Groups Actions in Neo-Riemannian Music Theory

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

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+ AE 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.

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. Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion The Oettingen/Riemann Tonnetz

11 61 8 3 10

2 9 4 11 6 1 R L 10 5 0 7 2 9 P 1 8 3 10 5 0

9 4 11 6 1 8

0 7 2 9 4 11

8 3 10 5 0 7 2

11 6 1 8 3 10 Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion The Torus Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion The Dual Graph to the Tonnetz

11 61 8 3 10 c E b C# a A e F# c# E 2 9 4 11 6 1 R L a B f# A e 10 5 0 7 2 9 P E b D a C 1 8 3 10 5 0 e G d F c C g B f A 9 4 11 6 1 8 c E b C# a 0 7 2 9 4 11 A e F# c# E 8 3 10 5 0 7 2 P L R

11 6 1 8 3 10 Figure: Douthett and Steinbach’s Graph. Figure: The Tonnetz. Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion The Dual Graph to the Tonnetz

A c E b C# a D a A e c# E F# b G a B f# A e f# e d E b D a C E C e G d F c c# B g F C g B f A a c c E b C# a F# A A e F# c# E B e E P L R b C# f Figure: Douthett and Steinbach’s Graph. Figure: Waller’s Torus. Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion Beethoven’s 9th, 2nd Mvmt, Measures 143-17 (Cohn)

R / L / R / L / R dd/ ?>=<89:;C ?>=<89:;a ?>=<89:;F ?>=<89:;d ?>=<89:;Bd♭dddddd ?>=<89:;g ddddddd ddddddd dddLdddd ddddddd ddddddd ddddddd dr d / L / R / L / R d/ ?>=<89:;E♭ ?>=<89:;c ?>=<89:;A♭ ?>=<89:;f ?>=<89:;D♭ddddddd ?>=<89:;b♭ R ddddddd ddddddd ddLddddd ddddddd ddddddd ddddddd dr dd / L / R / L / R / ?>=<89:;G♭ ?>=<89:;e♭ ?>=<89:;B ?>=<89:;g♯ ?>=<89:;E dddddd ?>=<89:;c♯ R ddddddd ddddddd Lddddddd ddddddd ddddddd ddddddd dr dddd L R L R A r / f ♯ / D / b / G / e ?>=<89:; R ?>=<89:; ?>=<89:; ?>=<89:; ?>=<89:; ?>=<89:; Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion Thus Far We have seen:

How to encode pitch classes as integers modulo 12, and consonant triads as 3-tuples of integers modulo 12

How the T /I -group acts componentwise on consonant triads

How the PLR-group acts on consonant triads

Duality between the T /I -group and the PLR-group

Geometric depictions on the torus and musical examples.

But most music does not consist entirely of triads! Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion Extension of Neo-Riemannian Theory

Theorem (Fiore–Satyendra, 2005)

Let x1,..., xn ∈ Zm and suppose that there exist xq, xr in the list such that 2(xq − xr ) 6= 0. Let S be the family of 2m pitch-class segments that are obtained by transposing and inverting the pitch-class segment X = hx1,..., xni. Then the 2m transpositions and inversions act simply transitively on S. Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion Extension of Neo-Riemannian Theory: Duality

Theorem (Fiore–Satyendra, 2005) Fix 1 ≤ k,ℓ ≤ n. Deﬁne

K(Y ) := Iyk +yℓ (Y )

Ti Y if Y is a transposed form of X Qi (Y ) := ( T−i Y if Y is an inverted form of X.

Then K and Q1 generate the centralizer of the T /I group of order 2m. This centralizer is called the generalized contextual group. It is dihedral of order 2m, and its centralizer is the mod m T /I-group. Moreover, the generalized contextual group acts simply transitively on S. Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion Subject of Hindemith, Ludus Tonalis, Fugue in E

Let’s see what this theorem can do for us in an analysis of Hindemith’s Fugue in E from Ludus Tonalis.

Subject: Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion The Musical Space S in the Fugue in E

Transposed Forms Inverted Forms P0 h2, 11, 4, 9i p0 h10, 1, 8, 3i P1 h3, 0, 5, 10i p1 h11, 2, 9, 4i Consider the P2 h4, 1, 6, 11i p2 h0, 3, 10, 5i P p four-note motive 3 h5, 2, 7, 0i 3 h1, 4, 11, 6i P4 h6, 3, 8, 1i p4 h2, 5, 0, 7i P0 = hD, B, E, Ai P5 h7, 4, 9, 2i p5 h3, 6, 1, 8i P6 h8, 5, 10, 3i p6 h4, 7, 2, 9i = h2, 11, 4, 9i. P7 h9, 6, 11, 4i p7 h5, 8, 3, 10i P8 h10, 7, 0, 5i p8 h6, 9, 4, 11i P9 h11, 8, 1, 6i p9 h7, 10, 5, 0i P10 h0, 9, 2, 7i p10 h8, 11, 6, 1i P11 h1, 10, 3, 8i p11 h9, 0, 7, 2i Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion

Q−2 Applied to Motive in Subject and I11-Inversion

Q−2 / Q−2 / Q−2 / ONMLHIJKP0 ONMLHIJKP10 ONMLHIJKP8 ONMLHIJKP6 Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion

Q−2 Applied to Motive in Subject and I11-Inversion

Q−2 / Q−2 / Q−2 / ONMLHIJKp11 ONMLHIJKp1 ONMLHIJKp3 ONMLHIJKp5 Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion

Product Network Encoding Subject and I11-Inversion

1 Q−2 / Q−2 / Q−2 / ONMLHIJKP0 ONMLHIJKP10 ONMLHIJKP8 ONMLHIJKP6

I11 I11 I11 I11 34 Q−2 / Q−2 / Q−2 / ONMLHIJKp11 ONMLHIJKp1 ONMLHIJKp3 ONMLHIJKp5

Our Theorem about duality guarantees this diagram commutes! Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion Self-Similarity: Local Picture

29 GFED@ABCG

T9 GFED@ABCE

I1 GFED@ABCA

I11 GFED@ABCD Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion Self-Similarity: Global Picture and Local Picture 8-11 Q−2 / Q−2 / Q−2 / WVUTPQRSRow 1 WVUTPQRSP5 WVUTPQRSP3 WVUTPQRSP1 WVUTPQRSP11

T9 T9 T9 T9 T9 57-60 Q−2 / Q−2 / Q−2 / WVUTPQRSRow 2 WVUTPQRSP2 WVUTPQRSP0 WVUTPQRSP10 WVUTPQRSP8

I1 I1 I1 I1 I1 34-37 Q−2 / Q−2 / Q−2 / WVUTPQRSRow 3 WVUTPQRSp11 WVUTPQRSp1 WVUTPQRSp3 WVUTPQRSp5

I11 I11 I11 I11 I11 1-4 Q−2 / Q−2 / Q−2 / WVUTPQRSRow 4 WVUTPQRSP0 WVUTPQRSP10 WVUTPQRSP8 WVUTPQRSP6 Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion Utility of Theorems

Again, our Theorem about duality allows us to make this product network.

More importantly, the internal structure of the four-note motive is replicated in transformations that span the work as whole. Thus local and global perspectives are integrated.

These groups also act on a second musical space S′ in the piece, which allows us to see another kind of self-similarity: certain transformational patterns are shared by distinct musical objects! Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion Summary

In this lecture I have introduced some of the conceptual categories that music theorists use to make aural impressions into vivacious ideas in the sense of Hume.

These included: transposition and inversion, the PLR group, its associated graphs on the torus, and duality. Introduction Transposition and Inversion The neo-Riemannian Group and Geometry Extension of neo-Riemannian Theory Hindemith, Fugue in E Conclusion Summary

We have used these tools to ﬁnd good ways of hearing music from Hindemith, the Beatles, and Beethoven. Some of these ideas would have been impossible without mathematics.

I hope this introduction to mathematical music theory turned your impressions of music theory into vivacious ideas!