Download Documentation for the Music Theory Ontology

Documentation for the Music Theory Ontology
Ontology Developed by:
Jordan Feenstra, Yonatan Schreiber
This Documentation by:
Yonatan Schreiber
Contents:
1. Scope
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3. Processes and Structures
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4. Implimentations
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5. Conclusion
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2. Definitions
1. Scope:
The Music Theory Ontology is intended to serve as a reference ontology that supports finegrained descriptions of musical phenomena. In particular we hope to enable the creation of rdf stores
that contain as much information about a given piece of music as would its score. If this is
accomplished, relations and restrictions axiomatized in the ontology can provide a robust set of
inferences, substantially in excess of what is encoded in the score, that will greatly facilitate analysis.
Our work thus far (and for the foreseeable future) focuses on traditional Western music theory,
which is a very large body of theory. We have attempted however, to define the classes at the top of
our hierarchy in terms broad enough to encompass alternative kinds of music and the bodies of theory
that are used to describe or analyze them.
The following example, drawn from http://www.musictheory.net, illustrates how we obtain our
desiderata:
The musical notation encodes information about the first four measures of an arrangement of
“O Canada”. Immediately beneath the staff is a series of roman numerals that encodes a harmonic
analysis of these measures. The subsequent bullet-pointed sentences describe how the analysis is
derived from the musical notation.
Our primary desideratum is thus to enable the creation of a set of rdf triples that encodes all of
the data encoded by the musical notation – the pitch, duration, and temporal position of each tone
notated on the staff. More complicated scores will encode additional kinds of information, such as
recommended instrumentation, dynamic (loudness) directives, tone emphasis, etc.
The harmonic analysis, and the observations and judgments from which it is derived, inform
further desiderata. The ontology must provide the classes necessary for making the sorts of assertions
typified by the observations and judgments in the example. The judgments are inferences based on
relations between the musical objects observed, so the ontology should define object properties that
characterize these relationships and assert axioms that characterize the inferences they justify. At
minimum, such restrictions should be inconsistent with incorrect analysis; ideally they should enable an
owl reasoner to infer analyses from a triple store representing a piece of music.
While a number of ontologies relevant to the domain of music currently exist, none of them has
developed the sorts of analytic tools we hope to provide. The Music Ontology1 in particular is broadly
used, but the terms it defines are suitable for describing only such external features of musical
compositions and performances as where, when, and by whom they were written or performed. This is
a valuable resource, but we feel that there is a need for means to describe music in greater internal
detail. We expect our ontology to be used in conjunction with a resource such as the Music Ontology
which describes the external features of individual performances, or the Audio Features Ontology2,
which can be used to describe recordings.
The Information Artifact Ontology3 is also closely related to our project. The Music Theory
Ontology will reuse some of its terminology and structure for describing musical information artifacts
such as scores and mp3’s. At present, we have developed only a sketch of this section of the ontology.
We feel that developing these terms is an important part of our project, because they are necessary for
enabling our ontology to be used in the description of music as it is encoded in such artifacts. Since it is
common for theorists to analyze a musical score, which is a material entity rather than a musical
process, the Music Theory Ontology must clarify the relationship between the object and the process.
The Music Theory Ontology is built on the basic categories profided by Basic Formal Ontology
(BFO)4. The remainder of this documentation assumes basic familiarity with BFO terms and distinctions.
2. Definitions:
In writing definitions it has been our goal at every point to leave room for alternative
understandings of music. One way to accomplish this is to define each term as broadly as possible. On
the other hand, a set of broadly defined terms is not conducive to the kind of precision required for the
fine-grained descriptive power we hope to achieve. Our approach has therefore been to define the
terms in the upper hierarchy broadly, while defining more narrowly those terms used specifically to
characterize the branch of theory we are currently developing.
1
http://www.musicontology.com
http://motools.sourceforge.net/doc/audio_features.html
3
https://code.google.com/p/information-artifact-ontology/
4
http://www.ifomis.org/bfo/
2
At the top of the hierarchy is, of course, “music” itself. How should this be defined? The
Merriam-Webster dictionary offers: “vocal, instrumental, or mechanical sounds having rhythm, melody,
or harmony”. This definition seems, prima facie, to be quite suitable. Since rhythm, melody, and
harmony are themselves terms our ontology must define, the breadth of this definition of music will
depend on the breadth of the definitions we offer for these subsidiary terms.
Two considerations ultimately caused us to reject this definition. Firstly, if music is defined in
terms of rhythm, melody, and harmony, then these terms must be defined without reference to music
in order to avoid circularity. Formulations that respect this restriction sacrifice the expressivity of the
subsidiary terms in favor of the expressivity of the higher-level term, which is contrary to the design
principle just stated.
Secondly, no matter how broadly we define rhythm, melody, and harmony, there will be
theorists who will argue that there are instances of music that fail to instantiate any of these. John
Cage, for instance, characterizes music as “purposeless play”, and is interested in “discovering means to
let sounds be themselves rather than vehicles for man-made theories or expressions of human
sentiments”5. Perhaps such an approach to music is inconsistent with any kind of analysis, and our
ontology will be unable to describe it despite our best efforts, but we nevertheless wish to avoid a
definition that excludes from the outset so influential a theorist.
The definition we have provisionally adopted is minimally expressive:
MusicEvent =def A temporal interval containing a musical process.
This says only that “music” is the superset of any and all processes that are musical. This union is
essentially arbitrary, which means that any theory of music that can benefit from ontological treatment
can be built directly into the Music Theory Ontology. It seems likely to us that our ontology contains
tools (in particular the development of structural process profiles, see below) that would be of use in
any such treatment.
We develop two kinds of musical processes: MelodyEvents, and HarmonyEvents. Definitions of
essential terms follow:
5
From his 1957 lecture “Experimental Music”, printed in: Cage, John. 1973. Silence: Lectures and Writings,
Wesleyan University Press Paperback
Sound =def Mechanical radiant energy that is transmitted by longitudinal pressure waves in a material
medium (as air) and is the objective cause of hearing.6
Tone =def A temporal part of a sound whose frequency remains within a restricted range for the duration
of the temporal interval that contains it.
Pitch =def The characteristic (BFO: ‘process profile’) of a tone determined by its frequency restriction.
Melody =def A musical process consisting in a sequence of continuous or discontinuous tones
commensurable with a given temporal interval (the beat).
Beat =def A temporal part of a musical process that serves as a unit for measuring durations of temporal
parts of the musical process.
Rhythm =def The temporal structure of a musical process.
HarmonyEvent =def A musical process that occurs within a temporal interval and that contains as proper
parts multiple tones such that more than one pitch is simultaneously instantiated.
Chord =def A HarmonyEvent consisting of at least three distinct pitches.
3. Processes and Structures:
The most significant challenge we faced in the development of an ontology of musical
phenomena is that the entities involved are for the most part not continuants, but occurrents;
music is not something that simply is, but rather something that happens, unfolding over time.
The appropriate BFO categorization of such entities is thus as processes. Proper description of
processes, however, is attended by significant difficulties.
When we describe continuant entities, we frequently do so by attributing properties to
them. This is one of the main functions of predication in natural language, and overwhelmingly
the primary function of predication in rdf. In describing, for instance, a red apple as sweet and
tart, we have attributed to the apple the color property of redness, and the flavor properties of
sweetness and tartness – the apple has_color red, has_flavor sweet, and has_flavor tart.
6
This definition is due to the Merriam-Webster dictionary. We do not assert “sound” in our ontology, hoping
instead to reuse an implementation from a physics ontology when one becomes available.
This sort of property attribution enables us to speak of things as changing. By
distinguishing an object from its properties, our talk can thus maintain the diachronic identity of
the object while adding or removing properties. Our apple, if baked, might lose its redness and
its tartness while nevertheless remaining the same apple. At some time t the apple has_color
red, has_flavor sweet, and has_flavor tart, and at some later time t’ the same apple (having
been baked) has_color brown, has_flavor sweet, and has_temperature hot. Without property
attribution, we would be able to say only that at t we have a sweetredtartapple, while at t’ we
have instead a sweetbrownhotapple.
Processes however, cannot be said to change, since they are themselves changes.
Because a process spans a temporal interval, it is not meaningful to speak of its diachronic
identity; if the same process exists at both t and t’, it is because the temporal interval occupied
by the process includes t and t’. Thus if two processes are qualitatively distinct, they are also
quantitatively distinct. So while it is appropriate to say of a baked apple that it
has_temperature_of 150° F, it is not correct to say of a running process that it has_speed_of 6
miles per hour. Rather, we must say that it is_a 6MilePerHourRunningProcess.7
Many of the assertions we want to be able to make about music seem to involve
property attribution. Consider the following examples (and their intuitive rdf predication): “this
tone’s pitch is F#” (has_pitch F#), or “this chord is in first inversion” (in_inversion firstInversion),
or “D is the tonic in this melody” (has_tonic D). Since tones, chords, and melodies are all
processes, none of these predications is legitimate.
Some of these problems can be addressed by means of process profiles, which have
been developed in BFO 2.0 as a tool for analyzing processes. The crucial thing about this
concept is that it lays out a parthood relation for processes that is distinct from temporal
parthood. Here is the elucidation of this concept provided in the BFO 2.0 documentation:
ELUCIDATION: b process_profile_of c holds when
7
For a full account of this see the BFO 2.0 specification, and also Smith, B. (2012), CLASSIFYING PROCESSES: AN
ESSAY IN APPLIED ONTOLOGY. Ratio, 25: 463–488
b proper_occurrent_part_of c & there is some proper_occurrent_part d of c which has
no parts in common with b & is mutually dependent on b & is such that b, c and d
occupy the same temporal region.8
The nature of this relation can be most readily grasped in connection with the relation
between processes and the continuant entities on which they depend. Because a process is a
change, it occurs in some (usually material) substrate. Since, for example, the musical
processes we are developing (melody and harmony, see above) are both defined in terms of
sounds, they are processes that depend on (and occur in) a volume of air in which the sound
waves propagate.
When a process occurs in a continuant substrate, it is often the case that multiple things
about the continuant are changing. Consider a running process. Such a process depends on
the body of an animal. In the course of such a process, several things about the animal change:
its spatial location, its body temperature, its heart rate, to name a few. These considerations
determine parts of the running process: process profiles. Thus, we have a spatial-locationchanging (velocity) process profile, a body-temperature-increasing process profile, and a heart
rate-increasing process profile. These are thus some possible process profiles of a running
process.
8
BFO 2.0 Specification, http://purl.obolibrary.org/obo/bfo/2012-07-20/Reference
There are a few straightforward applications of this notion to musical processes. A pitch
is a process profile of a tone; a melody has a beats-per-minute process profile. Once we had
resolved these applications however, there remained a number cases with respect to which it
was less apparent how the notion of a process profile should be employed. In the
characterization of keys and tonality, for example, how should an expression of the form “D is
the tonic in this melody” be asserted? We might paraphrase this loosely as “D is functioning as
the tonic” or “D assumes the role of the tonic”, but functions and roles are defined in BFO as
dependent continuants – properties of objects. As we have seen, these cannot be applied to
processes. But is it correct to say that “tonic” is a process profile of the tone that instantiates
the “D”? A tone is a sound, so it is a process that consists of the changes in the properties of a
volume of air. Which changes in the air can we point to as characterized by a “tonic” process
profile?
We gave serious consideration to an alternative approach that focused on musical
information artifacts (e.g. sheet music, tablature, chord charts, etc.) rather than musical
processes.9 This would have produced an ontology that would describe the information
artifacts that direct musicians to produce the musical processes. The foundational entities
would thus have been notes, rather than tones or pitches. Since notes are continuants (as
when they are instantiated as blots of ink on paper), we could employ the mechanisms of
property attribution to describe them. In addition, since music theorists speak of notes and
tones more or less interchangeably,10 this approach would have had the virtue of using terms in
familiar ways.
By defining the relations that hold between directive entities, the ontology would thus
characterize the production of musical processes, understood in terms of sounds, by means of
performance processes. Because the information artifacts provide sufficient conditions (in
conjunction with the musical training of the performers) for the production of the sounds,
9
We are indebted to Barry Smith for this suggestion, and for the consequent discussion of pattern instantiation
that informed the approach we ultimately adopted.
10
See for instance “Analyzing the notes and chords of a song is a major part of music theory” in our above example
from www.musictheory.net.
analogous relations would then hold between the sounds themselves. The musical notation
must be in some sense isomorphic with the actual musical phenomena in order to provide such
conditions. The ontology would thus still be describing the musical phenomena themselves.
Although these considerations are compelling, we feel that such an approach would
potentially have limited the usefulness of our ontology. The goal of usefulness beyond currently
anticipated applications, as an ontological design principle, entails that one should attempt to
define entities and their relations as closely as possible to the way they are understood to
actually exist. Otherwise it is less likely that the ontology will be suitable for use in applications
that have not occurred to its designers.
The insight that the musical notation is in some sense isomorphic with the musical
phenomena provides us with an alternative strategy. Such an isomorphism seems to consist of
patterns that are instantiated both by the notation and by the phenomena. A term such as
“tonic” is thus characterized by its position within the larger structure of tonality – its bearing of
a specific set of relations to the other tones in the melody.
The notion of entities that are determined only in the context of a larger pattern of
interdependent entities is developed by structuralist theories in the philosophy of mathematics
to describe mathematical objects. Based on this idea, we define “structural process profiles” as
process profiles that apply to processes in virtue of the relations between their parts. Our
tentative elucidation is as follows:
b structural_process_profile_of c iff: b process_profile_of c, & b has disjoint proper parts
all of which are mutually s-dependent & for all proper parts x, x’ of b there is some
relation R (R ≠ s-dependence) such that xRx’, & for all relations R (R ≠ s-dependence)
such that xRx’, there are some parts y, y’ of c such that yRy’.
This defines a structural process profile as a set of interdependent parts each of which bears
some specific relationship to each other part. The structural process profile is instantiated by a process
when the process itself has parts that bear all of the same relations to one another as do the structural
parts.11 Structural process profiles might be appropriate for describing a variety of processes in addition
to musical phenomena. Candidates we have considered include: a set of rules of order as instantiated in
a meeting process; it seems likely that the best way to describe clouds is in terms of processes, so
structural process profiles might be suitable for differentiating different kinds of clouds; various cooking
processes (as e.g. in sautéing, the heating processes of a surface, a volume of oil, and some vegetables
are all interrelated with the protein-denaturing process of the vegetables in a consistent way).
4. Implimentations:
The Music Theory Ontology implements structural process profiles to describe a variety of
theoretically abstract features of music. Before presenting the details of these implementations, it will
be helpful to consider our treatment of “pitch” and “interval”.
Intervals are asserted in the Music Theory Ontology as object properties that relate pitches to
one another, and are asymmetric (with the exception of the augmented fourth), irreflexive, and
functional. 12 pitches are asserted. Currently, these function as pitch-classes; the ontology does not yet
have a means of distinguishing between the same pitch at different octaves. Each pitch has an
associated axiom that relates it to every other pitch by means of the appropriate interval. Thus, for
example A has_interval_of_minor_sixth_to F. Each interval (with the exception of the augmented
fourth) is the inverse of its complimentary interval. Thus since A has_interval_of_minor_sixth_to F, F
has_interval_of_major_third_to A.
Mode, Scale, Key:
Modes are structural process profiles of MelodyEvents that have seven proper parts – the scale
degrees. Each of the seven modes has in addition the interval between each scale degree defined. A
mode, when instantiated, is the key of a melody and in each instance has a particular pitch serving as its
tonic.
Each key can thus be defined by means of an equivalence axiom such as the following, for A
major:
keyOf_A_major equivalentTo majorKey
11
The main problem with the current formulation of the elucidation is that it stipulates that the entire structure
must be instantiated. Since we want to be able to apply structural process profiles in cases where this restriction is
unmet (as e.g. in the melody of “Twinkle, Twinkle, Little Star”, in which the seventh scale degree is never
instantiated), reformulation is required.
and (structural_process_profile_of some
(('has proper occurrent part' some
(('has profile' some A)
and ('has profile' some Tonic)))
and (has_structural_profile some IonianMode)))
The import of this, roughly, is that the keyOf_A_major is the (specific) mode of any melody that
instantiates the structural process profile of the Ionian mode and includes tones that instantiate both
the pitch of A, and the Tonic relative to the structural process profile.
The axioms thus far asserted will not enable an owl reasoner to infer a key given a set of
rdf triples that describe the tones of a melody. A reasoner should, however, be able to correctly
associate each tone in the melody with its proper scale degree if given an explicit assertion of a key.
Currently, the ontology defines only major and minor modes and keys (and only five each
of these), but given this axiom scheme it should be a simple matter to extend the ontology to exhaust all
individual keys.
Chord, Triad, Diatonic Series
A chord is a structural process profile of a harmony event. Currently, triads are defined as a
subclass of chords, and seventh chords are defined as a subclass of triads. This is convenient because it
enables each subsequent chord type to inherit the properties of its superclass.
Triads have three proper parts – the root, the third, and the fifth. The relations between these
parts are, of course, the corresponding intervals, with minor triads having a minor third between the
root tone and the third tone, and major triads having a major third between these parts.
The instantiation of a chord is a HarmonyEvent, and individual chords are characterize by axioms
of such as the following for a D minor triad:
Dm subClassOf has_structural_profile some minorTriad
and 'has proper occurrent part' some
(('has profile' some D)
and ('has profile' some chordRoot))
Note that these axioms are much less strong than the equivalence axioms asserted for keys. We are
working on necessary and sufficient conditions for chords, but these will have to incorporate a notion of
tone overlap, which we have yet to successfully implement.
We have yet to define the diatonic series, but the approach we used for modes provides a
promising model. The biggest obstacle will be to describe chord inversions. As currently defined,
intervals hold between pitch-classes, rather than between specific octave subclasses of pitch-classes. So
for example, an instance of A6 has_interval_of_minor_third_to an instance of C3. While our current
system has benefits, it is clear it will have to be revised to accommodate this sort of consideration.
Rhythm
Our definition of rhythm (see above) implies our intention to implement rhythmic notions in
terms of structural process profiles, but we have yet to devise a satisfactory method for doing so. At
present, the term “Rhythm” is not asserted as an entity in our ontology.
Our immediate goal in this area is to develop restrictions that completely determine a tone’s
duration and temporal position (in terms of beats and measures) relative to the whole melody, and thus
to every other tone.
Our approach at present involves Allen’s Interval Algebra12, and we have incorporated the
temporal base relations employed by this system into our ontology as object properties. It may turn out
that this system is insufficient for our needs, since it does not include a robust notion of duration (it has
no relation more specific than “overlaps”). We are currently working with the conjecture that a
structure of beats (and probably beat-fractions) underlying a musical process can provide a means of
determining precise durations. Since the Allen’s Interval Algebra incorporates a decidable logic, we
hope that by using it we can build into our ontology a framework for describing temporal features of
music that is at once expressive and suitable for automated inference.
5. Conclusion
The Music Theory Ontology is currently at a very early stage of development. We feel that we
have solved a few of the most difficult problems involved in such a project, and that the Music Theory
Ontology is therefore heading in a promising direction. Immediate goals for the ontology are to develop
12
James F. Allen: Maintaining knowledge about temporal intervals. In: Communications of the ACM. 26 November
1983. ACM Press. pp. 832–843
the areas relevant to the temporal/rhythmic aspects of music, and the aspects relevant to musical
information artifacts, as described above.