Funcional automático
Harmonic Analysis
W.. Bas De Haas,∗ Jos´e Pedro Magalh ˜aes,†
Frans Wiering,∗ and Remco C. Veltkamp∗
∗Department of Information and
Computing Sciences
Universidad de Utrecht
P.O. Box 80.089
3508 TB Utrecht, Los países bajos
{W.B.deHaas, F.Wiering, R.C.Veltkamp}@uu.nl
†Department of Computer Science
Universidad de Oxford
Wolfson Building, Parks Road
Oxford OX1 3QD, Reino Unido
jpm@cs.ox.ac.uk
Abstracto: Music scholars have been studying tonal harmony intensively for centuries, yielding numerous theories
and models. Desafortunadamente, a large number of these theories are formulated in a rather informal fashion and lack
mathematical precision. In this article we present HarmTrace, a functional model of Western tonal harmony that
builds on well-known theories of tonal harmony. In contrast to other approaches that remain purely theoretical, nosotros
present an implemented system that is evaluated empirically. Given a sequence of symbolic chord labels, HarmTrace
automatically derives the harmonic relations between chords. Para esto, we use advanced functional programming
techniques that are uniquely available in the Haskell programming language. We show that our system is fast, easy to
modify and maintain, robust against noisy data, and that its harmonic analyses comply with Western tonal harmony
theory.
For ages, músicos, composers, and musicologists
have proposed theories regarding the structure of
music to better understand how it is perceived,
performed, and appreciated. En particular, tonal har-
mony exhibits a considerable amount of structure
and regularity. The first theories describing tonal
harmony date back at least to the 18th century
(Rameau 1722). Since then, a rich body of literature
that aims at explaining the harmonic regularities in
both informal and formal models has emerged (p.ej.,
Lerdahl and Jackendoff 1996). Such models have
attracted numerous computer music researchers
to investigate the automation of the analysis and
generation of harmony. Most of these theories,
sin embargo, have proven to be very hard to implement
(p.ej., Clarke 1986). We are not aware of a model that
has a working implementation that effectively ana-
lyzes tonal harmony and deals robustly with noisy
datos, while remaining simple and easy to main-
manchar, and scaling well to handle musical corpora of
considerable size. In this article we present Harm-
Trace (Harmony Analysis and Retrieval of Music
Computer Music Journal, 37:4, páginas. 37–53, Invierno 2014
doi:10.1162/COMJ a 00209
C(cid:2) 2013 Instituto de Tecnología de Massachusetts.
Publicado bajo una atribución Creative Commons-
No comercial 3.0 no portado (CC BY-NC 3.0) licencia
with Type-level Representations of Abstract Chord
Entidades), a system that meets these requirements
using state-of-the-art functional programming tech-
niques. HarmTrace allows us to easily adapt the
harmonic specifications, empirically evaluate the
harmonic analyses, and use these analyses for
tasks such as similarity estimation and automatic
annotation of large corpora.
The HarmTrace harmony model draws on the
ideas of Rohrmeier (2007, 2011). Rohrmeier mod-
eled the core rules of Western tonal harmony as a
(grande) context-free grammar (CFG, see Chomsky
1957). Más tarde, De Haas et al. (2009) implemented this
grammar and specifically tuned it for jazz harmony,
with the aim of modeling harmonic similarity.
The HarmTrace system transfers these ideas to a
functional setting, solving typical problems that
occur in context-free parsing (p.ej., the rejection of
pieces that cannot be parsed) and controlling the
number of ambiguous solutions. Because it relies
on advanced functional programming techniques
not readily available in most programming lan-
calibres, HarmTrace is inextricably bound to Haskell
(Peyton Jones 2003). Haskell is a purely functional
programming language with strong static typing.
It is purely functional because its functions, como
regular mathematical functions, are guaranteed to
De Haas et al.
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Cifra 1. A typical chord
secuencia (a) and its
harmonic analysis, como
generated by HarmTrace
(b). The chord labels are
printed below the score,
and the scale degrees and
functional analysis above
the score. Por simplicidad,
we ignored voice-leading.
Ton, Dom, and Sub denote
tonic, dominant, y
subdominant, respectivamente.
Ton
I
C
Piece
Ton
Dom
I
C
Sub
IV
F
Dom
V 7
V 7
G7
V / V
II 7
7
D
(b)
(a)
produce the same output when given the same
aporte. It is strongly typed because it enforces re-
strictions on the arguments to functions, and it does
so statically (es decir., at compilation time). Through
its main implementation, the Glasgow Haskell
Compiler (GHC, http://haskell.org/ghc), Haskell
offers state-of-the-art functional programming tech-
niques, like error-correcting combinator parsers,
type-level computations, and datatype-genericity
(polytypic functions), that are not available in any
other mainstream language. These features proved
to be essential to HarmTrace, as we will show.
Naturalmente, any Turing-complete language can
be used to implement a particular algorithm or
modelo; this also holds for implementing HarmTrace.
Most programming languages, sin embargo, do not offer
características, such as type-indexed computations, eso
make implementing HarmTrace much easier. Nosotros
prefer Haskell over, p.ej., C, Java, or LISP, porque
its strong type system gives us better guarantees
of correctness, makes the development process
clearer and easier, and allows for compiler type-
directed optimizations (p.ej., fusion laws). Otro
non-mainstream programming languages with type
computations exist, such as Coq and Agda, pero ellos
do not have practical library support.
Following Rohrmeier, a core assumption that
underlies our harmony model is that Western tonal
harmony is organized hierarchically and transcends
Piston’s table of usual root progressions (Piston 1991,
ch. 3, pag. 21). Como consecuencia, within a sequence
of chords some chords can be removed because of
their subordinate role, leaving the global harmony
structure intact, whereas removing other chords
can significantly change how the chord sequence
is perceived. This is illustrated in the sequence
displayed in Figure 1a: the D7 chord in this sequence
can be removed without changing the general
structure of the harmony, although removing the G7
or the C at the end would cause the sequence to be
perceived very differently. This implies that within
a sequence not all chords are equally important, y
must be organized hierarchically. This hierarchical
organization is reflected in the tree structure shown
in Figure 1b. The subdominant F has a subordinate
role to the dominant G7, which is locally prepared
by a secondary dominant D7. The tonic C releases
the harmonic tension built up by the F, D7, and G7.
As in our earlier work with Rohrmeier (De Haas
et al. 2009), the development of HarmTrace has been
driven by its application in content-based music-
information retrieval (MIR, Downie 2003) investigación.
Within MIR, the notion of musical similarity plays
a prominent role because it allows ordering musical
pieces in a corpus. Using such an ordering, uno
can retrieve harmonically related pieces, like cover
songs, classical variations, or all blues pieces in a
cuerpo. For performing such searches, a measure of
harmonic similarity is essential. Because our system
captures the global and local relations between
acordes, it can be used to improve systems that can
benefit from this contextual information. Tiene
been shown that analyzing the hierarchical relations
between chords in a sequence improves the quality
of a harmonic-similarity measure in retrieval tasks
(De Haas et al. 2009, 2011; Magalh ˜aes and De Haas
2011). Another application area for HarmTrace is
automatic chord transcription (De Haas, Magalh ˜aes,
and Wiering 2012). This is a common MIR task
that aims at recognizing a chord sequence from the
38
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audio signal. For every beat position, the chords that
match the spectrum well are selected from a chord
dictionary. Después, the HarmTrace model is
used to select the candidate sequence that best fits
the modeled rules of tonal harmony. Chordify, un
Internet startup, uses HarmTrace to make automatic
chord transcription accessible to the general public
(http://chordify.net, see also De Haas et al. 2012).
The application to MIR explains some of the
choices made in the development of HarmTrace. En
particular, because a large corpus of chord sequences,
mainly from the jazz repertoire, is available for
retrieval tasks, the harmony model exhibits a bias
towards jazz harmony. In this article we describe the
musical aspects of HarmTrace; its Haskell-specific
implementation aspects are described elsewhere
(Magalh ˜aes and De Haas 2011).
A fully functional model of tonal harmony that
can quickly analyze chord sequences offers several
other benefits. Musicologists study the harmonic
structure of pieces and annotate them by hand. Este
is a time-consuming enterprise, especially when
large corpora are involved. With the automatic
annotation techniques that we present here, este
can be done quickly, even for large corpora possibly
containing errors. Además, HarmTrace could
aid in (automatic) composition by generating se-
quences of chords, generating harmonically realistic
continuations given a sequence of chords, or auto-
matically harmonizing a melody (Koops, Magalh ˜aes,
and De Haas 2013).
This article starts by discussing a relevant
selection of the large body of existing literature on
harmony theory and modeling in the next section.
Después, we explain our harmony model,
followed by an evaluation of some detailed, ejemplo
analyses created by this model. Próximo, we show that
HarmTrace can deal with large amounts of noisy
datos. Finalmente, we conclude the article by discussing
the limitations of our system and differences from
other models.
Trabajo relacionado
The 19th and 20th centuries have yielded a wealth
of theoretical models of Western tonal music; tonal
harmony, En particular, has received considerable
atención. Most theories that describe the relation-
ships between sequential chords capture notions of
order and regularity; some combinations of chords
sound natural whereas others sound awkward (p.ej.,
Rameau 1722). These observations led music the-
orists to develop ways to analyze the function of a
chord in its tonal context (p.ej., Riemann ca. 1895).
Desafortunadamente, the majority of these theories are
formulated rather informally and lack descriptions
with mathematical precision. In this section we give
a condensed overview of the theories that played
an important role in the formation of the harmony
model we present in this article.
Lerdahl and Jackendoff’s Generative Theory of
Tonal Music (GTTM, 1996) is a seminal work that
further formalized the ideas of Schenker (1935).
GTTM structures Western tonal compositions
by defining recursive, hierarchical dependency
relationships between musical elements using
well-formedness and constraint-based preference
normas. The GTTM framework distinguishes four
kinds of hierarchical structure: meter, grouping,
time-span reduction, and prolongational reduction.
Although GTTM can be considered one of the
greatest contributions to music theory and music
cognition of the last few decades, implementing
the theory is difficult because the often vague and
ambiguous preference rules lead to a wide range
of possible analyses (Clarke 1986; Temperley 2001,
ch. 1; Hamanaka, Hirata, and Tojo 2006).
The recursive formalization proposed by Ler-
dahl and Jackendoff suggests a strong connection
between language and music. Many other authors
have argued that tonal harmony should be organized
in a hierarchical way similar to language, conduciendo a
numerous linguistically inspired models since the
1960s (Roads 1979). One of the pioneers to propose
a grammatical approach to harmony was Winograd
(1968). More recently, Steedman (1984, 1996) modificación-
eled the typical twelve-bar blues progression with a
categorial grammar; Chemillier (2004) elaborates on
these ideas by transferring them to a CFG. Similarmente,
Pachet (1999) proposes a set of rewrite rules for
jazz harmony comparable to Steedman’s grammar.
Pachet furthermore shows that these rules can be
learned from chord sequence data in an automated
De Haas et al.
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moda. Además, quasi-grammatical systems
for Schenkerian analysis have been proposed re-
cently (Marsden 2010). Además, Choi (2011)
developed a system for analyzing the harmony of
jazz chord sequences; this system identifies common
harmonic phenomena, like secondary dominants
and tritone substitutions, and labels the chords in-
volved accordingly. The relation between music and
language is not merely a theoretical one; a growing
body of evidence, also from neuropsychology and
neuroimaging, suggests that music and language
are more closely related than was previously be-
lieved (patel 2003). An in-depth overview of the
animated debate on the relation between music
and language is beyond the scope of this article,
sin embargo.
The generative formalism proposed by Rohrmeier
(2007, 2011), which the HarmTrace model draws
extensively from, expands upon many of these
earlier approaches in a number of ways. Rohrmeier
gives an encompassing account of how relation-
ships in tonal harmony can be modeled using a
generative CFG with variable binding. His gram-
mar models form, phrasing, theoretical harmonic
función (Riemann ca. 1895), prolongation of scale
degrees (Schenker 1935; Lerdahl and Jackendoff
1996), and modulation. Rohrmeier’s model differs
from earlier grammatical formalisms in various
maneras. Steedman’s approach (1984, 1996) uses seven
context-sensitive rules (with variations) to explain
the structure of blues progressions. Steedman’s
grammar does not go beyond the jazz blues domain,
sin embargo, and it remains unclear how it could be
adapted to support other styles. Rohrmeier’s for-
malism also differs from GTTM: The latter aims
at describing the core principles of tonal cognition,
and harmony is covered mainly as a prolongational
fenómeno, whereas Rohrmeier’s formalism de-
scribes the structure of tonal harmony from a music-
theoretical perspective with concrete context-free
normas. Rohrmeier acknowledges that a full account
of tonal harmony would require a large number
of varying style-specific rules, and his formalism
aims to capture only the core rules of Western tonal
harmony.
De Haas et al. (2009) performed a first attempt
to implement the ideas of Rohrmeier. Aunque el
results were promising, the context-free parsing
techniques used in that work hampered both
theoretical as well as practical improvements. Primero,
a sequence of chords that did not precisely match
the context-free specification was rejected and no
explanatory information was given to the user.
Por ejemplo, appending one awkward chord to an
otherwise grammatically correct sequence of chords
forced the parser to reject the complete sequence, no
returning any partial information about what it had
parsed.
Segundo, musical harmony is ambiguous and
chords can have multiple meanings depending on
the tonal context in which they occur. Esto es
reflected in all the grammatical models discussed
earlier. A major drawback of CFGs is that they are
very limited in ways of controlling the ambiguity
of the specification. It is possible to use rule-
weightings and to set low weights to rules that
explain rare phenomena. This allows for ordering
the ambiguous solutions by the total relevance of
the rules used. This does not overcome the fact that,
for some pieces, the number of parse trees grows
exponentially as function of the number of input
acordes.
Finalmente, writing context-free rules by hand is
a tedious and error-prone enterprise, especially
because the grammatical models can become rather
grande. Por ejemplo, a rule generalizing over an
arbitrary scale degree has to be expanded for each
scale degree, I, II, III, etc.. Por eso, some form of
high-level grammar generation system is needed to
allow for generalizing over scale degree and chord
tipo, and to control conditional rule execution.
Another important model that has influenced the
development of HarmTrace is that of Temperley
(2001) and Temperley and Sleator (1999). They give
an elaborate formal account of Western tonal music,
and also provide an efficient implementation. Este
rule-based system, which is partly inspired by
GTTM, can perform an analysis of the chord roots
and the key given a symbolic score, but does not
formalize the hierarchical relations between chords.
Our system continues where Temperley’s left off:
Based on the available chord and key information of
the piece, we model the global and local dependency
relations between these chords.
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Cifra 2. An example of a
diatonic cycle-of-fifths
progression in C major.
The leaves represent the
input chords, using the
syntax of Harte et al.
(2005), and the internal
nodes denote the
harmonic structure. El
Vd /X nodes represent
diatonic fifth successions.
Ton
I
«C:maj»
Sub
IV
«F:maj»
Ton
I
«C:maj»
Piece
Dom
Sub
Vd /II
·
Dom
Dom
V 7
metro
II
«GRAMO:7»
metro
VI
«D:mín.»
·
Vd /VI
·
metro
III
«A:mín.»
Vd /III
metro
VII
«B:hdim»
«mi:mín.»
The HarmTrace System
In this section we explain how we model the
regularities and hierarchical dependencies of tonal
harmony. HarmTrace transfers the ideas developed
by De Haas et al. (2009) to a setting based on
functional programming. Whereas the contributions
of the majority of models we discussed in the
previous section are purely theoretical, we present
a system that can be evaluated empirically and is
usable in practice. This comes at a price, sin embargo:
Our present model does not support full modulation,
es decir., modulation to every possible key. The model
can only handle change of mode—going from major
to minor or vice versa—without changing the root
of the key. (We also use the term parallel keys
to describe this relationship.) Como consecuencia,
this requires the model to have information about
the key of the piece. También, because we mainly use
jazz-oriented input data in this article, nosotros también
include some specifications that describe typical
phenomena in jazz harmony. Cifra 2 shows an
example analysis as produced by HarmTrace. El
chord labels that were used as input are the leaves
of the tree, and the internal nodes represent the
harmonic relations between the chords.
Music, and harmony in particular, is intrinsi-
cally ambiguous; certain chords can have multiple
meanings within a tonal context. Although a model
of tonal harmony should reflect some ambiguity,
defining many ambiguous specifications can make
the number of possible analyses grow exponen-
tially for certain chord progressions. In most of
the ambiguous cases it is clear from the context
which of the possible solutions is the preferred one,
sin embargo. Por eso, we can select the favored analysis
by constraining the application of the specification
leading to the undesired analysis. We are able to do
so because we rely on strongly typed data structures,
which allow us to formulate our rules in a very
precise manner. In cases where it is less clear from
the context which solution is preferred, we accept a
small number of ambiguous analyses.
The HarmTrace system explores the relations
entre (generalized) algebraic data types and
context-free production rules. A CFG defines a
idioma: Given a set of production rules and a set
of words (or tokens), it accepts only combinations of
tokens that are valid sequences of the language. El
notion of an algebraic datatype is central in Haskell.
Similarly to a CFG, a datatype defines the structure
of values that are accepted. Por eso, a collection of
datatypes can be viewed as a very powerful CFG:
The type-checker accepts a combination of values
if their structure matches the structure prescribed by
the datatype, and rejects this combination if it does
no. Within HarmTrace, the datatypes represent
the relations between the structural elements in
tonal harmony, and the chords are the values. Un
important difference between a CFG and a Haskell
datatype, sin embargo, is that datatypes provide more
much modeling freedom and control, especially
the generalized algebraic datatypes (Schrijvers
et al. 2009) that we use. These datatypes allow us to
restrict the number of applications of a specification,
constrain the conditions for application, and order
the specifications by their importance. Además,
they allow for the definition of specifications that
only apply to certain keys or modes, the exclusion of
the application of transposition functions to specific
scale degrees (p.ej., Spec. 18, described later), y el
preference of certain specifications over others. Para
technical details, we refer to Magalh ˜aes and De Haas
(2011). The source code is available online at
http://hackage.haskell.org/package/HarmTrace-0.6.
A Model of Tonal Harmony
We now elaborate on how our harmony datatypes
estan organizados. Haskell knowledge is not required to
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understand our model: We use a simplified syntax
to describe the datatype specifications that is in-
spired by the syntax used to describe CFGs. Users
of HarmTrace need only know Haskell if they wish
to change the model in any way; simply using
the current model to obtain analyses can be done
without any Haskell knowledge. En el siguiente
subsections we group related datatype specifications
(abbreviated as Spec.), and explain their musical
interpretation and background. Primero, we introduce
a variable M for the mode of the key of the piece,
which can be major or minor. The mode variable
is used to parametrize all the specifications of our
harmonic datatype specification; some specifica-
tions hold for both modes (inidicated by M), mientras
other specifications hold only for the major or minor
mode (respectively indicated by “Maj” and “Min”).
Similarly to a CFG, we use a | to denote alterna-
tives, y un + to represent optional repetitions of a
datatype.
Functional Categories
1. PieceM → Func+
2. FuncM → TonM | DomM M ∈ {Comandante, mín.}
3. DomM → SubM DomM
METRO
Spec. 1–3 define that a valid chord sequence, PieceM,
consists of at least one and possibly more func-
tional categories. A functional category classifies
chords as being part of a tonic (TonM), dominant
(DomM), or subdominant (SubM) estructura, dónde
a subdominant structure must always precede a
dominant structure. These functions constitute
the top-level categories of the harmonic analy-
sis and model the global development of tonal
tension: A subdominant builds up tonal tension,
the dominant exhibits maximum tension, y el
tonic releases tension. The functional categories
capture the idea proposed by Riemann (ca. 1895)
that every chord within a tonality can be reduced
to one of these harmonic functions. The order of
the dominants and tonics is not constrained by
el modelo, and they are not grouped into larger
phrases.
From Functions to Scale Degrees
| IMajIV MajIMaj
| I m
MinIV m
MinI m
mín.
| VM
| III m
Comandante
| (cid:2)III Maj
M ∈ {Comandante, mín.}
4. Ton Maj → IMaj
5. Ton Min → I m
mín.
6. DomM → V 7
METRO
7. Dom Maj → VII m
Comandante
8. Dom Min → (cid:2)VII Min
9. SubM → II m
METRO
10. SubMaj → IV Maj
11. SubMin → IV m
mín.
| III m
MajIV Maj
| (cid:2)III MinIV Min
Spec. 4–11 translate the tonic, dominant, y
subdominant datatypes into scale-degree datatypes.
Apart from a few minor differences, Spec. 4–11
follow Rohrmeier’s (2007) modelo. A tonic translates
to a first degree, a dominant to a fifth degree, y el
subdominant to a fourth degree in both major and
minor keys. We denote scale-degree datatypes with
Roman numerals, but because our model jointly
specifies datatypes for major as well as minor mode,
we deviate from notation that is commonly used
in classical harmony and represent scale degrees
as intervals relative to the diatonic major scale.
Por ejemplo, III m unequivocally denotes the minor
chord built on the note a major third interval above
the key’s root and does not depend on the mode of
the key. Similarmente, a (cid:2)III denotes a major chord built
on the root a minor third interval above the key’s
raíz.
A scale degree datatype is parametrized by a
mode, a chord class, and the scale degree, es decir.,
the interval between the chord root and the key
raíz. The chord class categorizes scale degrees as
one of four types of chords (denoted in lowercase
with superscripts) and is used to constrain the
application of certain specifications, p.ej., Spec. 16–17
(defined later). The four classes are major (No
superscript), minor (metro), dominant seventh (7), y
diminished (0). Chords in the minor class contain
a minor or diminished triad and can have possibly
altered or non-altered additions, except for the
diminished seventh. Chords categorized as major
contain a major triad and can be extended by
non-altered additions, with the exception of the
dominant seventh chord (with additions). Chords
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Computer Music Journal
of the dominant class have a major triad and a
minor seventh, or an augmented triad. Chords in
the dominant class can be extended with altered
or non-altered notes. Finalmente, the diminished class
contains only the diminished seventh chord. El
usage of chord classes allows us to define certain
specifications that only hold for dominant chords,
whereas other specifications might hold only for
minor chords, etc..
Tonics can furthermore initiate a plagal cadence
(Spec. 4–5). We deliberately chose to model the
plagal cadence with scale degrees and not with
Sub and Ton because this reduces the number of
possible analyses. Also a SubM can translate into
a II m
M because of its preparatory role, y el
dominant translates into the seventh scale degree,
Comandante ((cid:2)VII Min in minor). Similarmente, we could have
VII m
chosen to model the SubM to translate also to VI m
Comandante
((cid:2)VI Maj in minor). We chose to solve this by creating
specifications for chains of diatonic fifths instead
(Spec. 18–19; ver, por ejemplo, Cifra 2).
The TonM translating into III m
Comandante ((cid:2)III Min in minor)
is perhaps the most unexpected transformation (y
we deviate from Rohrmeier’s 2007 model here).
Often the third degree can be explained as either a
preparation of a secondary dominant (Spec. 17), como
being part of diatonic chain of fifths (Spec. 19), or as
supporting the subdominant (Spec. 10–11). In certain
progressions, sin embargo, it cannot be explained by
any of these specifications, and is best assigned a
tonic function because it has two notes in common
with the tonic and does not create strong harmonic
tension.
From Scale Degrees to Chords
→ «C:maj» | «C:maj6» | «C:maj7» |
12. IMaj
«C:maj9» | . . .
13. I m
mín.
«C:mín.(13)» | . . .
→ «C:mín.» | «C:min7» | «C:min9» |
14. V 7
Comandante
→ «GRAMO:7» | «GRAMO:7(b9,13)» |
«GRAMO:(#11)» | «GRAMO:7(#9)» | . . .
→ «B:dim(bb7)»
15. VII 0
METRO
Finalmente, scale degrees are translated into the
actual surface chords, which are used as input
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for the model. The chord notation used is that of
Harte et al. (2005). The conversions are trivial and
illustrated by a small number of specifications above,
but the model accepts all chords in Harte’s syntax.
The model uses a key-relative representation; en
Spec. 12–15 we used chords in the key of C. Por eso,
a IMaj translates to the set of C chords with a
major triad, optionally augmented with additional
chord notes that do not make the chord minor or
dominant. Similarmente, V 7
Maj translates to all G chords
with a major triad and a minor seventh, etc.. A
simplify the treatment of chords differing only by
these additional chord notes, we cluster chords
with the same class and same scale degree (p.ej.,
«C:min7» «C:min9») in one datatype.
Secondary Dominants
16. XC
METRO
17. X7
METRO
→ V /X7
→ V /Xm
M XC
METRO
M X7
METRO
C ∈ {∅, metro, 7, 0}
X ∈ {I , (cid:2)II , II , . . . , VII }
Besides these basic elements of tonal harmony,
we distinguish various scale-degree substitutions
and transformations. For this we introduce the
function V /X which transposes an arbitrary scale
degree X a fifth up. Herewith, Spec. 16 accounts
for the classical preparation of a scale degree by its
secondary dominant, stating that every scale degree,
independently of its mode, chord class, and root
interval, can be preceded by a chord of the dominant
class, one fifth up. In the case of chord class, nosotros
denoted the independence of a chord class with
the chord class variable C. The empty symbol, ∅,
represents the major class (no superscript). (Nota
that in all specifications that contain transposition
funciones, terms like Y /XC
interpreted as (Y /X)C
METRO.) Similarmente, every dominant
scale degree can be prepared with the minor chord
one fifth above (Spec. 17). These two specifications
together allow for the derivation of the typical
and prominently present ii-V-I progressions in jazz
harmony. (The ii-V-I progression is a very common
cadential chord progression in jazz harmony, pero
it also occurs in other musical genres. The I-
chord can practically be an arbitrary chord in a
sequence which is preceded by its relative secondary
M should always be
De Haas et al.
43
dominant a fifth interval up, the V, and its relative
subdominant another fifth interval up, the ii.)
The explicit annotation of these structures is
very common in jazz analysis and education (ver,
por ejemplo, Jaffe 1996), and also Rohrmeier (2007,
2011) incorporates Spec. 16 in his model. Spec. 16–17
interfere with Spec. 4–11, sin embargo, causing multiple
analiza. Because we prefer, p.ej., a II m, V 7, y yo
to be explained as Sub, Dom, and Ton, we constrain
the application of Spec. 16 y 17 to the cases where
Spec. 4–11 do not apply. The rationale behind this
choice is more of a pragmatic than of a musical
naturaleza, and the user of the model might also decide
on a different specification preference.
Diatonic Chains of Fifths
18. Xm
Comandante
19. XMin
→ Vd /Xm
→ Vd /XMin XMin
Maj Xm
Comandante
The model also accounts for the diatonic chains of
fifths in major (Spec. 18) and minor (Spec. 19) keys.
These diatonic-chain specifications are necessary
to explain the typical cycle-of-fifths progressions:
I IV VII m III m VI m II m V 7 I (ver figura 2 para el
major case, as well as the Autumn Leaves example
in the next section). These strong chord progressions
are very commonly found throughout Western tonal
música. También, Rohrmeier’s (2011) model has a rule
dedicated specifically to this progression. To reduce
multiple ambiguous solutions, we constrain the
major-key specification to apply only to the minor
acordes, because I , IV , and V 7 translate directly to
Ton, Sub, and Dom, respectivamente. Similarmente, Spec. 19
captures the same phenomenon in a minor key.
Aquí, we restrict the application of the specification
only to the major chords because, de nuevo, I m, IV m, y
V 7 translate directly to Ton, Sub, and Dom in minor.
We implemented one exception to the diatonic
minor specification that allows the (cid:2)VI to precede
the II m in minor. Aquí, the major chord precedes a
minor chord. See the E(cid:2)(cid:3) Am7(cid:2)5 in the analysis of
Autumn Leaves, later in this article.
Tritone Substitution
→ (cid:2)V /X7
METRO
20. X7
METRO
The harmony model in HarmTrace allows for
various scale-degree transformations. Every chord of
the dominant class can be transformed into its tri-
tone substitution with Spec. 20. This specification
uses another transposition function (cid:2)V /X which
transposes a scale degree X a diminished fifth—a
tritone—up (or down). The tritone substitution spec-
ification allows for the derivation of progressions
with a chromatic baseline, p.ej., Am G(cid:4)7 GRAMO, and is
very common in jazz harmony (ver, por ejemplo,
Jaffe 1996). También, a tritone specification was earlier
used in a similar model in De Haas et al. (2009).
Because we want the application of the Spec. 16–
20 to terminate, we limit the number of possible
recursive applications of these rules (see the Parsing
sección).
Diminished Sevenths
→ (cid:2)III /X0
METRO
→ I I(cid:2)7(cid:2)9/X0
METRO
21. X0
METRO
22. X7
METRO
Diminished seventh chords can have various roles
in tonal harmony. An interesting characteristic of
these chords—consisting only of notes separated by
minor third intervals—is that they are completely
symmetrical. Por eso, a diminished seventh chord
has four enharmonic equivalent chords that can be
reached by transposing the chord by a minor third
with the transposition function (cid:2)III /X (Spec. 21).
En general, we treat diminished seventh chords as
dominant-seventh chords with a (cid:2)9 and no root
nota, which is a common interpretation of this
chord (Saslaw 2013). Por ejemplo, in a progression
Am7 A(cid:2)0 G7, the A(cid:2)0 closely resembles the G7(cid:2)9,
because a G7(cid:2)9 chord consists of G, B, D, F, y
A(cid:2), and an A(cid:2)0 chord consists of A(cid:2), B, D, y
F. This similarity is captured in Spec. 22, dónde
(cid:2)II /X transposes a scale degree one semitone up.
Similarmente, by combining secondary dominants
(Spec. 16) with Spec. 22, p.ej., F E(cid:2)0 ( ≈ D7(cid:2)9) GRAMO, y
an application of Spec. 21, p.ej., F F(cid:4)0 (≈E(cid:2)0) GRAMO, nosotros
can account for most of the ascending diminished
chord progressions. Within harmony theory, este
phenomenon is usually labeled as VII /X, where F(cid:4)0
would be the VII /V (in C major) in the previous
ejemplo. Because our model can explain it without
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a separate VII /X specification, there is no need to
add one.
Parallel Keys and the Neapolitan
23. FuncMaj → FuncMin
24. FuncMin → FuncMaj
25. SubMin → (cid:2)II Min
We support borrowings from the parallel key by
changing the mode but not the root of the key in
the FuncM datatype (Spec. 23 y 24). Aunque el
parallel keys are often considered rather distantly
related (there is a difference of three accidentals in
the key signature), borrowing from minor in major
occurs frequently in jazz harmony, and sometimes
in classical music. These specifications account,
por ejemplo, for the picardy third—ending a minor
piece on a major tonic. The actual implementation
of Spec. 23 y 24 differs marginally to overcome
endless recurring transitions between major and
minor. Finalmente, the Neapolitan chord (cid:2)II Min is
categorized as being part of a SubMin structure
(Spec. 25), which is also reachable in a major key
through Spec. 23. Although it might be considered
an independent musical event, it often has a
subdominant function (Drabkin 2013).
The datatype specifications that we have
presented in this section match the Haskell code
closely. Sin embargo, to maintain clarity, alguno
minor implementation details were omitted; estos
can be found in the real datatype specifications of
el modelo, es decir., the Haskell code.
Modulation
HarmTrace supports chord borrowing from parallel
keys, and its secondary dominant specifications
can be used to explain local key changes, pero
does not support modulating to all possible keys.
Although adding a modulation specification to the
model is straightforward, this would quickly lead to
an avalanche of ambiguous analyses. Por ejemplo,
all specifications of the model are currently
parametrized by a mode, which can be major,
minor, or mode agnostic. Extending this parameter
to contain the key of the piece, in line with
Rohrmeier’s (2011) modelo, is problematic: incluso con
a constrained modulation specification that allows
modulation only to specific other keys, and restricts
the number of modulations, the total number of
ambiguous analyses quickly explodes, given the
rules of the previous section. If a user wishes to
have such a modulation specification (and this is
very well possible from a specification point of
vista), we recommend using a much smaller model.
Such a model should not, por ejemplo, contener
rules for explicitly labeling chains of secondary
dominants, but a secondary dominant would be
analyzed as a local modulation (like Schenkerian
tonicization). Even with a much smaller model,
sin embargo, keeping the number of different analyses
under control might be challenging. Por eso, nosotros
chose to first explore the usability of HarmTrace
without modulation, keeping the model fast and
flexible.
Otro, more practical, solution is using external
key information, either obtained from key signatures
in the score, or by applying an automatic key-finding
algoritmo. Within the audio domain, key-finding
algorithms have improved considerably in the
last few years, providing annotations of both the
global key and key changes. Within the symbolic
domain, the key is often annotated explicitly
or can be derived using symbolic key-finding
algoritmos (p.ej., Temperley 2001). Información
about the key can then be used to segment the
chord sequences into sections that contain only
a single key, and HarmTrace can be applied to
these sections individually. Por ejemplo, in earlier
work we applied HarmTrace to improve harmonic
similarity estimation, using information about
key that was available in the data (De Haas et al.
2011). We have also used HarmTrace for a chord
transcription task, using a key-finding algorithm
(De Haas, Magalh ˜aes, and Wiering 2012). The key-
finding algorithm used in this latter work was
similar to that used by Temperley (2001, ch. 7).
Various different approaches to key finding exist
(for a fairly recent overview, see Noland and Sandler
2009). These concrete applications of HarmTrace
show that the availability of key data does not have
to hamper practical use.
De Haas et al.
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Parsing
Once we have a formal specification as a datatype,
the next step is to define a parser that transforms
textual chord labels into values of our datatype.
Writing a parser that parses labels into our datatype
would normally mean writing tedious code that
closely resembles the datatype specification. En
Haskell however, we can use datatype-generic pro-
gramming techniques (Jeuring et al. 2009) to avoid
writing most of the repetitive portions of the code.
(These techniques are not to be confused with regu-
lar polymorphism, as in Java generics.) Además,
we derive not only the parser automatically, pero
also a pretty-printer for displaying the harmony
analysis in tree form, and functions for comparing
these analyses. This makes the development and
fine-tuning of the model much easier, as only the
datatypes have to be changed, and the code adapts
itself automatically. The technical details of the
implementation of our model, and of the generic-
programming techniques we use, are covered in an
earlier publication (Magalh ˜aes and De Haas 2011).
Because music is an ever-changing, culturally
dependent, and extremely diverse art form, nosotros
cannot hope to model all valid harmonic relations in
our datatype. Además, song data may contain
mistakes or mistyped chords, perhaps feature
extraction noise, or malformed data of dubious
harmonic validity. In HarmTrace we use a parsing
library featuring error-correction: Chords that do
not fit the structure are automatically deleted
or preceded by inserted chords, according to the
datatype structure (Swierstra 2009).
The error-correction process uses heuristics to
find a good parse tree in a reasonable amount of
tiempo. When faced with a chord that does not fit
the harmony model, it will consider all possible
combinations of deletion and insertion of chords (arriba
to a fixed depth of three steps) to adapt the chord
sequence to the model. In this way, the simplest
corrections (involving the fewest insertions or dele-
ciones) are chosen. We could also assign different costs
to each specification, in order to prefer some rules
over others. En la práctica, we did not find this neces-
sary. The order in which the rules are specified also
matters, as earlier rules take precedence over later
normas; we use this fact to guide the correction process.
For most songs, parsing proceeds with no correc-
ciones, or only very few. Songs with a very high error
ratio denote multiple modulations, bad input, o un
wrong key assignment. Tenga en cuenta que, depending on the
severity of “unexpectedness” of a chord, there might
be multiple error-corrections necessary to create a
valid analysis (p.ej., one deletion and two insertions).
In our model, one particular parameter has a
large influence on the parsing and error-correction
proceso. This parameter controls the number of
recursive applications of the specifications for
secondary dominant and the like (Spec. 16–20). Él
must be set carefully, because setting it to a very low
value will lead to bad analyses, and setting it to a high
value will make the internally generated model very
grande, resulting in increased error-correction times
and often sub-optimal error-correction solutions.
For the examples and results in this article we have
used values between five and seven.
Example Analyses
In this section we demonstrate the analytic power
of the HarmTrace system. The input presented to
HarmTrace consists of a sequence of plain-text chord
labels in the syntax defined by Harte et al. (2005), y
the output consists of a parse tree similar to those
often used in natural language processing. In these
árboles, the input chord labels are the leaves, cual
are subsequently grouped into scale degrees, escala-
degree transformations, and functional categories,
and finally collected in a Piece node, as prescribed
by the rules of the model. The notation used in
the parse trees is identical to the notation used to
describe the datatype specifications in the previous
sección. Musical harmony can be very ambiguous,
and sometimes multiple analyses are defensible.
A strength of HarmTrace is that, by changing the
specifications in the previous section, it can be
easily adapted to meet the musical intuitions of its
users.
We start by analyzing the B section of the
Autumn Leaves jazz standard as found in The
Real Book (Sher 1988). The parse tree of this piece
in the key of G minor is depicted in Figure 3.
Within the piece, the three A0 D7 Gm sequences
translate into subdominant, dominant, and tonic
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Cifra 3. The harmonic
analysis of the B section of
the jazz standard Autumn
Leaves, as derived by
HarmTrace. Para
convenience, we left out
the mode subscript Min
and printed the chord
labels as commonly found
in lead sheets. The key of
the piece is G minor. El
(cid:3) denotes an added major
seventh.
Cifra 4. The automatic
analysis of the first two
phrases of J. S. Bach’s Ich
dank’ dir schon durch
deinen Sohn, BWV 349, en
the key of F major.
Cifra 5. An analysis of
Blue Bossa in C minor.
Dom
Ton
Dom
Sub
II m
Dom
I m
Sub
V 7
Gm7
IV m
Piece
Ton
I m
Dom
Dom
Ton
Sub
Dom
I m
Sub
Dom
Gm7
Vd /II ·
II m
V 7
Gm
Am7(cid:2)5
D7
Cm7
Vd /II ·
II m
V 7
V /(cid:2)VI 7
(cid:2)VI
Am7(cid:2)5
D7(cid:2)13
Vd /(cid:2)VI ·
(cid:2)VI
Am7(cid:2)5
D7
V /(cid:2)III m
(cid:2)III 7
mi(cid:2)Δ
V /(cid:2)III 7
(cid:2)III
mi(cid:2)Δ
V /(cid:2)VII 7
(cid:2)VII m
B(cid:2)7
(cid:2)VII 7
B(cid:2)Δ
F7
Fm7
IV 7
C9
Cifra 3
Piece
Piece
Ton
Dom
Ton
Dom
Ton
Dom
I
F
Sub
Dom
Vd /II ·
II m
Gm7
VI m
Dm
V
C
Sub
IV
B(cid:2)
I
F
V /V 7
II(cid:2)7(cid:2)9/II 7
V
C
I
F
(cid:2)III 0
(cid:2)V 0
B0
Cifra 4
Dom
Ton
Dom
I m
Sub
Dom
Ton
I m
Dom
Ton
Dom
Sub
Dom
I m
Sub
Dom
V /IV 7
II(cid:2)7(cid:2)9/I 7
IV
B(cid:2)
Sub
Dom
V /IV 7
IV
V
Cm6
metro
IV
Fm7
Sub
metro
II
Dom
Cm6
V /(cid:2)II
7
(cid:2)II
Sub
Dom
Cm6
V 7
V /(cid:2)VI
metro
7
(cid:2)VI
D(cid:2)Δ
metro
II
V 7
(cid:2)II 0
III 0
A0
B(cid:2)
C6
I 7
F7
V /I m
V m
Cm
Dm7(cid:2)5
G7(cid:3)5(cid:3)9
metro
(cid:2)III
mi(cid:2)m7
A(cid:2)7
Dm7(cid:2)5
G7(cid:3)5(cid:3)9
Cifra 5
V 7
G7
categories (Spec. 9, 6, y 5, respectivamente), forming
ii-V progressions, towards the tonic of the piece.
The second and third Dom branches of Piece display
different types of descending fifth movements,
both of which build up tension towards a D7: El
progressions from F to B(cid:2) are labeled either as a
secondary dominant or as a diatonic descending
fifth, depending on whether the chord is dominant
or major (Spec. 16 o 19). In the current analysis,
the Cm chord is viewed only as a subdominant of
the Gm chord. Además, the Cm could have
been seen as being part of a ii-V-I motion to the E(cid:2)(cid:3)
chord. HarmTrace misses this additional analysis,
sin embargo, because the maximal number of recursive
applications has been reached.
Although the model has a bias towards a jazz
repertoire, it can be used to analyze Bach chorales as
Bueno. En figura 4 we present the HarmTrace analysis
of the first nine measures of Bach’s harmonization
of the chorale melody Ich dank’ dir schon durch
deinen Sohn. The key of the piece is F major. Después
an introduction of the tonic, a diatonic chain of
fifths prepares the dominant, C, which subsequently
resolves to the tonic. The next branch prepares
a C with a B0, which is VII /V . As explained in
the previous section, the B0 is enharmonically
equivalent to A(cid:2)0 (I I I(cid:2)0) which is very similar to
a G7(cid:2)9 (denoted by I I(cid:2)7(cid:2)9/II , Spec. 22), which is in
turn the V/V, preparing the C chord in the key of F.
The final branch creates some harmonic movement
around the B(cid:2) by preparing the second B(cid:2) con un
VII /IV . The VII /X derivation is identical to this
uno, but transposed down a whole step. The fragment
concludes with a descending fifth preparation of the
subdominant, followed by the dominant.
En figura 5 we show the analysis of another
well-known standard, Blue Bossa. The progression
De Haas et al.
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Cifra 6. Two analyses of a
phrase from Tebe Poem by
Dmitry Bortniansky
(1751–1825). comparamos
an analysis adapted from
Rohrmeier (2011),
appearing above the score
(a), with an analysis
produced by HarmTrace
(b). The nodes labeled TR
in Rohrmeier’s analysis
indicate “tonal regions,” a
concept not included in
our model.
Cifra 7. An excerpt of the
analysis of It Don’t Mean a
Thing (If It Ain’t Got That
Balancearse).
Piece
Ton
Dom
Dom
Ton
I
C
Sub
Dom
V /V 7
V
I
V /IV
7
IV
V /V 7
V
V /III
metro
((cid:2)II (cid:2)9)/II
7
GRAMO
C
I 7
C7
F
7
II
GRAMO
V /VI
7
metro
VI
0
(cid:2)III
7
D
7
III
Am
IV (cid:3)0
E7
F(cid:3)0
(a)
(b)
Piece
Dom
Ton
Dom
Dom
Ton
V /V 7
V 7
I
Sub
Dom
V /V 7
V /II m
VI m
II 7
C7
Gm
GmΔ
Gm7
Gm6
F7
B(cid:2)6
V /IV 7
IV
V /V 7
V 7
V /II m
V /I m
I 7
mi(cid:2)6
V /II m
II 7
F7
V /VI 7
VI m
B(cid:2)7
V m
Fm7
C7
VI m
Gm7
III 7
D7
Gm
GmΔ
Gm7
Gm6
V 7
I
F7
B(cid:2)6
II 7
C7
Cifra 6
V /VI 7
III 7
D7(cid:3)5
Cifra 7
starts by introducing the tonic, Cm, followed by a
perfect cadence. The B-part displayed in the second
Dom branch shows a ii-V-motion to the Neapolitan
(cid:2)II (Spec. 25) and is followed by a ii-V-I to Cm.
Cifra 6 displays the score and two analyses of
an extract from Tebe Poem, by Dmitry Bortniansky.
The analysis in Figure 6a is the theoretical anal-
ysis proposed by Rohrmeier (2011). Aunque el
notation used by Rohrmeier differs slightly from
the notation used in this article, the analyses are
clearly similar. There are also some differences.
Rohrmeier connects the tonic, dominant, and sub-
dominant nodes in tonal regions whereas HarmTrace
(see Figure 6b) does not. We elaborate on this issue
in the Discussion section. Another difference in
derivation is that because we treat the F(cid:4)0 (≈D7(cid:2)9) como
a V /V , the E7 and Am are analyzed as being part of
a larger chain of fifths.
En figura 7 we show the HarmTrace analysis of
the jazz standard It Don’t Mean a Thing (If It Ain’t
Got That Swing). The analysis shows how similar
Gm chords are grouped under one VI m node. Él
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Cifra 8. Two examples
that illustrate error
correction: excerpts from
the jazz standard Someday
My Prince Will Come (a)
and the B section of the
standard There Is No
Greater Love (b).
Ton
I
B(cid:2)Δ
Piece
Dom
Piece
Dom
Ton
Sub
Dom
Sub
Dom
I
V /II 7
II m
Sub
Dom
Vd /II m
Vd /II m
II m
V 7
B(cid:2)Δ
V /VI 7
VI 7
Cm7
V /II 7
II m
V 7
V /VI 7
VI m
V /VI 7
VI m
Inserted
F7
III 7
IV
G7(cid:3)5
D7(cid:3)5
Deleted
VI 7
G7(cid:3)5
mi(cid:2)Δ
(a)
Cm7
F7
V /III m
III 7
Gm6
V /III m
III 7
Gm6
VII m
D7
VII m
D7
Am
7(cid:2)5
Am7(cid:2)5
(b)
furthermore illustrates how the Sub and Dom nodes
are prepared by chains of secondary dominants.
We conclude this section with two small exam-
ples that contain error corrections. These corrections
should not be interpreted as corrections of musical
errores, but they illustrate the behavior of HarmTrace
in cases where harmonies cannot be explained by
el modelo. The example in Figure 8a is an excerpt
of the jazz standard Someday My Prince Will Come,
Figure 8b is taken from the jazz standard There Is
No Greater Love. In Someday My Prince Will Come,
the model cannot explain the E(cid:2)(cid:3) where it occurs.
Because the D7 can immediately resolve to the G7,
the parser deletes the E(cid:2)(cid:3). The model specification
does not allow a Sub to translate into a VI m scale
degree. Adding such a specification would cause a
large number of ambiguous solutions, if the diatonic
fifth specification (Spec. 19) were not constrained.
Por lo tanto, in Figure 8b, the model needs a diatonic
chain of fifths to explain the VI m and the parser
solves this by inserting a II m. Corrections like the
ones in Figure 8 represent typical examples of error
corrections in HarmTrace.
Experimental Results
To demonstrate that HarmTrace can be efficiently
and effectively used in practice, we evaluate its
parsing performance on two chord sequence data
conjuntos: a small data set we have used in earlier work
(De Haas et al. 2009), and a larger data set (usado
in De Haas, Veltkamp, and Wiering 2013). Nosotros
refer to these two data sets as small and large,
respectivamente. The small data set contains 72 chord
sequences that describe mainly jazz pieces. El
large data set contains 5,028 chord sequences
that describe jazz, latín, and pop pieces, también
as a few classical works. Both data sets consist
of textual chord sequences extracted from user-
generated Band-in-a-Box files that were collected
on the Internet. (Band-in-a-Box [Gannon 1990]
is a commercial software package that generates
accompaniment, given a chord sequence.) Para el
extraction of the plain-text chord labels we have
extended software developed by Mauch et al. (2007).
To our knowledge, the large data set is the largest
data set of symbolic chord sequences available to
the research community, as of June 2013.
The small data set contains a selection of pieces
that were checked manually and “make sense”
harmonically, and the large data set includes many
songs that are harmonically atypical. This is because
the files are user-generated, so they contain peculiar
and unfinished pieces, wrong key assignments, y
other errors; it can therefore be considered “real
world” data. También, the large data set contains
pieces that modulate, and even some pieces that
might be considered atonal (p.ej., John Coltrane’s
Giant Steps). We deliberately chose to use a “real
De Haas et al.
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Mesa 1. Parsing Results
Data Set
Deletions
Insertions
Corrections
Chords
Parse Time
Total Time
pequeño
grande
0.83
3.38
2.79
9.85
3.63
13.24
42.49
62.05
10.00
76.53
0.72
384.81
Numbers of deletions, insertions, and total corrections per song; total number of chords per song; parsing time per song (in msec);
total parsing time (in sec).
world” data set to show that HarmTrace is robust
against noisy data, offers good performance in terms
of parsing speed, and still delivers analyses that
make sense.
Parsing Results
When parsing the data we measure the number
of parsed chords, deleted chords, and inserted
acordes, as well as parsing time. These numbers are
summarized in Table 1. Both runs were performed
on the same Intel Core 2 6600 machine running
en 2.4 GHz with 3 GB of random-access memory
compiled using GHC version 7.0.3.
On the small data set the HarmTrace model
performs very well. The songs are parsed quickly
y, on average, fewer than one chord per song
is deleted. También, fewer than three insertions are
necessary, on average, for a piece to parse. It would
have been possible to adapt the model in such way
that the small data set would parse without any
errores, as was done by De Haas et al. (2009). Nosotros elegimos
to accept this small number of error corrections and
keep our grammar small and easy to comprehend.
The data set is parsed within a second.
For the large data set, the parsing time per
song increases considerably, due to a larger number
ambiguous solutions and increased error-correction.
El 5,028 chord sequences are still parsed rea-
sonably quickly, en 6 mín. 25 segundo. The number of
error corrections increases considerably, pero el
parser never crashes or refuses to produce valid
producción. The higher number of error corrections
is expected, because this data set contains songs
with modulations, atonal harmonies, and a variety
of errors. Still, HarmTrace keeps the number of
deleted chords under 6 percent of the total chords
parsed.
When we compare the parsing results of the
small data set with the results of an older, Java-
based parser (De Haas et al. 2009), we notice that
HarmTrace is much faster. The Java-based parser
took more than 9 min to parse this data set. Nosotros
cannot compare the parsing results of the large
data set because the majority of the pieces are
rejected by the grammar used by the older parser.
This emphasizes how important the error-correction
el proceso es. Even with error correction, HarmTrace
parses the large data set faster than the Java-based
parser parses the small data set.
Discusión
We have presented HarmTrace, a system that au-
tomatically analyses sequences of musical chord
labels. Implementing our system in Haskell has
proven to be a profitable decision, given the advan-
tages of error-correcting parsers and datatype-generic
programming. We have shown that HarmTrace can
handle corpora of considerable size, parses chord
progressions quickly, and is robust against noisy
datos. HarmTrace currently does not support full
modulation, but it can explain changes between
parallel major and minor keys. That the lack of
full modulation does not have to hamper prac-
tical application is shown by the application of
HarmTrace to practical MIR tasks like automatic
harmonic-similarity estimation, chord transcrip-
ción, and automatic harmonization. Aunque el
model presented has a bias towards jazz harmony,
we have shown that it can be used to analyze some
classical works as well.
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If we compare HarmTrace to other models of
tonal harmony, we notice various differences.
The theoretical work of Steedman (1984), para
instancia, focuses only on the structure of the (muy
particular) style of twelve-bar blues, whereas our
model aims to formalize the core of tonal harmony
with a bias towards jazz harmony, including the
twelve-bar blues. Although our work draws on
the work of Rohrmeier (2007, 2011), hay
also considerable contrasts. The most pronounced
difference from Rohrmeier’s CFG is that the latter
features modulation and tonicization. By tonicizing
to a local tonic, chords are analyzed with respect
to that local tonic. Como consecuencia, his approach
can explain secondary dominants by tonicization,
whereas the HarmTrace model uses a more jazz-
oriented approach to harmony by identifying ii-V-I
motions, (p.ej., En figura 3). Por ejemplo, en un
progression in the key of C major, after moving
to the subdominant, F can be viewed as a local
tonic, allowing the derivation of Gm and C as
local subdominant and local dominant. A benefit of
Rohrmeier’s approach is that it is also possible to
derive B(cid:2) C as a local subdominant/dominant pair.
A specification for this would be easy to add to the
HarmTrace model. Implementing both tonicizations
and secondary dominants, as Rohrmeier suggests,
would be problematic, sin embargo, because both
rules explain the same phenomena. Después de todo, el
preparation of F by C can be explained both by
the tonicization rules as well as by the secondary
dominant rules. This would inevitably lead to an
explosion of ambiguous solutions for a harmony
progression featuring secondary dominants.
Another difference is that Rohrmeier groups
tonics and dominants into higher-order phrases or
functional regions. He acknowledges that these
rules are highly ambiguous, but chooses to keep
them for theoretical completeness. El problema
is that the harmonic information alone generally
does not provide enough information for determin-
ing phrase boundaries. Por ejemplo, it is unclear
whether Ton Dom Ton represents a half-cadence
phrase followed by a tonic (Ton Dom) (Ton), o un
introduction of the tonic followed by a perfect-
cadence phrase (Ton) (Dom Ton). We believe that
such clusterings should be done in a post-processing
step, based on metrical positions and phrase-length
constraints.
On the whole, when we compare HarmTrace to
other models of tonal harmony, we observe that
most models remain purely theoretical. Esto es
regrettable, because although theoretical work can
yield valuable insights, having an implementation
of a model allows it to be evaluated empirically
and used in practice. HarmTrace has been clearly
demonstrating its practical use by its application to
harmonic similarity estimation, chord recognition,
and automatic harmonization. As we have seen in
this article, it may take state-of-the-art programming
techniques to create a model that is maintainable,
has expressive power, and yet remains fast. Nosotros
are confident that HarmTrace will contribute to
new insights in the modeling of harmonic analysis,
and that it will prove useful in practical music
applications.
Expresiones de gratitud
W.. Bas De Haas was funded by the Netherlands
Organization for Scientific Research, NWO-VIDI
grant 276-35-001. Jos ´e Pedro Magalh ˜aes was sup-
ported by EPSRC grant EP/J010995/1. Frans Wiering
and Remco C. Veltkamp are supported by the FES
project COMMIT/. We thank Martin Rohrmeier for
the inspiring discussions and ideas about modeling
harmony. We thank Anja Volk and two anonymous
reviewers for comments on a draft version of this
artículo.
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