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Half-Diminished Functions and Transformations in Late Romantic MusicAuthor(s): Richard BassSource: Music Theory Spectrum, Vol. 23, No. 1 (Spring 2001), pp. 41-60Published by: University of California Press on behalf of the Society for Music TheoryStable URL: http://www.jstor.org/stable/10.1525/mts.2001.23.1.41 .

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Half-Diminished Functions and Transformations in Late Romantic Music

Richard Bass

Discussions of harmonic function and harmonic transformationshare not only an indebtedness to the theoretical work of HugoRiemann, but also a tendency to focus on the relationships amongmajor and minor triads. The study of harmonic function typicallybegins with the construction of triads on scale degrees, followedby the grouping of those triads into tonic, dominant, and subdomi-nant “families.” Generalizations can then be made about the placeof individual triads within harmonic progressions that express atonality.1 By contrast, neo-Riemannian transformational modelsfocus on the organization of triads into algebraic groups and onthe use of certain voice-leading paradigms to connect them.2

Chord sevenths can be treated as simple appendages to triads.For example, the functions of seventh chords on V and ii are es-sentially the same as those of triads on those scale degrees; and atransformation such as Leittonwechsel (“leading-tone exchange”),

which connects a minor triad to its submediant major by a half-step move in one voice, may be applied without regard for thepresence of a seventh in either chord.3 But seventh chords canalso be considered independently of their triadic bases. With re-gard to harmonic function, such a view directs attention to the dissonance created by the presence of the seventh—what Schoen-berg called a “force of necessity” that requires resolution. 4

Approaching seventh chords as discrete harmonic entities also fa-cilitates an understanding of the ambiguities inherent in theirvoice-leading tendencies (for example, the enharmonic reinterpre-tation of a dominant-seventh harmony as an augmented-sixthchord). In late-Romantic music, the voice-leading implications as-signable to dissonant chords of all kinds become less predictable,and the number of plausible successor-harmonies (those that re-solve the dissonance as well as those that do not) can be quite large.

In neo-Riemannian theory, the independent treatment of sev-enth chords has led to the formulation of new transformationalsystems and networks not connected to those based on triadic re-lationships. The absence of lines-of-communication between tri-adic and seventh-chord networks presents a special problem in the application of neo-Riemannian transformations to music that,

3This situation can be observed in Cohn 1996, 14–5, Example 2, which con-cludes with a progression from G*s minor to E7.

4Schoenberg 1978, 81.

1The term “function” was � rst used in this way by Riemann, although simi-lar views of harmonic relationships can be found in the writings of numerousother late-nineteenth- and early-twentieth-century theorists. For a detailed studyof this approach to harmony and its application to late Romantic music, seeHarrison 1994, especially Part 2, “A Historical Account of Harmonic Functionand Dualism.”

2The theoretical foundations for these and other sorts of transformations arepropounded in Lewin 1987. Triadic transformations that grow more directly outof Riemann’s theories include Hyer 1995, and Cohn 1997. See also Cohn 1998.Cohn 1996 has particular relevance for the present study and is discussed inmore detail below.



42 Music Theory Spectrum

despite potentially high levels of chromaticism, is fundamentallytonal.5

Another, more speci� c problem is that neo-Riemannian theoryrelies heavily on pitch-class set theory to constitute the objects of its study. This leads to the combining of inversionally relatedseventh chords (i.e., dominant and half-diminished sevenths, bothof which belong to set-class 4-27[0258]) into a single network,thereby establishing an equivalency between the two types that, in functional terms, is not re� ected in the music. The dominantseventh belongs to a group of chord types—major and minor tri-ads, plus dominant and diminished sevenths—that hold privilegedstatus attributable to established conventions of consonance/dissonance treatment, voice leading, and harmonic progression.The preeminence of these constructions is con� rmed by the sheerfrequency of their occurrence in tonal music. The half-diminishedseventh, on the other hand, belongs to a second group of chordsthat also includes the augmented triad and the “French” sixth.These sonorities can be regarded as “late bloomers”: althoughthey become more prevalent in the later nineteenth century, wherethe desire for an ever-expanding harmonic palette leads com-posers to exploit more fully the potential of formerly rare and spe-cialized harmonies, these chords never achieve the status of thosein the � rst group. While there is considerable justi� cation—bothin the history of music theory and in the history of compositionalpractice—for combining the inversionally related major andminor triads (set-class 3-11[037]) within a single network, theubiquity of the dominant seventh chord as well as its adaptabilityto so many different harmonic situations argues against consider-ing it together with the more restricted half-diminished chord inanything short of an atonal environment.

The goals of the present study are to move beyond the empha-sis on triadic relations that has thus far characterized the study of

late-Romantic harmony and, at the same time, to provide practicalanalytic applications that demonstrate how voice-leading transfor-mations operate in conjunction with harmonic functions. For thesake of clarity and simplicity, especially with regard to the trans-formational aspects of analysis, I adopt an approach that, by re-cent neo-Riemannian standards, is rather limited. Whereas trans-formational models for seventh chords proposed by other writerssuch as Adrian Childs, Edward Gollin, Jack Douthett and PeterSteinbach entail complex, three-dimensional constructs that areintriguing from a purely theoretical perspective, I contend that atwo-dimensional model may have more immediate analytic rele-vance.6

The functional and constructional properties of the half-diminished seventh chord (considered independently of its inver-sional partner, the dominant seventh) make it an ideal object forcloser examination in the harmonically problematic repertoire oflate nineteenth-century music. In the following discussion, I ex-plore the approaches composers took that ultimately liberatedhalf-diminished chords from their diatonic origins and promotedthem to higher standing in the chromatic sphere. These ap-proaches are of two types: the � rst entails the adaptation of thehalf-diminished sonority to a variety of extended harmonic func-tions, and the second focuses on associations among differenthalf-diminished chords based on minimal voice-leading distance.The applicability of the two approaches, individually and in com-bination, is illustrated in musical examples taken from the worksof a diverse group of composers as well as in the analysis of acomplete piece by Scriabin.

5Such investigations of seventh-chord transformations have been under-taken only recently; see Childs 1998, Gollin 1998, and Douthett and Steinbach1998.

6For Childs 1998, Gollin 1998, and Douthett and Steinbach 1998, the neces-sity of a three-dimensional model in each case derives from the inclusion of theseventh as an equal member of the chord, plus the assumption of equivalencebetween inversionally related chord-types. The model used in Cohn 1996 (seehis Figure 1, p. 17) is two-dimensional because it deals only with triads. A simi-lar, two-dimensional model for seventh chords is possible only by excluding inversional equivalence.



Half-Diminished Functions and Transformations in Late Romantic Music 43

The simplest way to describe the function of a half-diminishedseventh chord is by its resolution. For example, a half-diminishedseventh can occur diatonically as a leading-tone harmony in majormode (viiø7) or as a supertonic harmony in minor (iiø7), and it canbe assumed that when one of those scale-degree names is used, itis because the harmonic context implies that the half-diminishedchord will resolve “normally” to a tonic- or dominant-functionedchord, respectively.

As a leading-tone harmony, the half-diminished chord is func-tionally related to the dominant seventh and the fully-diminishedseventh, but it is less potent than either of them. Because the half-diminished chord can also occur as supertonic harmony, it lacksthe de� nitive tendency of the dominant seventh.7 At the sametime, it does not possess the symmetrical properties, and conse-quently the harmonic � exibility, of the fully-diminished seventh.8

In addition, viiø7 is modally restricted due to the false relation ofthe tritone in its resolution to a minor triad, which grows out ofthe atypical mode mixture in which 3̂ follows 6̂, as seen inExample 1(a). It must be carefully voiced even in the major-moderesolution as well, to avoid the potential parallel � fths, illustratedat (b). This is especially true in second inversion, where 2̂ in thebass must progress to 3̂ rather than to 1̂ in order to prevent objec-tionable parallels. The comparatively meager status of the half-

diminished seventh chord is largely attributable to these factors.9

It is only in the more densely chromatic music nearer the end ofthe tonal era that this sonority � nally gains admission as a fullpartner in the fellowship of harmonies.

In addition to its more frequent appearances as viiø7, the half-diminished seventh chord also takes on the four types of extendedfunctions illustrated in Example 2. The resolution of the “half-diminished dominant seventh” chord, shown at (a), endows it withdominant function, even though the chord can be understood as asupertonic iiø7 in a different key. This progression initiates the fa-mous theme from Rachmaninov’s second Piano Concerto, shownin Example 3. The subdominant “half-diminished added-sixthchord” shown in Example 2(b) is equivalent in pitch-class contentto a supertonic seventh, but it is distinguished by its plagal reso-lution to the tonic (with 2̂ rising to 3̂). It ultimately becomes anineteenth-century harmonic cliché, and appears commonly in cadential prolongations, such as at the end of Brahms’s DieMainacht, shown in Example 4.

The “half-diminished augmented-sixth” and “half-diminisheddiminished seventh” chords given in Examples 2(c) and (d) areidenti� ed not by scale-degree function, but rather by the resolu-tion of their most characteristic intervals. Both chords have the

7The major triad with minor seventh is commonly referred to as the “domi-nant seventh” chord for the very reason that it occurs diatonically almost exclu-sively on the � fth scale degree. (An exception is the subtonic seventh in minor,which nevertheless assumes the function of a dominant resolving to the mediantharmony.) The only ambiguity associated with the dominant seventh is its occa-sional reinterpretation as an augmented sixth chord as noted above, but the en-harmonic nature of this reinterpretation gives it a striking chromatic effect.

8The set 4-28[0369] expressed by the fully-diminished seventh chord hasthe highest degree of symmetry of any tetrachord, mapping onto itself at T0, T3,T6, T9, T0I, T3I, T6I and T9I. Through enharmonic reinterpretation, it can serveas viio7 in four different keys (major and minor modes), and this � exibility iscompounded by the additional possibilities of secondary functions and so-called “common-tone” resolutions. By contrast, the half-diminished chord mapsonto itself only under the identity operation, T0.

9Although diatonic in minor mode, iiø7 is not modally restricted and canalso be found in major-mode progressions. A paradigmatic example of this mix-ture occurs in m. 3 of Schumann’s “Ich grolle nicht” from Dichterliebe, op. 48.

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Example 1. Voice-leading problems in resolving viiø7

(a) false relation (b) parallel � fths




same pitch-class content, but the augmented-sixth type resolves toa major harmony, whereas the diminished-seventh variety is usedin minor-mode resolutions. The augmented sixth at (c) arisesthrough the reinterpretation of the minor seventh interval in thehalf-diminished chord (assuming stacked-third construction) sothat F-E becomes F-D . Its resolution therefore is to the third of the following C-major triad, E, as opposed to the customaryresolution to the � fth scale degree (i.e., the root of the dominantharmony) or, in other circumstances, to the tonic.10 Example 5, an excerpt from the second movement of Rachmaninov’s PianoConcerto No. 2, shows an augmented sixth A-F spelled enhar-monically but resolving nevertheless to the third (G ) of the E-major tonic harmony. The half-diminished seventh in Example2(d) involves a similar recasting and inversion of the minor thirdA -C as a diminished seventh (B-A ). The presence of the thirdof the ensuing C-minor harmony as a common tone makes for aprogression weaker than those found in the other parts of Example2. This half-diminished seventh type appears as the opening ges-ture of Scriabin’s Prelude, op. 27, no. 1, which will be discussedin detail later in this study. (See Example 16 below.)

While these categories account for many of the extended, non-diatonic functions of half-diminished chords, they by no meanscomprise an exhaustive survey of the tonal circumstances underwhich these chords may be found.11 The seventh of viiø7, for example, frequently acts as a non-essential note, moving down by

10Schubert’s works are often cited as an early source of prominent, tonic-resolving augmented sixths (i.e., augmented sixths “of I”); see, for example, theconclusion of the � rst movement of his Piano Sonata in A, D. 959, or the StringQuartet in C, D. 46 (IV), mm. 427–8. The augmented-sixth chord in which thecharacteristic interval resolves to the third of the following harmony is dis-cussed in Harrison 1995, 185, in relation to its occurrence in Richard Strauss’sTill Eulenspiegel, mm. 47–9. This example is cited also in Gauldin 1997, 435,where it is termed the “Till sixth.”

11An extended function not shown in Example 2 is the resolution of the“half-diminished augmented sixth” chord with the outward expansion of thecharacteristic interval to the root of the following harmony. For an illustration,see the comments on Franck’s Piéce hèroique in Harrison 1995, 185–6, where

Example 2. Extended half-diminished functions with resolutions to C

(a) half-diminished (b) half-diminished (c) half-diminished (d) half-diminisheddominant seventh added sixth augmented sixth diminished seventh

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Example 4. Brahms: Die Mainacht, op. 43, no. 2, conclusion




step to create a V65 .12 Also, as with other chord types, the half-

diminished sonority may be used to create a unique affect, as inthe Vergessenheit motive from Wagner’s Götterdämmerung,where a C ø 7 alternates with a C-minor harmony above a G in thebass voice.13 The chord may also arise somewhat accidentally as abyproduct of voice-leading directed toward a larger harmonicgoal. Finally, the resolution of a half-diminished chord may beelided if it proceeds to another chromatic harmony, thus effectinga replacement of its functional implications. Treatments such asthese, which obviate the sense of harmonic function, are commonin late-Romantic music. Therefore, when half-diminished chordsoccur with some regularity in a passage, a section, or even a com-

plete piece, they may take on new signi� cance in their own rightas characteristic sonorities of the music. And when they occur indirect succession or in close proximity to one another, they tend tobe related in ways that are not necessarily functionally consistent,or even functionally relevant. A more penetrating explanation oftheir relatedness, and one that even � lls in lacunae left by an ex-panded functional approach, can be found in a group structure thatmodels voice-leading transformation s among half-diminishedchords.

The possibilities for functional connections between half-diminished chords are so limited that it is extremely rare for twoof them to occur consecutively within tonal harmonic progres-sions. One situation involves two leading-tone harmonies in a descending-� fth relationship in which the � rst half-diminishedchord is secondary and the second acts as a dominant substitute(viiø7/V–viiø7–I).14 Another is the mixed-mode progression iiø7–viiø7, which is highly irregular because it necessitates an awkwardchange from 6̂ to 6̂ that weakens the directed motion to 5̂, as illustrated in Example 6, the opening progression of RichardStrauss’s Von den sieben Zechbrüdern. Its role in this song is motivic, however, as the conspicuous, “lustige” ascent from C toC here and elsewhere in the song seems to parallel the text’seventual transformation of the seven brothers into gushing foun-tains.15

The concluding progression from Scriabin’s Left-Hand Noc-turne, shown in Example 7, lies closer to the outer perimeters oftonal-functional practice. In this case, chromatic voice-leading incontrary motion interpolates a half-diminished augmented-sixthchord into the resolution of the leading-tone harmony. The harmonic

an enharmonic F ø7 progresses to an F minor triad. A similar resolution can befound in Wagner, Götterdämmerung, act I scene 3, mm. 628–9, where the Dand E in Eø7 are treated enharmonically as an augmented sixth and move out-ward to the root of a major triad on E .

12See, for example, Schubert’s Piano Sonata in A, D. 664 (II), mm. 3–4.13Throughout this study, the pitch-class content of a speci� c half-diminished

chord is de� ned by its conventional, “non-functional” chord symbol, irrespec-tive of the chord’s spelling or inversion. That is, a note name followed by thehalf-diminished sign (ø) refers to the pitch classes of the customary “stacked-third” arrangement above that note, but that designation is not intended toimply a fundamental bass note or a particular harmonic function.

14See, for example, Rachmaninov, Piano Concerto No. 3 (I), mm. 107–8 (inB ); here the half-diminished sevenths are incorporated into a chromatically descending bass line E-E -D (viiø7/V–viiø4

3–V/vi), with the V/vi harmony at thebeginning of m. 108 leading to a brief tonicization of C minor (V/V–iiø 4

2–V6

5–V7–i).15I am grateful to Walter Everett for calling my attention to this text-music

relationship.

Example 5. Rachmaninov: Piano Concerto no. 2, op. 18 (II), mm.154– 6 (reduction)

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reduction beneath the score reveals a close voice-leading relation-ship between the � rst two chords: two of the four pitch classesmove by half-step and the other two are common tones. In termsof overall voice-leading distance, this connection is the closestpossible relationship that can obtain between half-diminishedchords.16 The same type of connection exists between the half-diminished chords in Example 6, even though they are not in thesame transpositional relationship (T9 as opposed to T6) as those

in Example 7. Among the other transpositions of a given half-diminished chord, only T3 is in the same voice-leading relation asT6 and T9, described above. The remaining transpositions fall intotwo additional categories: either all voices move in parallel half-steps (T1,TB), or else one pitch class is retained as a common tonewhile two voices move by half-step and one by whole-step (T2,T4,T5,T7,T8,TA). The total number of chromatic steps involved inboth these types of transformations is four.

Richard Cohn has shown that major and minor triads can begrouped according to this kind of “optimally parsimonious”voice-leading into hexatonic systems. In addition, he has exam-ined the transformational relationships between discrete hexatonicsystems and has also explored the potential analytical relevance ofsuch an approach.17 Cohn’s work in this area therefore providesmuch of the essential theoretical underpinning for the presentstudy, which relies on similar principles to group half-diminishedchords. There are, however, some signi� cant differences betweenthe two approaches. First, whereas Cohn divides the twenty-fourmajor and minor triads into four systems of six members each, I group the twelve half-diminished chords into three systems of four members. The pitch-class content of these systems is not hexatonic (set-class 6-20[014589]), but rather octatonic (8-28[0134679A]).18 In addition, while the closest connectionswithin triadic hexatonic systems are “maximally smooth,” mean-ing that only one half-step move is required for the transformation(e.g., E minor to C major, or C major to C minor), the minimum16The relevance of this point derives from the basic tenet that in moving

from one harmony to another each voice will take the shortest route to a mem-ber of the next chord. Schoenberg 1978, 39 refers to this phenomenon as the“law of the shortest way.” Cohn 1997, 2, writes about this principle of parsimo-nious voice-leading that “conjunct voice-leading in general, and semitonalvoice-leading in particular, are enduring norms through an impressive range ofchronological eras and musical styles.” These voice-leading transformations arealso based on the assumption that such close connections are perceived evenwhen at the musical surface they may occur in different registers or may be ex-changed between voices. The speci� c relationship here has been de� ned for-mally as P2,0 (i.e., two voices move by half-step and none by whole-step, whileall other voices remain as common tones) in Douthett and Steinbach 1998,243–4.

17Cohn 1996, 13–25. The expression “optimally smooth” is from Cohn1997, 12. For an analytical demonstration of these triadic relationships, seeCohn 1999, 213–32.

18This difference between systems is traceable to one that is more funda-mental: the inherent inapplicability of Riemannian transformations like “paral-lel,” “relative,” and “Leittonwechsel ” to connections among seventh chords ofany type. For explanations of these terms, see Hyer 1995, 106 ff. An indepen-dent network for voice-leading transformations among dominant and half-diminished seventh chords belonging to a single octatonic collection is given asFigure 6 in Childs 1998, 188.

Example 6. R. Strauss: Von den sieben Zechbrüdern, op. 47, no. 5,mm. 1–3

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voice-leading distance between half-diminished chords is twohalf-steps. Finally, the closest relationships within an octatonicsystem are not just successive, but completely uniform: the mini-mum voice-leading distance holds not only for adjacent chords,but for all members of the same system.

The three half-diminished octatonic systems, termed OS-I, OS-II, and OS-III, are given in Example 8, in which arrows indicateall chords related by this minimum voice-leading distance.19 Eachsystem can also be seen as a T3-cycle (and the octatonic collec-tions could be generated as well with T3-cycles of major or minor

triads), but the basis of these systems in the present case is voice-leading, and the resultant octatonic pitch-class content is of onlyperipheral importance. Example 9 linearizes the voice leading be-tween chords in OS-I as an illustration of the transformations thatcan occur within each system. The progression in Example 9(a)proceeds counterclockwise around the top circle shown in Example8; it is designated the ic 4-2 transform, in which the two pitchclasses in each chord related by interval-class 4 move by half-stepin opposite directions to new pitch classes related by interval-class2.20 One of these notes then forms a new ic-4 relationship with oneof the other voices, and the process continues until the pitchclasses of the � rst chord are reproduced, thus completing the cycle19In this study, systems are named for the octatonic collections expressed by

their total pitch-class content and numbered according to the lowest-numberedpitch-classes they contain. Therefore OS-I expresses the octatonic collectionthat contains C and C (0 and 1), OS-II is the octatonic collection with C and D(0 and 2), and OS-III has C and D (1 and 2). The chord that appears at the“top” of each system is the one in that system that is built on the lowest num-bered pitch class.

20Childs 1998, 185, de� nes this procedure (i.e., the ic 4-2 transform as wellas the ic 5-5 transform discussed here) more generally as a “C transform,” re-ferring to the contrary motion between the two moving voices. His “S trans-form,” which involves similar motion between two voices, converts a half-diminished chord into a dominant seventh and vice versa.

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after four transformations. The ic 2-4 transform simply reversesthe process, moving clockwise around the circle. Example 9(b)shows two successions of chords that lie opposite one another onthe circle. In this ic 5-5 transform, contrary motion between pitchclasses related by interval-class 5 produces another form of thesame interval class (i.e., a perfect � fth contracts to a perfectfourth, or a fourth expands to a � fth).

These transformations offer a means of approaching those rareinstances in which half-diminished chords occur consecutively, ameans that complements the expanded functional view offered

earlier. Therefore, in the tonally mercurial passage represented inExample 10, the progression from Fø7 to Dø7 is more easily under-stood as an ic 4-2 transform than as iiø6

5–viiø65 within a toniciza-

tion of an E major that never materializes. As useful as a transfor-mational approach is in this situation, it is of greater value indealing with associations among half-diminished chords that donot occur in direct succession. The property of minimum voice-leading distance is a connective force that transcends the proce-dures involved in simply getting from one chord to the next. Thereis, thus, a special kind of relationship among the members of each

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Example 8. Half-diminished octatonic systems

OS-I

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system that manifests itself in the ways composers tend to employhalf-diminished chords in late-Romantic music. Two tendencies inparticular can be observed with some consistency in this reper-toire. The � rst entails the proximate grouping of half-diminishedchords that belong to the same system, and the second involvesthe movement from one system to another according to some dis-tinct process or compositional scheme across larger musical spans.

Example 11, a chord series from the accompaniment to a songby Richard Strauss, illustrates consistency of half-diminishedgroupings in a short passage that is otherwise harmonically (aswell as registrally) disjunct. The corresponding vocal text de-scribes the title character, Herr Lenz, snatching � owers from agarden to make a bouquet for his sweetheart. Three different half-diminished chords occur in the harmonic succession, and all of

(a) ic 4-2 transform

B 21 A A A 11 A4 2

G 11 G 21 F F F4 2 4 2

E E 11 E 21 D D4 2

C C C 11 C 21 B

(b) ic 5-5 transform

B 21 A 11 B

5 G 5 G 5 G 5

E 11 E 21 E

C C C

A A A

G 21 G 11 G

5 E 5 E 5 E 5

C 11 D 21 C

Example 9. Voice-leading transformations, OS-I




them (C ø7, A ø7, Eø7) belong to OS-III. Their close transforma-tional relationship therefore serves as a unifying feature of the� rst part of the excerpt, despite the apparent randomness of HerrLenz’s avaricious (habgierig) plucking (an act also represented vi-sually by the pianist’s hand-crossings). Only toward the end of thephrase does the text reveal the purpose behind this behavior ( fürsein Mädel ), and this is supported by a strong cadence on thedominant. The last of the three half-diminished chords is the ele-ment that links the tonally chaotic succession of mm. 21–5 to thefunctional harmonic progression that eventually brings about sta-bility in mm. 26–8.

Systematic half-diminished groupings are also featured in thefamous opening sequence (mm. 1–11) of the Tristan Prelude in away that helps to explain the form of the third “Tristan chord” (m.10), which is incongruent with the transpositional projection ofthe sequence, seen in Example 12. That is, two exact transposi-tions of the original Tristan chord (Fø7, m. 2) in ascending minorthirds would yield the A ø7 at m. 6 followed by Bø7 at m. 10—allbelonging to OS-II. The substitution of Dø7 (in a different inver-sion) at m. 10 breaks the pattern and leads to a dominant seventhon B.21 A remarkable aspect of this substitution, however, is thatCø7 is not used to precede the B dominant seventh, which might

have been expected given the pattern of the previous progressions(Fø7 to E7 and A ø7 to G7); instead, it is another member of theOS-II group.22 Although the regularity of the sequential pattern isdisturbed, the use of half-diminished chords from a single systemremains consistent.23

The use of chords from a single system also characterizes situ-ations in which half-diminished harmonies appear in rudimentarylinear series, as in Example 13, a reduction of the passage imme-diately preceding the recapitulation in the � rst movement ofDvorÏ ák’s “New World” symphony. This excerpt contains threeconsecutive linear-chromatic series. The � rst, mm. 250–1, beginswith a C-major triad and moves through triads on D and D; thesecond, mm. 253–7, consists of half-diminished chords that willbe examined below; and the third, mm. 261–9, consists of domi-nant seventh chords in 4

3 position with roots on A, B , and � nallyB, which, as a functional dominant, sets up the beginning of therecapitulation in E minor at m. 273. The series of half-diminishedharmonies extends chromatically from C to D . Both of theseframing harmonies belong to OS-I, and they are also sustainedthrough four measures each, in contrast to the one-measure dura-tion of the intervening chords on C and D. The half-diminishedseries occupies a structurally central position in this passage aswell, because the three series combine to produce two overlappingunfoldings of the tritone C-F —the � rst one in the highest voice

21The “expected” Bø7 does appear at m. 89, however, at the correspondingpoint in the restatement of the opening material; see Example 15 below.

22Another discussion of this passage based on voice-leading transformationsis given in Lewin 1996, 207–8, which describes the transformation betweenseventh chords where two voices remain as common tones and two move byhalf-step as a “DOUTH2 relation” (after Jack Douthett). Lewin identi� es occur-rences of this relationship among both the half-diminished and dominant sev-enth chords that appear in this passage, but he does not consider their organiza-tion according to octatonic systems.

23I have dealt in detail with the sequential aspects of this passage elsewhere.See Bass 1996, 279–84. That study also summarizes the tonal and motivic fea-tures of the disruption in the sequential pattern. The view of the third Tristanchord offered there, which combines the mechanics of real sequence with afunctional interpretation of the harmonic progression, is synergistic with thepresent explanation.

Example 10. Wagner: Götterdämmerung, Act I, Scene 1, mm. 173–7(harmonic reduction)

&

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c

c

wwb

ww

wwn

ww

ww

wwbb

wwb

ww

wwn

ww

wwb

wwbb

wwbn

ww

wwb

wwb

wwbb

wwbb

Fø7

ic 4-2 transform

4 2

Dø7

( )

( )




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#

#

42

42f

(habgierig)œœœœ#-

Œ

Œœœœœ#-

C # ø7

œœœœn -

Œ

Œ œœœœn -

œœœœ##-

Œ

Œœœœœ

###-

A# ø7

OS-III

œœœœnn-

Œ

Œ œœœœnn-

œœœœ### -Œ

Œœœœœbnn-

Eø7

iiø7

?

œœœœbnn -œ œn œ œ# œœ

Œœœœœ

8V 6

4

œ œ œ œ.

‰jœœ#

Jœœœœ ‰ ‰ Jœ

7

35

œœœ Jœœ ‰

œœœ

#Jœœœ ‰

I

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86

86

jœ

Ó

.œ œ jœ

Ó

.œ# œjœ.2#

..22

jœ# œ jœ ‰ ‰.œn Jœ

..œœ# Jœœ ‰ ‰

Œ . Œ jœ

Ó

.œ# œ jœn

Ó

.œ œjœ.2#

..22b

jœ# œ jœ ‰ ‰.œn Jœ

..œœ Jœœ ‰ ‰

Œ jœ.œ

Ó

Jœ œb Jœ œ

Ó

.œ œ œ# œ

.2#

.œ œ jœ.2

Ujœ# œ# jœ ‰ ‰.œ Jœ

..œœ# Jœœ ‰ ‰U

Fø7 A bø7 Dø7

OS-II

Example 11. R. Strauss: Herr Lenz, op. 37, no. 5, mm. 21–8 (piano)

Example 12. Wagner: Prelude to Tristan und Isolde, mm. 1–11

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#

#

250m. 251 253 256 257 261 269 273

Recap.

(f: i V)

( )

i

ÏÏÏbn ú Ïb Ïn úb Ïn Ïn ú# ÏÏÏ#nn ÏÏÏnbb ÏÏÏ#nn úúúnnnÏÏ ÏÏb n ÏÏn# ÏÏbb ÏÏn n ÏÏb ÏÏ# n

Ïn Ï Ïn Ï Ï# ú Ï# Ïn ú# Ïn Ïn ú# únÏb Ïne: V7Cø7 C #ø7

Dø7 D #ø7

OS-I

Example 13. DvorÏ ák: Symphony no. 9 (I), mm. 250–73, harmonic reduction




(violins), the second in the bass (trombones followed by cellos)—and it is the overlapped portions of the two lines that are harmo-nized by the half-diminished chords.

The lengthy succession of half-diminish ed sonorities fromPuccini’s La Bohème exhibits similar groupings within segments,but also re� ects a larger-scale progression involving all three octa-tonic systems. As seen in Example 14, the voices move in parallel65 chords throughout these measures and proceed mostly by half-steps. Certain chords receive emphasis as the boundaries of thevarious ascending and descending lines, and some of these spansare subdivided at the halfway point by means of other types ofemphasis. The initial directed motion is an ascending octave (mm.145–7) framed by Aø7 chords, with the central D ø7 falling on thedownbeat of m. 146. The line changes direction at m. 147 and de-scends a minor third to F ø7. The next two spans are ascendingand descending tritones between F ø7 and Cø7, subdivided by Aø7 on the downbeats of the intervening measures as before. Allthe emphasized harmonies to this point are members of OS-I.Following the downbeat of m. 153, however, there is a repeatedunfolding of the tritone E ø7 to Bø7, comprising a transient shift toOS-II, which precedes a lengthier segment that emphasizes OS-III. This last system is articulated by an octave ascentbounded by Gø7. The octave is bisected by two C ø7 chords thatinitiate a series of lower-neighbor gestures embellishing this climactic � nal ascent.

The large-scale progression in this excerpt from La Bohèmesuggests an effort to group half-diminished chords within sys-tems as part of a compositional strategy. That is, wherever half-diminished harmonies become conspicuous sonorities in a pieceor passage, they are likely to be organized into different octatonicsystems according to established aesthetic principles like com-pletion (the representation of all three systems) or statement–contrast–restatement (setting up one system and returning to it following the introduction of one or both of the others). This strat-egy can be observed over the course of the Tristan prelude. Aschematic representation of half-diminished sonorities in the piece

is shown in Example 15. As discussed above, the opening empha-sizes half-diminished harmonies belonging to OS-II, but the satu-ration of half-diminished sonorities abates considerably in mm.13–79: in this span the only prominent harmonies of this type aremembers of OS-I (D ø7 in mm. 23 and 25, and Cø7 in mm. 78 and79), and they are not as proximate as the half-diminished chordsin the opening passage.24 A return to OS-II takes place with thereintroduction of Fø7 in mm. 80–1, anticipating the restatement ofthe opening material beginning at m. 82. A second departure fromthe OS-II group occurs with the introduction of a single OS-IIIchord (Gø7) at m. 97 and the insertion of one more member ofOS-I in m. 99, which precedes the � nal, partial restatement of theopening material and its attendant OS-II harmonies.

The primacy of OS-II in the Tristan prelude grows out of itsconsistent and exclusive coupling with the prelude’s initial mo-tive, its immediate composing-out in sequence, and its subsequentrestatements. The half-diminished sonorities that appear in themiddle portion of the piece are members of a different system(OS-I), which fosters a sense of arrival upon the return of OS-IIharmonies in association with the restatement of the prelude’sopening idea (mm. 80 ff.); but the subsequent appearance of anOS-III harmony indicates the intentional incorporation of at leastsome representativ e of all three octatonic systems in order toachieve a greater sense of completion. This kind of aesthetic bal-ance, as well as the deliberate distribution of harmonies belongingto the different systems, is suggested also by the last interjectionof an OS-I sonority before the inevitable return of OS-II in theconcluding gesture.

In other late-Romantic works, schemes related to the interplayof half-diminished harmonies and their octatonic systems may

24Excluded from this list are those half-diminished constructions that occur� eetingly as the momentary con� uence of melodic motion, like the Aø7 on thefourth sixteenth note of m. 24 or the D ø7 on the second eighth of m. 43. TheA ø7 on the second half of m. 70 is excluded as well, although it is somewhatmore conspicuous than the others.




assume enough importance to serve as one of the principal com-ponents of a musical structure. These schemes can be integratedvery successfully, for instance, with the kind of transpositionalpatterning often found in the tonal works of Scriabin—a techniquedescribed by James Baker as “one of the composer’s primarymeans of reaching the main harmonic goals of his tonalschemes.”25 The following analysis of Scriabin’s Prelude, op. 27,

no. 1, in G minor examines the interaction of half-diminishedfunctions and transformations within a self-contained tonal struc-ture. Example 16 gives the score of the piece; Example 17 pre-sents a formal analysis, and Examples 18–20 contain analyticalgraphs of speci� c sections.

The piece is in rounded binary form with a coda, and half-diminished chords are prevalent in all the related four-measurephrases a1, a2, a3, and a4, the � rst two of which are analyzed inExample 18. Although most of phrase a2 is a simple transposition25Baker 1986, 8.

Example 14. Puccini: La Bohème, Act II, mm. 145–67, half-diminished chord groupings

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octave divided at the tritone

145

úúúbÏÏÏ

bbb

ÏÏÏnnn

ÏÏÏbb ÏÏÏ

n#n

ÏÏÏnnb

ú Ïb Ïn Ïb Ïn Ï

ÏÏÏ##n

ÏÏÏnnb

ÏÏÏ##n

ÏÏÏn#n

ÏÏÏnnb

ÏÏÏ##n

Ï# Ï Ï# Ïn Ïb Ïn

147 úúúnnb ÏÏÏ

##n

ún Ï

OS-I

minor 3rd

ÏÏÏnnb úúú

n#

Ïb ú

149 ÏÏÏnb ÏÏÏ

##n

Ïb Ïn

tritone divided at the minor 3rd

ÏÏÏnb ÏÏÏ

##n

ÏÏÏn

Ï Ï# Ï

tritone divided at the minor 3rd

151 úúúbb ÏÏÏ

nn

ÏÏÏbbb

úb Ï Ïb

ÏÏÏnb ÏÏÏ

##

ÏÏÏnnb

Ï Ïn Ïb

153úúú#

ú

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?

153

úúú# úúú##

ÏÏÏn#

ú ú# Ï

OS-II

tritone

ÏÏÏnnb

ÏÏÏbbb

úúúnnn

Ïb Ïb ún

155ÏÏÏ# úúú##

ÏÏÏn#

Ï ú# Ï

tritone

ÏÏÏnnb

ÏÏÏbbb

úúúnnn

Ïb Ïb ún

157ÏÏÏ#úúú##

Ï ú#

ÏÏÏ##

ÏÏÏnb ÏÏÏ

##n

Ï Ï# Ï#

159 ÏÏÏnn ÏÏÏ

###

Ï Ï#&

OS-III

160ÏÏÏ# ÏÏÏ###

Ï Ï#

ÏÏÏ#ÏÏÏ

nb

Ï Ï

octave divided at the tritone

162 ÏÏÏ##n

ÏÏÏnnb

Ï# Ïn

ÏÏÏ##n

ÏÏÏnb

Ï# Ï

164 ÏÏÏbb ÏÏÏ

nb

Ïb Ï

ÏÏÏbb ÏÏÏ

n#n

Ïb Ïn

166–7úúúb

úb

Aø7 D #ø7 Aø7 F#ø7 Aø7 Cø7 Aø7 F#ø7

C#ø7E#ø7 Bø7 E#ø7 Bø7 Gø7 C#ø7 Gø7




of a1 a fourth lower, the overall tonal motion of a1 is to a half cadence on the dominant (D) in m. 4, while a2 moves to an au-thentic cadence on the relative major (B ) in m. 8. Both a1 and a2

are characterized melodically by a descending stepwise line.Each of these phrases begins with one half-diminished sonority

and incorporates others as well. Some of the half-diminishedchords function conventionally as leading-tone or supertonic har-monies (viiø4

3, mm. 2, 6, and 7; iiø65 , m. 3); but the chords that ini-

tiate each phrase, and the one that begins the second half of a1

(m. 3), are of the functionally weak “half-diminished diminishedseventh” variety, which was illustrated in Example 2(d) above.The chord in m. 1 resolves to the tonic, and the one in m. 5 pro-gresses similarly to the dominant minor. The chord at the begin-ning of m. 3 is the same as the one in m. 5, but its implied func-tion is not realized; instead of resolving directly, it moves througha iiø6

5 and a “German” 53 chord on the way to the half cadence.The harmonic rationale for this commixture of diatonic, sec-

ondary, and extended-function half-diminished harmonies is pro-vided by an emergent cyclic arrangement of octatonic systems.The symbols above Example 18 identify the half-diminishedchords and indicate their OS-af� liations. Phrase a1 establishes the

primacy of OS-I, represented by Cø7 and Aø7. Gø7 at the beginningof m. 3 introduces OS-II, which becomes the principal system ofphrase a2. This phrase in turn introduces a single member (Dø7) ofOS-III, implying the continuation of the pattern and the comple-tion of a simple cyclic process in the following phrase (i.e., OS-IIIwith the appearance of one member of OS-I). This third phrase,however, does not materialize in mm. 9–12; instead, the introduc-tion of contrasting material interrupts the progress of the cycleand postpones its completion.

In the central portion of the piece (section B, mm. 9–22), fully-diminished seventh chords become the prevailing sonority andhalf-diminished chords occur only incidentally as non-essentialharmonies. This section prolongs B through four measures andthen begins a dramatic octave ascent (reversing the descendinglines of the � rst two phrases), most of which proceeds in a seriesof outer-voice tritones. Shown in Example 19, this progression isaccompanied by an accelerando and crescendo that continuethrough the dominant preparation of G minor and reaches a cli-max with the restatement of the opening material at m. 23.Phrase a3 (mm. 23–6) is essentially the same as a1, but with muchgreater dynamic intensity and textural density. The only structural

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Ã Ãm. 2 6 10 12

opening statement

úú## úú# úú# úú#

úú úúnb úún úú

23 25 78 79

úú## úú##úúbb úúbb

úú#n úú# úúbn úúb

80–82 83 86 89 95

restatement

úúbb úú## úú# úún úúnb

úú

bn úú úúnb úúnn úún

97 99

úúbúú

úúbúú

bn

102 105

partial restatement

úú#b úú#úú úúb

Fø7 A bø7 Dø7 Dø7 D#ø7 D#ø7 Cø7 Cø7 Fø7 Fø7 A bø7 Bø7 Dø7 Gø7 Aø7 Fø7 A bø7

OS-II OS-IIOS-I OS-II

OS-IOS-III

Example 15. Half-diminished chords in the Tristan Prelude




Example 16. Scriabin: Prelude, op. 27, no. 1, score

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89

89F

œ.

œ.

Œ

Œ

œ .œ.

œ .œ œ œ.

œ.œ ..œœ œœ Jœœ Jœ

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nn

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nn

œ .œ.

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œn.

.œ œbJœ ..œœœ Jœ .œ

.

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jœœ

œ jœ œ jœ œ œ œ.œœ Jœ

..œœ ..œœ ..œœ

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.œ œbJœœ Jœ .œ ..œœ

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‰ ‰ œ œb..22bb œb Jœ.œ

.2b

.2‰œb œb œ œ Å œ œ

jœ.œ

œ .œb œb .œ œ œ œ..22b ..œœ

.2 .œb œ .œ œ œ œ œ.œ

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12

œ jœ .œ ‰ œœ

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.2 Œ ‰.œ œ Jœ Jœ ‰ ‰

cresc. poco a poco

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Rœœœn J

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Å œœ>

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‰ ‰

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œœ

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jœœ.œb .œ .œ

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jœ jœ

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Patetico q . = 92



Example 16 [continued ]


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bb

accel.16 .

.œœ

œœ

nn œ

œ##

œœ

.œ# œœ Å Rœœœ#n J

œœœ? &

œœœ# œœ

nn

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Å œœ>

jœœ>

‰ ‰

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bb

jœœ

nn

œœ

##

jœœ

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jœœ‰

œ œ‰

œ œ œœb œœ œœ

‰ œœ

##

œœœ ‰œœœn

œœ

œœ>

œœ

nn>

œœ>

œœ

nn

jœœ

##

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##

jœœ

œœ

jœœ

bb

‰œ œ

‰œ œ œœnn œœ œœ

‰ œœ

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19 œœ

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.

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.

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jœœœœ œœ œœ œœ œœ œœ œœ œœ œœ

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Jœœ

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jœœ

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Example 16 [continued ]

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bbf

26

.

.22

‰ œœ

.œœ

nn

.

.œ# œ Jœ Jœ

œ jœ œ jœ jœ ‰ ‰..

œœ

œœ Jœ Jœ

œœ

bb .

.œœ

œœ

.

.œœ

œœ œœœ

.œœ

...œœn ..œœ J

œœ œ

œœœnœœ>

œœœb œœœ jœœ

œœœ œœœœœ

œœ .

.œœ

œœ

.

.œœ

œœ

jœœœœb J

œœn ..œœ ..œœ

œœœb œœ>

œœœnœœœ jœ

œ

œœœ jœœ

.

.22

œœ

jœœ..22n# ..œœ

....œœœœn# .

.œœ>

...œœœ

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bb

bbp

mesto30

œœ

jœœ

nn

.

.œœ

‰ œ œnœœ# Jœ

œ jœ œ jœ jœ ‰ ‰..

œœ..

œœ

œb œ œ .œ œ œ œ..œœb n ..œœ Jœœ œœb

œ Jœb .œ .œ

œ œ œ .œ œ jœœœb Jœœn ..œœ ..œœ

œJœ .œ .œ

.2 œ jœ.œb œ# Jœ ..œœ#.2#

.2 .œ

.2nŒ ‰

.2

U

.2ggggggŒ ‰.

.22u

Example 17. Scriabin: Prelude, op. 27, no. 1, formal divisions

measure 1 5 9 23 27

Section A B (A9) CodaPhrase a1 a2 a3 a4 a4

g: ®V ®B :I B ®g:V i®V i ®I

an attenuation of the dramatic energy built up in the middle sec-tion and continued throughout phrase a3 and, � nally, a satisfyingtreatment of the problematic opening harmony are not accom-plished until the coda, which consists of the two structurallyequivalent a4 phrases (mm. 27–30 and 31– 4). The � rst phrase ofthe coda retains the cadential tonic bass (G) as a pedal tone andreduces the dynamic level from fff to f. The � nal phrase, how-ever, is the ultimate denouement of the prelude’s dramatic con-tour: the dynamic is p, the dotted rhythms are discontinued, andthe texture is four voices without octave doublings. As indicatedin Example 20, the opening chord in phrase a4 (F ø7) is the long-delayed representative of OS-III that resumes the cyclic processinitiated in section A. No other members of OS-III appear in thisphrase, thus reinforcing the unique importance of the Fø7 chord.Its signi� cance is not only structural, but also referential, because

difference between the two phrases is the cadence on the tonicthat is appended to a3, which has the melodic 2̂–1̂ in an innervoice.

Phrase a3, then, accomplishes the requisite thematic restate-ment and tonal closure typical of rounded binary forms, but itdoes not address some of the other musical issues in the piece. Aresumption of the cyclic arrangement of half-diminished chords,




it replicates the famous Tristan chord in its original register andspacing. Occurring precisely at this moment, and marked “mesto,”this quintessentially Romantic sonority is a poignant indicationthat Scriabin based the structure of this piece not just on the pro-gression of tonal and thematic events, but also on long-range rela-tionships among half-diminished chords.

The � nal half-diminished chord in the piece is the penultimateharmony (m. 33), which duplicates the pitch-class content of theopening sonority and completes the large-scale OS-cycle. Thiscyclic overlap occurs only in the second statement of the a4 phrase,because the corresponding harmony in the � rst statement (m. 29) isfully diminished, recalling the characteristic sonority of the B sec-tion. In addition, the � nal half-diminished chord resolves moreconclusively than does its counterpart sonority in m. 1. There, Bwas a common tone as part of a “half-diminished diminished sev-enth” formation illustrated in Example 2(d); but here, at the end,B is enharmonically changed to A and ascends by half-step intoa concluding tonic harmony with Picardy third, changing thechord function to a “half-diminished augmented sixth,” illustratedin Example 2(c).

The coda of this prelude does more than simply summarizemusical events and relationships presented earlier in the piece;more signi� cantly, it also completes a cyclic process that is interwoven with, but ultimately independent of, the tonal processthat reaches closure earlier, with the cadence in m. 26. Both thethematic organization and the dynamic shape of the prelude fol-low the cyclic process closely, indicating that an accurate under-standing of the piece relies on the recognition not only of half-diminished functions, but also of half-diminished voice-leadingtransformations and their octatonic systems.

The inadequacy of assigning functional labels to chromaticharmonies has long been recognized: such labels become either anexercise in arbitrary categorization or, at the other extreme, a cata-logue of miscellaneous harmonic “devices.” The study of transfor-mations based on minimal voice-leading distance is a relatively

Example 18. Scriabin: Prelude, op. 27, no. 1, mm. 1–8 (phrases a1 anda2), linear-harmonic structure

phrase a1

phrase a2

&

?

bb

b b

Ï ÏjÏ Ï Ï Ï Ï ÏÏÏ

ÏÏ ÏÏ Ï ÏÏ

ÏÏ ÏÏb Ï Ï Ïn ÏÏ Ï ÏÏ

Ï ÏÏ Ï ÏÏÏ ÏÏÏ#

Ï# Ïn Ï# Ï Ï Ïn Ï Ï# Ï Ï Ï Ïb Ïn Ïn Ï# Ï

Cø7 Aø7

Gø7Aø7(OS-II)

OS-I

to phrase a2

g: (øo7) i viiø43 I6 (øo7) iiø6 (Gr

5)5 3V

III

&

?

bb

b b

Ï ÏjÏ Ï Ï Ï Ï Ï Ï Ï ÏÏÏ

n ÏÏ ÏÏ ÏÏn

ÏÏn ÏÏb Ï Ïb Ïn ÏÏ Ï ÏÏ Ïb Ï Ï Ïn ÏÏbb ÏÏ

Ï# Ïn Ï# Ïn Ï Ïn Ïb Ï# Ï Ï Ï Ïb Ïn Ïb Ï Ï

Gø7 Eø7 Dø7

OS-II

(øo7) i viiø43 I6 I6

6 – 5 IIII

(OS-III)

viiø43

IV

(g:) v

iv6 V7

III




recent undertaking, but one that has already begun to enjoy suc-cess in elucidating some of the complexities of triadic relations inWestern music of the late nineteenth and early twentieth centuries.What is not yet clear, however, is which models for seventh-chordtransformations will prove most informative.

The analysis I have presented here takes a step outside thesphere of triadic relations and employs a hybridized approach ininvestigating the use of half-diminished seventh chords in late-Romantic harmonic practice. Many composers of the period de-veloped an increased fondness for the half-diminished chord, butthey continued to use it within an overarching tonal context. I believe, then, that we can learn more about the late-Romantic harmonic language by keeping both functional and transforma-tional considerations in mind, and it seems likely that we couldpro� tably approach the role of other non-triadic constructions insimilar ways. Further research in this direction might also focuson groupings of half-diminished chords in the works of othercomposers (Debussy, for instance) that move more deliberately

&

?

bb

b b

19m. 9 13 15 20 21 22

outer-voice tritones

g: III iv V

64

53

6

whole-tone

( )

ÏjÏ Ï

jÏ ÏjÏ Ïn

jÏ Ï#jÏn Ï#

jÏn Ïb Ïn Ï Ï# Ï jÏb ÏÏb Ï Ïb Ïn Ïb Ï# Ïb Ïn Ïn

Ïb Ïb Ïb Ï Ïb Ï Ïn ÏbÏ Ïb Ï

#6

Example 19. Scriabin: Prelude, op. 27, no. 1, mm. 9–22 (section B), linear-harmonic structure

Section B

&

?

bb

b b

(OS-I)OS-III

g: iv

Ï ÏjÏb Ï Ï Ï Ï ÏÏÏnb Ï Ï Ï Ï ÏnÏÏb n ÏÏn ÏÏb ÏÏb Ï ÏÏb# Ï

Ï Ïb Ï Ï Ï Ï Ï Ï Ï Ï

Fø7 Cø7

(øo7) V42 vii o6

5 ib 6 – 5

(ø+6) In

Example 20. Scriabin: Prelude, op. 27, no. 1, mm. 31–4 (phrase a4),linear-harmonic structure

phase a4




away from established tonal practices of the eighteenth and nine-teenth centuries, or on the relationship between half-diminishedoctatonic systems in late-Romantic music and the more explicit oc-tatonicism operative in numerous twentieth-century compositions.

LIST OF WORKS CITED

Baker, James M. 1986. The Music of Alexander Scriabin. NewHaven: Yale University Press.

Bass, Richard. 1996. “From Gretchen to Tristan: The ChangingRole of Harmonic Sequences in the Nineteenth Century.” 19thCentury Music 19: 263–85.

Childs, Adrian P. 1998. “Moving Beyond Neo-Riemannian Triads:Exploring a Transformational Model for Seventh Chords.”Journal of Music Theory 42: 181–93.

Cohn, Richard. 1996. “Maximally Smooth Cycles, HexatonicSystems, and the Analysis of Late Romantic Triadic Progres-sions.” Music Analysis 15: 9– 40.

———. 1997. “Neo-Riemannian Operations, Parsimonious Tri-chords, and Their Tonnetz Representations.” Journal of MusicTheory 41: 1–66.

———. 1998. “Introduction to Neo-Riemannian Theory: A Surveyand Historical Perspective.” Journal of Music Theory 42: 167– 80.

———. 1999. “As Wonderful as Star Clusters: Instruments forGazing at Tonality in Schubert.” 19th Century Music 22: 213–32.

Douthett, Jack, and Peter Steinbach. 1998. “Parsimonious Graphs:A Study in Parsimony, Contextual Transformations, and Modesof Limited Transposition.” Journal of Music Theory 42: 241–63.

Gauldin, Robert. 1997. Harmonic Practice in Tonal Music. NewYork: Norton.

Gollin, Edward. 1998. “Some Aspects of Three-DimensionalTonnetze.” Journal of Music Theory 42: 195–206.

Harrison, Daniel. 1994. Harmonic Function in Chromatic Music:A Renewed Dualist Theory and an Account of Its Precedents.Chicago: University of Chicago Press.

———. 1995. “Supplement to the Theory of Augmented SixthChords.” Music Theory Spectrum 17: 170–95.

Hyer, Brian. 1995. “Reimag(in)ing Riemann.” Journal of MusicTheory 39: 101–38.

Lewin, David. 1987. Generalized Musical Intervals and Transfor-mations. New Haven: Yale University Press.

———. 1996. “Cohn Functions.” Journal of Music Theory 40:181–216.

Schoenberg, Arnold. 1978. Theory of Harmony (Harmonielehre).Translated by Roy E. Carter. Berkeley: University of Cali-fornia Press.

Soderberg, Stephen. 1998. “The T-hex Constellation.” Journal ofMusic Theory 42: 207–18.

ABSTRACTTheoretical discussions of harmonic function and of neo-Riemannianvoice-leading transformations have centered on triadic relationships. Trans-formational models for seventh chords have been proposed, but withoutdemonstration of their usefulness in the analysis of complete musicalstructures. This study focuses on half-diminished seventh chords indepen-dently of their inversional relatives, the dominant sevenths, and offers ananalytic model that is directly applicable to the nineteenth-century reper-toire. In late-Romantic practice, half-diminished chords are adaptable to avariety of extended harmonic functions, and they can also be organizedinto three groups of four members each, within which they are associatedby minimal voice-leading distance. The total pitch content of each groupexpresses an octatonic collection, and in works where half-diminishedchords appear with some regularity, there are two tendencies that can beobserved with regard to their usage: � rst, the proximate grouping of mem-bers of the same system, and second, changes from one system to anotheracross larger spans in some systematic way. These functional and trans-formational relationships are illustrated in examples from the works of adiverse group of composers, including Wagner, Richard Strauss, DvorÏ ák,Puccini, and Rachmaninov, and in the analysis of a complete piece byScriabin.



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