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A Quick Tour of Rothenberg's Musical Pattern Recognition Model
The model concerns a code that can be used to extract information from
large stimulus spaces despite the limitations of the human memory
system...
We have hypothesized that when a listener is presented with a
series of unfamiliar tonal stimuli, he must mentally construct
a reference frame, P, to which all such stimuli are referred.
Many proper P maysatisfy this requirement. If the stimuli are
sufficiently unfamiliar (as when one listens to music of an
alien culture) many repeated hearings may be necessary during
which a listener replaces a familiar P with one more appropriate
for classifying the stimuli heard. The cardinality of the
constructed P will depend upon the numbers of distinctions
required by the particular musical language or, if the stimuli
are not musical, upon the fineness of discrimination required by
the recognition task to be performed.
Once an appropriate P has been found, the next step is to locate a
stimulus within it...
Given any "interval" (pair) in P, the listener is able to
recognize the possible positions its elements might occupy in P.
In effect, given any pair of elements he can mentally supply
(interpolate) a possible set of remaining elements of P which
satisfy the equivalence class and tuning he has learned. Such
an interpolation becomes unique after a sufficient number of
elements of P are heard. This is equivalent to the
identification of x (key) in a give P(x).
For example, the major triad [C E G] is not a sufficient subset
of the C-major scale since there are two other major scales
(namely G and F) in which it occurs. But C-major is the only
major scale which contains the four notes [G B D F], so the
"dominant seventh chord" is a sufficient subset of C-major. A
minimal set is a sufficient set with no sufficient subsets.
Thus, [G B D F] is not a minimal set, but [G B F] is, as there
is no major scale except C which contains [G B F], while each of
its proper subsets [G B] [G F] [B F] are included in some other
major scale (G, F, and F# respectively).
It is straightforward to verify that sufficient (and therefore
minimal) sets are invariants of equivalence. They depend only
on the equivalence class, not on the particular tuning.
A measure is developed for this locating...
Consider a language whose alphabet consists of n letters (or
phonemes). How many distinct words can be formed using this
alphabet? Certain restrictions may exist which can limit the
sequences of letters that can occur. The more distinct words
that can be formed whose length is less than or equal to some
maximal value, the more efficient the alphabet is said to be.
A similar situation applies when "words" are formed from
sequences of intervals. Since interval sequences are formed
from tone sequences, we consider sequences of the elements of
some P. Also, since no new intervals are formed when an element
is repeated, only non-repeating sequences will be considered.
Since we are here concerned only with properties deriving from
the structure of P, we will use the following criterion for the
termination of a "word" (other criteria apply when "motifs" are
considered): When all remaining elements of P are determined by
a sequence of some of its elements, the addition of elements
will impart no further information of this type, and the "word"
will be considered terminated. That is, any sequence will be
considered complete as soon as a sufficient set occurs in it.
We now ask, given a particular equivalence class, how many
distinct words can be formed using k elements where k varies
from 1 to n. Consider all non-repeating sequences of n points
(there are n! such sequences). Let S_i be the number of
elements in each such sequence which must appear before a
sufficient set is encountered. The F(P) is defined as the
average,
n!
Sigma(S_i) / n!
i=1
F(P) may be interpreted as the average number of elements in a
non-repeating sequence of n elements of P(x) required to
determine the key, x. Efficiency, E, is defined as F(P)/n and
Redundancy, R, as 1 - F(P)/n. Both numbers lie between 0 and 1.
It should be noted that this kind of efficiency and redundancy
differs from the meanings these terms assume in information
theory. The distinction is important and applies to alphabets
in spoken natural languages as well as to musical scales. The
redundancy of information theory refers to a redundancy in the
message, not in the code. In our discussion here, that property
of the code which determines whether efficient messages _can_ be
constructed (if such are desired) is considered. This property
is inherent in the code itself, and does not apply to the
message.
Rothenberg outlines six scenarios...
Scale type Stability Efficiency
(a) proper high high
(b) proper high low
(c) proper low high
(d) proper low low
(e) improper ---- high
(f) improper ---- low
And discusses them...
Notice that in Figure 1, all scales in 12tET with which we are
most familiar (the major, minor, and Chinese pentatonic) conform
to situation (a). In fact, the major scale (of which the
"natural" minor is a mode) is far higher in both stability and
efficiency than any other 7-tone scale. Next among 7-tone
scales is the "melodic" minor (2,2,2,2,1,2,1). The Chinese
pentatonic stands out among scales of 5 and 6 tones.
However, situation (b) applies to many scales with which we are
familiar, such as the "whole-tone" and "12-tone" scales. Note
that while these are strictly proper scales, from the hearing of
a sufficient set (any element) alone, it is not possible to code
the elements of P into scale degrees. That is, although PxP is
coded by the proper mapping, there is no way to index elements
of P except by arbitrary choice. __Thus, since in these cases
intervals are coded but tones are not, composition with these
scales must involve relations which make use of motivic
similarities rather than relations between scale degrees.__
Hence the tone row basis of 12-tone music (which is essentially
motivic in concept) is not surprising. An examination of
Debussy's whole-tone piano prelude "Violes" show similar motivic
dependency.
Now consider improper scales. PxP is not coded except by the
employment of proper subsets or a fixed tonic. __Hence
information is primarily communicated by the scale degrees.__
Thus it is important that P be coded as quickly as possible,
which is indicated by a high redundancy (low efficiency) as in
case (f). It would be expected that scales characterized by
case (e) would be extremely difficult to use, except when the
tonic is fixed by a drone or similar device and, in fact, we
have not discovered such scales in any musical culture examined
thus far. In general, the use of motivic sequences on different
scale degrees of improper scales would not be expected (except
within proper subsets of such scales), and this is strongly
supported by examination of Indian and other music using
improper scales.
We would also expect that proper scales characterized by low
stability would tend to be used as improper scales, so that case
(c) would resemble case (e), and (d) resemble (f), and similar
remarks apply.
Makes cross-cultural observations...
In Java there exist two scale systems, "Slendro" and "Pelog",
each containing a variety of scales. It has been observed that
all scales in the "Slendro" class are strictly proper and that
all in the "Pelog" class are improper. In a study conducted
with the assistance of Mr. Surya Brata of the Ministry of
Education and Culture, Jakarta, the uses of these scale systems
were observed to be in accord with the predictions of this
model.
Here are references to the Javanese study...
Kunst, J. (1949), _Music in Java_, The Hague; Martinus Nijhoff.
Hood, M. (1954), _The Nuclear Theme as a Determinant of Patet in
Javanese Music_, Groningen, Djakarta: J.B. Wolters.
Hood, M. (1966), "Slendro and Pelog Redefined", Selected
Reports, Institute of Ethnomusicology, University of California
at Los Angeles.
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