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An Idiom-independent Representation of Chords for
Computational Music Analysis and Generation
Emilios Cambouropoulos
Maximos Kaliakatsos-Papakostas
Costas Tsougras
School of Music Studies,
Aristotle University of Thessaloniki
emilios@mus.auth.gr
School of Music Studies,
Aristotle University of Thessaloniki
maxk@mus.auth.gr
School of Music Studies,
Aristotle University of Thessaloniki
tsougras@mus.auth.gr}
ABSTRACT
In this paper we focus on issues of harmonic representa-
tion and computational analysis. A new idiom-
independent representation is proposed of chord types
that is appropriate for encoding tone simultaneities in any
harmonic context (such as tonal, modal, jazz, octatonic,
atonal). The General Chord Type (GCT) representation,
allows the re-arrangement of the notes of a harmonic
simultaneity such that abstract idiom-specific types of
chords may be derived; this encoding is inspired by the
standard roman numeral chord type labeling, but is more
general and flexible. Given a consonance-dissonance
classification of intervals (that reflects culturally-
dependent notions of consonance/dissonance), and a
scale, the GCT algorithm finds the maximal subset of
notes of a given note simultaneity that contains only con-
sonant intervals; this maximal subset forms the base upon
which the chord type is built. The proposed representa-
tion is ideal for hierarchic harmonic systems such as the
tonal system and its many variations, but adjusts to any
other harmonic system such as post-tonal, atonal music,
or traditional polyphonic systems. The GCT representa-
tion is applied to a small set of examples from diverse
musical idioms, and its output is illustrated and analysed
showing its potential, especially, for computational music
analysis & music information retrieval.
1. INTRODUCTION
There exist different typologies for encoding note simul-
taneities that embody different levels of harmonic infor-
mation/abstraction and cover different harmonic idioms.
For instance, for tonal musics, chord notations such as the
following are commonly used: figured bass (pitch classes
denoted above a bass note – no concept of ‘chord’),
popular music guitar style notation or jazz notation (abso-
lute chord), roman numeral encoding (relative to a key)
[1]. For atonal and other non-tonal systems, pc-set theo-
retic encodings [2] may be employed.
A question arises: is it possible to devise a ‘universal’
chord representation that adapts to different harmonic
idioms? Is it possible to determine a mechanism that,
given some fundamental idiom features, such as pitch
hierarchy and consonance/dissonance classification, can
automatically encode pitch simultaneities in a pertinent
manner for the idiom at hand?
Before attempting to answer the above question one
could ask: What might such a ‘universal’ encoding sys-
tem be useful for? Apart from music-theoretic interest
and cognitive considerations/implications, a general
chord encoding representation may allow developing
generic harmonic systems that may be adaptable to di-
verse harmonic idioms, rather than designing ad hoc sys-
tems for individual harmonic spaces. This was the prima-
ry aim for devising the General Chord Type (GCT) repre-
sentation. In the case of the project COINVENT [3], a
creative melodic harmonisation system is required that
relies on conceptual blending between diverse harmonic
spaces in order to generate novel harmonic constructions;
mapping between such different spaces is facilitated
when the shared generic space is defined with clarity, its
generic concepts are expressed in a general and idiom-
independent manner, and a common general representa-
tion is available.
In recent years, many melodic harmonisation systems
have been developed, some rule-based [4,5] or evolution-
ary approaches that utilize rule based fitness evaluation
[6, 7] others relying on machine learning techniques like
probabilistic approaches [8,9] and neural networks [10],
grammars [11] or hybrid systems (e.g. [12]). Almost all
of these systems model aspects of tonal harmony: from
“standard” Bach–like chorale harmonisation [4,10]
among many others) to tonal systems such as “classic”
jazz or pop ([9,11] among others). These systems aim to
produce harmonizations of melodies that reflect the style
of the discussed idiom, which is pursued by utilising
chords and chord annotations that are characteristic of the
idiom. For instance, the chord representation for studies
in the Bach chorales include usually standard Roman
numeral symbols, while jazz approaches encompass addi-
tional information about extensions in the guitar style
encoding.
For tonal computational models, Harte’s representa-
tion [13] provides a systematic, context-independent syn-
tax for representing chord symbols which can easily be
written and understood by musicians , and, at the same
time, is simple and unambiguous to parse with computer
programs. This chord representation is very useful for
annotating manually tonal music - mostly genres such as
pop, rock, jazz that use guitar-style notation. However, it
Copyright: © 2014 Emilios Cambouropoulos et al. This is an open-
access article distributed under the terms of the Creative Commons
Attribution License 3.0 Unported, which permits unrestricted use, distri-
bution, and reproduction in any medium, provided the original author
and source are credited.
cannot be automatically extracted from chord reductions
and is not designed to be used in non-tonal musics.
In this paper, firstly, we present the main concepts
behind the General Chord Type representation and give
an overall description, then, we describe the GCT algo-
rithm that automatically computes chord types for each
chord, then, we present examples form diverse music
idioms that show the potential of the representation and
give some examples of applying statistical learning on
such a representation, and, finally, we will discuss prob-
lems and future improvements.
2. REPRESENTING CHORDS
Harmonic analysis focuses on describing the harmonic
content of pitch collections/patterns within a given music
context in terms of harmonic labels, classes, functions
and so on. Harmonic analysis is a rather complex musi-
cal task that involves not only finding roots and labelling
chords within a key, but also segmentation (points of
harmonic change), identification of non-chord notes, met-
ric information and more generally musical context [14].
In this paper, we focus on the core problem of labelling
chords within a given pitch hierarchy (e.g. key); thus we
assume that a full harmonic reduction is available as in-
put to the model (manually constructed harmonic reduc-
tions).
Our intention is to create an analytic system that may
label any pitch collection, based on a set of user-defined
criteria rather than on standard tonal music theoretic
models or fixed psychoacoustic properties of harmonic
tones. We intend our representation to be able to cope
with chords not only in the tonal system, but any harmon-
ic system (e.g. octatonic, whole-tone, atonal, traditional
harmonic systems, etc.).
Root-finding is a core harmonic problem addressed
primarily following two approaches: the standard stack-
of-thirds approach and the virtual pitch approach. The
first attempts to re-order chord notes such that they are
separated by (major or minor) third intervals preserving
the most compact ordering of the chord; these stacks of
thirds can then be used to identify the possible root of a
chord (see, for instance, recent advanced proposal by
[15]). The second approach, is based on Terhard’s virtual
pitch theory [16] and Parncutt’s psychoacoustic model of
harmony [17]; it maintains that the root of a chord is the
pitch most strongly implied by the combined harmonics
of all its constituent notes (intervals derived from the first
members of the harmonic series are considered as ‘root
supporting intervals’).
Both of these approaches rely on a fixed theory of
consonance and a fixed set of intervals that are consid-
ered as building blocks of chords. In the culture-sensitive
stack-of-thirds approach, the smallest consonant intervals
in tonal music, i.e. the major and minor thirds, are the
basis of the system. In the second ‘universal’ psychoa-
coustic approach, the following intervals, in decreasing
order of importance, are employed: unison, perfect fifth,
major third, minor seventh, and major second. Both of
these approaches are geared towards tonal harmony, each
with its strengths and weaknesses (for instance, the se-
cond approach has an inherent difficulty with minor har-
monies). Neither of them can be readily extended to other
idiosyncratic harmonic systems.
Harmonic consonance/dissonance has two major
components: Sensory-based dissonance (psychoacoustic
component) and music-idiom-based dissonance (cultural
component)[18]. Due to the music-idiom dependency
component, it is not possible to have a fixed universal
model of harmonic consonance/dissonance. A classifica-
tion of intervals into categories across the dissonance-
consonance continuum can be made only for a specific
idiom. The most elementary classification is into two
basic categories: consonant and dissonant. For instance,
in the common-practice tonal system, unisons, octaves,
perfect fifths/fourths (perfect consonances) and thirds and
sixths (imperfect consonances) are considered to be con-
sonances, whereas the rest of the intervals (seconds, sev-
enths, tritone) are considered to be dissonances; in poly-
phonic singing from Epirus, major seconds and minor
sevenths may additionally be considered ‘consonant’ as
they appear in metrically strong positions and require no
resolution; in atonal music, all intervals may be consid-
ered equally ‘consonant’.
Let’s examine the case of tonal and atonal harmony;
these are probably as different as two harmonic spaces
may be. In the case of tonal and atonal harmony, some
basic concepts are shared; however, actual systematic
descriptions of chord-types and categories are drastically
different (if not incompatible), rendering any attempt to
‘align’ two input spaces challenging and possibly mis-
leading (Figure 1). On one hand, tonal harmony uses a
limited set of basic chord types (major, minor, dimin-
ished, augmented) with extensions (7ths, 9ths etc.) that
have roots positioned in relation to scale degrees and the
tonic, reflecting the hierarchic nature of tonal harmony;
on the other hand, atonal harmony employs a flat mathe-
matical formalism that encodes pitches as pitch-class sets
leaving aside any notion of pitch hierarchy, tone centres
or more abstract chord categories and functions. It seems
as if it is two worlds apart having as the only meeting
point the fact that tones sound together (physically sound-
ing together or sounding close to one another allowing
implied harmony to emerge).
Figure 1. Is mapping between ‘opposing’ harmonic
spaces possible?
Pc-set theory of course, being a general mathematical
formalism, can be applied to tonal music, but, then its
descriptive potential is mutilated and most interesting
tonal harmonic relations and functions are lost. For in-
stance, the distinction between major and minor chords is
lost if Forte’s prime form is used (037 for both - these
two chord have identical interval content), or a dominant
seventh chord is confused with half-diminished seventh
(prime form 0258); even, if normal order is used, that is
less general, for the dominant seventh (0368), the root of
the chord is not the 0 on the left of this ordering (pc 8 is
the root). Pitch-class set theory is not adequate for tonal
music. At the same time, the roman-numeral formalism is
inadequate for atonal music as major/minor chords and
tonal hierarchies are hardly relevant for atonal music.
In trying to tackle issues of tonal hierarchy, we have
devised a novel chord type representation, namely the
General Chord Type (GCT) representation, that takes as
its starting point the common-practice tonal chord repre-
sentation (for a tonal context, it is equivalent to the stand-
ard roman-numeral harmonic encoding), but is more gen-
eral as it can be applied to other non-standard tonal sys-
tems such as modal harmony and, even, atonal harmony.
This representation draws on knowledge from the domain
of psychoacoustics and music cognition, and, at the same
time, ‘adjusts’ to any context of scales, tonal hierarchies
and categories of consonance/dissonance.
At the heart of the GCT representation is the idea that
the ‘base’ of a note simultaneity should be consonant.
The GCT algorithm tries to find a maximal subset that is
consonant; the rest of the notes that create dissonant in-
tervals to one or notes of the chord ‘base’ form the chord
‘extension’. The GCT representation has common char-
acteristics with the stack-of-thirds and the virtual pitch
root finding methods for tonal music, but has differences
as well (see section 4.3). Moreover, the user can define
which intervals are considered ‘consonant’ giving thus
rise to different encodings. As will be shown in the next
sections, the GCT representation encapsulates naturally
the structure of tonal chords and at the same time is very
flexible and can readily be adapted to different harmonic
systems.
3. THE GENERAL CHORD TYPE REPRE-
SENTATION
3.1 Description of the GCT Algorithm
Given a classification of intervals into conso-
nant/dissonant (binary values) and an appropriate scale
background (i.e. scale with tonic), the GCT algorithm
computes, for a given multi-tone simultaneity, the ‘opti-
mal’ ordering of pitches such that a maximal subset of
consonant intervals appears at the ‘base’ of the ordering
(left-hand side) in the most compact form. Since a tonal
centre (key) is given, the position within the given scale
is automatically calculated.
Input to the algorithm is the following:
• Consonance vector: The user defines which intervals
are consonant/dissonant through a 12-point Boolean
vector of consonant (1) or dissonant (0) intervals. For
instance, the vector [1,0,0,1,1,1,0,1,1,1,0,0] means
that the unison, minor and major third, perfect fourth
and fifth, minor and major sixth intervals are
consonant – dissonant intervals are the seconds,
sevenths and the tritone; this specific vector is
referred to in this text as the common-practice
consonance vector.
• Pitch Scale Hierarchy: The pitch hierarchy (if any) is
given in the form of scale tones and a tonic (e.g. a D
maj scale is given as: 2, [0,2,4,5,7,9,11], or an A
minor pentatonic scale as: 9, [0,3,5,7,10]).
• Input chord: list of MIDI pitch numbers (converted to
pc-set).
GCT Algorithm (core) - computational pseudocode
Input: (i) the pitch scale (tonality), (ii) a vector of the
intervals considered consonant, (iii) the pitch class set
(pc-set) of a note simultaneity
Output: The roots and types of the possible chords de-
scribing the simultaneity
1. find all maximal subsets of pairwise consonant
tones
2. select maximal subsets of maximum length
3. for all selected maximal subsets do
4. order the pitch classes of each maximal subset in
the most compact form (chord ‘base’)
5. add the remaining pitch classes (chord ‘exten-
sions’) above the highest of the chosen maximal
subset's (if necessary, add octave - pitches may
exceed the octave range)
6. the lowest tone of the chord is the ‘root’
7. transpose the tones of the chord so that the low-
est becomes 0
8. find position of the ‘root’ in regards to the given
tonal centre (pitch scale)
9. endfor
The GCT algorithm encodes most chord types ‘correctly’
in the standard tonal system. In example 1, Table 1 the
note simultaneity [C,D,F#,A] or [0,2,6,9] in a G major
key is interpreted as [7,[0,4,7,10]], i.e. as a dominant sev-
enth chord (see similar example in Section 3.3).
However, the algorithm is undecided in some cases,
and even makes ‘mistakes’ in other cases. In most in-
stances of multiple encodings, it is suggested that these
ideally should be resolved by taking into account other
harmonic factors (e.g., bass line, harmonic functions,
tonal context, etc.). For instance, the algorithm gives two
possible encodings for a [0,2,5,9] pc-set, namely minor
seventh chord or major chord with sixth (see Table1, ex-
ample 2); such ambiguity may be resolved if tonal con-
text is taken into account. For the [0,3,4,7] pc-set with
root 0, the algorithm produces two answers, namely, a
major chord with extension [0,[0,4,7,15]] and a minor
chord with extension [0,[0,3,7,16]]; this ambiguity may
be resolved if key context is taken into account: for in-
stance, [0,4,7,15] would be selected in a C major or G
major context and [0,3,7,16] in a C minor or F minor
context. Symmetric chords, such as the augmented chord
or the diminished seventh chord, are inherently ambigu-
ous; the algorithm suggests multiple encodings which can
be resolved only by taking into account the broader har-
monic context (see Table1, example 3). Since the aim of
this algorithm is not to perform sophisti
!
!
Example 1
Example 2
Example 3
Tonality - key
Cons. Vector
Input
pc-set
G: [7, [0, 2, 4, 5, 7, 9, 11]]
[1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0,
0]
[60, 62, 66, 69, 74]
[0, 2, 6, 9]
C: [0, [0, 2, 4, 5, 7, 9, 11]]
[1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0]
[50, 60, 62, 65, 69]
[0, 2, 5, 9]
C: [0, [0, 2, 4, 5, 7, 9, 11]]
[1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0]
[62, 68, 77, 71]
[2, 5, 8, 11]
Maximal subsets
Narrowest range
Add extensions
Lowest is root
Chord in root position
Relative to key
[2, 6, 9]
[2, 6, 9]
[2, 6, 9, 12]
2 (note D)
[2, [0, 4, 7, 10]]
[7, [0, 4, 7, 10]]
[2, 5, 9] and [5, 9, 0]
[2, 5, 9] and [5, 9, 0]
[2, 5, 9, 12] and [5, 9, 0, 14]
2 and 5 (notes D & F)
[2, [0, 3, 7, 10]] & [5, [0, 4, 7, 9]]
[2, [0, 3, 7, 10]] & [5, [0, 4, 7, 9]]
[2, 5], [5, 8], [8, 11], [2, 11]
[2, 5], [5, 8], [8, 11], [2, 11]
all rotations of [2,5,8,11]
2,5,8,11 (resp. for each rotation)
[X,[0,3,6,9]], where X∈{2,5,8,11}
[X,[0,3,6,9]], where X∈{2,5,8,11}
Extra
steps:
Subset overap
Base in scale
[2, [0, 3, 7, 10]]
[11,[0,3,6,9]]
Table 1. Examples of applying the GCT algorithm.
cated harmonic analysis, but rather to find a practical and
efficient encoding for tone simultaneities (to be used, for
instance, in statistical learning and automatic harmonic
generation – see end of Section 4), we decided to extend
the algorithm so as to reach in every case a single chord
type for each simultaneity (no ambiguity).
!
GCT Algorithm (additional steps) - for unique encoding
If more than one maximal subsets exist:
• Overlapping of maximal subsets: create a sequence of
maximal subsets by ordering them so as to have maxi-
mal overlapping between them and keep the maximal
subset that appears first in the sequence (chord's base)
• Chord base notes are scale notes: prefer maximal subset
that contains only pcs that appear in the given scale (to-
nal context) – i.e. avoid non-scale notes in the chord
base (this rule is rather arbitrary and is under considera-
tion)
• if neither of the above give a unique solution, chose one
encoding at random
Additional adjustment: for dyads, in a tonal context, pre-
fer perfect fifth over perfect fourth, and prefer seventh to
second intervals
!
The additional steps select chord type [2, [0,3,7,10]] in
example 2, Table1 (maximal overlapping between two
maximal subsets), and [11, [0,3,6,9]] in example 3, Table
1 (last pitch-class is Ab that is a non-scale degree in C
major).
3.2 Formal description of the Core GCT Algorithm
The proposed algorithm for extracting the computation of
GCT receives a simultaneity of pitches that are trans-
formed into pitch classes and produces a chord type rela-
tive to a key, namely the root, the base and the extension,
which specify qualitative information about the chord
that more precisely describes this simultaneity. A detailed
description of the algorithm follows, based on an exam-
ple input simultaneity. Suppose that the input set of notes
results in the pc-set [0, 2, 6, 9], which could be described
as a D major chord with minor seventh regarding the to-
nal music environment – described by the υ = [1, 0, 0, 1,
1, 1, 0, 1, 1, 1, 0, 0] consonance vector. Therefore, the
algorithm should produce an output in the form: [r, [b],
[e]] = [2, [0, 4, 7], [10]].
By utilising the input pc-set and given a consonance
vector that represents a selected music idiom, a binary
matrix is constructed that is denoted as B. Each row and
column of B represents a pitch class of the input chord,
while a matrix entry is 1 or 0, signifying whether the pair
of row and column pcs are consonant or dissonant respec-
tively – according to the current consonance vector.
Strictly, if the consonance vector is denoted as υ and the
input pcset as p, then!∀!i, j!∈!{1, 2, . . ., length(p)}
!
(1)
where the function length(x) return the length of vector x.
The B matrix in the discussed example, where p = [0, 2,
6, 9], is the following:
!
!
(2)
Afterwards, a tree is constructed for each of the rows of
B. The root node of these trees is the pitch class that cor-
responds to the respective row, while their branches from
leaves to nodes include pitch classes that are pairwise
consonant (according to υ). The construction of the tree
that corresponds to the i–th element of p, is implemented
by recursively traversing B in a depth–first–search (DFS)
fashion, beginning from the i–th row and following the
paths ‘circumscribed’ by the occurrences of units. Such a
traversal is exhibited in Table 2 for the second row of the
current example’s B matrix. This step’s outcome is a col-
lection of trees, each of which corresponds to a row of B.
The trees of the current example are shown in Table 3.
Τable 2. The steps of the algorithm when scanning the path of the second row.
!
Table 3. All the trees for the current example. The max-
imal path is highlighted with boldface typesetting.
After the application of the above procedure, the paths
from root to leaves with maximal length are kept either as
the output chord candidates, or for further processing in
the steps described in the remaining of this section. In the
current example there is a single maximal path ([2, 6, 9]),
which is highlighted with boldface typesetting (Table 3).
After the longest path has been extracted, the pitch clas-
ses that constitute it, are recombined in their most com-
pact form, which in the current example is [2, 6, 9] (unal-
tered). The pitch class 0 of the initial [2, 6, 9] pc-set is
considered as an extension. Thereby, the simultaneity [0,
2, 6, 9] is circularly shifted to [2, 6, 9, 12], disregarding
the fact that pitch classes can take integer values between
0 and 11. In turn, [2, 6, 9, 12] is transformed to the fol-
lowing [r, [b], [e]] denotation: [2, [0, 4, 7], [10]]. This
denotation clarifies that the simultaneity [0, 2, 6, 9] is
actually a major chord (base [0, 4, 7]) with a minor sev-
enth (extension [10]) and fundamental pitch class 2, (i.e.
D7). As the tonal context is given as input, for instance G
major key, the absolute chord type [2, [0,4,7,10]] (i.e. D7
chord) is converted to relative chord type, i.e.,
[7,[0,4,7,10]] which means dominant seventh in G major.
This is equivalent to the roman numeral analytic types.
3.3 An example analysis with GCT
An example harmonic analysis of a Bach Chorale phrase
illustrates the proposed GCT chord representation (Figure
2). For a tonal context, chord types are optimised such
that pcs at the left hand side of chords contain only con-
sonant intervals (i.e. 3rds & 6ths, and Perfect 4ths & 5ths).
For instance, the major 7th chord is written as [0,4,7,10]
since set [0,4,7] contains only consonant intervals where-
as 10 that introduces dissonances is placed on the right-
hand side – this way the relationship between major
chords and major seventh chords remains rather transpar-
ent and is easily detectable. Within the given D major
key context it is simple to determine the position of a
chord type in respect to the tonic – e.g. [7,[0,4,7,10]]
means a major seventh chord whose root is 7 semitones
above the tonic, amounting to a dominant seventh. This
way we have an encoding that is analogous to the stand-
ard roman numeral encoding (Figure 2, top row). If the
tonal context is changed, and we have a chromatic scale
context (arbitrary ‘tonic’ is 0, i.e. note C) and we consid-
er all intervals equally ‘consonant’, we get the second
GCT analysis in Figure 1 which amounts to normal or-
ders (not prime forms) in a standard pc-set analysis – for
tonal music this pc-set analysis is weak as it misses out
important tonal hierarchical relationships (notice that the
relation of the dominant seventh chord type to the plain
dominant chord is obscured). Note that relative ‘roots’ to
the ‘tonic’ 0 are preserved as they can be used in harmon-
ic generation tasks.
!
Figure 2 Chord analysis of a Bach Chorale phrase by
means of traditional roman numeral analysis, pc-sets
and two versions of the GCT algorithm.!
For practical reasons of space in the musical illustrations,
the form [r,[b],[e]] is not preserved: the base and exten-
sion is concatenated and brackets are omitted. For in-
stance: [7,[0,4,7],[10]] may be depicted as 7,[0,4,7,10] or
even as 7.04710.
4. HARMONIC ENCODING & ANALYSIS
WITH THE GCT
The GCT algorithm has been applied to tonal extracts
from standard tonal pieces, such as Bach Chorales, but
additionally it has been tested out on harmonic structures
from diverse harmonic idioms. Some examples are pre-
sented below to give an idea of the potential of the GCT
representation. Strong points of the encoding are given
along with weaknesses. Some aspects of the analysis are
difficult to judge in some idioms and further study in
required.
4.1 GCT Encoding Examples
In common-practice tonal music, GCT works very
well. Mistakes are sometimes made in case of symmetric
chords such as the diminished seventh chord or the aug-
mented triad. In the case of the half diminished seventh
chord GCT ‘prefers’ to label it as a minor chord with
added sixth instead of a diminished chord with minor
seventh. Chords that include chromatic notes such as the
German sixth, Italian sixth, Neapolitan sixth are encoded
consistently even though not necessarily coinciding with
analytic interpretations by theorists (the French sixth is
more tricky as it is a symmetric chord and GCT finds two
equally prominent ‘roots’).
Below, a number of examples are presented that illus-
trate the application of the GCT algorithm on diverse
harmonic textures. The first example (Figure 3) is taken
from the first measures of Beethoven’s Moonlight Sona-
ta. In this example, GCT encodes classical harmony in a
straightforward manner. All instances of the tonic chord
inverted or not (i.e., C# minor) are tagged as 0,[0,3,7] and
[10] is added when the 7th is present; the dominant sev-
enth is 7,[0,4,7,10] and it appears once without the fifth
[7]; the fourth chord is a Neapolitan sixth and it is encod-
ed as 1,[0,4,7] which means major chord on lowered se-
cond degree (Db major chord in the C# minor key).
Figure 3 Beethoven, Sonata 14, op.27-2 (reduction of
first five measures). Top row: roman numeral harmonic
analysis; bottom row: GCT analysis. GCT successfully
encodes all chords, including the Neapolitan sixth chord
(fourth chord).
In the example of Figure 4 a tonal chord progression by
G. Gershwin is presented. Chromaticism is apparent in
this passage. The GCT ‘agrees’ with the roman numeral
analysis of the excerpt including the Italian sixth chord
that is labelled as 8,[0,4,10], and it even labels the chord
that was left without a roman numeral tag by the analyst
(see question mark) encoding it as a minor chord with
sixth on the flattened sixth degree (Gb-Bbb-Db-Eb)
(Note: actually it could be even encoded as a half-
diminished 7th on the fourth degree Eb-Gb-Bbb-Db).
Figure 4. G. Gershwin, Rhapsody in Blue (reduction of
first five measures). Top row: roman numeral harmonic
analysis; bottom row: GCT analysis. GCT successfully
identifies all chords (see text).
Figure 5 illustrates an Early Renaissance example of
fauxbourdon by G. Dufay. Parallel motion of voices is
typical in this idiom. The GCT labels correctly all dyads
and triads, taking into account musica ficta that produces
rather unusual chord progressions in regards to standard
tonal harmony.
In Figure 6 an example from the polyphonic
singing tradition of Epirus is presented. This very old 2-
voice to 4-voice polyphonic singing tradition is based on
the anhemitonic pentatonic pitch collection and more
specifically the pentatonic minor scale that functions as
source for
Figure 5. G. Dufay’s Kyrie (reduction) - first phrase in
A phrygian mode that exemplifies parallel motion in
fauxbourdon and a phrygian cadence (early Renais-
sance). GCT correctly identifies and labels the open
fifths as well as the triadic chords.
both the melodic and harmonic content of the music. A
unique harmonic aspect of these songs is the unresolved
dissonances (major second and minor seventh intervals)
at structurally stable positions of the pieces (e.g. cadenc-
es). In the example two GCT versions are presented: the
first (top row) depicts the encoding for the standard con-
sonance vector and the second (bottom row) presents the
GCT labelling that considers additionally major seconds
and minor sevenths as ‘consonant’ (it is the same as for
the ‘atonal’ consonance vector as no minor seconds and
major sevenths exist in the idiom). It is interesting to note
that for the standard consonance vector almost all chords
have the drone tone as their root. On the other hand, in
the second encoding different relations between chords
become apparent (e.g. 10,[0,2,5] and 10,[0,2,5,7]) and
also an oscillation of the chord ‘root’ between the tonic
and a note a tone lower is highlighted. Polyphonic songs
from Epirus are the focus of a different study [19].
!
Figure 6 Excerpt from a traditional polyphonic song
from Epirus. Top row: GCT encoding for standard
common-practice consonance vector; bottom row: GCT
encoding for atonal harmony – all intervals ‘consonant’
(this amounts to pc-set ‘normal orders’)
4.2 Learning and generation with GCT
In a current study, the GCT representation has been uti-
lised in automatically analysing and encoding scores (ac-
tually, harmonic reductions of scores) from diverse idi-
oms, and then employing this extracted information for
melodic harmonisation. In [20] the authors discuss the
utilization of a well–studied probabilistic methodology,
namely, the hidden Markov model (HMM) methodology,
in combination with constraints that incorporate fixed
beginning and ending chords and intermediate anchor
chords. To this end, a constrained HMM (CHMM) is
developed, which allows the manual insertion of interme-
diate chords, providing alternative harmonisations that
comply with specific constraints.
The reported results indicate that the CHMM method,
harnessed with the novel General Chord Type (GCT)
algorithm, functions effectively towards convincing me-
lodic harmonisations in diverse idioms. In Figures 7 & 8,
two examples of melodic harmonisation are illustrated for
a Bach chorale melody and for a traditional melody from
Epirus. In both cases, the system has been trained on a
corpus of harmonic reductions of pieces in the idiom,
and, then, used to generate new melodic harmonisations.
The results are very good: the Bach chorale harmonisa-
tion is typical of the style and at the same time not trivial
(uses secondary dominants that enrich the harmonisa-
tion); the Epirus melody harmonisation is close to the
style of polyphonic singing (if additional melodic and
rhythmic elements were added the phrase would become
rather typical of the idiom).
Figure 7. Automatically generated GCTs for a Bach
Chorale melody employing a HMM for fixed bounda-
ries (first and last chords are given). Voice leading has
been arranged manually.
Figure 8. Automatically generated GCTs for an Epirus
melody (reduced version) employing a HMM for fixed
boundaries. Voice leading has been arranged manually.
4.3 Discussion and future development
The current version of GCT encodes only the chord type
and the relative position of its ‘root’ to the local tonic of a
given scale. However, it can readily be extended to in-
corporate explicit information on chord inversions (i.e.
bass note position), on scale degrees (chromatic notes
that do not belong to the current scale can be tagged so
that indirectly scale degrees are indicated), and, even, on
voice-leading (for instance, motion of bass, or even for
note extensions that may require resolution by down-
wards step-wise motion). A rich chord representation
should embody such information.
The organisation of tones by GCT for the ‘standard’
consonance vector gives results quite close to those pro-
duced by the stack-of-thirds technique, as implicit in the
latter is consonance of thirds and fifths (as two thirds sum
up to a fifth). Some difference are:
• the stack-of-thirds approach usually requires
traditional note names (that allow enharmonic
spellings) whereas the GCT is based on pitch classes
(no direct explicit link to a scale). For instance, GCT
considers the chord CEG# or CEAb ([0,4,8]) as
consonant since its intervals are pairwise consonant1,
1 Question: why is the augmented triad considered dissonant
when all its tones are pairwise consonant?
i.e. two 4 semitone intervals (major thirds) and one 8
semitone interval (minor sixth or augmented fifth)
with root any one of the three tones; stack-of-thirds
determines C as the root in the first case and Ab in the
second case. The GCT algorithm misses out on
sophisticated tonal scale information but is still
informative at the same time being simpler, and easier
to implement.
• in the standard consonance vector version of GCT,
diminished fifths are not allowed whereas in the
stack-of-thirds approach all fifths are allowed. For
instance, the root of the half-diminished chord BDFA
is B according to the stack-of-thirds whereas GCT
considers D as the root and B as a sixth above the root
(DFAB), i.e. diminished triads are not consonant
chords according to CGT. Of course, the consonance
vector in GCT may be altered so that the tritone is
also consonant in which case the two approaches are
closer.
• the stack-of-thirds method allows empty third
positions in the lower part of the stack whereas GCT
always prefers to have a compact consonant set of
pitches at the bottom. For instance, a chord
comprising of notes: CEFG ([0,4,5,7]) will be
arranged as FCEG by the stack-of-thirds technique
and CEGF ([0,4,7,17]) by GCT.
In relation to the virtual pitch root finding method, the
proposed approach differs in that minor thirds are equally
consonant to major thirds allowing equal treatment of
major and minor chord (as opposed to the virtual pitch
approach that is biased towards major thirds due to the
structure of the harmonic series).
It is also possible to redesign the GCT algorithm alto-
gether so as to make use of non-binary conso-
nance/dissonance values allowing thus a more refined
consonance vector. Instead of filling in the consonance
vector with 0s and 1s, it can be filled with fractional val-
ues that reflect degrees of consonance derived from per-
ceptual experiments (e.g., [21]) or values that reflect cul-
turally-specific preferences. Such may improve the algo-
rithm’s performance and resolve ambiguities in certain
cases (future work).
5. CONCLUSIONS
In this paper a new representation of chord types has been
presented that adapts to diverse harmonic idioms allow-
ing the analysis and labelling of tone simultaneities in
any harmonic context. The General Chord Type (GCT)
representation, allows the re-arrangement of the notes of
a harmonic simultaneity such that idiom-specific types of
chords may be derived. Given a consonance/dissonance
classification of intervals (that reflects culturally-
dependent notions of consonance/dissonance), and a (set
of) scales, the GCT algorithm finds the maximal subset of
notes of a given note simultaneity that contains only con-
sonant intervals; this maximal subset forms the basis up-
on which the chord type is built. The proposed represen-
tation is ideal for hierarchic harmonic systems such as the
tonal system and its many variations, but adjusts to any
other harmonic system such as post-tonal, atonal music,
or traditional polyphonic systems.
The GCT representation was applied to a small set of
examples from diverse musical idioms, and its output was
presented and analysed showing its potential use, espe-
cially, for computational music analysis and music in-
formation retrieval tasks. The encoding provided by GCT
is not always correct according to the interpretation given
by music theorists, but, at least, it is consistent (i.e. a cer-
tain chord will always be encoded the same way) render-
ing it adequate for machine learning and generation (e.g.
melodic harmonisation) where music theoretical correct-
ness is not so important. Sometimes GCT ‘uncovers’
chordal relations that are obscured by notation and en-
harmonic spellings, and may assist a musician in harmon-
ic analysis. Overall, the proposed encoding seems to be
promising and potentially useful in computational music
applications.
Acknowledgments
The project COINVENT acknowledges the financial sup-
port of the Future and Emerging Technologies (FET)
programme within the Seventh Framework Programme
for Research of the European Commission, under FET-
Open grant number: 611553. Special thanks are due to
Andreas Katsiavalos for preparing the harmonic dataset
that has been used in the music examples in the paper.
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