Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion Groups Actions in neo-Riemannian Music Theory Thomas M. Fiore with A. Crans and R. Satyendra http://www.math.uchicago.edu/~fiore/ Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion Introduction Mathematics is a very powerful descriptive tool in the physical sciences. Similarly, musicians use mathematics to communicate ideas about music. In this talk, we will discuss some mathematics commonly used by musicians. Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion Our Focus Mathematical tools of music theorists: Transposition Inversion The neo-Riemannian PLR-group Its associated graphs An Extension of the PLR-group by Fiore-Satyendra. We will illustrate this extension with an analysis of Hindemith, Ludus Tonalis, Fugue in E. Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion References Material from this talk is from: 1 Thomas M. Fiore and Ramon Satyendra. Generalized contextual groups. Music Theory Online, 11(3), 2005. 2 Alissa Crans, Thomas M. Fiore, and Ramon Satyendra. Musical actions of dihedral groups. American Mathematical Monthly, In press since June 2008. Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion What is Music Theory? Music theory supplies us with conceptual categories to organize and understand music. David Hume: impressions become tangible and form ideas. In other words, music theory provides us with the means to find a good way of hearing a work of music. Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion Who Needs Music Theory? Composers Performers Listeners Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion The Z12 Model of Pitch Class C B 0 C#/D 11 1 A#/B 10 2 D We have a bijection A 9 3 D#/E between the set of pitch Z classes and 12. 8 4 G#/A E 7 5 G 6 F F#/G Figure: The musical clock. Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion Transposition The bijective function T Z Z n : 12 → 12 G G G E♭ 7 7 7 3 Tn(x) := x + n 7 7 7 T−4(7) is called transposition by musicians, or translation F F F D by mathematicians. 5 5 5 2 5 5 5 T−3(5) T−4(7) = 7 − 4 = 3 T−3(5) = 5 − 3 = 2 Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion Inversion The bijective function In : Z12 → Z12 In(x) := −x + n is called inversion by musicians, or hC, Gi hC, F i reflection by mathematicians. h0, 7i hI0(0), I0(7)i h0, 7i h0, 5i I0(0) = −0 = 0 I0(7) = −7 = 5 Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion The T /I -Group C B 0 C#/D Altogether, these 11 1 transpositions and A#/B 10 2 D inversions form the T /I-group. A 9 3 D#/E This is the group of 8 4 G#/A E symmetries of the 7 5 12-gon, the dihedral G 6 F group of order 24. F#/G Figure: The musical clock. Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion Major and Minor Triads Major and minor triads are very The set S of consonant triads common in Major Triads Minor Triads C f Western music. = h0, 4, 7i h0, 8, 5i = C♯ = D♭ = h1, 5, 8i h1, 9, 6i = f ♯ = g♭ D = h2, 6, 9i h2, 10, 7i = g D♯ = E♭ = h3, 7, 10i h3, 11, 8i = g♯ = a♭ C-major = hC, E, Gi E = h4, 8, 11i h4, 0, 9i = a = h0, 4, 7i F = h5, 9, 0i h5, 1, 10i = a♯ = b♭ F ♯ = G♭ = h6, 10, 1i h6, 2, 11i = b G = h7, 11, 2i h7, 3, 0i = c c-minor = hG, E♭, Ci G♯ = A♭ = h8, 0, 3i h8, 4, 1i = c♯ = d♭ A = h9, 1, 4i h9, 5, 2i = d = h7, 3, 0i A♯ = B♭ = h10, 2, 5i h10, 6, 3i = d♯ = e♭ B = h11, 3, 6i h11, 7, 4i = e Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion Major and Minor Triads C B 0 C#/D The T /I -group acts on the 11 1 set S of major and minor A#/B D triads. 10 2 A 9 3 D#/E T1h0, 4, 7i = hT10, T14, T17i 8 4 = h1, 5, 8i G#/A E 7 5 G 6 F I0h0, 4, 7i = hI00, I04, I07i F#/G = h0, 8, 5i Figure: I0 applied to a C-major triad yields an f -minor triad. Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion Neo-Riemannian Music Theory Recent work focuses on the neo-Riemannian operations P, L, and R. P, L, and R generate a dihedral group, called the neo-Riemannian group. As we’ll see, this group is dual to the T /I group in the sense of Lewin. These transformations arose in the work of the 19th century music theorist Hugo Riemann, and have a pictorial description on the Oettingen/Riemann Tonnetz. P, L, and R are defined in terms of common tone preservation. Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion The neo-Riemannian Transformation P We consider three The set S of consonant triads functions Major Triads Minor Triads C f P, L, R : S → S. = h0, 4, 7i h0, 8, 5i = C♯ = D♭ = h1, 5, 8i h1, 9, 6i = f ♯ = g♭ D = h2, 6, 9i h2, 10, 7i = g Let P(x) be that D♯ = E♭ = h3, 7, 10i h3, 11, 8i = g♯ = a♭ triad of opposite E = h4, 8, 11i h4, 0, 9i = a type as x with the F = h5, 9, 0i h5, 1, 10i = a♯ = b♭ first and third F ♯ = G♭ = h6, 10, 1i h6, 2, 11i = b notes switched. G = h7, 11, 2i h7, 3, 0i = c For example G♯ = A♭ = h8, 0, 3i h8, 4, 1i = c♯ = d♭ Ph0, 4, 7i = A = h9, 1, 4i h9, 5, 2i = d A♯ = B♭ = h10, 2, 5i h10, 6, 3i = d♯ = e♭ P(C-major) = B = h11, 3, 6i h11, 7, 4i = e Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion The neo-Riemannian Transformation P We consider three The set S of consonant triads functions Major Triads Minor Triads C f P, L, R : S → S. = h0, 4, 7i h0, 8, 5i = C♯ = D♭ = h1, 5, 8i h1, 9, 6i = f ♯ = g♭ D = h2, 6, 9i h2, 10, 7i = g Let P(x) be that D♯ = E♭ = h3, 7, 10i h3, 11, 8i = g♯ = a♭ triad of opposite E = h4, 8, 11i h4, 0, 9i = a type as x with the F = h5, 9, 0i h5, 1, 10i = a♯ = b♭ first and third F ♯ = G♭ = h6, 10, 1i h6, 2, 11i = b notes switched. G = h7, 11, 2i h7, 3, 0i = c For example G♯ = A♭ = h8, 0, 3i h8, 4, 1i = c♯ = d♭ Ph0, 4, 7i = h7, 3, 0i A = h9, 1, 4i h9, 5, 2i = d A♯ = B♭ = h10, 2, 5i h10, 6, 3i = d♯ = e♭ P(C-major) = c-minor B = h11, 3, 6i h11, 7, 4i = e Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion 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 Lh0, 4, 7i = h11, 7, 4i 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 Rh0, 4, 7i = h4, 0, 9i R(C-major) = a-minor. Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion Minimal motion of the moving voice under P, L, and R. C C B 0 B 0 11 1 C#/D 11 1 C#/D A#/B 10 2 D A#/B 10 2 D A 9 3 D#/E A 9 3 D#/E 8 4 8 4 G#/A E G#/A E 7 5 7 5 G 6 F G 6 F F#/G F#/G PC = c LC = e C B 0 11 1 C#/D A#/B 10 2 D A 9 3 D#/E 8 4 G#/A E 7 5 G 6 F F#/G RC = a Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion Example: The Elvis Progression The Elvis Progression I-VI-IV-V-I from 50’s Rock is: R / L / R◦L◦R◦L / L◦R / ONMLHIJKC ONMLHIJKa ONMLHIJKF ONMLHIJKG ONMLHIJKC Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion Example: The Elvis Progression The Elvis Progression I-VI-IV-V-I from 50’s Rock is: R / L / R◦L◦R◦L / L◦R / ONMLHIJKG ONMLHIJKe ONMLHIJKC ONMLHIJKD ONMLHIJKG Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion Example: “Oh! Darling” from the Beatles L / R / P◦R◦L / ONMLHIJKf ♯ ONMLHIJKD ONMLHIJKb ONMLHIJKE E+ A E Oh Darling please believe me f ♯ D I’ll never do you no harm b7 E7 Be-lieve me when I tell you b7 E7 A I’ll never do you no harm Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion 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.