Finding Cadences through
Boundary Entropy
Matthias Mauch
Daniel Müllensiefen
Geraint Wiggins
Simon Dixon
Abstract
The identification of recurrent patterns in chord progressions into meaningful units is a useful
tool for a variety of tasks in automatic music processing and analysis, e.g. for the
identification of song sections and the induction of song structure, sytle identiication and
analysis, cover song detection, and the modelling of harmonic expectations. We propose an
unsupervised approach that identifies cadences (i.e. recurrent patterns that indicate
harmonic closure) from entropy profiles of chord progressions from symbolic harmony
annotations for pop songs. Results indicate a large overlap between the most highly ranked
progressions of 3 subsequent chords and well-known cadences from music theory.
Further research will go into segmentation of chords into meaningful episodes, n-gram
models and semantic clustering.
We map chords with and without extensions onto the chord
classes major (without dominant chords), minor, dominant,
diminished, augmented, suspended and neutral (≈ open fifth).
Consecutive chords that are equal (after this procedure) are
joined. At present we ignore bass notes.
• Transposition to start on
<http://purl.org/ontology/chord/
Retrieve
chords from transcribed
symbol/Ds:min7(*b3,9)/5>
chord
collections in RDF
a chord:Chord;
chord:root [ a chord:Note;
• Beatleschord:modifier
Transcriptions
chord:sharp;
all studio albums, 180 songs (Harte et al., 2005)
chord:natural <http://
purl.org/ontology/chord/note/D>
];
MMA
file collection
•
• Goldsmiths’
Extracted Labels
chord:bass [ a chord:ScaleInterval;
Several file types can be converted to the Chord Ontology (Sutton et
chord:degree 5 ];
al., 2007) RDF format, which our Java code reads in. A conversion
tool for Band-in-a-Box files will soon be available and will allow
access to a large number of freely available, manual transcriptions
from the internet.
2 Pre-Processing & Trie
• Reduce chords to classes
1 Chord Transcriptions
Trie of depth 3+1, level
5 in pale grey. Numbers
in brackets indicate how
many sequences go
through the node
Bb Eb C F Bb6.
C
We make the assumption of invariance to transposition, i.e. we
regard the chord changes C-F and D-G as equivalent. That’s
done to every sub-sequence.
• Insert into Trie
We use a tree structure known as a trie or n-gram tree (see right)
into which we insert all sub-sequences of a song collection to
depth n+1. Every sequence is stored only once.
F(3)
C F D G C .
C F D G
C A D
C F
C
C(5)
C .
G .
Bb.
F .
C .
D(1)
G(1)
A(1)
Bb(1)
D(1)
G(1)
C(1)
3 Results: “Cadences”
•Calculate Boundary Entropy of 3-grams
We adapt an approach devised by Cohen & Adams (2001) for segmentation of text into
episodes. Given one particular chord 3-gram we consider the chord that immediatly follows it,
i.e. the fourth chord. Intuitively, if the 3-gram is in the middle of a cadence, the fourth chord will
nearly
always be the same. If the 3-gram constitutes the last
three chords of a cadence, since it occurs in different
songs or before different parts of the same song,
more different chords will occur and the distribution of
chords is flatter. A measure for the flatness of a
distribution is the entropy, and it can be easily
calculated from the trie.
Dm G C
Dm
G
...
•Rank 3-grams
The only thing left to do is to rank the chord 3-grams according to their entropy value.
Almost all of these 3-grams
correspond to meaningful
harmonic "episodes" as they
appear in harmonic cadences.
Most of them have proper names
in music theory,
e.g.: 1. "2-5-I" cadence; 2."Perfect
cadence"; 3. "Plagal cadence", 6.
"Authentic cadence".
Highest ranking boundary entropy 3-grams over
180 Beatles songs (possible functional interpretation in
brackets)
1. 0 MIN → 5 MAJOR → 10 MAJ (ii-V-I)
2. 0 MAJ → 2 MAJOR → 7 MAJ (IV-V-I)
3. 0 MAJ → 5 MAJOR → 0 MAJ (I-IV-I)
4. 0 MAJ → 0 MINOR → 7 MAJ (IV-iv-I)
5. 0 MAJ → 5 MAJOR → 10 MAJ (II-V-I)
6. 0 MAJ → 7 MAJOR → 0 MAJOR (I-V-I)
7. 0 DOM → 5 MAJOR → 10 MAJOR (II7-V-I)
References
Cohen, P. & Adams, N. (2001). An Algorithm for Segmenting
Categorical Time Series into Meaningful Episodes. Lecture Notes
in Computer Science 2189, Springer, 198-207
Harte, C., Sandler, M., Abdallah, S. & Gomez, E. (2005).
Symbolic representation of Musical chords: A Proposed Syntax for
Text Annotations. ISMIR Proceedings.
Sutton, C., Raimond, Y., Mauch, M. (2007): The OMRAS2 Chord
Ontology,
http://motools.sourceforge.net/chord_draft_1/chord.html