Algebraic compatibility of Riemannian operators’ voice-leading properties
Maxx Cho
Department of Mathematics/Department of Music, Swarthmore College
Introduction
Transformation theory is a branch of music theory that studies how
musical objects, such as tonal areas, rhythm, or chords can transform from
one to another. Neo-riemannian music theory, named after the music
theorist Hugo Riemann, studies the transformation of chords through the
so-called neo-Riemannian operators. These neo-Riemannian operators
form a group, called the Riemannian group.
In the early formulations of Riemannian operators, they were defined by
their voice-leading properties. In other words, they were defined by how
individual notes move from one chord to the next. In particular, the
Riemannian group is generated by the maximally smooth voice-leading
operations on the set class of major and minor triads. Unfortunately, this
has the disadvantage of not easily being generalizable to arbitrary pc-set
classes.
Some have suggested a root-intervallic approach to Riemannian operators.
Unfortunately, this has the disadvantage of abandoning the operators’
voice-leading properties. In particular, isomorphic chord progressions on
different pc-set classes have different common-tone retention.
In our paper, we show that there exists “dual” pc-set classes, on which
isomorphic chord progressions have same common-tone retention. We do
this by defining the voice-leading automorphism on the Riemannian
group, which preserves common-tone retention.
Elementary results in Riemannian
music theory
Retention operators
We now proceed with an example to demonstrate the voice-leading
automorphism.
Throughout this paper, γ shall denote a pc-set class of 24 elements. Also,
, which is a coordinate system used to represent elements of γ, as we now
define.
Definition 1.1/1.2 [Operators on Coordinate Space]
(1) The transposition operator
is given by
Once a coordinate system has been chosen, this induces operators directly
on the space downstairs, as the above diagram shows. We shall denote this
group downstairs by .
(2) The inversion operator
is given by
Definition 1.3 [Coordinate System on pc-Set Class]
We choose to represent elements of γ with Γ via a bijection
called the coordinate system, such that
We remark that such a coordinate system is necessary, as chords in γ do
not have the notion of a “root” in general.
One of the most important tenets of Western music up to the 19th century
is minimal voice-leading. In particular, retention notes are an important
aspect of a chord progression.
Definition 3.1 [Retention Value] Let γ be a pc-set class with coordinate
system f. Consider some
.
(1) Suppose that
for all
. Then n is called the
retention value of C and is denoted by
.
(2) The retention value of some
relative to f is
, where
Let
, denoted as
, where
The equal-tempered 12 chromatic pitches of Western music is represented
by
.
Theorem [The Riemannian Group]
, the set of all Riemannian operators, form a
group under function composition. This group is isomorphic to
.
, which
C-major triad = {0,4,7}
Next, we define two musically significant operators on pc-sets:
(1) The transposition operator , which is given by
{0,4,7}
{11,4,7}
The Riemannian group is actually the normal subgroup of a larger class of
operators, called uniform triadic transformations.
(11,-)
(0,+)
The pc-set class containing {0,4,7}
and the pc-set class containing {0,1,4}
are dual. Thus, there exists an
automorphism on the Riemannian
group that preserves common-tone
retention.
(5,-)
g
g
{0,1,4}
{5,4,1}
Recent advancements
Consider some Riemannian operator U, which induces an operator G on γ.
Now suppose that we apply the same G (that is, with the same retention
value) to a different pc-set class γ’. This, in turn, induces a Riemannian
operator U’. When is the mapping
an automorphism? This
question is important as such an automorphism would imply that
isomorphic chord progressions have similar voice-leading.
Definition 4.1 [Dual pc-Set Classes] Let γ and γ’ be pc-set classes of
trichords. Also, fix some
and
. Now, suppose
that
and
. Then γ and γ’ are dual if
and
.
I have recently begun computing dual pc-set classes for tetrachords. In
particular, the set class of dominant seventh/half-diminished seventh
chords and the set class containing {0,2,3,6} are dual in that they have
algebraically compatible voice-leading.
DUAL
{0,2,3,6}
C7={0,4,7,10}
Literature cited
Cohn, Richard. Neo-riemannian operations, parsimonious trichords, and their “tonnetz”
representations. Journal of Music Theory 41(1):1-66, 1997.
Hook, Julian. Uniform Triadic Transformations. Journal of Music Theory 46(1/2):57-126,
2002.
Hook, Julian. Cross-type transformations and the path consistency condition. Music Theory
Spectrum 29(1):1-40, 2007.
Hyer, Brian. Reimag(in)ing Riemann. Journal of Music Theory 39(1):101-138, 1995.
Since the Riemannian operators act on Γ directly, its action on γ will vary
depending on the choice of the coordinate system f. The following theorem
shows that the action of a Riemannian operator on γ through a coordinate
system can be preserved via an automorphism.
Theorem 2.1 [Preservation Automorphism]
Let f and g be coordinate systems on a pc-set class γ. Then there exists an
automorphism Ω on
such that for all
,
(0,+)
f
Voice-leading automorphism
Music theory preliminaries
A pitch-class set (abbreviated as “pc-set”) is simply a subset of
could represent a chord, a melodic line, or a scale.
In this case, the voice-leading
automorphism sends the generators
and
. Thus,
f
Definition [Riemannian Operators]
Let
, denoted as
We review basic standard music theory terminology, for the benefit of the
musically uninitiated.
Example
Please refer to the paper for a full list of works cited.
Here is the central result of this paper:
(2) The inversion operator , which is given by
In other words, the following diagram commutes:
Both of the operators defined above have the property that they preserve
all of the intervals present in the pc-set, thus preserving the sound of the
chord.
Now define an equivalence relation on the set of all pc-sets with the same
cardinality as follows:
if
or
for some n.
Such an equivalence class is called a pc-set class.
For example, the set of all major and minor triads forms a pc-set class.
In general, the preservation automorphism is an inner automorphism.
The automorphism can be computed explicitly by using the modulation
function, which measures the difference between two coordinate systems f
and g. Please refer to the paper for details.
Theorem 4.2 [Voice-Leading Isomorphism] Let γ and γ’ be dual pc-set
classes, with corresponding groups and . Then there exists a nontrivial
group isomorphism
such that
Corollary 4.5 [Voice-Leading Automorphim] Let γ and γ’ be dual pc-set
classes with coordinate systems f and f’, respectively. Then there exists a
unique automorphism
such that
Acknowledgments
I would like to thank the 2008 summer REU at UNCA, where this research was
conducted. In particular, I would like to thank professors Sam Kaplan, Dave Peifer, and
Patrick Bahls. I would also like to thank mathematical music theorists Julian Hook
(University of Indiana) and Jon Kochavi (Swarthmore College) for reading my paper and
providing helpful suggestions.
Please contact hcho1@swarthmore.edu for a copy of the paper.