturmian Involution and Scale/Fundament Conversion DGG Ionian GGD DG~G Locrian G~GD Dorian D~G~G~ % DG~G~ Aeolian G~G~D~ D~G~G G~G~D Phrygian Scale pattern Fundamentpattern G~GD~ GGD~ Lydian Mixolydian % D~GG

  35. -b -b a a Hypo-Ionian Ionian y x y x y x y x y x y x x y a a b a a a b b b a b b a b -y x -y x

  36. Twisted Adjoint of a Well-formed Word x y | x y x yy = D~GG(x |y ) a a b a | a a b = GGD(a |b ) F C G D A C F Bb E D C B A G F E substitute x with |w*|y and y with -|w*|x substitute a with |w*|b and b with -|w*|a 4 -3 |4 -3 4 -3 -3 2 2 -5 2 | 2 2 -5 calculate partial sums calculate partial sums 0 4 1 5 2 6 3 (0) 0 2 4 -1 1 3 5 (0) create word as single loop starting from the maximal value.(go left = x, go right = y) create word as single loop starting from the maximal value.(go left = a, go right = b) 0 2 4 -1 1 3 5 0 4 1 5 2 6 3 a x y a b x a y a x a y b y

  37. Twisted Adjoint of a Well-formed Word xxyxy | xy= GD~D~ (x |y ) b b a |b b a b = DDG(a | b) substitute x with |w*|y and y with -|w*|x substitute a with |w*|b and b with -|w*|a 33-43-4 | 3-4 -2 -2 5 |-2 -2 5 -2 calculate partial sums calculate partial sums 0 3 6 2 5 1 4 (0) 0 -2 -4 1 -1 -3 2 (0) create word as single loop starting from the maximal value.(go left = x, go right = y) create word as single loop starting from the maximal value.(go left = a, go right = b) 0 -2 -4 1 -1 -3 2 0 3 6 2 5 1 4 (0) x b x b a y x b b y a x b y

  38. Twisted Adjoint w*of a Well-formed Word w w=xxyxy xy w*= b b a b b a b Remarks: The twisted adjoints mimic the involution of Special Sturmian morphisms on the images of xy (and ab): f*(xy) = f(xy)* and f*(ab) = f(ab)* (2) Standard words are mapped to Christoffel words and vice versa This property is music-theoretically particularly illuminating. It distinguishes Aeolian (Minor) from the other modes.

  39. b b b b a b a a a a a a b b b a a a b a a a a b b a b b b “Fifth”- generated Standard Modes x|y D(x|y) = yx|y DG(x|y) = yx|yxy DGG(x|y) = yx|yxyxy DGGD(x|y) = yxyxyyx|yxyxy Thanks: Tomi Kärki !

  40. aaba -b-a-a aaba aaba -b-a-a -b-a-a -b-a-a -b a y x Question: Should the pythagoreantone system be studied interms of the free group F2 ? x y y x y x y x y (-b-a-aaab)a(-b-a-aaab)a(-b-a-aaab)a-b-a-a a-b≠ (aa)a-b(-a-a)

  41. What does that mean? It is not possible to circularly embed the fundamental line into the scale and vice versa without making a“mistake”. The elementary melodic step and the (compound)fundamental step differ by a conjugation. Likewize the elementary fundamental progression and the (compound) melodic fifth differ by a conjugation. There is a simpler scale, where this problem does notoccur: The Tetrade (“Footprint”). a =yx b = - y x =ba =- yyx = x

  42. -b a a -b y x y y x y x y x y a a b a a a b b a b -y x -y x

  43. a -b Standard Fundamental Tetrade y x y F FundamentDivider C G Minor Third Subsitution D b a b -y x

  44. Coda: Roger “DJ Christoffel” Costa (Barcelona)

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