paraphrase of some Rothenberg

The model concerns a code that can be used to extract information from huge
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 may
satisfy 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.

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.

Does anybody know anything about these references?

-C.