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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) 1996 Society for Music Theory
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| Volume 2, Number 1     January, 1996     ISSN:  1067-3040   |
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  All queries to: mto-editor@boethius.music.ucsb.edu or to
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AUTHOR: Rothgeb, John
TITLE: Re: Eytan Agmon on Functional Theory
KEYWORDS: harmony, function, scale degree, Riemann, Schenker
John Rothgeb
Binghamton University
Department of Music
Binghamton, NY 13902-6000
rothgeb@bingsuns.cc.binghamton.edu
ABSTRACT: A recent article by Eytan Agmon proposes a
modified version of the theory of harmonic functions
promulgated by Hugo Riemann.  It is argued here that the
proposed theory is superfluous unless the Schenkerian
conception of scale degree is trivialized beyond
recognition, and that (in any case) the reduction of seven
independent scale degrees to only three categories cannot be
reconciled with certain palpable musical effects.
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[1] Eytan Agmon's recent article "Functional Harmony
Revisited"(1) proposes a theory of functional meaning for
harmonic degrees related to that of Hugo Riemann but
differing from it by virtue of "greater theoretical rigor
and the removal of arbitrary features" (211).  As Agmon
explains, "the hallmarks of functionalism are: (1) the
characterization of individual chords as tonic (T),
subdominant (S), or dominant (D) in function; and (2) the
notion that the so-called primary triads, I, IV, and V
somehow embody the essence of each of these functional
categories."  These are the characteristics of functionalism
that Agmon wishes to preserve.(2)
====================================================
1.*Music Theory Spectrum* 17/2 (Fall, 1995), 196-214.
2.Among the features he is willing to abandon is Riemann's
notion of *Scheinkonsonanz*, the idea that secondary triads
are merely "apparent" consonances, each being accompanied
always, if only tacitly, by an associated "characteristic
dissonance."
=======================================================
[2] Agmon begins by situating functional theory in a larger
intellectual context, presenting it as, in effect, a special
case of what is known as *prototype theory*: "Indeed, one
way of stating the core idea of the present article is:
given a separation of chord progression from harmonic
function, the notions *function *and *primary triad* are
fully reducible to *category* and *prototype*, respectively"
(199).(3)  Note well what is stated here as a "given":
harmonic function is conceived as entirely separable from
chord progression.  We shall return to this presupposition
shortly.
=======================================================
3.Later (202), by analogy, "functional strength" is said to
reduce to "prototypicality."
=======================================================
[3] Agmon's theory is admirable in its simplicity.  Its
principal components are (1) a definition, based on note-
content, of *degree of triadic similarity* for diatonic
triads; (2) the specification of three "principles" that
"uniquely select the triads I, IV, and V as prototypes of
three harmonic categories. . ." (201); and (3) two
additional "principles" that "determine the additional
members of each category and their respective
prototypicalities."  Of the five "principles," two that
select prototypes and one that selects additional members
are described as "self-evident," a status that might be
granted one of the other two as well--namely the principle
that prototypes must not be maximally similar to each
other.(4)  The remaining principle, however--the
*principle of symmetry*, which states that the graphic
symmetry of Agmon's Fig. 2c, quoted here as example 1, "must
not be violated"--is not self-evident, nor is its
necessity established by Agmon on any persuasive independent
basis.
=======================================================
4.Agmon puts this too strongly in his statement that
"prototypes must be maximally dissimilar to each other . .
." (201).  I and IV, for example, are both prototypes, but
their degree of similarity by his measure is "intermediate"
rather than "minimal"; this must surely entail that their
degree of dissimilarity is less than maximal.
=======================================================
[4] The diagram in example 1 shows the harmony of II as
standing within the subdominant function, but also, just
slightly, within the dominant.  Agmon puts this into words
at a later point in his essay with the following assertion:
"although the function of II is primarily subdominant, a
weak dominant function nevertheless exists" (206).  Does
this mean that II can be both subdominant and dominant at
the same time? Apparently not: ". . .the dominant function
of II, I believe, may be felt in certain contexts where II
(or II^6^) is followed by I (or I^6^)."  Thus the previously
"given" separation of chord function from harmonic
progression is, at least in this instance, retracted.  As we
shall see later, it must be retracted not in this one
instance only, but, indeed, across the board.
[5] Probably the central core of functional theory (the part
that Agmon wishes to retain), unlike many of Riemann's
ideas, was not merely a case of "theory for the sake of
theory," but was rather a well-meaning attempt to respond to
problems posed by a wide variety of perceptual phenomena.
Let us examine one such phenomenon in detail.  Consider the
final cadence of Schumann's "Am Kamin" from *Kinderszenen*,
shown in example 2.  The three-note penultimate chord c - e
- a contains the notes of a III in F major, but has the
"aura" or in Agmon's term the "essence" of the dominant;
this elusive "aura" is, I suspect, what is sought to be
represented by the word "function" in functional theory,
most of whose practitioners would here assign the symbol D
for dominant.  Henceforth in this review I shall (in most
cases) enclose in quotation marks any Roman numeral that
represents literal pitch content but not "aura," which
latter entity will be designated by Roman numeral without
quotation marks.  Thus in Schumann's cadence, the "III"
*means* V.
[6] Why *does* this "III" *mean* V?  At least two reasons
can be adduced from the structure of Schumann's phrase,
which is displayed in example 3.  First, and probably most
important, the treble voice negotiates a fifth-progression
(a re-drawing, here in the coda, of the Urlinie descent).
The last passing tone in that fifth, the g of the
penultimate bar, moves to f at the end; the note a that
intervenes, far from obliterating the g, takes on a
subordinate role as a kind of embellishment or enhancement
of the fundamental stepwise progression g - f. To be exact,
it serves as what is variously termed an escape tone or an
incomplete upper neighbor, but is perhaps still more
precisely understood as an anticipation of the third of the
coming tonic harmony.  In any case, g implicitly but
effectively remains present as the fifth of c. The third-
space delineated by the succession a - f associates weakly
with the preceding one from b-flat to g (see the brackets).
[7] Secondly, an independent force is at work here, one that
may as well be called harmonic syntax, or the syntax of
scale-degree progression.  The simplified harmonic basis of
the passage is shown in example 4a.
[8] We may perhaps agree with functional theory that "the
so-called primary triads, I, IV, and V" are indeed primary
in some meaningful sense.  Example 4a is based on the
succession of these primary harmonies prescribed by the most
fundamental principle of their syntax: that of progression
by fifth.(5)  (Example 4b explains the origin of Schumann's
II as the result of extending the bass of IV and letting the
treble anticipate the fifth of the coming V.)  The
implication of the first falling fifth, f - b-flat, which
could cast doubt on the identity of the tonic, is set right
by the second one, c - f.  Given the construction of the
bass, people who hear musically will have a strong
predilection to hear this final fifth as representing V - I
even though its penultimate member does not bear the 5/3
sonority which alone would provide full congruence between
scale-degree meaning and vertical chord.  Add to this
predilection the melodic factors described above and the
penultimate chord c - e - a is heard as unmistakably
expressing the "aura" of the dominant.  *It is a harmonic
realization of the dominant scale degree*.  This means that
the note a *in no sense functions as a harmonic root* here.
The chord under discussion is *not* an inversion of an a-
minor triad. Although it contains the notes of the triad of
the mediant, if the Roman numeral as an analytic symbol is
to reflect aural qualities of the music as heard by a
perceptive listener rather than merely the appearance of the
notation, the Roman numeral III cannot, without the
accompanying "shudder-quotes" I have used here, accurately
be applied to it.
=======================================================
5.Schenker occasionally speaks evocatively of the
"Quintengeist der Stufen."  Agmon acknowledges "the
privileged status of certain root relationships, most
notably by descending fifth" (211).
=======================================================
[9] When I say that in this case "III" *means* V, the word
*means* may be explicated as "constitutes, or is included
within, a harmonic expression of."  "III" may equally well
*mean* I; "II" may *mean* IV; indeed, instances of "X"
*meaning* Y are legion in the repertoire of tonal music, and
virtually no a-priori limits can be set on the ranges of 'X'
and 'Y'.(6)
=======================================================
6.Certain limits would probably stand up under scrutiny.
Although "II" can, under certain circumstances, be the sole
constituent of an expression of IV, it is doubtful that "I"
and V, for example, could be so related.
=======================================================
[10] This peculiarity of the relationship of momentary note-
content to harmonic entities was grasped, to a large extent,
by Heinrich Schenker as early as 1906: ". . . not every
triad must be considered as a scale-step. . . ."(7)
Schenker's subsequent work might be construed as a massive
effort to explicate this perception, whose most concise and
complete modern verbal formulation has been provided by Carl
Schachter:  "There is no such 'thing' as a I chord in C
major, but only an idea that can find expression through the
notes C, E, and G in any kind of simultaneous blending,
through intervals created by two of these notes, through the
note C alone, through such combinations as C - E-flat - G, C
- E - G - A, and C - E - G - B-flat, through melodic lines
of the most various shapes, through whole constellations of
contrapuntal lines and chord successions controlled by the
note C."(8)  This improves on Schenker's clairvoyant but
underexplicit 1906 formulation in its recognition that the
relation between vertical note-combination and scale degree
is even far less intimate than Schenker's early statement
might suggest: altogether, it is not a note-combination but
an "idea" (or "aura" or "essence") that is designated by the
properly applied Roman numeral.
=======================================================
7.Heinrich Schenker, *Harmony* (ed. O. Jonas, trans. E.M.
Borghese; Chicago: University of Chicago Press, 1954), 139.
8.Schachter, "Either/Or," in H. Siegel, ed, *Schenker
Studies* (Cambridge: Cambridge University Press, 1990), 166.
=======================================================
[11] Unfortunately, however, this is not what Roman numeral
and scale degree mean to Agmon.  About Brahms's Intermezzo
Op. 117, No. 2, he writes that ". . . in the consequent
phrase, which begins in m. 9, the opening II^6^ chord
concludes the dominant prolongation which begins in m. 6 . .
." (208).  The chord referred to is the last chord of bar 9,
and it does indeed occur within a prolongation of the
dominant.(9)  To call it II^6^ -- merely to use the Roman
numeral in this way -- is to devalue a profoundly meaningful
analytic symbol by turning it into a mere mechanical
reduction of a trivial transliteration of note-content.
Agmon, it is clear, speaks the language of the completely
conventional harmony textbook.
=======================================================
9.The identical 6/3 chord in the upbeat to bar 1 harmonizes
a passing tone within a tonic prolongation.  Agmon applies
to it as well the functional symbol D--surely a gross
overburdening with harmonic significance of one of the most
elemental of contrapuntal phenomena: parallel motion in 6/3
chords.
=======================================================
[12] The theory of tonal music is thus effectively deprived
of the scale degree in Schenker's visionary conception of
it. Once the scale degree has been so devalued to identity
with vertical note-content, as it invariably is in Agmon's
article, the inevitable, and completely unacceptable, result
is a theoretical void.(10)  There is simply no longer any
available theoretical correlate for certain palpable musical
realities.
=======================================================
10.Riemann, of course, had to be completely innocent of
Schenker's breakthrough, and thus he cannot be accused of
having trivialized an earlier important theoretical insight.
Today matters are different, and it is discouraging that the
best insights in our discipline remain incompletely
understood.
=======================================================
[13] In certain contexts, function-theoretic analysis fills
this void in a way that is perhaps not objectionable.
Agmon's example 4a is given here as example 5.  It is quoted
by Agmon from Aldwell and Schachter's harmony textbook,(11)
with only the replacement of the latter's (completely
sufficient) "IV - I" by "S - T."  In such a case, functional
theory could be regarded as relatively benign.  There would
be no substantive objection to the replacement of the
symbols; after all, "IV" and "subdominant" are
interchangeable for almost all purposes.  For its raison
d'etre, however, functional theory would still be indebted
only to the trivialization of scale degree and Roman numeral
just described.
=======================================================
11.Edward Aldwell and Carl Schachter, *Harmony and Voice
Leading* (second edition, New York: 1989), 392.
=======================================================
[14] Functional theory goes much further, though, because it
insists that (for example) II *always* represents one of the
primary categories--usually S, less often D (see example
1).  How plausible is this claim when it is applied to
actual music--for example, to the opening bars of Haydn's
piano sonata Hob. 52, Finale (see example 6)?  The music of
bar 1ff. composes out the tonic scale degree, the I.
Beginning at the end of bar 8, the same diminution is
applied to the second scale degree, the II.  Clearly, Haydn
has moved up a step.  What is to be gained by insisting that
the resulting F-minor area stands for anything but scale
degree II--by claiming that it represents, for example,
the subdominant?  The justification for such a claim might
argue from the fact that this harmony, like the subdominant
so often, moves to V; if so, then all pretense of treating
function apart from progression would have to be
renounced.(12)  In some cases--chiefly when 2^ appears in
an inner voice, which is hardly the case here--it might be
said that II "sounds" (somewhat) like IV (at least to most
undergraduage students); this by no means justifies
depriving II of its independent place as a harmony in the
key.
=======================================================
12.For all that the similar behavior of II and IV is
conventional wisdom, analytic theory and analytic insight in
fact gain nothing by its affirmation, which is merely a
compromise convenient for certain pedagogical purposes.
=======================================================
[15] Worse still, to assert that this II "is" a version of
the subdominant would lead to certain bizarre results. It
would mean, for example, that the relation between the root
of the F-minor harmony in bar 9ff. of example 6 and the root
of the initial tonic is primarily to be understood as a
fifth rather than a second; and moreover, that the harmonic
progression in bars 16-17 is by second rather than by fifth.
This is where functional theory ceases to be benign and
becomes pernicious.
[16] Thus "II" *need not* represent IV.  It *may* do so, of
course, as has long been understood.  In case it does, the
explanation is to be sought in domains other than harmonic
theory.  A careful consideration of such a "II" *in its
context* will show that its constituent scale degree 2^ has
a linear mission (e.g. as a passing or neighboring
note).(13) Such a mission excludes any interpretation of 2^
as a harmonic root, and thus excludes an interpretation of
such a "II" as II.(14)
=======================================================
13.Here I draw attention particularly to Agmon's statement,
quoted earlier, that ". . .the dominant function of II. . .
may be felt in *certain contexts*. . ." (emphasis added).
It is not only "II", however, but *every* chord whose
meaning is ascertainable *only* with reference to all
features of its context.
14.It does not, however, exclude the possibilty that IV may
appear to "turn into" II in the course of prolongation. Such
a reinterpretation exploits the equivocality of the 6/3
chord, in which the 6 may in principle represent either a
linear element or the root of an inverted chord.  When such
a 6 is so reinterpreted as a root, it may be reincarnated as
the bass of a 5/3.  See Schenker, *Free Composition* (trans.
E. Oster; New York: Longman, 1979), p. 90, "Addition of a
Root."  Much complexity is added to this topic (a full
treatment of which exceeds the scope of this review) by the
relation between structural levels: a note that arises in
the background by a linear process may become a "root" in
the foreground.  This is fully analogous to the notion of
"key areas" as foreground "illusions."
=======================================================
[17] Agmon had no choice, therefore, but to retract (in at
least one instance) his postulate that "harmonic function"
can be treated apart from "chord progression."  He
represents the case in which he does retract it as special,
but in fact the circumstances that lead him to take "chord
progression" into consideration there are completely general
and equally present everywhere in music.  And it is not
merely "chord progression" but voice leading, meter and
rhythm, motif, and in brief all aspects of what Schenker
called *Auskomponierung* that must be considered in
assessing the function--harmonic and otherwise--of
chordal entities as they occur in music.  Functional theory
remains a superfluous appendage so long as we do not discard
what has been learned about music thus far.
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