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M U S I C T H E O R Y O N L I N E
A Publication of the
Society for Music Theory
Copyright (c) 1994 Society for Music Theory
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| Volume 0, Number 9 July, 1994 ISSN: 1067-3040 |
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All queries to: mto-editor@husc.harvard.edu
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AUTHOR: Rahn, Jay
TITLE: From Similarity to Distance; From Simplicity to
Complexity; From Pitches to Intervals; From Description to
Causal Explanation
KEYWORDS: similarity, proximity, structure, simplicity,
complexity, pitch, interval, psychoacoustics, Gestalt,
behaviorism, psychology, perception, logic, definition,
theorem, proof, postulate, John Rahn, Jay Rahn, Nelson
Goodman, nominalism, individual, part, whole, platonism,
number, set, description, explanation
Jay Rahn
York University (Canada)
Atkinson College
Fine Arts Department
4700 Keele Street
North York, Ontario M3J 1P3
relation@vm1.yorku.ca
ABSTRACT: Words and phrases specifying similarity and
distance abound in musical discourse. This essay explores
such pitch-predicates as "matches", "is in the vicinity of, and
"is closer to ... than to". Sought here are ways in which pitch
and pitch-interval predicates might be inter-connected logically.
Within a nominalist framework, an orderly progression from pitch-
to pitch-interval-predicates can be proven and interpreted
in terms of simplicity. Also indicated are connections
between the present formulation and issues of (i) dimension-
ality and explanation in psychoacoustics, Gestalt psychology,
and behaviorism, and (ii) nominalism and the "numerological
fallacy" in music theory.
KEY TO LOGIC SYMBOLS:
(x)(....) = for any thing, x, ....
.... <--> .... = .... if and only if ....; .... iff ....
.... v .... = .... and/or ....; (vel; inclusive or)
.... . .... = both .... and ....
-.... = not ....; it is not the case that ....
.... --> .... = if ...., then ....
(:Ex)(....) = there is at least one thing, x, such that ....
...+... = ...plus...; all the parts of individual..., as well
as all the parts of individual... (including any parts
common to both ... and ...); the sum of individuals ...
and ... (which is, itself, an individual)
[0] INTRODUCTION
[0.0] Words and phrases specifying similarity and distance
abound in musical discourse. For example: (i) two notes or
tones (or even passages or pieces) are often characterized as
matching, being the same, like each other, or similar in pitch
(or pitch-structure), and (ii) pairs of sounds are frequently
described as differing, contrasting, or being remote, far
apart, or distant -- all these, in varying degrees or amounts
(e.g., somewhat, extremely, etc.). Between twosomes that are
precisely the same in some respect, and those which are
exceedingly distant, one recognizes intermediate cases. One
asserts, for instance, that two sound-events, a and b, are (i')
next, or adjacent, to each other (i.e., neighbours, in some
sense), or (i'') closer, or nearer, to one another (i.e., than a
and/or b are/is to yet another sound-event, c).
[0.1] "Common sense" acknowledges as orderly, or linear, a
progression from sameness (i), through adjacency (i'), to
relative proximity (i''), and onward to remoteness (ii). Things
that do not match, differ. Of those that differ, the closest are
those that are adjacent. And among non-adjacent things, one
can distinguish degrees of distance. Despite the continuity of
this progression, common sense also acknowledges that
sameness is "qualitative," whereas distance is "quantitative."
Nonetheless, one might contend that things which are
relatively similar are also relatively close in some
"dimension" (e.g., pitch), and that, within a single dimension,
far-apart things are very dissimilar.
[0.2] Particular traditions of musical thought (e.g., in
psychoacoustics) comprehend similarity and distance as polar
opposites along single continua (e.g., scaled or gradated in
difference-limens or mels: cf. Stevens and Davis 1983, 76-98).
By contrast, other lines of musical inquiry (e.g., inspired by
Gestalt psychology: cf. Koehler 1947, 84-85 and 117-18)
dichotomize in this regard, understanding similarity (or
sameness) and proximity (or distance) as distinct principles
of perceptual organization or grouping.
[0.3] The present study probes the notion that, in such a
musical dimension as pitch, there might be continuity between
similarity and distance. The investigative strategy undertaken
here is to pursue, until an impasse is reached (if at all), the
possibility that there might be an unbroken progression from
similarity to distance. In the development that follows, I try
to clarify ways in which one might distinguish (or not) pitches
from intervals. Attempted is the formulation of a common
groundwork for dealing with similarity and distance in pitch.
As far as possible, pitch- and interval-predicates are defined
in terms of a single, maximally economical, basic, primitive
predicate, so that (i) necessary, logical connections between
predicates can be proven, or (but only if need be) postu-
lated -- albeit, one seeks, in a maximally coherent manner, and
(ii) stages or gradations between similarity and distance (e.g.,
comprising next-to and closer-to relations) can be identified
clearly. A further tactic adopted in this account construes
predicates of similarity and distance in terms of simplicity
and complexity (i.e., "structure") and emphasizes, in this
connection, relations between parts and wholes and between
statements that can be true of a single thing and statements
that only can be true of more than one thing.
[0.4] The formulation of pitch that follows does not involve
numerical modeling (i.e., as such). In this way, an effort is
made here to prevent, or at least forestall, systematically and
fundamentally, lapses in discourse that might produce instan-
ces of the "numerological fallacy." As John Rahn notes at the
outset of his exposition of the "integer model of pitch" (1980,
19 -- John Rahn's emphases asterisked: *...*):
... all sorts of things can be proven true of integers -- see any
book on number theory. It does *not* follow that, because we
are using integers to name pitches (or grapes, [etc.]), all those
things that are true of integers are going to be true of pitches
(or grapes, [etc.]). We must carefully determine the limits of
similarity between integers (with their structure) and pitches
(with their possible structures). To do otherwise would be to
fall into the *numerological fallacy*.
[0.5] To this end, instead of being framed in terms of numbers,
the following formulation is cast in terms of things that are
neither numbers nor sets; that is, what follows is cast in a
nominalist outlook (for which, see Goodman 1966) and accor-
dingly framed in terms of "individuals," as well as predicates
that specify relations between, or among, such individuals. In
this sense, general music-theoretical traditions that form an
immediate background to the present study are Aristoxenian
rather than Pythagorean, nominalist rather than platonist.
[0.6] An idea that guides the present exposition is that two
things might constitute a relatively simple whole, and, corres-
pondingly, be relatively similar, or close, to one another, to
the extent that they are described in terms that can be used to
describe a single thing. Initial stages of the subsequent ac-
count involve relations of matching, adjacency, and proximity,
and explore a sense in which these correspond, respectively, to
situations of increasing complexity.
[1] MATCHING
[1.0] If two things match in pitch, they form a simpler pair
than if, all other aspects being the same, they had differed in
pitch. The criterion, or benchmark, is a single, inherently
pitched thing, which (i) cannot differ, in pitch, or pitch-wise,
from itself, and (ii) if inherently pitched, necessarily is the
same, in pitch, as itself. Concerning (ii), one can define an
inherently pitched thing as follows (1):
DEFINITION: (x)(IPx <--> xPRx)
For any thing, x, x is inherently pitched if and only if x is
pitch-related to x (i.e., to itself).
========================================================
(1) In this and following formulations, such a phrase as "is
inherently pitched" can be replaced by such phrases as "is
heard as inherently pitched", "is heard as being inherently
pitched", etc. -- see below. Note also that inherent pitchedness
here differs from pitchedness, which, in Jay Rahn 1992, 165,
is a "property" of any thing, x, if and only if there is at least
one thing, y, such that x is pitch-related to y. The subsequent
theorems claimed in the latter study which depend on this
definition of pitched (P) things can be proven for inherently
pitched (IP) things.
========================================================
[1.1] The two-place predicate "is pitch-related to" can be
defined in the following way:
DEFINITION: (x)(y)(xPRy <--> xAHy v yAHx)
For any thing, x, and any thing, y, x is pitch-related to y if
and only if x is at least as high as y and/or y is at least as
high as x (cf. Jay Rahn 1992, 164-65).
[1.2] The predicate "matches, in pitch," can be defined as
follows:
DEFINITION: (x)(y)(xMPy <--> xAHy . yAHx)
For any thing, x, and any thing, y, x matches, in pitch, y if
and only if x is at least as high as y and y is at least as high
as x (cf. ibid., 167).
[1.3] From these definitions, one can prove the following
theorem:
THEOREM: (x)(IPx <--> xMPx)
Proof: (x)(IPx <--> xPRx)
(x)(xPRx <--> xAHx v xAHx)
(x)(xAHx v xAHx <--> -(-xAHx . -xAHx))
(x)(-(-xAHx . -xAHx) <--> -(-xAHx))
(x)(-(-xAHx) <--> xAHx)
(x)(xAHx <--> xAHx . xAHx)
(x)(xAHx . xAHx <--> xMPx) -- i.e., MP is reflexive for any
inherently pitched thing.
[1.4] The other portion (i) of the criterion arises directly
from the following definition for pitch-difference:
DEFINITION: (x)(y)(xDPy <--> xPRy . -xMPy)
For any things, x and y, x differs, in pitch, from y if and only
if x is pitch-related to y and x does not match, in pitch, y (cf.
ibid., 172).
[1.5] As a relation, pitch-matching is proven (above) to be
reflexive for any inherently pitched thing whatever. As well,
pitch-matching has been proven, in an earlier study, to be
symmetric for any pair of things whatever (i.e., pitched or non-
pitched, inherently so, or not: ibid., 168):
THEOREM: (x)(y)(xMPy <--> yMPx)
[1.6] Additionally, it can be proven that if any two things, x
and y, match, in pitch, then each is pitch-related to the other
even if "they" are precisely the same thing (i.e., even if x=y):
THEOREM: (x)(y)(xMPy --> xPRy . yPRx)
[1.7] If, amidst the "booming, buzzing confusion" of Nature, one
acts in an AH-manner, that is, if one "hears" certain things "as"
being at least as high as others (or even as themselves), an
immediate consequence of the energy expended in such an act
of hearing is that one's world divides into things that are
entirely unpitched and things that are inherently, and/or non-
inherently, pitched. The single inherently pitched things are
heard as matching themselves in pitch. Such singletons consti-
tute a template, or model, for pitch-simplicity, or pitch-
singleness; pitch-matching necessarily holds within inherently
pitched things (i.e., "severally", e.g., within each of the two
inherently pitched things of a pair) but might not hold between
or among them (i.e., "jointly", e.g., between the two inherently
pitched things of such a pair). Two inherently pitched things
that match pitchwise constitute a simpler pair than two that do
not match in pitch, at least with respect to pitch, all other
factors being equal.
[1.8] "A pitch" can be regarded merely as a sum of all, and only,
certain things that match each other in pitch (cf. ibid., 165 on
"pitch-identity wholes" and Goodman 1966 on sums of indivi-
duals, which, as sums, are, themselves, individuals). Unless
there were at least one instance of non-matching, or diffe-
rence, in pitch, between two of them, then all pitched things
would be heard as matching in pitch. In such a situation of
universal non-differentiation in pitch (i.e., obtaining between
any and all pitched things), all pitched things would be heard
as "parts" of "a" single "pitch". That is, only "one pitch" would
be heard and would comprise all, and only, the pitched things.
[1.9] Behaviorally, however, it is generally advantageous for a
listener that hears pitchwise to hear with optimum pitch
acuity (e.g., relative to an immediate biological niche), that
is, to hear as few things as possible as matching in pitch. To be
sure, each inherently pitched thing necessarily matches itself
in pitch. But in such an instance, pitch-matching is just
another name for pitched-ness. Pitch-matching relations are not
effectively significant, or important, for a listener, unless
they hold between non-identical things (i.e., between, for in-
stance, acts of hearing, x and y, where -(x=y)).
[2] VICINITY
[2.0] In the present formulation, every inherently pitched thing
is regarded as being "in its own pitch vicinity." As well, all
things that match each other in pitch, whether inherently
pitched or not, are held to be pitch "neighbors." This general
sense of pitch-neighborhood or -vicinity is conveyed as follows:
DEFINITION: (x)(z)(xIPVz <--> xNMTJHTz v zNMTJHTx)
For any things, x and z, x is in the pitch-vicinity of z
if and only if x is no more than just higher than z and/or z is
no more than just higher than x.
DEFINITION: (x)(z)(xNMTJHTz <--> xAHz . -(:Ey)(xHTy . yHTz))
For any things, x and z, x is no more than just higher than z if
and only if x is at least as high as z and there is no thing, y,
such that x is higher than y and y is higher than z.
DEFINITION: (x)(y)(xHTy <--> xAHy . -yAHx)
For any things, x and y, x is higher than y if and only if x is
at least as high as y and y is not at least as high as x.(2)
========================================================
(2) As defined here, IPV is more general (i.e., is less determi-
nate) than NP ("is next, in pitch, to"), as defined in ibid.,
172. E.g., for any things, x and y, xIPVy might hold even if
xHTy, but xHTy excludes the possibility of xNPy.
========================================================
[2.1] It can be shown that xMPz is a special case of xIPVz only
if there is no "intervening" thing, y. Such a situation would
arise in an instance of "Shepard's tones" (cf. Shepard 1964),
where, arranged in "descending" semitones, the first might be
heard as matching the thirteenth, and yet as higher than the
twelfth, which would be heard as higher than the thirteenth.
Important to emphasize is that the "illusion" of Shepard's tones
depends on temporal succession and is not merely a matter of
pitch, as might be a (hypothetical) Shepard's "sonority". In the
present formulation, matching in a vicinity is linear, not
cyclic, and a pitch-vicinity comprises not only pitch-proximity,
but concomitant temporal closeness too.
[2.2] If xIPVz and x does not match z pitchwise, then xHTz or
zHTx. Such cases are similarly linear, and involve always an
HT-relation. Such an HT-relation constitutes a significant arti-
culation in the continuity from matching to distance, for no
thing whatever can be higher than itself. In this way, vicinity-
relations straddle singleness and multitude.
[2.3] The present sort of distinction, i.e., between matching and
vicinity, arises "for free," as it were, once such a predicate
as AH is let loose in the world. The differences in definition
between matching and vicinity involve, in their respective
formulations, merely differences in their patterning of quan-
tifiers, conjunctions, modifiers, individual-variables, and the
AH predicate. If one hears in an AH manner, opportunities to
make such a distinction can arise (if the world is, in fact,
truly characterized according to first-order logic).
[2.4] Defining vicinity relations widens the net of simplicity
that can be caught in a formulation. Single inherently pitched
things (even single inherently pitched sums of individuals,
which are themselves individuals) supply a criterion for
asserting the simplicity of pairs of things, whether the things
are individuals and/or sums of individuals, and whether the
pairs are, pitch-wise, both matching and neighbors, or merely
neighbors. The relatively weak, indeterminate specification
that pitch-vicinity things (x and z in the definition) need
merely be pitch-related (insofar as xAHz and/or zAHx), rather
than, say, different in pitch (i.e., by virtue of one being
higher than the other), widens the net considerably.
[3] INTERIORITY
[3.0] Concerning the following succession of letters: a b c,
one can say, informally, that a is closer to b than (it, i.e., a,
is) to c, and conversely, that c is closer to b than to a. As
well, informally, b is closer to a than a is to c, b is closer
to c than c is to a, b is closer to the sum of a and c (i.e.,
a+b) than a is to c, and so forth. Abbreviating the first two
statements as aCb, and cCb,a, respectively, one can characterize
closeness relations in the larger succession: a b c d, as follows:
aCb,c; aCb,d; aCc,d; bCc,d; cCb,a; dCb,a; dCc,a; dCc,b; aCb+c,d,
and dCb+c,a.
[3.1] A corresponding pitch-predicate, "is, in pitch, at least
as close to ... as to", can be defined as follows:
DEFINITION: (x)(y)(z)(xPACy,z <--> xPOSy,z)
For any things, x, y, and z, x is, in pitch, at least as close to
y as (x is) to z if and only if x is, in pitch, on the opposite
side of y from z.
DEFINITION: (x)(y)(z)(xPOSy,z <--> xHSy,z v zHSy,x)
For any things, x, y, and z, x is, in pitch, on the opposite side
of y from z if and only if x is on the high side of y from z
and/or z is on the high side of y from x.
DEFINITION: (x)(y)(z)(xHSy,z <--> xAHy . yAHz . xAHz)
For any things, x, y, and z, x is on the high side of y from z if
and only if x is at least as high as y, y is at least as high as
z, and x is at least as high as z.(3)
========================================================
(3) As defined here, PAC is more general than NICP ("is non-
intervallically closer, in pitch, to ... than to") in ibid.,
172-73.
========================================================
[3.2] Pitch-closeness and pitch-sidedness of this sort can be
formulated in terms of things that comprise, or include,
pitchwise, other things (or themselves), as follows:
DEFINITION: (x)(y)(z)(x+zCPy <--> xPOSy,z)
For any things, x, y, and z, the sum of x and z comprises, pitch-
wise, y if and only if x is, in pitch, on the opposite side of y
from z.
[3.3] The partially-ordered character of pitch-comprising
relations can be provided for in terms of pitch-interiority (or
pitch-insideness):
DEFINITION: (w)(x)(y)(z)(x+yPIw+z <--> w+zCPx . w+zCPy)
For any things, w, x, y, and z, the sum of x and y is, pitchwise,
inside, or interior to, the sum of w and z if and only if the sum
of w and z comprises, pitchwise, x, and the sum of w and z
comprises, pitchwise, y.
[3.4] As well, one can prove that things which form pitch-,
pitch-matching, and pitch-vicinity relations constitute
pitch-interiority relations, but not necessarily *vice versa*.
For example, within a pitch-interiority framework, possible situ-
ations involve xHTy, yHTz, and xHTz, and xHTy, yHTz, and xMPy
(the latter corresponding to a moment of "dis-illusion-ment" in
hearing Shepard's tones).
[4] POSSIBLE STEPS TOWARD INTERVAL PREDICATES
[4.0] Interiority relations exhaust the farthest reaching possi-
bilities for specifying degrees of pitch-distance within the
AH-dimension. Whereas one can acknowledge (i.e., for any w, x,
y, and z -- see above) that w+z is more inclusive than w+y or
x+z, one cannot specify, in the most general way, whether w is
more distant from y than x is from z, or w is farther from x
than y is from z, or w forms a larger interval with x than x
forms with y, etc., except, for example, by (i) specifying that
all non-matching vicinity-pairs are equidistant (i.e., taking
non-matching vicinity as the "unit" or "degree" of pitch
gradation), or (ii) resorting to a predicate other than AH.
[4.1] Plausible predicates to perform such functions include "is
at least as large as", "is at least as much larger than ... as
... is than", "is at least as much higher than ... as ... is
than", and "is at least as much larger than its next to largest
part, as ... is than its next to largest part" -- cf. Jay Rahn
1994b, where the arguments might be individual-variables (e.g.,
w, x, y, z, above) or sums of individual-variables, which are
themselves individual-variables (e.g., w+x, w+y, ..., above).
Each such predicate can be regarded as introducing a novel
"dimension" into a formulation (e.g., a dimension of pitch-inter-
val, pitch-proportion, or pitch-proportionateness -- as distin-
guished from pitch). And each can yield matching, vicinity, and
interiority relations in its respective dimension.
[4.2] Alternatively, one could "count" overlapping vicinity-
pairs by, for example, (i) defining a special case of pitch-
vicinity, namely, discrete pitch neighbor-hood:
DEFINITION: (x)(y)(xDPNy <--> xIPVy . xDPy)
For any things, x and y, x is a discrete pitch-neighbor of y
if and only if x is in the pitch vicinity of y, and x differs, in
pitch, from y, and (ii) applying the label "pairwise twofold" to
the sums of such discrete-vicinity pairs as w+x and x+y (i.e.,
w+x+y) and x+y and y+z (i.e., x+y+z), the label "pairwise three-
fold" for the sum of such a pair as w+x, x+y, and y+z (i.e.,
w+x+y+z), etc. Such a formulation could suffice in certain situa-
tions (e.g., where the semitone functioned as the DPN unit, in
total-chromatic pieces, or passages). However, much music, if not
most, is not totally chromatic. Instead, diatonic and pentatonic
works, for example, are generally "gapped" (i.e., relative to the
twelve-semitone collection, or aggregate). In order to specify
that, for instance, e-f was half as large as f-g, or as much
smaller than f-g as g-b-flat was than b-flat-d, would require
such a postulate as the following:(4)
POSTULATE: (x)(y)(z)(xDPNy . yDPz . zPOSy,x . -yDPNz <-->
(:Ey')(y'POSy,x . zPOSy',y . yDPNy'))
========================================================
(4) Postulates are regarded here as asserting the existence of
at least one thing, whereas definitions do not make such an
ontological claim (cf. Goodman 1961, 6 -- or Goodman 1972,
343-44).
========================================================
[4.3] Nonetheless, among things that are partially ordered in
pitch, one can specify certain degrees of proximity based on
discrete-vicinity, or "unit," relations -- for example, as
follows:
DEFINITION: (x)(y)(z)(x+zJLPy+z <--> xDPNy . zPOSy,x)
For any things, x, y, and z, the sum of x and y is just (i.e., by
one "unit") larger, in pitch, than the sum of y and z if and only
if x is a discrete pitch-neighbor of y, and z is, pitchwise, on
the opposite side of y from x.
[5] CHALLENGES OF NOMINALISM
[5.0] As Nelson Goodman indicates (1966, 41), "To reconstruct
in the language of individuals [i.e., as distinguished from sets
and numbers, as such] all of mathematics that is *worth
saving* is a formidable task that need not concern us here. It
will be enough to consider typical arithmetical statements
used in ordinary discourse [my emphasis]." At the conclusion of
his subsequent preliminary survey of ways in which mathema-
tical statements can be "de-numerated", and "dis-membered" (my
terms -- to which one could add "de-generated"), Goodman
stresses (1966, 45) that the "effort to carry out a construc-
tive nominalism is still so young that no one can say exactly
where the limits of translatability lie. We have seen above
that some statements that look hopelessly platonistic yield to
nominalistic translation and the full resources available to the
nominalist have not by any means been fully exhausted as yet."
[5.1] More recent studies have attempted non-numerical ren-
derings of mathematics on quite a large scale (notably, Field
1980 and Hellman 1989, the former not without controversy).
However great their eventual success might be, attempts at
modeling music by means of nominalistic formulations, or,
alternatively, by means of the mathematics of sets and
numbers, should be assessed not only systematically and
philosophically, but also by considering seriously what is
"worth saving" in accounts of music. Arguably, pitchedness
(inherent or not), matching, vicinity, and interiority, none of
which presumes an intervallic dimension, i.e., distinct from
the AH-dimension, are worth saving. Nonetheless, after more
than two millenia of music theory, it is still not entirely clear
just what else is worth saving for a theory of pitch -- or how
it might best be saved.
[6] FROM DESCRIPTION TO CAUSAL EXPLANATION
[6.0] Quite surely, insights of Gestalt psychology into musical
structure are worth saving in music theory. However, whereas
Gestalt approaches provide valuable descriptions of musical
activity, the explanatory status of such accounts is generally
questionable (cf. Skinner 1974, 29 and 71-75 on "topography"
or mere description, as contrasted with causal explanations of
behavior). Nevertheless, the Gestalt principles of similarity
and proximity, drawn here into a single account, can be
outfitted with causal force by construing as reinforcing the
sorts of simplicity considered in the present study (cf. also
Jay Rahn 1994a).
[6.1] For example, things (i.e., stimuli) that are heard as
matching pitchwise can be considered to constitute the
immediate reinforcers of acts (i.e., responses) of hearing
things, in general, in an AH-manner. By virtue of being heard as
matching in pitch, such stimuli as x and y immediately become
a pair of things that have been heard as matching pitchwise.
Such a pair is a reinforcing stimulus, or reinforcer, for the
relevant acts of "hearing as." Such acts can be designated x'
and y', as in the following, behavioral postulate:
POSTULATE: (x)(y)(xHAHy <--> (:Ex')(:Ey')(x'Hx . y'Hy . x'AHy'))
For any things, x and y, x is heard as being at at least as high
as y if and only if there is at least one thing, x', and there
is at least one thing, y', such that x' is a hearing of (i.e.,
an act of hearing, or an auditory response to) x, y' is a
hearing of y, and x' is at least as high as y'.
[6.2] The latter postulate distinguishes between stimuli and
responses and can be considered to occupy a territory that
straddles descriptive music theory and the causal formulations
of behaviorism. In this postulate, the HAH- and AH-predicates
are morphologically identical. Each merely orders pairs of
things; that is, neither involves presumptions, or axioms, of
reflexivity, symmetry, etc., and both presume only that the
distinction between xy and yx might be significant within its
respective (HAH- or AH-) "dimension". Accordingly, one can
reason about HAH-things in a manner quite parallel to ways, out-
lined above, for reasoning about AH-things.
[6.3] The preceding postulate can be replaced and extended in
significance by the following:
POSTULATE: (x)(y)(xMPy <-->
(:Ex')(:Ey')(x'SHx . y'SHy . x'HAHy' . y'HAHx' . x'+y'Rx+y))
For any thing, x, and any thing, y, x matches, in pitch, y if and
only if there is at least one thing, x', and there is at least
one thing, y', such that x' stimulates the hearing of x, y'
stimulates the hearing of y, x' is heard as being at least as
high as y', y' is heard as being at least as high as x', and the
sum of x' and y' reinforces the sum of x and y.
[6.4] Fashioning nominalistically an account of reinforcement
presents formidable challenges (cf. Jay Rahn 1993). Among
these is distinguishing between earlier and later instances, or
portions, of stimuli and responses. That reinforcement often
develops in a curvilinear manner can also be problematic. Just
as one can eat too much, beyond a certain point one can be
satiated by repetition and other kinds of similarity. As bore-
dom sets in, sorts of stimuli that formerly had been reinfor-
cing become aversive. One way of preventing something from
becoming "too much of a good thing" involves rendering it less
thing-like: less simple, less "singular". Between the extreme
possibilities of a world where (i) every inherently pitched
thing differed in pitch from every other and (ii) every (inhe-
rently or non-inherently) pitched thing matched pitchwise
every other (including itself), lies a region in which music has
specialized. Moreover, music has specialized in a world where
salience is sought and reinforced.
[6.5] Stimuli that are heard as pitchwise matching constitute
reinforcers of acts of pitchwise hearing. Stimuli that escape or
"e-lude" the net of matching-relation simplicity might be caught
by the discrete-vicinity net. Those that elude discrete-vicinity
might be trapped by relations of interiority, which can, in turn,
vary in their degrees of simplicity. Moreover, interiority-
simplicity can been shown to abound generally in the middles of
things, i.e., of individual-sums. That the highest degree of pro-
portionateness between things that differ in (e.g., pitch-inter-
vallic) size arguably obtains between a thing and its (precise)
half indicates the possibility of coherent continuity in a pro-
gression of simplicity reinforcement that might extend beyond AH-
relations into the "truly intervallic". And that highest and
lowest things stand out, are salient, or "edgy", not only derives
from the general paucity of adjacency or interiority relations
in which they participate but also renders them "excellent,"
"privileged," "prime" candidates for "resolution," that is, for
being heard, by way of other, simplifying relations, as (proper)
parts of relatively simple, reinforcing wholes (e.g., as the
soprano-bass "skeleton" of a complex texture, or as the "exo-
skeleton" (my term) of a "contour" -- on which see Morris 1993).
REFERENCES CITED
Field, Hartry. 1980. Science without Numbers: A Defence of
Nominalism. Princeton, NJ: Princeton University Press.
Goodman, Nelson. 1961. Science and Simplicity. Washington, D.C.:
Voice of America (repr. *In* Nelson Goodman. 1972. Problems and
Projects. Indianapolis: Bobbs-Merrill. 337-46).
______. 1966. The Structure of Appearance. Indianapolis: Bobbs-
Merrill.
Hellman, Geoffrey. 1989. Mathematics without Numbers: Towards a
Modal-Structural Interpretation. Oxford: Clarendon Press.
Koehler, Wolfgang. 1947. Gestalt Psychology. New York: Liveright
(repr. New York: New American Library).
Morris, Robert D. 1993. "New Directions in the Analysis of Musi-
cal Contour." Music Theory Spectrum 15, 2: 205-28.
Rahn, Jay. 1992. "An Advance on a Theory for All Music: At-Least-
As Predicates for Pitch, Time, and Loudness." Perspectives of New
Music 30, 1: 158-83.
_____. 1993. "A Nominalist Formulation of Basic Terms in Radical
Behaviorist Theory." unpub. ms. Toronto: the author (formulations
distributed in 10pp. handout of Jay Rahn. "Imaginary Entities in
a Phenomenal Theory of Music." paper presented at Annual Confe-
rence of Society for Music Perception and Cognition. University
of Pennsylvania. Philadelphia. June).
______. 1994a. "Outline of a Causal Theory of Music." paper pre-
sented at Graduate Music Theory Colloquium, Eastman School of
Music. Rochester. February.
______. 1994b. "A Non-Numerical Predicate of Wide Applicability
for Intervallic Relations in Music." paper presented at the Sim-
posion International de Muzicologie: Muzica si Matematica, Bucha-
rest. May.
Rahn, John. 1980. Basic Atonal Theory. New York: Longman.
Shepard, Roger N. 1964. "Circularity in Judgments of Relative
Pitch." Journal of the Acoustical Society of America 36, 2346-53.
Skinner, B.F. 1974. About Behaviorism. New York: Vintage.
Stevens, Stanley Smith and Hallowell Davis. 1983. Hearing: Its
Psychology and Physiology. 2nd ed. New York: American Institute
of Physics for the Acoustical Society of America.
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