Introduction and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
What is Mathematical Music Theory?
An Introduction via Perspectives on Consonant
Triads
Thomas M. Fiore
http://www-personal.umd.umich.edu/~tmfiore/
Introduction and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
What is Mathematical Music Theory?
Mathematical music theory uses modern mathematical structures
to
1
analyze works of music (describe and explain them),
2
study, characterize, and reconstruct musical objects such as
the consonant triad, the diatonic scale, the Ionian mode, the
consonance/dissonance dichotomy...
3
compose
4
...
Introduction and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
What is Mathematical Music Theory?
Mathematical music theory uses modern mathematical structures
to
1
analyze works of music (describe and explain them),
2
study, characterize, and reconstruct musical objects such as
the consonant triad, the diatonic scale, the Ionian mode, the
consonance/dissonance dichotomy...
3
compose
4
...
Introduction and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
Levels of Musical Reality, Hugo Riemann
There is a distinction between three levels of musical reality.
Physical level: a tone is a pressure wave moving through a
medium, “Ton”
Psychological level: a tone is our experience of sound,
“Tonempfindung”
Intellectual level: a tone is a position in a tonal system,
described in a syntactical meta-language, “Tonvorstellung”.
Mathematical music theory belongs to this realm.
Introduction and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
Work of Mazzola and Collaborators
Mazzola, Guerino. Gruppen und Kategorien in der Musik.
Entwurf einer mathematischen Musiktheorie. Research and
Exposition in Mathematics, 10. Heldermann Verlag, Berlin,
1985.
Mazzola, Guerino. The topos of music. Geometric logic of
concepts, theory, and performance. In collaboration with
Stefan Göller and Stefan Müller. Birkhäuser Verlag, Basel,
2002.
Noll, Thomas, Morphologische Grundlagen der
abendländischen Harmonik in: Moisei Boroda (ed.),
Musikometrika 7, Bochum: Brockmeyer, 1997.
These developed a mathematical meta-language for music theory,
investigated concrete music-theoretical questions, analyzed works
of music, and did compositional experiments (Noll 2005).
Introduction and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
Lewin
Lewin, David. Generalized Musical Intervals and
Transformations, Yale University Press, 1987.
Lewin, David. Musical Form and Transformation: 4 Analytic
Essays, Yale University Press, 1993.
Lewin introduced generalized interval systems to analyze works of
music. The operations of Hugo Riemann were a point of
departure. Mathematically, a generalized interval system is a
simply transitive group action.
Introduction and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
II. Basics
Introduction and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
The Z12 Model of Pitch Class
B
We have a bijection
between the set of pitch
classes and Z12 .
A#/B
C
0
1
11
C#/D
2 D
10
3 D#/E
A 9
G#/A
4 E
8
5
7
G
6
F#/G
F
Figure: The musical clock.
Introduction and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
Transposition and Inversion
We have the musical operations transposition (rotation)
Tn : Z12 → Z12
Tn (x) := x + n
and inversion (reflection)
In : Z12 → Z12
In (x) := −x + n.
These generate the dihedral group of order 24 (symmetries of
12-gon).
Introduction and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
III. Characterization of the Consonant Triad
Introduction and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
Major and Minor Triads
Major and minor
triads are very
common in
Western music.
(Audio)
C -major = hC , E , G i
= h0, 4, 7i
c-minor = hG , E ♭, C i
= h7, 3, 0i
The set S of consonant triads
Major Triads Minor Triads
C = h0, 4, 7i h0, 8, 5i = f
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♭
E = h4, 8, 11i h4, 0, 9i = a
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
G ♯ = A♭ = h8, 0, 3i h8, 4, 1i = c♯ = d♭
A = h9, 1, 4i h9, 5, 2i = d
A♯ = B♭ = h10, 2, 5i h10, 6, 3i = d♯ = e♭
B = h11, 3, 6i h11, 7, 4i = e
Introduction and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
Major and Minor Triads
The T /I -group acts on the
set S of major and minor
triads.
B
A#/B
C
0
1
11
C#/D
2 D
10
3 D#/E
A 9
T1 h0, 4, 7i = hT1 0, T1 4, T1 7i
= h1, 5, 8i
G#/A
4 E
8
5
7
I0 h0, 4, 7i = hI0 0, I0 4, I0 7i
= h0, 8, 5i
G
6
F#/G
F
Figure: I0 applied to a C -major triad
yields an f -minor triad.
Introduction and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
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 and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
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
Ph0, 4, 7i =
P(C -major) =
The set S of consonant triads
Major Triads Minor Triads
C = h0, 4, 7i h0, 8, 5i = f
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♭
E = h4, 8, 11i h4, 0, 9i = a
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
G ♯ = A♭ = h8, 0, 3i h8, 4, 1i = c♯ = d♭
A = h9, 1, 4i h9, 5, 2i = d
A♯ = B♭ = h10, 2, 5i h10, 6, 3i = d♯ = e♭
B = h11, 3, 6i h11, 7, 4i = e
Introduction and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
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
Ph0, 4, 7i = h7, 3, 0i
P(C -major) = c-minor
The set S of consonant triads
Major Triads Minor Triads
C = h0, 4, 7i h0, 8, 5i = f
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♭
E = h4, 8, 11i h4, 0, 9i = a
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
G ♯ = A♭ = h8, 0, 3i h8, 4, 1i = c♯ = d♭
A = h9, 1, 4i h9, 5, 2i = d
A♯ = B♭ = h10, 2, 5i h10, 6, 3i = d♯ = e♭
B = h11, 3, 6i h11, 7, 4i = e
Introduction and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
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 and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
Minimal motion of the moving voice under P, L, and R.
B
C
0
B
1 C#/D
11
A#/B 10
4 E
3 D#/E
G#/A 8
4 E
5
7
G
2 D
A 9
3 D#/E
G#/A 8
6
F#/G
5
7
G
F
PC = c
6
F#/G
LC = e
B
1 C#/D
A#/B 10
2 D
A 9
C
0
11
C
0
11
1 C#/D
A#/B 10
2 D
A 9
3 D#/E
8
4 E
G#/A
7
G
5
6
F#/G
RC = a
F
F
Introduction and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
Uniqueness of the Consonant Triad
Theorem (Richard Cohn, 1997)
The consonant triad is uniquely characterized by parsimonious
voice leading. Namely, suppose h0, x, y i is a 3-note chord in Z12 ,
and we generate its orbit under the dihedral group. Then the P, L,
R operations associated to the orbit of h0, x, y i admit parsimonious
voice leading if and only if h0, x, y i is a consonant triad.
Remarkable: this result is completely independent of the acoustic
properties of the consonant triad!
Introduction and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
Example: The Elvis Progression
The Elvis Progression I-VI-IV-V-I from 50’s Rock is:
HIJK
ONML
C
R
HIJK
/ONML
a
L
HIJK
/ONML
F
R◦L◦R◦L
ONML
/HIJK
G
L◦R
ONML
/HIJK
C
Introduction and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
Example: “Oh! Darling” from the Beatles
HIJK
ONML
f♯
L
HIJK
/ONML
D
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
R
HIJK
/ONML
b
P◦R◦L
OHIJK
/ NML
E
Introduction and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
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 and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
IV. Geometry of Consonant Triads
Introduction and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
The Tonnetz Graph and its Dual are Tori
6
11
8
1
c
10
3
9
2
5
10
1
0
8
4
9
P
8
0
8
E
0
4
9
11
3
Figure: The Tonnetz.
10
f#
G
d
B
L
A
e
F
c
E
C
A
f
C#
F#
e
P
a
a
b
E
C#
c#
D
g
A
2
B
b
c
7
F#
e
C
1
5
1
5
e
a
9
2
10
2
10
6
7
1
6
11
6
11
3
11
4 L
3
7
0
8
R
b
E
A
a
c#
E
R
Figure: Douthett and Steinbach’s
Graph.
See Crans–Fiore–Satyendra, Musical Actions of Dihedral Groups,
American Mathematical Monthly, 2009.
Introduction and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
The Dual Graph to the Tonnetz on a Torus
c
A
E
b
B
f#
e
F#
a
E
b
C
g
e
c
P
D
a
B
f
e
A
c#
a
D
E
b
f#
c
C
c#
B
g
a
c
F#
A
E
e
C#
B
E
R
Figure: Douthett and Steinbach’s
Graph.
d
E
A
a
G
e
C
C#
F#
L
A
F
b
E
e
A
a
c#
d
G
C#
b
f
Figure: Waller’s Torus.
F
Introduction and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
Beethoven’s 9th, 2nd Mvmt, Measures 143-175 (Cohn)
?>=<
89:;
C
R
89:;
89:;
/ ?>=<
/ ?>=<
B♭
dddddd g
d
d
d
d
d
d
d
d
d
d
d
L dddddddd
ddddddd
d
d
d
d
d
d
d
ddddddd L
R
L
R
dr dddddd / ?>=<
?>=<
89:;
89:;
89:;
89:;
89:;
89:;
/ ?>=<
/ ?>=<
/ ?>=<
/ ?>=<
c
f
E♭
A♭
D♭
ddddddd b♭
d
d
R
d
d
d
d
d
d
d
d
d
L dddddddd
ddddddd
d
d
d
d
d
d
ddddddd L
R
L
R
dr dddddd / ?>=<
?>=<
89:;
89:;
89:;
89:;
89:;
89:;
/ ?>=<
/ ?>=<
/ ?>=<
/ ?>=<
g♯
B
E
G♭
e♭
ddddddd c♯
d
R
d
d
d
d
d
d
d
d
d
d
L dddddddd
ddddddd
d
d
d
d
d
d
dd
ddddddd L
R
L
R
?>=<
89:; dr dddd / ?>=<
89:;
89:;
89:;
89:;
89:;
/ ?>=<
/ ?>=<
/ ?>=<
/ ?>=<
A
R
R
89:;
/ ?>=<
a
f♯
L
89:;
/ ?>=<
F
D
R
89:;
/ ?>=<
d
b
L
G
e
Introduction and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
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!
A mathematical perspective helps here.
Introduction and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
V. Extension of Neo-Riemannian Theory
Introduction and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
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 , . . . , xn i. Then the 2m transpositions
and inversions act simply transitively on S.
Introduction and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
Extension of Neo-Riemannian Theory: Duality
Theorem (Fiore–Satyendra, 2005)
Fix 1 ≤ k, ℓ ≤ n. Define
K (Y ) := Iyk +yℓ (Y )
Qi (Y ) :=
(
Ti Y if Y is a transposed form of X
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 and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
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 and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
The Musical Space S in the Fugue in E
Consider the
four-note motive
P0 = hD, B, E , Ai
= h2, 11, 4, 9i.
Transposed Forms
P0 h2, 11, 4, 9i
P1 h3, 0, 5, 10i
P2 h4, 1, 6, 11i
P3
h5, 2, 7, 0i
P4
h6, 3, 8, 1i
P5
h7, 4, 9, 2i
P6 h8, 5, 10, 3i
P7 h9, 6, 11, 4i
P8 h10, 7, 0, 5i
P9 h11, 8, 1, 6i
P10
h0, 9, 2, 7i
P11 h1, 10, 3, 8i
Inverted Forms
p0 h10, 1, 8, 3i
p1 h11, 2, 9, 4i
p2 h0, 3, 10, 5i
p3 h1, 4, 11, 6i
p4
h2, 5, 0, 7i
p5
h3, 6, 1, 8i
p6
h4, 7, 2, 9i
p7 h5, 8, 3, 10i
p8 h6, 9, 4, 11i
p9 h7, 10, 5, 0i
p10 h8, 11, 6, 1i
p11 h9, 0, 7, 2i
Introduction and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
Q−2 Applied to Motive in Subject and I11-Inversion
HIJK
ONML
P0
Q−2
HIJK
/ONML
P10
Q−2
HIJK
/ONML
P8
Q−2
HIJK
/ONML
P6
Introduction and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
Q−2 Applied to Motive in Subject and I11-Inversion
HIJK
ONML
p11
Q−2
HIJK
/ONML
p1
Q−2
ONML
/HIJK
p3
Q−2
OHIJK
/ NML
p5
Introduction and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
Product Network Encoding Subject and I11-Inversion
1
HIJK
ONML
P0
Q−2
I11
34
ONML
HIJK
p11
ONML
HIJK
/ P10
Q−2
I11
Q−2
OHIJK
/ NML
p1
HIJK
ONML
/ P8
Q−2
I11
Q−2
NML
/OHIJK
p3
HIJK
ONML
/ P6
I11
Q−2
NML
/OHIJK
p5
Our Theorem about duality guarantees this diagram commutes!
Introduction and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
Self-Similarity: Local Picture
29
@ABC
GFED
G
T9
GFED
@ABC
E
I1
GFED
@ABC
A
I11
GFED
@ABC
D
Introduction and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
Self-Similarity: Global Picture and Local Picture
8-11
PQRS
WVUT
Row 1
PQRS
WVUT
P5
T9
T9
57-60
PQRS
WVUT
Row 2
PQRS
WVUT
P2
I1
Q−2
PQRS
WVUT
p11
I11
I11
1-4
PQRS
WVUT
Row 4
PQRS
WVUT
P0
Q−2
T9
Q−2
I1
34-37
PQRS
WVUT
Row 3
PQRS
/ WVUT
P3
PQRS
/ WVUT
P0
Q−2
Q−2
Q−2
PQRS
/ WVUT
P10
Q−2
PQRS
/ WVUT
p3
Q−2
Q−2
PQRS
/ WVUT
P8
I1
Q−2
PQRS
/ WVUT
p5
I11
I11
WVUT
/ PQRS
P8
PQRS
/ WVUT
P11
T9
I1
I11
WVUT
/ PQRS
P10
Q−2
T9
I1
PQRS
/ WVUT
p1
PQRS
/ WVUT
P1
Q−2
PQRS
/ WVUT
P6
Introduction and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
Audio: Hindemith, Ludus Tonalis, Fugue in E
Introduction and Basics
The Consonant Triad
Geometry of Triads
Extension of Neo-Riemannian Theory
Symmetries of Triads
VI. The Topos of Triads and the PLR-Group (Fiore–Noll 2011)