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groupe
représentation
et traitement
des
connaissances
CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE
31, chemin Joseph Aiguier
F-13402 MARSEILLE
CEDEX
9 (France)
MODELLING MUSIC WITH GRAMMARS:
FORMAL LANGUAGE REPRESENTATION IN THE BOL
PROCESSOR
Jim Kippen & Bernard Bel
Abstract
Improvisation in North Indian tabla
drumming is similar to speech insofar as it is bound to an
underlying system of rules determining correct sequences. The parallel
is further reinforced by
the fact that tabla music
may be represented with an oral notation system used for its
transmission and, occasionally, performance. Yet the rules
are implicit and available only
through the musicians’ ability
to play correct sequences and recognise incorrect ones. A
linguistic model of
tabla improvisation and evaluation derived from pattern languages and
formal grammars has been
implemented in the Bol Processor, a software system used in
interactive fieldwork with expert musicians. The paper demonstrates the ability
of the model to
handle complex structures by taking real examples from the
repertoire. It also questions the
relevance of attempting to model irregularities encountered in actual performance.
Keywords
Formal grammars, patterns, membership test, rhythm, tabla, drumming, ethnomusicology
GRTC / 318 / octobre 1988
Jim Kippen & Bernard Bel
A. Marsden
& A. Pople (eds.):
Computer Representations and Models in Music
, London,
Academic Press, 1992, pp.207-38.
† This research has been partly funded by the Leverhulme Trust.
** Engineer in Computer Science
Groupe Représentation et Traitement des Connaissances
Centre National de la Recherche Scientifique
Marseille (France)
E-mail: grtc@frmop11.bitnet (EARN)
* Research Fellow
Department of Social Anthropology and
Ethnomusicology
The Queen's University of Belfast (U.K.)
E-mail: eihe4874@uk.ac.qub.v1 (JANET)
Modelling music with grammars:
formal language representation in the Bol Processor
†
Jim Kippen
*
and Bernard Bel
**
Introduction
The very nature of the music of the
tabla
(North Indian two-piece tuned drum set) suggests
that
linguistics may be an effective analytical approach to the investigation
of its structure, and
grammars
the most appropriate models. An oral notation system using verbal symbols called
bols (from the Urdu/Hindi bolna ‘to speak’) is used for its transmission and, occasionally,
performance:
dha, ti, ge, na, tirakita (trkt), dhee, tee, ta, ke
etc. Bols are quasi-
onomatopoeic
mnemonics that represent drum-strokes, although their aesthetic and technical
‘meanings’ (they have no semantic meaning per se) are nearly always context-bound
with no
one-to-one
correlation between a stroke and its name (Kippen 1988a:xvi-xxiii). The
equivalence of the verbal and
musical representations is suggested by the fact that the word
‘bol’ refers both to a spoken syllable and a drum stroke.
A large part of the tabla repertoire involves improvisation. Any
sequence of strokes may be
considered as a finite string of symbols whose organisation is related to some implicit
formal
system. Automatic sequences (i.e. sequences generated
by finite-state automata) share
properties that place them
somewhere between periodicity and chaos, yet closer to periodicity.
Since both strict and approximate
periodicity are essential features in music, it is realistic to
think
that strings of musical events may be appropriately represented with automata or formal
grammars. In other words, the most fundamental
reason for us to view formal language
models as relevant models of music lies in properties that are intrinsic to music, rather than in
analogical links between music and natural languages.
The following is an example of a compositional type known as
qa‘ida
,
the ‘theme and
variations’ form
par excellence
. It should be read linearly from left to right, and each group
represents a beat comprising four units (
trkt
is a compound stroke of two units).
dhatidhage
nadhatrkt
dhatidhage
dheenagena
dhatidhage
nadhatrkt
dhatidhage
teenakena
tatitake natatrkt tatitake
teenakena
dhatidhage
nadhatrkt
dhatidhage
dheenagena
Set
in sixteen beats, this qa‘ida comprises four four-beat statements of a basic ‘sentence’. In
accordance with the voiced/unvoiced structure of the metric cycle, the
first and fourth
statements are not inflected while part of
the second and all of the third undergo
transformations:
dha/ta, ge/ke, dhee/tee
. Variations are
derived from the ‘theme’, and
involve the repetition, permutation,
or substitution of bols. Students are taught that changes
occurring in the first half of the structure must be reflected in the second; variations,
too, are
subject to voiced/unvoiced transformations (see Kippen
1987:180-81), as can be seen in the
following three variations (changes have been italicised; hyphens in the third
variation represent
silences of one unit, though effectively they elongate the preceding syllable):
† This research has been partly funded by the Leverhulme Trust.
** Engineer in Computer Science
Groupe Représentation et Traitement des Connaissances
Centre National de la Recherche Scientifique
Marseille (France)
E-mail: grtc@frmop11.bitnet (EARN)
* Research Fellow
Department of Social Anthropology and
Ethnomusicology
The Queen's University of Belfast (U.K.)
E-mail: eihe4874@uk.ac.qub.v1 (JANET)
Kippen-Bel: Modelling music with grammars: formal language representation in the
Bol
Processor
dhatidhage
nadhatrkt
dhatidhage
dheenagena
dhatrktdha
tidhagena
dhatidhage
teenakena
tatitake natatrkt tatitake
teenakena
dhatrktdha
tidhagena
dhatidhage
dheenagena
dhatidhatr
ktdhatidha
trktdhage nadhagena
dhatidhage
nadhatrkt
dhatidhage
teenakena
tatitatr
kttatita
trkttake
natakena
dhatidhage
nadhatrkt
dhatidhage
dheenagena
dhagenadha trktdhage
nadhatrkt
dhatrktdha
tidha-dha tidhagena
dhatidhage
teenakena
takenata
trkttake
natatrkt
tatrktta
tidha-dha tidhagena
dhatidhage
dheenagena
This paper traces the development of a formal-language representation for tabla music
and its
implementation in a computer system. The discussion will take as
its starting point the musical
examples given above. It should be noted that the
grammatical models described below follow
an approximate chronological order
of development, even though many formal representations
were evolved simultaneously.
Pattern grammars
Variations obey strict rules of
construction, but these rules are rarely expressed formally by
traditional musicians; rather, an implicit model is transmitted by means of
sequences of positive
instances of the
‘language’. Negative instances composed by their students during the course
of training are rejected or corrected. In a performance, a musician normally creates
between six
and
ten variations for any qa‘ida. These explore only a tiny subset of all possible variations,
and thus it is unrealistic to
attempt to elicit a complete set as this could run to many thousands
of correct pieces. Therefore, when
beginning this research we considered that a more prudent
approach would be to construct a descriptive
generalisation based on observed recurrent
sequences of bols and patterns. For instance, it may be seen in
the above variations that
dhatidhagedheenagena
occurred periodically and may thus be
attributed a label A8 (the
numbers
attached to labels denote the lengths (in time units) of the terminal strings they
represent). Furthermore, the inner structure
of each variation may be represented as a
pattern
(a string of constants and variables — see Bel 1988:1-2). It is possible to
preserve these
structures because derivations in
pattern languages can be represented in a special grammatical
format. For instance, a production rule that we notate
A —> (=B) (:B)
implies that both derivations of B must be identical. In this pattern format,
(=) denotes a
referential sequence and
(:) a copy. Several levels of brackets can also be embedded to
represent complex patterns. An asterisk ‘*’ indicates that the
bracketed expression immediately
following it should be derived as a string of unvoiced bols. Thus
the initial formal
representation we developed, as applied to the first variation of the qa‘ida, was written:
3
Kippen-Bel: Modelling music with grammars: formal language representation in the
Bol
Processor
(=dhatidhage
nadhatrkt
dhatidhage
dheenagena)
(=dhatrktdha
tidhagena)
(=dhatidhage
teenakena)
*(:tatitake natatrkt tatitake
teenakena)
(:dhatrktdha
tidhagena)
(=dhatidhage
dheenagena)
This was then reduced to the more concise expression
(=A16) (=V8) (=A'8) *(:A16) (:V8) (=A8)
in which A16, A8, and A'8 were fixed patterns, and V8 a permutation of bols.
A pattern grammar
which described the three variations presented was:
GRAM#1
[1] S
—> (=A16) (=V8) (=A'8) *(:A16) (:V8) (=A8)
[2] S —> (=V16) (=A'16) *(:V16) (=A16)
[3] S —> (=V24) (=A'8) *(:V24) (=A8)
[4] A16
—> dhatidhagenadhatrktdhatidhagedheenagena
[5] A'16
—> dhatidhagenadhatrktdhatidhageteenakena
[6] A8
—> dhatidhagedheenagena
[7] A'8
—> dhatidhageteenakena
in which the first three rules indicated options for the transformation
of a starting symbol S.
This grammar was
implemented in a knowledge-based system called the
Bol Processor
(BP).
The computer software comprises an editor designed
to represent bols, variables, and
conventional symbols (brackets, asterisk, etc,) as tokens. The dependancy (=B)(:B)
is taken
care of by pointers so that any modification in (=B) is reflected identically in (:B). A
special
table is also defined that indicates the voiced/unvoiced mapping. The ‘active’ element of the
BP
is an
inference engine
that uses an enumerative/stochastic process
to generate sentences
derived from the grammar, and a
membership algorithm
to check whether
or not a sentence
entered into the editor
is consistent with the grammar. In the generative process (synthesis or
modus ponens
) the
BP is able either to enumerate all sentences of the language or to generate
one sentence randomly. For modus ponens, the order in which rules are placed (e.g.
as shown
in GRAM#1) is not important. This type
of representation in which the specification of the
problem (‘infer any
sentence from this set of rules’) can clearly be separated from the method
of solution (‘select candidate rules and rewrite the current work
string’) is known as
declarative
.
The derivation of sentences may be
viewed as a number of distinct stages involving several
transformational grammars (
subgrammars). Breaking a
process into several subprocesses is
basically a
procedural
approach. The word
‘transformational’ is borrowed from formal
language theory
(Bel 1987a:356), not linguistics. In a subgrammar there is more than one
starting symbol, and symbols
that are terminal to that subgrammar may become the starting
symbols of the next subgrammar(s)
to be applied, and so on. Subgrammars subsequent to
GRAM#1 should describe acceptable derivations of the terminal symbols V8, V16, and V24.
Context-sensitive derivations in synthesis
Based on observations of collected
data, and our understanding of the often incomplete and
ambiguous statements of musicians, we hypothesized that V8, V16, and V24 were strings
of
‘words’ which could be listed as follows:
4
Kippen-Bel: Modelling music with grammars: formal language representation in the
Bol
Processor
V3 = dhagena
V2 = trkt
V1 = dha, ti, -
To describe all possible sequences we wrote:
V8
—> V V V V V V V V
V16
—> V V V V V V V V V V V V V V V V
V24
—> V V V V V V V V V V V V V V V V V V V V V V V V
V —> V1
V V
—> V2
V V V
—> V3
It appeared that any arrangement of V2 and V3
was possible, but that the choice of derivations
for V1 was highly context-sensitive. Three conditions were established based on
both aesthetic
and technical (i.e. fingering) criteria:
(1) there should be no more than two consecutive
dha;
(2) there should be no more than two consecutive ‘-’;
(3)
ti
should not appear before or after
ti
or
trkt
.
Therefore a subgrammar describing acceptable derivations of V1 was written:
kt V1 tr —> kt dha tr
kt V1 ti
—> kt dha ti
kt V1 - —> kt dha -
kt V1 ) —> kt dha )
kt V1 V1 —> kt dha V1
na V1 tr —> na dha tr
na V1 ti —> na dha ti
na V1 - —> na dha -
na V1 ) —> na dha )
na V1 V1 —> na dha V1
ti V1 tr
—> ti dha tr
ti V1 ti
—> ti dha ti
ti V1 - —> ti dha -
ti V1 ) —> ti dha )
ti V1 V1 —> ti dha V1
- V1 tr —> - dha tr
- V1 ti —> - dha ti
- V1 - —> - dha -
- V1 ) —> - dha )
- V1 V1 —> - dha V1
(= V1 tr —> (= dha tr
(= V1 ti —> (= dha ti
(= V1 - —> (= dha -
(= V1 V1 —> (= dha V1
dha V1 dha
—> dha ti dha
dha V1 - —> dha ti -
dha V1 ) —> dha ti )
dha V1 V1 —> dha ti V1
na V1 dha
—> na ti dha
na V1 -
—> na ti -
na V1 )
—> na ti )
na V1 V1
—> na ti V1
- V1 dha —> - ti dha
- V1 -
—> - ti -
- V1 )
—> - ti )
- V1 V1
—> - ti V1
(= V1 dha
—> (= ti dha
(= V1 - —> (= ti -
(= V1 V1
—> (= ti V1
dha V1 tr —> dha - tr
dha V1 dha
—> dha - dha
dha V1 ti —> dha - ti
dha V1 ) —> dha - )
dha V1 V1 —> dha - V1
kt V1 tr
—> kt - tr
kt V1 dha —> kt - dha
kt V1 ti
—> kt - ti
kt V1 )
—> kt - )
kt V1 V1
—> kt - V1
na V1 tr —> na - tr
na V1 dha
—> na - dha
na V1 ti
—> na - ti
na V1 ) —> na - )
na V1 V1 —> na - V1
ti V1 tr
—> ti - tr
ti V1 dha —> ti - dha
ti V1 ti
—> ti - ti
ti V1 )
—> ti - )
ti V1 V1
—> ti - V1
(= V1 tr —> (= - tr
(= V1 dha
—> (= - dha
(= V1 ti
—> (= - ti
(= V1 V1 —> (= - V1
In this subgrammar all possible left and right
contexts were accounted for. Brackets generated
by GRAM#1 were retained to mark the beginning and end of a string.
V1 was also considered
to be a right context since it was
obvious that derivations of V8, V16, and V24 could have
resulted in strings of consecutive V1s. Apart from the additional bulk and complexity, the
main
defect of such a subgrammar was that it had both left and right contexts:
therefore, the
membership test of a sentence, although
decidable, was costly in computation time. Evidently
this kind of description was as unsatisfactory as it was user-unfriendly.
In the search for an alternative formal expression capable of satisfying the set of three
simple
criteria listed above, we introduced
negative contexts
. For example, the first condition was
written:
5
Kippen-Bel: Modelling music with grammars: formal language representation in the
Bol
Processor
#dha V1 —> #dha dha
meaning ‘V1 may be rewritten as
dha
unless it is preceded by
dha
’.
Since this type of rule
invoked the left context
exclusively, it followed that strings should be constructed from left to
right — more particularly, for any rule applied,
the
leftmost
occurrence of V1 should be
considered. To indicate this, an instruction ‘LEFT’ was
placed in the rule. The derivations of
V1 yielding
dha and ‘-’ were then:
[1]
LEFT #dha V1
—> #dha dha
[2]
LEFT #- V1
—> #- -
Derivations yielding
ti
, on the other hand, were too complex to represent in
a single production
rule. We chose not to implement conjunctive expressions in negative contexts, such as
[#kt and #ti] V1 #tr
—> [#kt and #ti] ti #tr
which expresses the third condition. Here it should be noted that
tr
cannot be a left context as
it
is always followed by
kt
, and
ti
cannot appear as a right context since V1s are derived from
left
to right. Instead we opted to represent positive contexts. Using positive contexts to
express
the third condition resulted in rules:
[3] LEFT
dha V1 #tr
—> dha ti #tr
[4] LEFT
na V1 #tr
—> na ti #tr
[5] LEFT - V1 #tr —> - ti #tr
[6] (= V1 #tr —> (= ti #tr
A new
problem arose because any negative context could be instantiated with V2 or V3. For
example, a feasible derivation of
dhagena V1 V2 was dhagenati V2
(using rule 4) and then
dhagenatitrkt
, which was incorrect but nevertheless possible since the
derivation of V2 was
context-free. Therefore,
as a general procedure it was clear that any variable or terminal used
as a right context should not be transformed in the
same subgrammar in which it appeared as a
context. And so we arrived at the more
concise grammar listed below. For the sake of clarity
the first six rules were written in two distinct subgrammars: GRAM#2 which
is context-free,
and GRAM#3 which is length-decreasing (type 0).
GRAM#2
[1] V8
—> V V V V V V V V
[2] V16
—> V V V V V V V V V V V V V V V V
[3] V24
—> V V V V V V V V V V V V V V V V V V V V V V V V
GRAM#3
[1] V —> V1
[2]
V V
—> trkt
[3]
V V V
—> V3
GRAM#4
[1] V3 —> dhagena
[2] LEFT #dha V1 —> #dha dha
[3] LEFT #- V1 —> #- -
[4] LEFT
dha V1 #tr
—> dha ti #tr
6
Kippen-Bel: Modelling music with grammars: formal language representation in the
Bol
Processor
[5] LEFT
na V1 #tr
—> na ti #tr
[6] LEFT - V1 #tr —> - ti #tr
[7] (= V1 #tr —> (= ti #tr
Membership test (1): RND grammars
The new grammar worked in synthesis, but for
it to be considered complete and consistent the
parsing of new sentences
had to be taken into account. In order to achieve rapid membership
tests, the parsing had to be deterministic and bottom-up
(data-driven) because a deterministic
procedure cannot backtrack should an unacceptable state be reached.
The main advantage of
such a description
was that it could easily be performed by hand. Bottom-up parsing implied
that subgrammars were to be considered in reverse order (4, 3, 2, 1).
The premises and
conclusions of the rules were then swapped. This produced a
dual grammar
where instead of
a
single arrow (—>), a double arrow (<—>) was used to notate
the possibility of a derivation in
both directions. The single arrow
was retained for rules relevant only in synthesis or analysis
(e.g. in the ‘absorption’ subgrammar below, under
Constructing stress-sensitive grammars
).
A string was accepted if its derivation
by the dual grammar yielded a starting symbol.
Evidently, many derivations were
possible, and so when several rules were candidates (i.e.
their conclusions were substrings of the work string) there had to be a
decision procedure to
determine which was to be prioritised. Therefore in subgrammars
where the order in which
symbols were to be rewritten in the work string was not specified, the rule selected was
the one
appearing nearest the bottom of the
list of rules. Such grammars were known as RND, or
‘random’, subgrammars. The following example shows
the successful parsing of
-ti-trktdhatrkt
in which rules used in the derivation have been specified in the left margin:
(= -ti-trktdhatrkt)
Step 1 G#4 [6]
(= - V1 -trktdhatrkt)
...
2 G#4 [3]
(= - V1 V1 trktdhatrkt)
...
3 G#4 [3]
(= V1 V1 V1 trktdhatrkt)
...
4 G#4 [2]
(= V1 V1 V1 trkt V1 trkt)
...
5 G#3 [2]
(= V1 V1 V1 trkt V1 V V)
...
6 G#3 [2]
(= V1 V1 V1 V V V1 V V)
...
7 G#3 [1]
(= V1 V1 V1 V V V V V)
...
8 G#3 [1]
(= V1 V1 V V V V V V)
...
9 G#3 [1]
(= V1 V V V V V V V)
...
10 G#3 [1]
(= V V V V V V V V)
...
11 G#2 [1] (= V8)
... successful.
Given the candidate rule, there were several occurrences
(steps 2, 5, 7, 8, and 9) of the
conclusion in the premise. Each time, the rightmost occurrence was selected.
The
parsing algorithm was dependent upon the order in which rules appeared in each
subgrammar. For instance,
if the work string had contained
trktdhagena
, then rule 2 of G#4
would have yielded
trkt
V1 gena
and consequently
gena
would never have been transformed.
Since dha was a substring of dhagena, the
rule producing
dhagena
had to be considered first.
This reflected a simple pattern recognition strategy that ‘large chunks should be
recognised
first’, or precisely:
‘Chunk’ rule
For any pair of rules p
i
—> q
i
and p
j
—> q
j
in the same subgrammar,
if q
i
is a substring of
q
j
then i < j.
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Kippen-Bel: Modelling music with grammars: formal language representation in the
Bol
Processor
Consequently the positions of rules 1 and 2 in G#4 needed to be swapped.
(It should be noted
that the grammar given in Kippen (1987:185,191) was written
before the introduction of the
chunk rule.)
Although V1s were derived from left to right in synthesis, they were not
necessarily recognised
from
right to left in analysis. This was due to the fact that in the RND parsing algorithm the
order
of rules had priority over the position of the derivation. The shortcoming of this
algorithm was evident in the parsing of (= --
dhatrktdhatrkt
),
which, according to our initial
hypothesis at least, was an incorrect sentence:
(= --dhatrktdhatrkt)
G#4 [3]
(= V1 -dhatrktdhatrkt)
G#4 [3]
(= V1 V1 dhatrktdhatrkt)
G#4 [2]
(= V1 V1 dhatrkt V1 trkt)
G#4 [2]
(= V1 V1 V1 trkt V1 trkt)
G#3 [2]
(= V1 V1 V1 trkt V1 V V)
G#3 [2]
(= V1 V1 V1 V V V1 V V)
G#3 [1]
(= V1 V1 V1 V V V V V)
G#3 [1]
(= V1 V1 V V V V V V)
G#3 [1]
(= V1 V V V V V V V)
G#3 [1]
(= V V V V V V V V)
G#2 [1] (= V8)
... successful.
In fact, the RND parsing algorithm yielded incorrect membership tests in most
context-sensitive
grammars.
Yet a simple technique was soon developed to determine the construction of
sentences from left to right
with constraints on left and right contexts. This we termed the
sliding marker
method. A sliding marker grammar equivalent to the previous one was written:
GRAM#2
RND
[1] V8
<—> V V V V V V V V
[2] V16
<—> V V V V V V V V V V V V V V V V
[3] V24
<—> V V V V V V V V V V V V V V V V V V V V V V V V
GRAM#3
RND
[1] LEFT V <—> V1
[2] LEFT
V V
<—> trkt
[3] LEFT
V V V
<—> V3
[4] (= #M <—> (= M #M
GRAM#4
RND
[1]
M V3
<—> dhagena M
[2]
M trkt
<—> trkt M
[3]
#dha M V1
<—> #dha dha M
[4]
#- M V1
<—> #- - M
[5]
dha M V1 #tr
<—> dha ti M #tr
[6]
na M V1 #tr
<—> na ti M #tr
[7]
- M V1 #tr
<—> - ti M #tr
[8]
(= M V1 #tr
<—> (= ti M #tr
[9]
#M M )
<—> #M )
8
Kippen-Bel: Modelling music with grammars: formal language representation in the
Bol
Processor
In synthesis, a
sliding marker M was created by rule G#3 [4]. Note that the negative context
#M prevented the rule from being self-embedding, i.e. from generating an infinite
number of
Ms. This marker was erased (in synthesis) or created (in analysis) by rule
G#4 [9]. So, using
the
sliding marker method, a derivation sequence yielding
dhatrktdhatidhatrkt
was the
following:
(= V8)
G#2 [1]
(= V V V V V V V V)
G#3 [1]
(= V1 V V V V V V V)
G#3 [2]
(= V1 tr kt V V V V V)
G#3 [1]
(= V1 tr kt V1 V V V V)
G#3 [1]
(= V1 tr kt V1 V1 V V V)
G#3 [1]
(= V1 tr kt V1 V1 V1 V V)
G#3 [2]
(= V1 tr kt V1 V1 V1 tr kt)
G#3 [4]
(= M V1 tr kt V1 V1 V1 tr kt)
G#4 [3]
(= dha M tr kt V1 V1 V1 tr kt)
G#4 [2]
(= dha tr kt M V1 V1 V1 tr kt)
G#4 [3]
(= dha tr kt dha M V1 V1 tr kt)
G#4 [5]
(= dha tr kt dha ti M V1 tr kt)
G#4 [3]
(= dha tr kt dha ti dha M tr kt)
G#4 [2]
(= dha tr kt dha ti dha tr kt M)
G#4 [8]
(= dha tr kt dha ti dha tr kt)
In analysis, the same sequence was followed in reverse order.
Membership test (2): LIN grammars
The derivation
just shown in synthesis mode is known as a
leftmost
derivation: a leftmost
derivation forces the rewriting
of the symbol appearing at the leftmost position of the work
string. Similarly, a rightmost
derivation
was used to parse sequences. The definition of a
leftmost/rightmost derivation applies well to context-free grammars, i.e. grammars in
which the
premise of each rule contains only one variable. Yet in context-sensitive grammars
the problem
consists in deciding
whether or not the string of symbols to be rewritten includes the context.
A general format for context-sensitive rules is
L C
R
—> L
C'
R
where L and R are left and right contexts respectively, and C is a string of variables rewritten
as
C'. Hart (1980:82) defined a canonic context-sensitive leftmost derivation in
strictly length
-
increasing
grammars, i.e. where C contains only one variable.
Bel (1987b:7ff) has adapted
Hart’s definition to rightmost derivations in length-decreasing grammars. Appendix 1 gives
the
formal definition of this derivation along with a list
of tests that a bottom-up parser must
perform in order to select the next rule to be applied.
In the BP fast parsing was achieved as a
result of a simplification of this algorithm, but at the cost
of some restrictions on grammars.
Two have already been encountered: the partial
ordering of rules (the ‘chunk’ rule), and the
restriction on transforming any symbol already functioning as a context
within the same
subgrammar. A third
restriction on overlapping patterns has been demonstrated in Bel
(1987a:358-59) and need not concern us here.
A subgrammar in which a context-sensitive rightmost
derivation was used in membership tests
was called a LIN grammar. This term was adopted in view of the fact that the LIN
membership
algorithm was needed
whenever right-linear subgrammars were used. It was generally
unnecessary to impose leftmost derivations in synthesis: when needed, these
were implemented
with sliding markers or
right-linear-left-context
rules (see below). Effectively, a LIN
grammar in synthesis was identical to a RND grammar in which all production
rules were
prefixed with LEFT. Consequently, the LIN version of the grammar became:
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GRAM#2
RND
[1] V8
<—> V V V V V V V V
[2] V16
<—> V V V V V V V V V V V V V V V V
[3] V24
<—> V V V V V V V V V V V V V V V V V V V V V V V V
GRAM#3
LIN
[1] V <—> V1
[2]
V V
<—> trkt
[3]
V V V
<—> V3
GRAM#4
LIN
[1] V3 <—> dhagena
[3] #dha V1 <—> #dha dha
[4] #- V1 <—> #- -
[5]
dha V1 #tr
<—> dha ti #tr
[6]
na V1 #tr
<—> na ti #tr
[7] - V1 #tr <—> - ti #tr
[8] (= V1 #tr <—> (= ti #tr
In this grammar, the sliding marker M
is no longer necessary because the analysis is forced to
rightmost derivations by the instruction LIN. It may also be noticed that rule
2 in GRAM#4
has become unnecessary.
Fixed patterns
By linking the original pattern subgrammar (GRAM#1) to GRAM#2, 3, and
4, a grammar was
obtained from which a large set of variations could
be generated. Unfortunately, however, the
chunk
rule was violated during parsing because some sequences of bols occurring in fixed
patterns A16, A8, etc. were recognised and transformed by GRAM#4, 3, and 2. For this
reason the four rules defining fixed patterns had to
be re-ordered and placed in a fifth
subgrammar:
GRAM#5
ORD
[1] A8
<—> dhatidhagedheenagena
[2] A'8
<—> dhatidhageteenakena
[3] A16
<—> dhatidhagenadhatrktdhatidhagedheenagena
[4] A'16
<—> dhatidhagenadhatrktdhatidhageteenakena
Here, ORD indicated that in synthesis the rules could
be applied in order (top to bottom) as no
alternative options were
possible. During analysis ORD subgrammars were treated identically
to RND subgrammars.
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Kippen-Bel: Modelling music with grammars: formal language representation in the
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Bol density
The vast majority of variations maintain the same bol density
established in the theme of the
qa‘ida.
However, during performance some musicians vary the bol density from variation to
variation, or even
within one variation. To indicate bol density in the BP’s editor, a slash
followed by an integer was
typed to specify the number of bols between two tabulations (beat
markers). When the tempo changed within the variation itself, the new (integer) value
appeared
as a terminal symbol. For
instance, in the following improvisation (a maverick variation in
view of the grammars listed above describing this qa‘ida)
dhatidhage
nadhatrkt
dhatidhage
dheenatrkt
dhadhatrkt
dhatidha-
dhatidhage
teena-ta
teena-ta
titakena tatitake
teenakena
/8
dhatidhagenadhatrkt
dhatidhagedheenagena
gena-dhatidhagena
dhatidhagedheenagena
the fourth line was executed at double
speed. Like terminals, density markers were
manipulated by production rules. Consider, for instance, GRAM#2 [26] in appendix 2:
GRAM#2 [26] S16 <—> /8+ A16 O16 ; /4
in which the fourth line of the example
above has been defined. ‘+’ and ‘;’ are context
markers; the sequence derived from A16 O16
(32 bols) is played at a density of eight bols per
beat, following which the density marker is reset to four bols per beat.
Templates
It
will be remembered that additional information, such as brackets and asterisks, was
incorporated into grammars in order to define
the structure of patterns as well as to provide
contexts in generation.
It followed, then, that sentences could only be parsed if they were
entered into the editor complete with structural information. Yet in
view of the fact that the BP
had to perform membership tests on large
amounts of data comprising examples to which it
was not always easy to assign a structure, this limitation was held to be unacceptable.
In
response to this, the
inference engine of the BP was modified to generate templates from a
grammar,
i.e. a list of possible structures in which each dot represented a terminal symbol of
one unit. Templates were enumerated and stored in the grammar file. For instance
[1] (=................)(=........)(=........)*(:................)(:........)(=........)
[2] (=...............)(=................)*(:................)(:................)
[3] (=....................)(=........)*(:........................)(=........)
were the three generated by the grammar
so far constructed for the qa‘ida in question. Bol
density markers also appeared in templates.
(See appendix 2: templates 4, 5 etc. include
sequences of six bols per beat, and 3, 6 etc. have sequences of eight bols per beat.)
This
development allowed us to dispense with structural information when entering data.
Consequently, in analysis the BP took a new sentence and
superimposed it on each template in
strict order. A
membership test was performed each time the sentence matched a template, so
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Kippen-Bel: Modelling music with grammars: formal language representation in the
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allowing for
any structural ambiguity to be assessed. It was important (especially when
inferring weights, see below under
The probabilistic model
) that the first template
producing a
successful parsing was the most
specific
, i.e. the one containing
the largest number of
brackets. This reflected the view that patterns fitting a production rule in
the pattern
subgrammar were not incidental.
Thus, templates had to be ordered from the most specific to
most general. This was achieved automatically once the rules that
produced them were ordered
accordingly. In appendix 3, it may be seen that rules GRAM#2 [4-5] are more specific
than [7-
8] although they may incidentally have
produced the same strings. It may also be noticed that
the partial ordering of rules from generic to
specific is not identical to the ordering imposed by
the ‘chunk’ rule, although it never contradicts with it.
Structural markers and wildcards
The templates shown in appendix 2 contain symbols (‘+’ and ‘;’) that played
a specific
structural role as contexts that indicated
precise metric positions in the sentence. For instance,
‘+’ was used to mark the beginning of a line of four beats (see templates 1 to 10).
However, if
it could be established that structures existed where two lines
merged and indivisible chunks of
bols could span the end of
one line and the beginning of the next, then the marker was
suppressed (see, for
example, templates 11 to 16 which were generated as derivations of L24
M8 in GRAM#2 [5] and [17]). Furthermore, ‘+’ was
used as a context in all production rules
defining cadential patterns (see GRAM#6). The offset from a position
marked ‘+’ was
determined by the number of wildcards ‘?’ (metavariables), each of
which could be instantiated
as a single variable or terminal (but
not
as a structural marker). For instance, in the following
rule
GRAM#6 [5] LEFT + ? ? ? ? ? ? ? ? A8-2 <—> + ? ? ? ? ? ? ? ? dhatidhagedheena
the variable A8-2 could only
be rewritten as
dhatidhagedheena
if it was located eight units to
the right of a ‘+’ marker.
In the grammar shown in appendix 2, ‘;’ has been used to
mark the final closing bracket of the
work string: in effect, the
end of a variation. A combination of metavariables and negative
context is notated ‘#?’. Used as a
left context, ‘#?’ would mark the very beginning of a work
string. This symbol denotes the absence of any context, be it a terminal,
variable, or structural
marker (bracket, equals, colon, semi-colon, asterisk, integer, plus,
etc.). As structural markers
were automatically inserted into input sentences during
template matching in membership tests,
there were few restrictions on their use in production rules. However, they had to be
employed
with great precision
because their inconsistent use resulted in too many distinct templates
(including, presumably, some that were unnecessary) and a much slower membership test.
A ‘complete’ grammar for this qa’ida
Until now we have discussed the construction
of a grammar based on three quite regular
variations of a qa‘ida. In this way it has been possible to demonstrate the principal stages in
the
development of formal representations for tabla music. However, it
should not be thought that
all improvisations based on
drum themes are equally regular. On the one hand, tabla teachers
usually demonstrate simple and systematic variations to their students
because their aim is to
pass on the essence of a system. Students remember
sets of ‘fixed improvisations’ and use
them as models on which to base their own improvisatory efforts. On the
other hand,
performance situations ‘offer musicians
greater licence to explore new channels of creativity
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Kippen-Bel: Modelling music with grammars: formal language representation in the
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and to stretch the limits of musical acceptability’ (Kippen 1988c:30). The variation given
above
under
Bol density
was taken from just such a performance: clearly there is no
duplication of
the material in the two halves, there is a change in density within the variation
itself, and the
usual cadential ending to the first half is modified.
Nevertheless, this piece was recognised by
knowledgeable listeners to be a masterful improvisation, and so we must conclude that if
our
models are intended to be truly representative of tabla playing, then they
too must be able to
cope with the complexity and structural variety contained in performances from
a range of
different social contexts. Therefore, any grammar should be seen only as a
hypothesis of
musical structure that is ‘complete’ for the data entered up
to that point but which makes no
claims to ‘completeness’ with regard
to as yet unexamined data. Experience has shown that a
grammar rarely remains unchanged following the processing of a new set of examples.
The grammar shown in appendix 2 was written
to account for a variety of very complex
structures explored during one performance of this qa‘ida by the expert musician
Ustad Afaq
Husain Khan of
Lucknow. Brief comments restricted to its salient features are as follows:
GRAM#2 is truly right-linear and
sets all structural markers (line markers, brackets, asterisks,
and tempo markers).
These are used only as contexts in subsequent subgrammars, and so
when the system is requested to generate templates it need only detail the language
generated by
GRAM#1 and 2, the alphabet of which
is {L16, L14, L12, L24, M16, M14, M40, O8, M18,
etc.}. Therefore, for each of these
variables the inference engine need only derive the
first
string of terminals using subgrammars 3, 4, and 5 and write into
the template a sequence of
dots representing the lengths of the
variables. GRAM#3 defines various subdivisions of L16,
L14,
etc. Its terminal alphabet comprises V30, V28, etc. (the starting symbols of GRAM#4)
and all variables A16, A'16, A16-2, etc. that denote
cadential patterns defined/recognised by
GRAM#6. A16 represents a voiced pattern of sixteen units whereas A'16 is its
partly unvoiced
transformation (see rules 18 and 19 in GRAM#6). A16-2 is a truncated A16 where the
last two
bols have been omitted. The same principles apply to A8 and A6. GRAM#4
generates a string
of Vs that are translated to bols in various
contexts in GRAM#5. Both these subgrammars are
LIN: the work string is transformed from left to
right in synthesis, and from right to left in
analysis. GRAM#5 [2] defines the leftmost gap that is always the
second bol in any section of
the sentence. This satisfies two hypothetical
conditions: (1) any section of a sentence may not
begin with a gap, and (2)
a gap is structurally linked to the bol immediately preceding it. The
ordering of rules in GRAM#5 has been determined by the chunk rule.
The parsing of the example given above under
Bol density
may be
found in Bel (1987b:19
-
20). The
test was positive when the sentence was matched against template 12. Templates 3
and 6 also matched the sentence but in both cases the
parsing led to a dead end. The total
matching of 21 templates took 1'43" on the Apple IIc.
However, this is unrepresentative
because membership tests were performed considerably more quickly with
regard to the vast
majority of grammars.
For instance, a grammar for a different qa‘ida that generated six
templates
may be found in Kippen and Bel (1989), in which the parsing of an input sentence
was completed in eleven seconds. It should perhaps be mentioned here that
the qa‘ida we have
dealt with in
this paper, a composition often referred to as the ‘King of qa‘idas’ by the
traditional tabla
players of Dehli where it originated (personal communication: Ustad Inam Ali
Khan, New Delhi, April 1984), offers
probably the widest scope of all for improvisation, and
the modelling of the performance by Ustad Afaq Husain Khan has resulted in the most
complex
grammar encountered to date. The
grammar would have been substantially more complex had
each and every variation
been included in the model (the improvisation extended to an
exceptional total
of twenty-nine variations for this qa‘ida). As suggested in appendix 2,
structures contained in the other variations would have been defined initially in
GRAM#1 [3]
and further additional rules.
Constructing context-sensitive grammars
In practice, GRAM#5 of the ‘complete’ grammar was easy to construct once
the structural
markers had been defined consistently
in GRAM#2 and typical divisions identified in
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Kippen-Bel: Modelling music with grammars: formal language representation in the
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GRAM#3. However, at one stage GRAM#5 [40] was
not included in the grammar when a
(correct)
V10 sequence
dhagedheenatidhagedheenage
was being analysed. The derivation
was:
(= dhagedheenatidhagedheenage)
G#5[21]
(= dhagedheenatidhage V V V)
G#5[9]
(= dhagedheenati V V V V V)
G#5[25]
(= V V V V ti V V V V V)
...
failed
The shortest unrecognised suffix was
tidhagedheenage
, and therefore a rule
V V V V V V
<—> tidhagedheenage
had to be added to the grammar. When the updated grammar was tried in synthesis mode,
a
derivation of V8 resulted in
dhatitidhagedheenage
,
which was considered to be incorrect
because of the two adjacent
ti
. Therefore we used the negative context
#ti
to yield the final
rule:
[40]
#ti V V V V V V
<—> #ti tidhagedheenage
If other incorrect left contexts had been found in productions using this rule, then
the derivation
would have been listed with all possible positive contexts:
dha V V V V V V
<—> dha tidhagedheenage
na V V V V V V
<—> na tidhagedheenage
etc...
(See, for instance,
rules 12, 13, 14 in the same subgrammar; note also how rule 12 reflects a
newly encountered derivation
where
kt
and
ti
are juxtaposed so disproving a previous
hypothesis.) The addition of
contextual constraints when a negative example is found may be
called a
specialisation
process, whereas adding a new
rule or suppressing a context is a
generalisation
process. The method described here
could be automated as it bears some
resemblance to grammatical inference
techniques that use the right formal derivatives of a
sample sequence (Fu & Booth 1975).
Constructing stress-sensitive grammars
We now consider another qa‘ida, this time of the Ajrara tradition (see Kippen 1988a:xi).
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Kippen-Bel: Modelling music with grammars: formal language representation in the
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Theme:
dhin--dhagena
dha--dhagena
dhatigegenaka
dheenedheenagena
tagetirakita
dhin--dhagena
dhatigegenaka
teeneteenakena
tin--takena
ta--takena
tatikekenaka
teeneteenakena
tagetirakita
dhin--dhagena
dhatigegenaka
dheenedheenagena
A few variations:
dhin--dhagena
dha--dhagena
dhatigegenaka
dheenedheenagena
tagetirakita
dhin--dhagena
dhatigegenaka
teeneteenakena
dheenedheenagena
teeneteenakena
tirakitatira
kitatirakita
tagetirakita
dhin--dhagena
dhatigegenaka
teeneteenakena
tin--takena
ta--takena
tatikekenaka
teeneteenakena
taketirakita tin--takena
tatikekenaka
teeneteenakena
dheenedheenagena
teeneteenakena
tirakitatira
kitatirakita
tagetirakita
dhin--dhagena
dhatigegenaka
dheenedheenagena
dhin--dhagena
dha--dhagena
dhatigegenaka
dheenedheenagena
tagetirakita
dhin--dhagena
dhatigegenaka
teeneteenakena
dhin--dhagena
dha-dha-dha-
dhagenadheen--
dhagenadha--
tagetirakita
dhin--dhagena
dhatigegenaka
teeneteenakena
tin--takena
ta--takena
tatikekenaka
teeneteenakena
taketirakita tin--takena
tatikekenaka
teeneteenakena
dhin--dhagena
dha-dha-dha-
dhagenadheen--
dhagenadha--
tagetirakita
dhin--dhagena
dhatigegenaka
dheenedheenagena
dhin--dhagena
dha--dhagena
dhatigegenaka
dheenedheenagena
tagetirakita
dhin--dhagena
dhatigegenaka
teeneteenakena
dheenedheenagena
dheenedha-dheene
dhatigegenaka
teeneteenakena
tagetirakita
dhin--dhagena
dhatigegenaka
teeneteenakena
tin--takena
ta--takena
tatikekenaka
teeneteenakena
taketirakita tin--takena
tatikekenaka
teeneteenakena
dheenedheenagena
dheenedha-dheene
dhatigegenaka
teeneteenakena
tagetirakita
dhin--dhagena
dhatigegenaka
dheenedheenagena
Our observations based
on several samples of variations (again from performances and
demonstrations by Ustad Afaq Husain Khan of Lucknow) suggested that
variable lines (shown
in italics) were constructed with chunks of
bols of lengths 3, 4, and 6 in permutations that we
presumed to be context-free. This view was reinforced
by the fact that no technical (i.e.
fingering)
difficulties were encoutered when chunks were arranged in any order. The
significant units were listed in the following
lexical
rules:
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Kippen-Bel: Modelling music with grammars: formal language representation in the
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Processor
RND
A3 <—> dhin--
A3 <—> dha--
A3 <—> dhagena
A4
<—> tirakita
A3 A3
<—> dhagenadhin--
A3 A3
<—> dhagenadha--
A6 <—> dha-dha-dha-
A6 <—> dha-ta-dha-
A6
<—> dheenedheenedheene
A6
<—> dheenedha-dheene
A6
<—> tagetirakita
A6
<—> dheenedheenagena
A6
<—> teeneteenakena
A6
<—> dhatigegenaka
In view of their
frequent occurrence in examples,
dhagenadhin-- and dhagenadha-- were
identified specifically so as to increase the likelihood that they would be generated as blocks.
A
grammar for defining all possible sequences in variable lines of 6, 12 or 24 units was:
LIN
B6
<—> A A A A A A
B12
<—> A A A A A A A A A A A A
B24
<—> A A A A A A A A A A A A A A A A A A A A A A A A
A A A
<—> A3
A A A A
<—> A4
A A A A A A
<—> A6
This grammar was correct in
the sense that any sequence of A3, A4, and A6 was a derivation
of B24 only if the sum of its metric values was
24. Yet incomplete derivations such as A3 A6
A3 A4 A3 A3 A A were also possible. Obviously not more than two isolated As
were found in
such sequences and so these were eliminated (or ‘absorbed’) in the
following subgrammar
where rules were taken in order (single arrows here indicate that
these rules were ignored in
analysis mode):
ORD
A A
—> A2
A3 A
—> A4
#
A3 A
—> A #A3
A6 A2
—> A4 A4
#
A6 A2
—> A2 #A6
Some of the pieces generated by this
grammar displayed irregularities in the accentuation. For
instance,
dhin
--tiraki tadhagenadhati
gegenaka
tira kitati
rakita
imposed a rhythm counter to the natural stresses of the beat and half-beat,
and was therefore
virtually impossible to recite or
perform at speeds normally employed by musicians (c.
mm=108-120, i.e.up to twelve bols per second). In
a four-beat string comprising 24 units,
primary accents fall on beats and half-beats: 1, 4, 7, 10, 13, 16, 19, and 22.
A cursory
analysis of variations created by musicians showed that in addition
to these divisions they
employed hemiolic rhythmic patterns beginning
on units 1, 7, and 13. This produces a series
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Kippen-Bel: Modelling music with grammars: formal language representation in the
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of secondary stresses on units 5, 9, 11, 15, 17, 21. The following
is a list of possible starting
positions for the blocks defined above:
A3: 1, 4, 7, 10, 13, 16, 19, 22
A4: 1, 5, 7, 9, 11, 13, 15, 17, 21
A6: 1, 4, 7, 10, 13, 16, 19
tagetirakita: 1, 4, 5, 7, 9, 10, 11, 13, 15, 16, 17, 19
The exceptional status of
ta
ge
ti
rakita
is due to the fact that it
was accentuated in two different
ways. Therefore it was labelled with a new variable: C6.
We developed a way to define
derivations of B24, B12, and B6 in a systematic way that took
into account acceptable starting positions. The grammar was right-linear:
LIN
B24 <—> A3 B21 ...(A3 in starting position: 24 - (3+21) +1 = 1)
B24 <—> A4 B20
B24 <—> C6 B18
B24 <—> A6 B18
B21 <—> A3 B18 ...(A3 in starting position: 24 - (3+18) +1 = 4)
B21 <—> A4 B17
...(cancelled: A4 in starting position 4)
B21 <—> A6 B15
B21 <—> C6 B15
B20 <—> A3 B17
...(cancelled: A3 in starting position 5)
B20 <—> A4 B16
B20 <—> A6 B14
...(cancelled: A6 in starting position 5)
B20 <—> C6 B14
etc...
This grammar could have generated a string A4 A4 A4 A4 A4 A4 whose
only derivation would
have been an unbroken series of
tirakita
s that musicians
would almost certainly have assessed
as incorrect. The solution to this problem may be found in subgrammar
3 of the final grammar
for this qa‘ida, given in appendix 3, where it may be seen that rules 28 to 31 were attributed
left
contexts. We term this kind
of subgrammar
right-linear-context-sensitive
. In the version
given here, the grammar shows a number of improvements over that published in Bel (1987a).
The probabilistic model
It should be noted that the grammars discussed in this
paper so far can only claim the status of
analytical models in view of
the fact that they were shown to have begun as initial hypotheses
and to have
developed as a result of our own intuitions, albeit based on considerable practical
experience of tabla and extensive analyses of the music. It is beyond
the scope of this paper to
enter into a detailed discussion of theoretical
and methodological aspects of this research: these
have been covered extensively in Kippen (1985, 1987, 1988b, 1988c). However, it should be
emphasized that a prominent
aim of the research has been to create a human-computer
interaction where musicians themselves respond
to the output of BP grammars, and grammars
are in turn
modified to account for the input of musicians. Thus, in theory at least, analytical
control over the models lies with the musicians, and it is they, not
us, who are the sole arbiters
of the correctness of computer-generated data.
A number of problems were encountered during actual interactions.
One was that sometimes a
grammar would reach a point of stagnation where computer-generated variations were
judged
to be neither very good nor incorrect. Consequently there was no simple way
of refining or
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Kippen-Bel: Modelling music with grammars: formal language representation in the
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improving the model. We felt that
a solution lay in attributing to each production rule a
coefficient of likelihood (or weight) where the probability that certain generative paths would
be
chosen in preference to others could be examined.
The probabilistic model that has been implemented on the BP is derived
from probabilistic
grammars/automata as defined by
Booth & Thompson (1973), the difference being that a
weight — within
the range [0,255] — rather than a probability is attached to every rule. The
rule probability is computed as follows: if
the weight is zero then the probability is zero; if the
weight is positive then the inference engine calculates the sum of weights of all
candidate
rules, and the rule probability is the ratio of its own weight to the
sum. Candidate rules are
those whose premise is a substring of the work string. Consider, for example,
GRAM#2 in
appendix 2
GRAM#2 [2] <100> S64 <—> L16 + S48
...etc.
GRAM#2 [12] <100> S16 <—> O16
GRAM#2 [13] <100> S50
<—> V10 A'8-2 * (= N18 +) + O16
...etc.
GRAM#2 [16] <100> S52 <—> M20 + S32
GRAM#2 [17] <100> S40 <—> M8 + S32
GRAM#2 [18] <100> S32
<—> * (=+ N16 +) + S16
...etc.
GRAM#2 [25] <100> S34
<—> * (= N18 +) + S16
GRAM#2 [26] <5> S16
<—> /8+ A16 O16 ; /4
and a work string containing S16. The sum of the weights of the
two candidate rules is 100+5
= 105. The probability of rule 26
is therefore 5/105 = 0.048. The advantage of weights over
probabilities is that they do not
presuppose the sum of the coefficients of all candidate rules to
be 1 — a condition that can only be satisfied in context-free grammars.
Weights (and their associated probabilities) have been used in modus
ponens to direct the BP’s
production along paths more likely to be followed
by musicians. In some context-free
grammars — those that fulfilled the
consistency
condition expressed by Booth
& Thompson
(1973:442) — they were used to compute
a
probabilistic sentence function
, i.e. a coefficient
representing the likelihood of occurrence of each sentence in the language.
Even in cases
where the likelihood sentence function
was not strictly probabilistic, it was used as an
approximate assessment of the consistency of the sentence with the grammar:
at least, any
sentence that required a rule
with weight <0> during its parsing was assigned likelihood zero.
Comparing the likelihoods obtained from parsing a
sentence on different templates helped to
determine decisions regarding ambiguous structures.
Another remarkable feature of consistent grammars is that rule probabilities can be
inferred
from a set of sentences (Maryanski & Booth 1977:525). The
algorithm implemented in the BP
is more powerful than the one devised by Maryanski and Booth, since
the latter required the
choice of a sample set in which
all rules had
been used. Given a grammar and a subset of the
language that this grammar generates (for instance a sample sequence taken from
a performance
of an expert musician), rule weights may be inferred as follows: let all
weights be reset to zero;
then analyse every sentence and increment by one unit the weights of all rules used in
the
derivation. The weights
displayed in the grammar in appendix 3 were inferred from the
analysis of thirteen examples. Consequently, it
produced variations that bore some
resemblance to those in the sample sequence.
Rules that were not used in this process, for
instance
GRAM#3 [11] or GRAM#1 [2], were scrutinised to see whether they were incorrect
or
whether they pointed to fragments of the language that had not yet been explored. To test
this, their weights were set to a high value so that the
BP was forced to generate sentences that
either had never been assessed or at least had not been considered by the informant at the time.
Using weighted rules resulted
in a marked improvement in the quality of the generated music.
This went a long way towards solving the problem
of musical credibility encountered in earlier
experiments, a problem that arose from the complete randomness of the generative
process and
the stagnation of grammars.
18
Kippen-Bel: Modelling music with grammars: formal language representation in the
Bol
Processor
Further developments
A more general theoretical model of BP grammars, called BP2,
is being developed and
implemented on a Macintosh computer. In
this new version, potentially any symbolic/numeric
musical code (e.g. MIDI code) can be used as a terminal
alphabet. The concept of
voiced/unvoiced transformations has been
generalised to any user-defined non-erasing
homomorphism — for instance tonal transposition can
be easily represented. Several
homomorphisms can be defined on the same alphabet.
The main
advantage of BP2 is its ability to represent the time structures of polyphonic music:
simultaneous events are listed in sets, each event may
in turn be a set of sub-events. A
program generating MIDI code sequences from BP polyphonic representations is
being
studied.
A free
copy of BP2 and the grammars described in this paper are available in Binhex
format on the E-mail network.
[Alan, plese set this bit as a footnote]
Conclusion
In this paper, we have attempted to show that formal models can be manipulated to
represent
highly elaborated musical concepts that underlie the
art of improvisation in tabla music.
Grammars are evidently easier to work on if the music is regular: i.e. if
it conforms to idealised
models such as those transmitted
to students in teaching situations. On the other hand, the
grammar shown in appendix 2 shows the extraordinary rise in complexity
when attempts are
made to account for the irregularities
encountered in actual performances. The unpredictability
of improvisation in this context serves only to compound
the problem, as there is no guarantee
that even a complex and reasonably comprehensive grammar will be adequate
to deal with
further
samples. Technically, however, the formal representations discussed here can be
adapted to express any structure likely to be played
by a tabla musician, whatever its degree of
complexity.
The major limitation
of the models, then, lies not in the formal representations themselves but
in the
transfer of knowledge from informants to the computer. Knowledge acquisition has
proved to be an
important consideration in artificial intelligence, and one that has produced no
satisfactory answers so far. The problem consists in the fact that expert systems like
the BP
represent knowledge at a low (non-hierarchical) theoretical level, and so it is impossible
to
separate general analytical statements from specific instances of facts. A discussion
of the
shortcomings of the
knowledge-acquisition strategy in the BP may be found in Kippen & Bel
(1989).
19
Kippen-Bel: Modelling music with grammars: formal language representation in the
Bol
Processor
References cited
Bel, Bernard
‘Grammaires de génération et de reconnaissance de phrases
rythmiques’, 6ème congrès
AFCET / INRIA
, Antibes, 1987a:353-66
‘Les grammaires et le moteur d’inférences du Bol
Processor’, note 237, Groupe
Représentation et Traitement des Connaissances, CNRS, Marseille, 1987b
‘Designing tools for knowledge representation in the anthropological study of
a musical
system’, unpublished, 1988
Booth, T.L. and R.A. Thompson
‘Applying
Probability Measures to Abstract Languages’,
IEEE Transactions on
Computers, Vol.C-22, n°5, 1973:442-50
Fu, K.S. and T.L. Booth
‘Grammatical Inference: Introduction and Survey - Part I’,
IEEE Transactions on
Systems,
Man and Cybernetics
, Vol. SMC-5, n°1, 1975:95-111
Hart, Johnson M.
‘Derivation
Structures for Strictly Context-Sensitive Grammars’,
Information and Control
45, 1980:68-89
Kippen, Jim
‘The dialectical approach: a methodology for the analysis of tabla music’,
International
Council for Traditional Music (UK Chapter) Bulletin
n°12, 1985:4-12
‘An
ethnomusicological approach to the analysis of musical cognition’,
Music Perception
Vol.5 Part 2, 1987:173-95
The Tabla of Lucknow: a
Cultural Analysis of a Musical Tradition
, Cambridge
University Press, 1988a
‘On the uses of
computers in anthropological research’,
Current Anthropology
Vol.29 Part
2, 1988b:317-20
‘Computers, fieldwork, and the problem
of ethnomusicological analysis’,
International
Council for Traditional Music (UK Chapter)
N°20, 1988c:20-35
Kippen, Jim and Bernard Bel
‘Can a computer help resolve the problem
of ethnographic description?’,
Anthropological
Quarterly
, forthcoming, 1989
Maryanski, F.J., and T.L. Booth
‘Inference of Finite-State Probabilistic Grammars’,
IEEE Transactions on
Computers,
Vol.C-26, n°6, 1977:521-36
[Some anomalies of this paper are corrected
in B.R. Gaine's paper ‘Maryanski's
Grammatical Inferencer’,
IEEE Transactions on Computers
, Vol.C-27, n°1, 1979:62-64]
20
Kippen-Bel: Modelling music with grammars: formal language representation in the
Bol
Processor
Appendix 1: context-sensitive canonic rightmost derivation
The formal definition
of the context-sensitive rightmost definition adapted from Hart (1980)
and used in the membership test of LIN grammars is
given below. Vertical bars denote the
lengths of strings:
Let G be a length-increasing grammar. The derivation in G:
W
0
=> W
1
=> ... => W
n
is
context-sensitive rightmost
iff:
∀i ∈
[0, n-1], W
i
= X
i
L
i
C
i
R
i
Y
i
W
i+1
= X
i
L
i
D
i
R
i
Y
i
when applying rule f
i
: L
i
C
i
R
i
-> L
i
D
i
R
i
,
and at least one of the two following conditions is satisfied:
(C1)
|C
i+1
R
i+1
Y
i+1
| > |Y
i
|
(C2)
|L
i+1
C
i+1
R
i+1
Y
i+1
| > |R
i
Y
i
|
In Hart’s definition (1980:82), |C
i
| = |C
i+1
| = 1, so that C1 may be written |R
i+1
Y
i+1
| ≥
|Y
i
|.
The diagram and commentary below illustrate conditions
C1
and
C2
:
Yi+1
Ri+1
Ci+1
Li+1
Xi
Li
Di
Xi+1
Xi+1
Li+1
Ci+1 Ri+1
Yi+1
Yi
Ri
Ri
YiCi
Li
Xi
Suppose conditions
C1
and
C2
are not satisfied. Since
C2
is not true, rule f
i+1
could
have been applied
before f
i
as L
i+1
C
i+1
R
i+1
would be
a substring of R
i
Y
i
. Besides, since
C1
is not true, applying rule
f
i+1
would only modify Y
i
without changing the context R
i
. In such a case the order in which f
i
and
f
i+1
are applied might have been inverted:
a change that would have been justified since all symbols
rewritten by f
i+1
are to the right of those rewritten by f
i
.
We now explain how
the inference engine of the BP handles ambiguity in bottom-up parsing.
Suppose that for a working string W
i
there are two candidate rules:
f
i
L
i
C
i
R
1
-> L
i
D
i
R
i
f'
i
L'
i
C'
i
R'
i
-> L'
i
D'
i
R'
i
where
X
i
L
i
C
i
R
i
Y
i
= X'
i
L'
i
C'
i
R'
i
Y'
i
= W
i
The selection criterion is the following:
f
i
will have priority over f'
i
if one of the following
conditions (in this order) is satisfied:
21
Kippen-Bel: Modelling music with grammars: formal language representation in the
Bol
Processor
(D1)
|X
i
L
i
C
i
| > |X'
i
L'
i
C'
i
|
(D2)
|X
i
L
i
C
i
R
i
| > |X'
i
L'
i
C'
i
R'
i
|
(D3)
|L
i
C
i
R
i
| > |L'
i
C'
i
R'
i
|
(D4)
i > i'
It can be proved (Bel 1987b:8) that once D1 has been
considered, D2 is no longer relevant and
ambiguity
may therefore be handled by D3 and D4. D3 makes a decision on the basis of the
length of the conclusions of the two rules, and D4 is a final arbitrary decision
that takes into
account the order in which the rules
appear in the grammar. To augment efficiency, D3 is not
considered by the inference engine of the BP, and therefore the test relies on the partial
ordering
of rules in the grammar (see ‘chunk’ rule).
22
Kippen-Bel: Modelling music with grammars: formal language representation in the
Bol
Processor
Appendix 2: Delhi qa‘ida (as performed by
Ustad Afaq Husain Khan of Lucknow
GRAM#1 [1] RND
GRAM#1 [2] <100> S <—> (=+ S64 ;)
GRAM#1 [3] <100> S <—> ...
Other patterns
-----------------------------------------------------------------------------------------------------------------------
GRAM#2 [1] LIN
GRAM#2 [2] <100> S64 <—> L16 + S48
GRAM#2 [3] <100> S64 <—> L14 S50
GRAM#2 [4] <100> S64 <—> L12 S52
GRAM#2 [5] <100> S64 <—> L24 S40
GRAM#2 [6] <100> S48 <—> M16 + S32
GRAM#2 [7] <100> S48 <—> M14 S34
GRAM#2 [8] <100> S48 <—> M40 O8
GRAM#2 [9] <100> S50 <—> M18 + S32
GRAM#2 [10] <100> S50 <—> M34 + S16
GRAM#2 [11] <100> S50 <—> M18 + * (=+ N14 ) O18
GRAM#2 [12] <100> S16 <—> O16
GRAM#2 [13] <100> S50 <—> V10 A'8-2 * (= N18 +) + O16
GRAM#2 [14] <100> S50 <—> M20 * (= N14 +) + O16
GRAM#2 [15] <100> S52 <—> V28 A8-2 O18
GRAM#2 [16] <100> S52 <—> M20 + S32
GRAM#2 [17] <100> S40 <—> M8 + S32
GRAM#2 [18] <100> S32 <—> * (=+ N16 +) + S16
GRAM#2 [19] <100> S32 <—> * (= /6+ V12 /4 A8 +) + O16
GRAM#2 [20] <100> S32 <—> * (= /6+ A16 V8 + /4) + O16
GRAM#2 [21] <100> S32 <—> * (= /6+ V24 + /4) + O16
GRAM#2 [22] <100> S32 <—> * (= /8+ N'16 + A16 + /4) + O16
GRAM#2 [23] <100> S32 <—> * (=+ N14 ) O18
GRAM#2 [24] <100> S34 <—> O34
GRAM#2 [25] <100> S34 <—> * (= N18 +) + S16
GRAM#2 [26] <5> S16 <—> /8+ A16 O16 ; /4
-----------------------------------------------------------------------------------------------------------------------
GRAM#3 [1] RND
GRAM#3 [2] <100> LEFT M16 <—> V16
GRAM#3 [3] <100> LEFT N18 <—> V18
GRAM#3 [4] <100> LEFT (=+ L16 <—> (=+ A16
GRAM#3 [5] <100> LEFT (=+ L16 <—> (=+ V16
GRAM#3 [6] <100> LEFT (=+ L14 <—> (=+ V10 A'6-2
GRAM#3 [7] <100> LEFT M14 <—> A'16-2
GRAM#3 [8] <100> LEFT (=+ L14 <—> (=+ A16-2
GRAM#3 [9] <100> LEFT (=+ L14 <—> (=+ A'16-2
GRAM#3 [10] <100> LEFT (=+ L12 <—> (=+ A16-4
GRAM#3 [11] <100> LEFT (=+ L24 <—> (=+ V24
GRAM#3 [12] <100> LEFT + M16 + * <—> + V10 A'6 + *
GRAM#3 [13] <100> LEFT + M16 <—> + A'16
GRAM#3 [14] <100> LEFT M8 + * <—> A'8 + *
GRAM#3 [15] <100> LEFT M16 + * <—> V8 A'8 + *
GRAM#3 [16] <100> LEFT M14 <—> V8 A'8-2
GRAM#3 [17] <100> LEFT M18 + * <—> V10 A'8 + *
GRAM#3 [18] <100> LEFT M18 + * <—> V18 + *
GRAM#3 [19] <100> LEFT M34 + <—> V28 A'6 +
GRAM#3 [20] <100> LEFT M34 + <—> V26 A'8 +
GRAM#3 [21] <100> LEFT M20 + * <—> V12 A'8 + *
GRAM#3 [22] <100> LEFT M20 <—> V20
GRAM#3 [23] <100> LEFT M40 <—> V8 A'8-2 V26
GRAM#3 [24] <100> LEFT * (= /8+ N'16 <—> * (= /8+ V16
GRAM#3 [25] <100> LEFT * (= /8+ N'16 <—> * (= /8+ N16
GRAM#3 [26] <100> LEFT N16 + <—> V12 A4 +
GRAM#3 [27] <100> LEFT N16 + <—> V8 A8 +
GRAM#3 [28] <100> LEFT N16 + <—> V10 C6 +
GRAM#3 [29] <100> LEFT * (=+ N16 <—> * (=+ A16
23
Kippen-Bel: Modelling music with grammars: formal language representation in the
Bol
Processor
GRAM#3 [30] <100> LEFT M14 <—> V8 A8-2
GRAM#3 [31] <100> LEFT * (=+ N14 <—> * (=+ V8 A8-2
GRAM#3 [32] <100> LEFT * (=+ N14 <—> * (=+ A16-2
GRAM#3 [33] <100> LEFT N14 + <—> V6 A8 +
GRAM#3 [34] <100> LEFT O8 ; <—> A8 ;
GRAM#3 [35] <100> LEFT O34 ; <—> V26 A8 ;
GRAM#3 [36] <100> LEFT O18 ; <—> V10 A8 ;
GRAM#3 [37] <100> LEFT O16 ; <—> V8 A8 ;
GRAM#3 [38] <100> LEFT + O16 ; <—> + A16 ;
-----------------------------------------------------------------------------------------------------------------------
GRAM#4 [1] LIN
GRAM#4 [2] <100> V30 <—> V V V28
GRAM#4 [3] <100> V28 <—> V V V26
GRAM#4 [4] <100> V26 <—> V V V24
GRAM#4 [5] <100> V24 <—> V V V V V20
GRAM#4 [6] <100> V20 <—> V V V18
GRAM#4 [7] <100> V18 <—> V V V16
GRAM#4 [8] <100> V16 <—> V V V V V12
GRAM#4 [9] <100> V12 <—> V V V10
GRAM#4 [10] <100> V10 <—> V V V8
GRAM#4 [11] <100> V8 <—> V V V6
GRAM#4 [12] <100> V6 <—> V V V V V V
-----------------------------------------------------------------------------------------------------------------------
GRAM#5 [1] LIN
GRAM#5 [2] <100> ? V <—> ? -
GRAM#5 [3] <100> V <—> dha
GRAM#5 [4] <100> V V <—> trkt
GRAM#5 [5] <100> V V <—> dheena
GRAM#5 [6] <100> V V <—> teena
GRAM#5 [7] <100> V V <—> dhati
GRAM#5 [8] <100> V V <—> gena
GRAM#5 [9] <100> V V <—> dhage
GRAM#5 [10] <100> + V V <—> +tidha
GRAM#5 [11] <100> - V V <—> -tidha
GRAM#5 [12] <100> kt V V <—> kttidha
GRAM#5 [13] <100> na V V <—> natidha
GRAM#5 [14] <100> ge V V <—> getidha
GRAM#5 [15] <100> - V V <—> -ti-
GRAM#5 [16] <100> kt V V <—> ktti-
GRAM#5 [17] <100> ge V V <—> geti-
GRAM#5 [18] <100> na V V <—> nati-
GRAM#5 [19] <100> V V V <—> dhagena
GRAM#5 [20] <100> V V V <—> teenake
GRAM#5 [21] <100> V V V <—> dheenage
GRAM#5 [22] <100> V V V <—> dhatrkt
GRAM#5 [23] <100> V V V <—> trktdha
GRAM#5 [24] <100> V V V V <—> tidhagena
GRAM#5 [25] <100> V V V V <—> dhagedheena
GRAM#5 [26] <100> V V V V <—> teena-ta
GRAM#5 [27] <100> #ti V V V V <—> #ti tidhatrkt
GRAM#5 [28] <100> V V V V <—> dheenagena
GRAM#5 [29] <100> V V V V <—> teenakena
GRAM#5 [30] <100> V V V V V <—> dhagenadheena
GRAM#5 [31] <100> V V V V V <—> dhagenateena
GRAM#5 [32] <100> V V V V V <—> dhagenadhati
GRAM#5 [33] <100> V V V V V <—> dhatrktdhati
GRAM#5 [34] <100> V V V V V <—> dhatidhatrkt
GRAM#5 [35] <100> V V V V V <—> dhatidhagena
GRAM#5 [36] <100> V V V V V V <—> dheenagedhatrkt
GRAM#5 [37] <100> V V V V V V <—> genagedhatrkt
GRAM#5 [38] <100> V V V V V V <—> dhagedheenagena
GRAM#5 [39] <100> V V V V V V <—> dhageteenakena
GRAM#5 [40] <100> #ti V V V V V V <—> #ti tidhagedheenage
GRAM#5 [41] <100> V V V V V V <—> teenakegenage
GRAM#5 [42] <100> V V V V V V V V <—> dhagenagenanagena
24
Kippen-Bel: Modelling music with grammars: formal language representation in the
Bol
Processor
GRAM#5 [43] <100> V V V V V V V V <—> dhatidhagedheenagena
GRAM#5 [44] <100> V V V V V V V V <—> dhatidhageteenakena
GRAM#5 [45] <100> V V V V V V V V A8 <—> dhatrktdhatidhatrkt A8
GRAM#5 [46] <100> V V V V V V V V A'8 <—> dhatrktdhatidhatrkt A'8
-----------------------------------------------------------------------------------------------------------------------
GRAM#6 [1] ORD
GRAM#6 [2] LEFT + ? ? ? ? ? ? ? ? ? ? A'6-2 <—> + ? ? ? ? ? ? ? ? ? ? dhageteena
GRAM#6 [3] LEFT + ? ? ? ? ? ? ? ? ? ? C6-2 <—> + ? ? ? ? ? ? ? ? ? ? dhagedheena
GRAM#6 [4] LEFT + ? ? ? ? ? ? ? ? ? ? A'6-2 <—> + ? ? ? ? ? ? ? ? ? ? dhageteena
GRAM#6 [5] LEFT + ? ? ? ? ? ? ? ? A8-2 <—> + ? ? ? ? ? ? ? ? dhatidhagedheena
GRAM#6 [6] LEFT A8-2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ; <—> dhatidhagedheena ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ;
GRAM#6 [7] LEFT + ? ? ? ? ? ? ? ? A'8-2 <—> + ? ? ? ? ? ? ? ? dhatidhageteena
GRAM#6 [8] LEFT + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? A'8-2 <—> + ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? dhatidhageteena
GRAM#6 [9] LEFT A'6 + <—> dhageteenakena+
GRAM#6 [10] LEFT A'8 + <—> dhatidhageteenakena+
GRAM#6 [11] LEFT A4 + <—> dheenagena+
GRAM#6 [12] LEFT C6 + <—> dhagedheenagena+
GRAM#6 [13] LEFT A8 + <—> dhatidhagedheenagena+
GRAM#6 [14] LEFT A8 ; <—> dhatidhagedheenagena;
GRAM#6 [15] LEFT + A16-4 <—> +dhatidhagenadhatrktdhatidhage
GRAM#6 [16] LEFT + A16-2 #ge <—>+dhatidhagenadhatrktdhatidhagedheena #ge
GRAM#6 [17] LEFT + A'16-2 #ke <—>+dhatidhagenadhatrktdhatidhageteena #ke
GRAM#6 [18] LEFT + A16 <—> +dhatidhagenadhatrktdhatidhagedheenagena
GRAM#6 [19] LEFT + A'16 <—> +dhatidhagenadhatrktdhatidhageteenakena
-----------------------------------------------------------------------------------------------------------------------
Templates:
[1] TEM
[2] (=+................+................+ * (=+................+) +................;)
[3] (=+................+................+ * (=+................+) +/8+................................;/4;)
[4] (=+................+................+ * (=/6+............/4........+) +................;)
[5] (=+................+................+ * (=/6+........................+/4) +................;)
[6] (=+................+................+ * (=/8+................+................+/4) +................;)
[7] (=+................+................+ * (=+..............) ..................;)
[8] (=+................+................................................;)
[9] (=+................+.............. * (=..................+) +................;)
[10] (=+................+.............. * (=..................+) +/8+................................;/4;)
[11] (=+................................+ * (=+................+) +................;)
[12] (=+................................+ * (=+................+) +/8+................................;/4;)
[13] (=+................................+ * (=/6+............/4........+) +................;)
[14] (=+................................+ * (=/6+........................+/4) +................;)
[15] (=+................................+ * (=/8+................+................+/4) +................;)
[16] (=+................................+ * (=+..............) ..................;)
[17] (=+................................................+................;)
[18] (=+................................................+/8+................................;/4;)
[19] (=+.............................. * (=..................+) +................;)
[20] (=+.................................. * (=..............+) +................;)
[21] (=+................................................................;)
25
Kippen-Bel: Modelling music with grammars: formal language representation in the
Bol
Processor
Appendix 3: Ajrara qa‘ida (as performed by
Ustad Afaq Husain Khan of Lucknow
GRAM#1 [1]
RND
GRAM#1 [2] <0>
S <—> S48
GRAM#1 [3] <13>
S <—> S96
-----------------------------------------------------------------------------------------------------------------------
GRAM#2 [1]
RND
GRAM#2 [2] <13>
S96 <—> (= F48 ) (= V24 ) F'24 *(: F48 ) (: V24 ) F24
GRAM#2 [3] <0>
S48 <—> (= V24 ) F'24 *(: V24 ) F24
GRAM#2 [4] <1>
V24 <—> (= V12 ) (: V12 )
GRAM#2 [5] <3>
V24 <—> (= V12 ) *(: V12 )
GRAM#2 [6] <0>
V24 <—> Q24
GRAM#2 [7] <2>
V24 <—> V12 V12
GRAM#2 [8] <7>
V24 <—> B24
GRAM#2 [9] <3>
V12 <—> (= B6 ) *(: B6 )
GRAM#2 [10] <5>
V12 <—> B12
-----------------------------------------------------------------------------------------------------------------------
GRAM#3 [1]
LIN
GRAM#3 [2] <3>
B3 <—> A3
GRAM#3 [3] <1>
B4 <—> A4
GRAM#3 [4] <6>
B6 <—> A6
GRAM#3 [5] <1>
B6 <—> C6
GRAM#3 [6] <3>
B24 <—> A3 B21
GRAM#3 [7] <0>
B24 <—> A4 B20
GRAM#3 [8] <4>
B24 <—> A6 B18
GRAM#3 [9] <0>
B24 <—> C6 B18
GRAM#3 [10] <3>
B21 <—> A3 B18
GRAM#3 [11] <0>
B21 <—> A6 B15
GRAM#3 [12] <0>
B21 <—> C6 B15
GRAM#3 [13] <0>
B20 <—> A4 B16
GRAM#3 [14] <0>
B20 <—> C6 B14
GRAM#3 [15] <2>
B18 <—> A3 B15
GRAM#3 [16] <0>
B18 <—> A4 B14
GRAM#3 [17] <4>
B18 <—> A6 B12
GRAM#3 [18] <1>
B18 <—> C6 B12
GRAM#3 [19] <0>
B16 <—> A4 B12
GRAM#3 [20] <0>
B16 <—> C6 B10
GRAM#3 [21] <2>
B15 <—> A3 B12
GRAM#3 [22] <0>
B15 <—> A6 B9
GRAM#3 [23] <0>
B15 <—> C6 B9
GRAM#3 [24] <0>
B14 <—> A4 B10
GRAM#3 [25] <0>
B14 <—> C6 B8
GRAM#3 [26] <4>
B12 <—> F'12
GRAM#3 [27] <3>
B12 <—> A3 B9
GRAM#3 [28] <0>
A3 B12 <—> A3 A4 B8
GRAM#3 [29] <1>
A6 B12 <—> A6 A4 B8
GRAM#3 [30] <0>
C6 B12 <—> C6 A4 B8
GRAM#3 [31] <0>
) B12 <—> ) A4 B8
GRAM#3 [32] <4>
B12 <—> A6 B6
GRAM#3 [33] <0>
B12 <—> C6 B6
GRAM#3 [34] <0>
B10 <—> A4 B6
GRAM#3 [35] <0>
B10 <—> C6 B4
GRAM#3 [36] <3>
B9 <—> A3 B6
GRAM#3 [37] <0>
B9 <—> A6 B3
GRAM#3 [38] <0>
B9 <—> C6 B3
GRAM#3 [39] <1>
B8 <—> A4 B4
GRAM#3 [40] <3>
B6 <—> A3 B3
-----------------------------------------------------------------------------------------------------------------------
26
Kippen-Bel: Modelling music with grammars: formal language representation in the
Bol
Processor
GRAM#4 [1]
RND
GRAM#4 [2] <4>
A3 <—> dhin--
GRAM#4 [3] <0>
A3 <—> dha--
GRAM#4 [4] <2>
A3 <—> dhagena
GRAM#4 [5] <1>
A3 A3 <—> dhagenadhin--
GRAM#4 [6] <7>
A3 A3 <—> dhagenadha--
GRAM#4 [7] <3>
A4 <—> tirakita
GRAM#4 [8] <2>
A6 <—> dha-dha-dha-
GRAM#4 [9] <1>
A6 <—> dha-ta-dha-
GRAM#4 [10] <2>
A6 <—> dheenedheenedheene
GRAM#4 [11] <1>
A6 <—> dheenedha-dheene
GRAM#4 [12] <8>
A6 <—> dheenedheenagena
GRAM#4 [13] <3>
A6 <—> teeneteenakena
GRAM#4 [14] <1>
A6 <—> dhatigegenaka
GRAM#4 [15] <2>
C6 <—> tagetirakita
-----------------------------------------------------------------------------------------------------------------------
GRAM#5 [1]
RND
GRAM#5 [2] <4>
F'12 <—> dhatigegenakateeneteenakena
GRAM#5 [7] <0>
Q24 <—> dhin--dhagenadha-
-
dhagenadhatigegenakadheenedheenagena
GRAM#5 [3] <12>
F24 <—> tagetirakitadhin--
dhagenadhatigegenakadheenedheenagena
GRAM#5 [4] <1>
F24 <—> tagetirakitagena-
dhagenadhatigegenakadheenedheenagena
GRAM#5 [5] <13>
F'24 <—> tagetirakitadhin--
dhagenadhatigegenakateeneteenakena
GRAM#5 [6] <0>
F'24 <—> tagetirakitagena-
dhagenadhatigegenakateeneteenakena
GRAM#5 [7] <13>
F48 <—> dhin--dhagenadha-
-
dhagenadhatigegenakadheenedheenagenatagetirakitadhin--dhagenadhatigegenakateeneteenakena
GRAM#5 [8] <0>
F48 <—> dhin--dhagenadha-
-
dhagenadhatigegenakadheenedheenagenatagetirakitagena-dhagenadhatigegenakateeneteenakena
-----------------------------------------------------------------------------------------------------------------------
27
Kippen-Bel: Modelling music with grammars: formal language representation in the
Bol
Processor
Modelling music with grammars:
formal language representation in the Bol Processor
Jim Kippen & Bernard Bel
Abstract
Improvisation in North Indian
tabla
drumming is similar
to speech insofar as it is bound to an
underlying system of rules determining correct sequences. The parallel
is further reinforced by
the fact that tabla music
may be represented with an oral notation system used for its
transmission and, occasionally, performance. Yet the rules
are implicit and available only
through the musicians’ ability
to play correct sequences and recognise incorrect ones. A
linguistic model of
tabla improvisation and evaluation derived from pattern languages and
formal grammars has been implemented in
the Bol Processor, a software system used in
interactive fieldwork with expert musicians. The paper demonstrates the ability
of the model to
handle complex structures by taking real examples from the
repertoire. It also questions the
relevance of attempting to model irregularities encountered in actual performance.
Keywords
1. Formal grammars
2. Patterns
3. Membership test
4. Tabla drumming
5. Ethnomusicology
28
