## Music Theory Online## The Online Journal of the Society for Music Theory |

Volume 7, Number 5, October 2001

Copyright © 2001 Society for Music Theory

[1] Held every fourth summer since 1993, this summer's Symposium on Neo-Riemannian Theory remained true to its tradition as a "working group." In particular, the Organizing Committee (David Clampitt, John Clough, Chair, Richard Cohn, and Jack Douthett) arranged for the papers to be distributed ahead of time among the presenters. As well, the scheduling of three successive non-parallel sessions followed by a fourth in which Richard Cohn responded to papers in the third session and David Clampitt moderated a discussion of the entire symposium by all participants ensured that colloquy was as intense and focused as possible throughout the meeting.

[2] The Symposium opened with __Julian (Jay) Hooks__' "Thirteen Ways of Looking
at the Schritt/Wechsel Group," a survey of terms and mathematical notations that
have been used to describe the Riemannian group of 24 triadic transformations (*R*)
which involve, respectively, no change of "mode," i.e., between major and minor
triads (*R+*), and change of mode (*R-*). Cast within John Clough's suggested
larger framework of the group *Q*, which comprises not only Hooks' UTTs (Unified
Triadic Transformations) but also inversions (T/I), Hooks' report also dealt with
recent cognate studies of non-triadic structures. In particular, Hook pointed out
that voice-leading properties are not intrinsic to the shared group structure of
triad and seventh-chord transformations, and that the combination of transposition,
T_{n}, and contextual inversion, *J*, which Lewin has applied to Stockhausen's
*Klavierstücke III*, can be replaced by a single, simply transitive group,
K12 (1,6), or even by the S/W group.

[3] __Jack Douthett__'s "4-Systems in Webern's *Concerto*, Op. 24" provided
the Symposium's most detailed analysis of a serial work. Re-casting the usual trichord/hexachord
approach to this piece in terms of Hook's UTTs and simply transitive groups, Douthett
defined "orbits" of a cyclic group of order 4, which result in GISs (Generalized
Interval Systems), specifically 4-systems, analogous to Richard Cohn's hexatonic
and octatonic systems, as well as "blocks" that give rise to other GISs, in this
instance "supersystems" of 4-, 6-, and 8-systems, analogous to Cohn's hyper-hexatonic
and hyper-octatonic systems.

[4] __Scott Murphy__'s study of "Some Intersections between Neo-Riemannian
Theory and Graph Theory" formulated neo-Riemannian networks as instances of graphs
or what he terms GRSs (Generalized Relational Systems), which are a kind of GIS
where int(a,b) = int(b,a) and the set IVLS has only two members, related and unrelated.
(Other cases he cited include interval cycles, row partitions, and most equivalence
relations.) In general, such graphs can be presented analytically or realized compositionally
in several ways. (E.g., all three components of a graph G<S,R> are determined by
any two components, i.e., in any of three ways: G,S; G,R; S,R). Accordingly, Murphy
focused on comparing and evaluating such alternatives. Emblematic of the oppositions
Murphy explored are maximally smooth cycles versus the over-determined triad on
one hand, and on the other, Childs' [0258] network and a region of Murphy's typology
where there are no GRSs.

[5] __Eytan Agmon__'s account of "The Multiplicative Norm and its Implications
for Set-Class Theory" problematized the often conflated concepts of equivalence
and relatedness among pitch and pitch-class sets and intervals. Introducing the
notions of "transpo-inversional" relatedness and a "normed interval/distance system,"
Agmon advanced definitions, propositions, lemmas and theorems to prove that transposition
preserves intervals, but inversion does not, and that T/I equivalence, though often
valid analytically, is ad hoc in all GISs other than the normed interval/distance
system. Agmon's concluding illustration showed that how the opening three melody
tones of Schoenberg's Op. 11/1 are connected with other sets of tones in the first
10 measures not through T/I equivalence, but rather via both pitch and pitch-class
set equivalence, transpositional equivalence, inversional relatedness (as distinguished
from equivalence!), and pitch-set distance-equivalence.

[6] __Jonathan Kochavi__'s treatment of "Parsimony and Contextuality of Diatonic
Sequences" first developed a calculus for comparing parsimony in various progressions
(between sets of equal cardinality), where degrees of parsimony are determined by
the number of common tones and the minimum total number of units (mod n) between
their registrally respective members (e.g., 0 3 6 and 0 6 9 would have 2 common
tones and a "total voice-leading displacement" or "VL-shift" of 6 = 0+3+3, or comparing
0 3 6 and 0 9 6, 6 = 0+6+0: an undecided distinction in Kochavi's formulation, which
allows registral "voice-crossing"). After advancing the theorems that any set forms
with its transposition up x units not only the same number of common tones but also
the same VL-shift as with its transposition down x units (e.g., 047 as compared
with L27 and 059, and that within a mod-7 system, progressions involving the usual
triads and seventh chords result in the same numbers of common tones and the same
VL-shifts (e.g., 024 vs 025 and 624—all mod 7). Kochavi went on to formulate a "sequence
succession operator" to deal with progressions comprising interlaced sequences of
sub-progressions, illustrating the correspondence between, e.g., __C__ A __D__
B __E__ … __C__ and __C__ F __D__ G __E__ …__C__ in mm. 214-21
and 254-58 of the Gloria in Beethoven's *Mass in C*.

[7] __Ramon Satyendra__ began the second session with "A Tonal *and*
Hexatonic Sonata Form: Shostakovich's *Piano Sonata No. 2* in B Minor, I,"
positing a "resolution" of the opening B-minor and E-flat major themes in the recapitulation,
in so far as their simultaneous combination completes the hexatonic collection (014589).
Understanding the Bm and EbM chords not only as salient foreground features that
function as linear connectives between tonal structures but also as upper structures
in their own right, Satyendra emphasized in his explicitly eclectic analysis the
primacy of well-formed (tonal) middleground counterpoint in contrast to the "consistent,"
"unifying," "dissonant coloration" of the polar triads.

[8] __Amy Shimbo__'s discussion of "Some Transformations between Triads and
Seventh Chords" focused on three "split functions" between a major or minor triad
(037) and a half-diminished or dominant seventh. In each, there are two common tones:
0368, T137, 0359. Surveying the recent literature on such progressions, Shimbo also
drew attention to Ziehn's 1912 account of "symmetrical inversion" and a related
passage in Tchaikovsky's *Pathétique* recently analyzed by Yellin and further
illustrated her typology by the first 11 measures of the *Liebestod*. Acknowledging
that if fully diminished and minor sevenths were included, her typology would specify
the same split relation between a particular common chord and three seventh chords
(e.g., above, not only T137, but also 0369 and 0358), Shimbo conjectured that "the
solution might be as simple as defining different varieties of transformation depending
on the chords being exchanged."

[9] Franck's *Piano Quintet in F minor*, I, mm. 26-37 and 90-102 served
as the main illustration for __Robert Cook__'s consideration of "Parsimony and
Extravagance": any transformations comprising, respectively, two common tones, and
three voices each moving a semitone. These provided the starting-point for Cook's
reflections on the interplay between naming transformations in response to the immediate
experience of listening and the construction and generalization of formal models.

[10] __Stephen C. Brown__'s account of "The Interaction of Ic1 and Ic5 in
Twentieth-Century Music" focused on one of the 15 possible dual interval spaces
considered in his 1999 dissertation, namely, a space where there are dimensions
for ic1 and ic5. Illustrating his talk with passages from Webern's Op. 5/4, Shostakovich's
*Sonata for Viola and Piano*, "Clashing Sounds" from Bartók's *Mikrokosmos*,
and Ruggles' *Evocations* no. 4, Brown paid special attention to the operation
of "interval exchange," which flips pc's around an ic1/ic5 axis. Although his analyses
emphasized linear progressions through one or both dimensions, Brown also noted
gaps in such processes.

[11] __John Clough__'s "Trichords and Transformations in Two Pieces from Gyorgy
Kurtag's *Kafka-Fragments*" considered "Penetrant Judisch" and "Die Guten gehn
im gleichen Schritt," as well as the *Quintetto per Fiati*, Op. 2/V, from the
perspective of the neo-Riemannian transformations L, R, and P on 014 and the diatonic
sets 037 and 025. Clough concluded by proving that for each of the 12 possible Wechsels,
there are 55 distinct sequences of 3 Wechsels that correspond to it (if pairs are
considered equivalent under reflection).

[12] The third session began with a joint presentation by __Fred Lerdahl__
and __Carol Krumhansl__ of "Modeling Tonal Tension and Attraction in Chromatic
Contexts." They construed tonal tension and relaxation in terms of harmonic motion
to and from the tonic, and to and from near or distant chords, whereas, for example,
tonal attraction to the tonic is high in the leading tone, and tonal attraction
to the leading tone low in the tonic. Lerdahl's forthcoming book *Tonal Pitch
Space* models these relations according to a multi-dimensional scaling in integers.
E.g., chord distance is measured by numbers of 3-degree transpositions (mod 7),
and distances between diatonic collections by numbers of 5-semitone transpositions
(mod 12). Such chords are located on toroidal surfaces comprising not only the usual
neo-Riemannian functions but also triadic/octatonic and triadic/hexatonic configurations.
Other tension values are weighted without such spatial modeling: e.g., if there
is a non-harmonic tone, add 1 if it is a seventh; otherwise, add 3 if diatonic,
and 4 if chromatic. Melodic, voice-leading, and harmonic attraction are quantified
by means of sums, products and ratios, and chord grammar, expressed in generative
trees, cross-cuts all these distinctions.

[13] Krumhansl's experimental report focused on the results of comparing components
of Lerdahl's quantitative models in various combinations with listeners' tension
responses. The latter involved moving a mouse horizontally on a computer screen.
For diatonic and chromatic versions of the Grail theme from *Parsifal*, and
mm. 1-6 and 7-12 of Messiaen's *Quartet for the End of Time*, V, correlations
between models and listeners' responses were uniformly quite high: R^{2}
ranged from 0.66 to 0.99, depending on which components of the models were combined.
The quantitative scaling results were presented not only statistically but also
graphically, the latter by means of color gradations in a two-dimensional display.
After the formal presentation, Krumhansl demonstrated how the quantitative models
and the subjects' responses can be compared moment-to-moment in a pair of constantly
shifting, real-time displays on a single PC monitor.

[14] Reporting "On Riemann's Theories of Dissonance as a Resource for Analysis,"
__Edward (Ed) Gollin__ explored an aspect of Riemann's *Skizze* that has
been largely neglected in favor of his Schritt-Wechsel system. In analyzing the
rondo theme of the Forlane from Ravel's *Tombeau de Couperin*, Gollin favored—as
arguably Riemann might have—an interpretation of the recurrent 0148 sonority in
terms of functional (e.g., subdominant and submediant) "transpositions" and chromatic
alteration of chord tones (cf. 0159). In a similarly "traditional, tonal" vein,
Gollin interpreted particular sonorities at the beginning of William Grant Still's
"Cloud Cradles" in terms of W3 and W7 transformations (e.g., 03478L as 047<->L38
and 0347 as 037<->047).

[15] In "Functional Fishing with Tonnetz: Toward a Grammar of Transformations
and Progressions," __Charles Smith__ argued that nineteenth-century music is
best understood in terms of chord progressions comprising tonic, dominant and dominant-preparation
functions, within the framework of a two-dimensional triadic Tonnetz. Illustrating
particular progressions by excerpts ranging from Beethoven to Rachmaninoff and Errol
Garner, Smith claimed, for example, that the dominant function depends solely on
the leading tone's inclusion in a chord (e.g., V, V7, bvi, vii, and iii (with the
possibility that a iii triad might be considered to serve a tonic function); a half-diminished
chord is a minor triad with the usual seventh as root and vice versa; and bIIb7
is a dominant, whereas bII is a dominant preparation.

[16] __Daniel Harrison__ concluded the final session with "Two Short Essays
on Neo-Riemannian Theory." In "The New Riemann: Same as the Old Riemann?" Harrison
critiqued the notion of transformation in terms of such oppositions as being-becoming,
subject-object, and dynamic-static. In "Some Hypotheses about Tonic and Antitonic
Trichords," he advanced the view that 026 comprises three kinds of discharge function
(e.g., FGB of G7 resolving to C, E, or Ab). These he described as "antitonic" rather
than dominant because, in each, two tones serve a dominant function (G-G, B-C; G/Fx-G#,
B-B; G-Ab, B-C) and one a subdominant function (F-E; F-E; F-Eb). Citing theorists
from Rameau to Persichetti and Ulehla, Harrison explored progressions among antitonic
trichords and their resolutions, favoring a view in which, e.g., whole-tone subsets
are understood as dynamic rather than static.

York University

Fine Arts Dept., Atkinson College

4700 Keele St.

Toronto, Ontario M3J 1P3

jayrahn@yorku.ca

Return to beginning of article

[1] Copyrights for individual items published in *Music Theory Online* (*MTO*)
are held by their authors. Items appearing in *MTO* may be saved and stored
in electronic or paper form, and may be shared among individuals for purposes of
scholarly research or discussion, but may *not* be republished in any form,
electronic or print, without prior, written permission from the author(s), and advance
notification of the editors of *MTO*.

[2] Any redistributed form of items published in *MTO* must include the
following information in a form appropriate to the medium in which the items are
to appear:

This item appeared in[ 3] Libraries may archive issues ofMusic Theory Onlinein [VOLUME #, ISSUE #] on [DAY/MONTH/YEAR]. It was authored by [FULL NAME, EMAIL ADDRESS], with whose written permission it is reprinted here.

This document and all portions thereof are protected by U.S. and international
copyright laws. Material contained herein may be copied and/or distributed for research
purposes only.

*prepared by*

*Brent Yorgason*

*Updated 18 November 2002
*