Yahoo Tuning Groups Ultimate Backup tuning Test: hard returns, 79 characters (paraphrase of some Rothenberg)

Sorry for the bandwidth folks, but I'd like to perform a test. Along the
way, anybody who cares should wind up with at least one copy of this
Rothenberg stuff that he can read and print with a clear conscience....

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.

>From: Carl Lumma <clumma@nni.com>

>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?

I know these references very well. I'm still not clear on what exactly
makes a scale "proper," but, as he says, both pelog and slendro encompass a
variety of scales. These "scales" can vary considerably from one set of
instruments to another, so it's difficult to make sweeping generalizations.

Kunst 1949 and Hood 1966 give some tuning data for certain gamelan. In his
pioneering ethnomusicological study of many aspects of Javanese music (not
just tuning), Kunst gives just the middle octave of a single instrument of
a select number of gamelan. Hood (his student) found this approach
insufficient because of the stretching and compression of octaves. That's
mainly what his 1966 article is about. The 1954 book (basically his
dissertation) is mostly an empirical study of how melodic formulae help
define the patet (roughly, "modes") of Javanese music. However, the largest
set empirical data of gamelan tunings is in:

Surjodiningrat, Wasisto, P. J. Sudarjana, and Adhi Susanto. Penjelidikan
dalam pengukuran nada gamelan-gamelan djawa terkemuka di Jogjakarta dan
Surakarta. Yogyakarta: Gadjah Mada University Press; translated by the
authors as: Tone measurements of outstanding Javanese Gamelan. Yogyakarta:
Gadjah Mada University Press, 1969, rpt. 1972, 1993.

One schema into which pelog tunings have been forced is that they are
tempered versions of a subset of 9TET. I have seen this hypothesis,
together with a kind of cycle of "fifths," proposed by both Javanese and
Western writers. Dan Wolf had some very interesting speculations along
these lines in a Xenharmonikon article, though I remain skeptical about the
9TET connection.

Though slendro approaches 5TET, very closely in some cases, the differences
are very deliberate and give each version of slendro a different quality.
Tuners and musicians describe these qualities as favoring one patet over
another, thus suggesting that the tunings are being tempered in various
ways to favor (or flavor) different modes, much as irregular temperaments
of the European Baroque favored different keys to a greater or lesser
degree.

Bill

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Test: hard returns, 79 characters (paraphrase of some Rothenberg)

🔗Carl Lumma <clumma@xxx.xxxx>

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