“Choral intonation practices: an
investigation and a model"
Johanna Devaney
1. Issues
2. Assumptions
3. Methodological Overview
4. Applications
5. Conclusions
6. References
9/08/06 Johanna Devaney - ISSSM 2006 1
johanna@music.columbia.edu
1. Issues
An a cappella’s choir’s tuning cannot be
consistently related to a single global
reference point
Local reference points are formed through a
combination of horizontal and vertical
musical factors
The weighting of the local reference points
may differ both from piece to piece and
within a single piece
9/08/06 Johanna Devaney - ISSSM 2006 2
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1. Issues
Consider a piece in C major that modulates to its
dominant, G major
How does B, the seventh of the home key, function
differently at various points in this piece?
B is the leading tone in C major (I) and the mediant in G
major (V)
In C Major (I) B is an unstable pitch, outside of the tonic
triad, that generally resolves up by step to the tonic (C)
In G major (V) B is a stable pitch within the key’s tonic triad
9/08/06 Johanna Devaney - ISSSM 2006 3
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2. Assumptions
The western tonal system is used as the primary
reference point for the harmonic/contrapuntal
activity in the chosen music
i.e. late pre-tonal early tonal western art music
Vertical tendencies are addressed in relation to the
harmonic series and theories of sensory
consonance (Helmholtz, Terhardt)
Horizontal tendencies are considered in terms of
recent theories of tonal tension and attraction
(Lerdahl, Larson)
9/08/06 Johanna Devaney - ISSSM 2006 4
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2. Assumptions
Intonation practices are assumed to be
intentional
The recordings selected to train the model are
considered to be “expert” renditions
The perceived F0 is assumed to be the
mean frequency over the duration of the
note or one beat in the current metrical
context, whichever is shorter
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3. Methodological Overview
Model of intonation
computational approach
based on frequency analyses of a number of choral
recordings
uses statistical machine learning techniques to
build a model of choral intonation practices from
the microtonal pitch variations in the recordings
provides the model with an objective basis
9/08/06 Johanna Devaney - ISSSM 2006 6
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3. Methodological Overview
Model of intonation
theoretical approach
draws on the various branches of music theory that
address harmonic and voice-leading practices,
music forces and expectation, acoustics and
psychoacoustics, and tuning, temperament, and
intonation
provides a contextual depth that empirical analysis
cannot produce
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3. Methodological Overview
Received
Recording Score
Music Theory
Data Collection
Data Correlation
Computational Theoretical
Approach Approach
Generalized Model
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3. Methodological Overview
Theories of sensory consonance make a
case for purely tuned vertical intervals on
psychoacoustic grounds, as there is a
greater coincidence of partials between them
than with tempered intervals.
Helmholtz on consonance and dissonance
(1863)
Terhardt on sensory consonance and tonal
significance/virtual pitch theory (1974, 1984)
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3. Methodological Overview
The horizontal aspects are a little more
complicated as there is no acoustical
template to refer to.
Theories of tonal tension offer tools with
which to address the horizontal dimension.
Lerdahl’s tonal pitch space theory (2001)
formalizes the tendency of a dissonant pitch
to resolve to a consonant neighbor
Attraction equation: s2/s1*1/n2
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3. Methodological Overview
Larson (2004, 2005) correlates the forces of gravity,
magnetism, and inertia explicitly into a single equation
F = wGG + wMM + wII
G has a value of 1 or 0
M = 1/dto2 - 1/dfrom2
I has a value of -1, 0, or 1
The tension models provide a means of exploring
intonational tendencies in linear pitch sequences by
providing representations of how the pitches within a
tonal system exist in relation to one another outside of
the harmonic context in which they occur
9/08/06 Johanna Devaney - ISSSM 2006 11
johanna@music.columbia.edu
3. Methodological Overview
Received
Recording Score
Music Theory
Data Collection
Data Correlation
Computational Theoretical
Approach Approach
Generalized Model
9/08/06 Johanna Devaney - ISSSM 2006 12
johanna@music.columbia.edu
4. Applications
This methodology allows for a detailed model to be
built from analyses of numerous recordings of real-
world performances
The resultant model may be useful in a number of
domains
informing composers about the intonation practices of
choirs
voice pedagogy for choral singing
creating intonationally accurate digital re-creations
music cognition - particularly in the study of expressivity
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5. Conclusions
The intersection of computational and
theoretical approaches offers a means of
addressing the potential conflicts between
horizontal and vertical intonation tendencies
The resultant model may serve a normalized
reference to intonation practices, which is
potentially useful in a number of domains
9/08/06 Johanna Devaney - ISSSM 2006 14
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6. References
Helmholtz, H. (1863). On the Sensation of Tone. Translated by A.J.
Ellis. (1954). New York, NY: Dover Publications.
Lerdahl, F. (2001). Tonal Pitch Space. Oxford: Oxford University
Press.
Larson, S. (2004). Musical Forces and Melodic Expectations:
Comparing Computer Models with Experimental Results. Music
Perception, 21:4, 457-498.
Larson, S. and VanHandel, L. (2005). Measuring Musical Forces.
Music Perception, 21:4, 457-498.
Terhardt, E. (1974). Pitch, Consonance and Harmony. Journal of
the Acoustical Society of America, 55, 1061-69.
Terhardt, E. (1984). The concept of Musical Consonance: A Link
between Music and Psychoacoustics. Music Perception, 1:3, 276-
295.
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