ricci non coinciding sequences (2011)

23
Society for Music Theory Non-coinciding Sequences Author(s): Adam Ricci Source: Music Theory Spectrum, Vol. 33, No. 2 (Fall 2011), pp. 124-145 Published by: University of California Press on behalf of the Society for Music Theory Stable URL: http://www.jstor.org/stable/10.1525/mts.2011.33.2.124 . Accessed: 13/06/2013 11:43 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . University of California Press and Society for Music Theory are collaborating with JSTOR to digitize, preserve and extend access to Music Theory Spectrum. http://www.jstor.org This content downloaded from 156.143.240.16 on Thu, 13 Jun 2013 11:43:36 AM All use subject to JSTOR Terms and Conditions

Upload: brianmoseley

Post on 18-Jan-2016

7 views

Category:

Documents


1 download

DESCRIPTION

Ricci-Non Coinciding Sequences

TRANSCRIPT

Page 1: Ricci Non Coinciding Sequences (2011)

Society for Music Theory

Non-coinciding SequencesAuthor(s): Adam RicciSource: Music Theory Spectrum, Vol. 33, No. 2 (Fall 2011), pp. 124-145Published by: University of California Press on behalf of the Society for Music TheoryStable URL: http://www.jstor.org/stable/10.1525/mts.2011.33.2.124 .

Accessed: 13/06/2013 11:43

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

University of California Press and Society for Music Theory are collaborating with JSTOR to digitize, preserveand extend access to Music Theory Spectrum.

http://www.jstor.org

This content downloaded from 156.143.240.16 on Thu, 13 Jun 2013 11:43:36 AMAll use subject to JSTOR Terms and Conditions

Page 2: Ricci Non Coinciding Sequences (2011)

124

Non-coinciding Sequences

adam ricci

This study considers a largely overlooked phenomenon in tonal music, the simultaneous pairing of two melodic sequences having different intervals of transposition; I term this phenomenon a non-coinciding sequence (in contrast to the more common coinciding sequence). In this essay I develop a typology of non-coinciding sequences and scrutinize numerous examples of them in art and pop-ular genres. Extending Allen Forte’s linear intervallic pattern, which models coinciding sequences, I group non-coinciding sequences by their configuration, an ordered list of their harmonic intervals, e.g., <8,10|10,12>. Configurations that permute (with certain restrictions) the same set of harmonic intervals belong to a single configuration class. I consider excerpts from the music of Rick Astley, Beethoven, Brahms, Chopin, Dvořák, Billy Joel, Nelly, Johann Strauss, Jr., Gwen Stefani, and Wagner to demonstrate the interaction of non-coinciding sequences with coinciding sequences and to identify suggestive connections between non-coinciding sequences and double counterpoint.

Keywords: sequence, melodic sequence, harmonic sequence, linear intervallic pattern, coinciding sequence, non-coinciding sequence, configuration, configuration class, double counterpoint, canon

Melodic sequences in tonal music occur in various contexts; sometimes a melodic sequence in one voice is unaccompanied by another within a different voice,

as in Example 1(a), in which a two-measure pattern in the top voice is transposed up by step. Melodic sequences in multiple voices can occur either successively (in an imitative texture) or simultaneously. When melodic sequences occur in multiple voices, each participating voice is usually transposed by the same interval. Example 1(b) illustrates successive melodic sequences; the two voices contain the same three-beat pattern and trans-pose that pattern up by step, but the lower one enters one beat later than the upper.1

Examples 1(c) and 1(d) contain simultaneous melodic se-quences. Because the outer voices are transposed by the same interval, the harmonic intervals in successive patterns remain the same, unlike those in Example 1(a), in which the descent of the lowest voice in mm. 3–4 results in parallel sixths rather than a wedge inward from a sixth to an augmented fourth as in mm. 1–2. The harmonic intervals in the two excerpts shown in Examples 1(c) and (d) are the same—each has a repeated 10–6 pattern—although their melodic intervals and intervals of

transposition differ. There is another important distinction be-tween these examples: in the former, all voices in the one-mea-sure pattern are successively transposed down by a third, generating a harmonic sequence.2 In the latter, however, only the outer voices are consistently transposed up by step.3 I will refer to the cases in which at least two simultaneous melodic sequences are transposed by the same interval as coinciding se-quences (CS), regardless of whether or not the sequence in-volves all voices. This definition is dependent on Allen Forte’s linear intervallic pattern (LIP): while a LIP and a harmonic sequence often proceed in lockstep, a LIP can persist despite alterations to a sequence.4

In this essay I concentrate on the relatively unusual case of two simultaneous melodic sequences that are transposed by dif-ferent intervals—what I term a non-coinciding sequence (NCS). As with CSs, accompanying voices need not contain melodic sequences. Often NCSs have accompanying sequential har-monic progressions, but sometimes an inner voice prevents the formation of a repeated pattern of root motions. For instance, Example 1(e) presents the NCS that closes each strophe of Brahms’s lied Wach’ auf, mein Hort. Because the outer voices are

I wish to thank Guy Capuzzo, Julian Hook, Bruce Moser, Jonathan Salter, and Dmitri Tymoczko for their invaluable comments on various drafts of this paper. I also thank Paul Duvall for his assistance with a mathematical proof in Appendix B. Earlier versions of this paper were presented at the annual meetings of Music Theory Southeast, 26–28 February 2009, Uni-versity of Central Florida, and the Society for Music Theory, 29 October– 1 November 2009, Montreal, Canada.

1 Example 1(b) may be viewed as consisting of simultaneous melodic se-quences as well: the harmonic intervals repeat every three beats, as shown between the staves, and the two voices as a unit are transposed up by step every three beats. Aldwell and Schachter (2003, 364) suggest an implied 32 meter here to explain the seemingly incorrect metric placement of the second and fourth 4–3 suspensions. The third and fourth voices, which enter in m. 10 and m. 12 (respectively), are omitted from the example.

2 I reserve the term “harmonic sequence” for this situation only, using Steven Laitz’s term “sequential progression” (2008, 525) for a repeating series of root motions without accompanying melodic sequences in every voice.

3 While there is a varied harmonic sequence in the second and third patterns (a diminished-seventh chord in the third pattern substitutes for the dominant-seventh chord of the second pattern), the harmonic functions in the first are different. A number of melodic sequences involving one or more voices and the harmonic archetype I–V | V–I are outlined in Gjerdingen (1986).

4 Forte and Gilbert (1982, 85) write “. . . the sequence is a melodic pattern in a single voice, which is repeated at different transpositions and in immedi-ate succession. . . . Such sequences may occur in connection with a linear intervallic pattern. . . . However, the melodic sequence is not a necessary condition for the linear intervallic pattern.” The term was introduced in Forte (1974).

MTS3302_02.indd 124 9/20/11 5:39 PM

This content downloaded from 156.143.240.16 on Thu, 13 Jun 2013 11:43:36 AMAll use subject to JSTOR Terms and Conditions

Page 3: Ricci Non Coinciding Sequences (2011)

non-coinciding sequences 125

transposed by different intervals, the harmonic intervals change from the first pattern to the second.5 NCSs most commonly contain the outer voices; this fact is not surprising, since their

registral positions best highlight their non-coincidence.6 Indeed, while any two voices may form an NCS, all of my ex-amples will involve the bass, and virtually all will include the soprano.

The purpose of my study of NCSs is twofold. The first is to outline a typology of NCSs based primarily upon their har-monic-interval content and then on their melodic-interval

5 Though it is conventional to label only compound seconds and thirds as ninths and tenths I label the compound fifth as a twelfth to better indicate the constant difference between the harmonic intervals in successive pat-terns. I will similarly convert simple intervals to their compound counter-parts (and vice versa) elsewhere in this essay. Since CSs by descending second are prominently featured in the rest of the song, this NCS can be heard in a larger sense as a variant of a CS.

6 Forte and Gilbert (1982, 84–85) define the linear intervallic pattern as oc-curring specifically between the outer voices of a passage.

� � � 34 � � � � � � � � � � � � � � � � � �

6 A4 6 6

� � � 34 � � � �� �� �� � � � �� �� �� �

��

↓ ↓↓↓ +1

V42 I6I6

↓ ↓

:B �

(a)

� � � � � � � � � � � � � � � � � �� � � �� � � �� � � �� �� �� � � � � �� �

4 3 -2 3 5 6 3 - 2 3 5 6 4 3- 2 3 5 6 3 - 2 3 5 6

� � � � � � � � � � �� � � � � � � � �� � � �� � � � � � � � � � � � �

�6- d5 4 6-- d5- 6- 5- 4 6-

+1+1 +1

+1 +1

(b)

� � � � � � � 24 �� �� �� �� �� ���� �

�����

�� ��

�� �� �

�����

�� ����

�� �� �

���

��

�� ��

��

10 6 10 6 10 6

� � � � � � � 24 ���� �

��� ���� �

��� ���� �

���

� � � �

�� �

�� �

� � � �

�� �

������

� � � �

� � �

–2 –2

V6I:G� vi iii6 I6IV

leggiero

(c)

� � � 34 � � � � �� � � � � � � � �� � � � � ������ � � ��� �� � � � �

10 6 10 6 10 6

� � � 34 �� ���� � �� ���

� ���

�� �

���

↓ ↓�

I6I:B� vii ii6V43

+1�

65 iiV 6

+1

+1

+1

(d)

�� 68

��� �

� ����� ���

���� � �������

����8 10 10 12

�� 68 � � �

�� �

� ��� � �

����

��

iiiG: ii

+1

–1

V4 3vi

(e)

example 1. Various contexts for melodic sequences (a) Melodic sequence in one voice in Mozart, Piano Sonata No. 4 in E b Major,K. 282, II, mm. 1–4; (b) Successive melodic sequences in J. S. Bach, Fugue in B b Minor, from The Well-Tempered Clavier, Book I,mm. 6–12; (c) Simultaneous melodic sequences with the same interval of transposition (with harmonic sequence) in Chopin, Étude in G b Major, Op. 25, No. 9, mm. 1–3; (d) Simultaneous melodic sequences with the same interval of transposition (without harmonic sequence) in Mozart, Fantasy in C Minor, K. 475, mm. 86–88; (e) Simultaneous melodic sequences with different intervals of transposition

in Brahms, Wach’ auf, mein Hort, WoO 33, No. 13, mm. 14–15

MTS3302_02.indd 125 9/20/11 5:39 PM

This content downloaded from 156.143.240.16 on Thu, 13 Jun 2013 11:43:36 AMAll use subject to JSTOR Terms and Conditions

Page 4: Ricci Non Coinciding Sequences (2011)

126 music theory spectrum 33 (2011)

content. The second is to suggest some of the compositional functions performed by NCSs by surveying examples from both art and popular genres.

Example 2 formalizes the difference between CSs and NCSs. It contains two transformation graphs, one each for an upper voice and lower voice. Arrows designate diatonic pitch transposi-tion up or down m and n steps; if m and n are equal, then a CS results, and the harmonic intervals in all patterns are identical; if m and n are unequal, then an NCS results. In the latter in-stance, the harmonic intervals in successive patterns differ by a constant, which is equal to the difference between the intervals of transposition in the upper and lower voice, respectively.7

Returning to Example 1(e), we can see why the harmonic inter-vals increase by two steps from the first pattern to the second: since the upper voice is transposed up by step (i.e., m = 1) and the lower voice is transposed down by step (i.e., n = –1), the harmonic intervals of the first pattern are augmented by two steps in the second (k = m – n = 1 – (–1) = 2).8

The model in Example 2 shows two pitches in each pattern and two patterns in total. What are the limits on the cardinality of the pattern and that of the sequence? With respect to har-monic sequences, there is an inverse relationship between the two: short patterns are likely to be subjected to more repetitions, whereas long patterns tend to be repeated fewer times. Various theorists have placed restrictions on the length of the pattern

7 The lower-case “t” for diatonic transposition follows Hook (2008) and the lower-case superscripted “p” for pitch transposition follows Rahn (1980). Elsewhere in this essay, I generally omit the “t” and “p” for convenience. That it is pitch transposition is important to remember. While the voice-leading prototype well known in jazz practice—in which an upper voice repeats interval-class 11s and the bass repeats interval-class 5s—may seem to be non-coinciding in the sense that the two voices exclusively feature different interval classes, it is in fact a coinciding sequence. Since the upper voice descends by half-step and the bass proceeds by alternating ascending fourths and descending fifths, the voices cohere into a pattern that is pitch transposed down by major seconds. Non-correspondence between the shortest unique series of root motions and the cardinality of the pattern also does not constitute non-coincidence in the sense meant here. For ex-ample, root motion by descending fifth can accommodate harmonic se-quence patterns of length two or three; the former is the sequence commonly known as “descending fifths,” while the latter, which features smooth voice leading throughout, can be found in the music of Chopin and Schubert. See Ricci (2002, 4–9) for a formalization of the relationship

between the shortest unique series of root motions and the cardinality of the pattern; for examples of the ascending-third sequence with parsimoni-ous voice leading between descending-fifth-related triads, see Chopin’s Nocturne in G Major, Op. 37, No. 2, mm. 7–9, and the first movement of Schubert’s Piano Sonata in B b Major, D. 960, I, mm. 165–72. Example 2 is suggestive of Klumpenhouwer networks: it models dual transposition, which O’Donnell (1998) and Buchler (2007) show to be equivalent to a strongly isographic K-net. Replacing the dotted lines with I-arrows would result in two K-nets, one for each pattern. Only in the case of an NCS (when m ≠ n) would the two networks interrelate members of different set classes.

8 The harmonic-interval-difference series will prove useful in connection with Example 7 and the associated proofs in Appendix A. The algebraic approach in this paper—including in particular the relating of melodic and harmonic intervals—has important precedents in Taneyev (1962) and Roeder (1989).

example 2. Transformation graph for a coinciding or non-coinciding sequence with pattern cardinality 2

MTS3302_02.indd 126 9/20/11 5:39 PM

This content downloaded from 156.143.240.16 on Thu, 13 Jun 2013 11:43:36 AMAll use subject to JSTOR Terms and Conditions

Page 5: Ricci Non Coinciding Sequences (2011)

non-coinciding sequences 127

and the length of the sequence. Most harmonic sequences con-tain at least three patterns; indeed, many scholars have sug-gested a three-pattern minimum. Walter Piston’s definition of a sequence explains the reason behind this mandate: “It is gener-ally agreed that a single transposition of a pattern does not con-stitute a full sequence . . . [rather,] three separate appearances . . . are necessary to show that the transposition interval is consistent” (italics mine).9 The requirement seems to be predicated on per-ception: after two patterns, the listener may ask, “Was that a sequence?”; only upon the arrival of the third pattern can she answer, “Yes, it is indeed a sequence.” Mark DeVoto proposes the term “half-sequence” for situations in which only two pat-terns appear,10 while Daniel Harrison prefers “pattern transpo-sition.”11 The norm for harmonic sequences is three patterns; however, I consider two patterns to be sufficient to define an NCS. Because there are a limited number of consonant har-monic intervals, and since the harmonic intervals of an NCS change from pattern to pattern, two-pattern examples are the rule. NCSs that incorporate more than two patterns almost al-ways include dissonant harmonic intervals, which—due to norms of dissonance treatment in tonal music—limit their via-bility. This scenario necessarily lends NCSs a certain fragility: they are not as likely to proceed beyond a second pattern as are CSs. But because many of the examples to be discussed here are either repeated (i.e., the entire NCS is repeated) or combined with CSs—and because the patterns are often sufficiently elab-orated, making them seem longer and more substantial—I be-lieve it makes sense to speak of these objects as sequences. In short, they sound enough like sequences to be referred to as such.

With regard to the content of a harmonic sequence’s pattern, many theorists have specified a minimum cardinality of two. For example, Richard Bass defines “a ‘pattern’ . . . [as] consist[ing] of a minimum of two different harmonies, because passages con-sisting of a single harmonic construction used at different trans-positional levels (e.g., parallel six-three triads or diminished seventh chords) are not inherently sequential,”12 and he cites Arnold Schoenberg’s work as a precedent for this view.13 As I have contended in previous work, I believe it is the absence of motion within the pattern as distinct from motion from pattern to pattern that motivates this stance: adjacent chords in such sequences are related by pitch transposition, entailing parallel motion in all voices.14 By definition, NCSs with one-chord patterns are not hampered by parallel motion, but the lack thereof makes NCSs with one-chord patterns sound even less like sequences.15

configurations, configuration classes, and realizations

The NCS in the Brahms lied in Example 1(e) uses exclusively the (compound counterparts of the) consonant harmonic inter-vals of a unison, third, and fifth. Counting only patterns contain-ing two distinct harmonic intervals, there are four permutations of such intervals (with repetition allowed) that will result in an NCS: <1,3|3,5>, <5,3|3,1>, <3,1|5,3>, and <3,5|1,3>. An NCS’s series of harmonic intervals will henceforth be called a configura-tion; the angle brackets indicate that the harmonic intervals are ordered, and the vertical line marks the boundary between pat-terns. These four configurations all belong to configuration class [1,3|3,5].16 The remaining eight permutations of {1,3,3,5} do not correspond to NCSs because, for each of these permutations, cor-responding harmonic intervals in successive patterns do not differ by the same constant. In all configurations that are part of the same configuration class, the harmonic intervals are related in particular ways: specifically, in order to generate one configura-tion from another, the intervals in every pattern must be equiva-lently permuted and/or retrograded. Thus, the maximum number of configurations in a configuration class is equal to 2(p!), where p is the cardinality of the pattern.17 Configuration class [1,3|3,5] is special in that it is the only two-pattern p = 2 configuration the harmonic intervals of which correspond to those of a root-posi-tion triad. Because the bass pitches can function as chord roots, NCSs exemplifying this configuration class can (and often do) support a corresponding sequential progression, a harmonic pro-gression the root motions of which match the melodic intervals in the lower voice. (Example 1[e] contains a sequential progres-sion; the bass line consists of chord roots.) There are three other diatonic configuration classes (p = 2) that contain exclusively con-sonant intervals: [1,3|6,8], [3,5|6,8], and [3,6|5,8].18 In order to accommodate a sequential progression, such configuration classes constrain one inner voice. For example, the configuration <3,6|5,8> would require that an inner voice a sixth above the bass accompany the second pattern’s octave in order to match the sixth above the bass in the model.19

9 Piston (1987, 317). 10 The term “half-sequence” first appears in the fourth edition of Piston’s

Harmony (1978), the first one to be edited by DeVoto. 11 Harrison (2003, 226). 12 Bass (1996, 266). 13 Schoenberg (1978, 283). 14 Ricci (2002, 13); Ricci (2004, 5). 15 Furthermore, the lack of any motion within the pattern means that such

NCSs must contain at least three patterns; otherwise any succession of two harmonic intervals would constitute an NCS.

16 The prime form of a configuration class is represented by ascending inte-gers both from pattern to pattern and—if possible—within each pattern, all surrounded by square brackets. In general, configurations (and configura-tion classes) will be represented by simple intervals, except where such rep-resentation obscures the harmonic-intervallic relationship between successive patterns.

17 Some configurations are invariant under certain permutations, in which case the number of configurations in the configuration class is less than 2(p!).

18 A common realization of configuration <8,6|5,3> is found in a schema identified in Gjerdingen (1986): upper voice 1–7|4–3 with lower voice 1–2|7–1. The upper voice’s pattern is transposed up three steps and the lower voice’s down one step. See in particular Gjerdingen’s Example 1 (27), mm. 9–12 of the Trio from Mozart’s Symphony in A Major, K. 114, III.

19 For this reason, harmonic intervals may not be directly translated into up-per-voice chord members. Even in the configuration class [1,3|3,5]—in which the upper voice often does consist of a root, chordal third, chordal third, and chordal fifth (in one of the four possible orderings)—inner voices may change the chord-member designation of the upper voice.

MTS3302_02.indd 127 9/20/11 5:39 PM

This content downloaded from 156.143.240.16 on Thu, 13 Jun 2013 11:43:36 AMAll use subject to JSTOR Terms and Conditions

Page 6: Ricci Non Coinciding Sequences (2011)

128 music theory spectrum 33 (2011)

Example 3 presents six NCSs that involve the bass line 1–4|5–1, including realizations of three of the four aforemen-tioned diatonic configuration classes—three realizations of [1,3|3,5] and one realization each of [1,3|6,8] and [3,6|5,8]. Upper voices with the same essential contour are superimposed to save space; each upper voice forms a different configuration in conjunction with the bass line. For example, the soprano voice of 3(a) is part of configuration <8,6|3,1>,20 while the alto is part of configuration <3,1|5,3>. Of the NCSs here, 3(c) is probably the rarest, as the parallel motion of the outer voices across the pattern boundary complicates the voice leading of inner voices.21

Example 4 elaborates the model in Example 3(b), using the lower two treble-staff voices of the model. The upper voice in the score traces a compound melody, the upper line of which, in conjunction with the bass line, realizes configuration <3,6|5,8>, as shown in the middleground reduction. The background re-duction removes the passing motion from the compound melo-dy’s lower line, showing how this line, in conjunction with the bass, realizes configuration <1,3|3,5>. Like the NCS in the Brahms lied (Example 1[e]), the pattern’s melodic intervals con-sist of a descending third in the upper voice paired with an as-cending fourth in the bass; the intervals of transposition differ, however: instead of an ascending and descending second, the upper and lower voices are transposed by a descending second and descending fourth, respectively.

Example 5 presents a special case of an NCS in which the upper and lower voices have the same melodic-interval content within the pattern.22 In each notated measure of the reduction (Example 5[a]), the voices move in parallel motion. The passage may be viewed in two ways: as a pair of overlapping NCSs, or as one larger NCS in which the interval of transposition in the upper voice is altered. Viewed from the former perspective, the passage illustrates two paradigmatic contrapuntal structures, each of which realize configuration class [3,3|6,6]: in the first, a double-neighbor motion in one voice is counterpointed by a scalar segment in the other; in the second, the two voices form a pair of overlapping voice exchanges.23

That these two voices have identical melodic content means that the separate transformation graphs for upper and lower voices (Example 2) may be joined together into a single one. Example 5(b) provides a transformation network that relates the second and third patterns. The new diagonal arrows indicate the double voice exchange: the pitch classes in the violin part reappear in the left-hand piano part and vice versa. The two parts exhibit double counterpoint at the (double) octave: the lower voice is transposed up an octave and the upper voice down an octave (a similar transformation network for the first two patterns would represent double counterpoint at the octave in those patterns).

Because each pattern contains only a single melodic interval, the double counterpoint is somewhat trivial. The relationships that this example addresses, however—in which the upper and lower voices contain the same melodic content within the pattern—will reappear later in a nontrivial sense. The trivial sense of the current example derives from the coextensive nature of the double-counterpoint segment and NCS pattern. When

20 There is some ambiguity with respect to the prime form of <8,6|3,1>’s configuration class: both [1,3|6,8] and [1,6|3,8] fulfill the dual conditions of ascending order both within the pattern and from pattern to pattern.

21 Measures 1–4 of the Adagio ma non troppo of Haydn’s String Quartet in B Minor, Hob. III/68, employs this realization of <5,3|3,1> with slight varia-tions of the pitch intervals: the cello plays ascending perfect fourths within the pattern and the first violin transposes its pattern by descending sixth rather than ascending third. There are consecutive perfect fifths between the cello and second violin in mm. 2–3. The NCS involving the lowest treble voice of Example 3(b) also demonstrates parallel motion at the pat-tern boundary; moreover, the skip down from the leading tone means that this voice cannot function as the soprano.

22 Example 1(b) presents a CS in which the upper and lower voices are trans-positionally related, but rhythmically displaced relative to each other.

23 Of the two sequences, the latter “double voice exchange” (and its retro-grade) is probably the more common: among many other examples that could be cited is Haydn’s Piano Sonata, Hob. XVI/37, I, mm. 3–4. See Wagner (1995, 162–72) for a discussion of the <10,10|6,6> succession and its connection with unfolding.

�� ��

�� ���� ��� ��� ��� ��� � � � �

8 6 3 1 5 1 7 33 1 5 3 3 6 5 8 5 3 3 1

1 3 3 5

�� � � � �

� � � �� � � �

–1

(a) (b) (c) (rare)

–1 +2

–3 –3 –3

G: V(7)IVii6

IVii6

IVii6

VI I I I I IV

example 3. Non-coinciding sequences sharing the bass line 1–4–5–1

MTS3302_02.indd 128 9/20/11 5:39 PM

This content downloaded from 156.143.240.16 on Thu, 13 Jun 2013 11:43:36 AMAll use subject to JSTOR Terms and Conditions

Page 7: Ricci Non Coinciding Sequences (2011)

non-coinciding sequences 129

�34

� � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �

�34 �

��� ��� ���� ��� �

��� ��� ���� ���

�����

�����

�������� �

��� ��� ���� ���

��

� � � �

C: V7ii6I I

(a) piano reduction

(b) middleground reduction ( = 1 measure of )� 34

� � � � � �� � � � � � � �3 6 5 8

� � � �� �

� � � � � �� � � �3 6 5 81 3 3 5

� � � �� �

–1

–3

(c) background reduction

–3

–1

example 4. Johann Strauss, Jr., Emperor Waltz, Op. 437, mm. 92–99

�� � � � � � � � � � � �

3 3 6 6 10 6 8

�� � � � � � � � � � � �

Violin

Piano (L.H.)

(a) reduction

(b) transformation network for m. 189

188 189 190

+1 +2

–2 –2

10

+2D5 C� 5 F � 5 E5

6 6 10 10

F E4 D4 C�4 �4–2

–7

+7

example 5. Beethoven, Violin Sonata in A Minor, Op. 23, II, mm. 188–90

MTS3302_02.indd 129 9/20/11 5:39 PM

This content downloaded from 156.143.240.16 on Thu, 13 Jun 2013 11:43:36 AMAll use subject to JSTOR Terms and Conditions

Page 8: Ricci Non Coinciding Sequences (2011)

130 music theory spectrum 33 (2011)

the double-counterpoint segment and NCS pattern have differ-ent cardinalities—and in particular when these cardinalities are different but proximate—a rich interaction between the two structures results. Example 6 illustrates a much-elaborated real-ization of configuration class [3,3|6,6]. The pattern is four mea-sures in length and elaborated with incomplete neighbors, passing tones, and arpeggiation. The surface-level incomplete neighbors (INs) are circled and labeled on the piano reduction in Example 6(a). As can be seen more clearly in the middle-ground reduction in Example 6(b), the C5 and B4 in the upper voice of the first pattern are passing in terms of the underlying ii6 harmony; in the second pattern, the As in both voices are passing in light of the underlying V harmony, and the F4 forms a passing chordal seventh.24 Example 6(c) takes the beginning and endpoints of each pattern as the structural template. This first-species model realizes the same configuration as m. 189 of the Beethoven violin sonata (Example 5[b]), but it features both

different melodic intervals within the pattern and distinct inter-vals of transposition in the two voices. While the Strauss ex-cerpt contains much more of the rhetoric of a sequence—given the greater duration of the pattern—the structural parallels be-tween the two are significant. Ultimately most of the NCSs I analyze in this study fall midway between the Beethoven and Strauss examples in terms of their complexity.25

Thus far, I have presented two cases in which the same con-figuration has been exemplified by different combinations of melodic intervals and/or intervals of transposition: the first drawn from the Brahms lied (Example 1[e]) and the initial excerpt from Strauss’s “Emperor Waltz” (Example 4[c], bass plus alto), and the second from the Beethoven violin sonata (Example 5[b]) and the second excerpt from the “Emperor Waltz” (Example 6[c]). I use the term realization to indicate the particular way in which a configuration is fleshed out; each realization is defined by its melodic intervals. Example 7 lists the twenty-five first-species realizations of configuration

24 It is easy to see why there are no passing tones in the bass in the first pat-tern: the first passing tone would result in a dissonant fourth, the harmonic meaning of which would be unclear. (It would be possible to place both the G2 and A2 on the third beat, although that would force an interruption of the accompaniment pattern.) At the same time, the omission of these em-bellishments points the way to a first-species model: the bass in the first pattern is functionally a cantus firmus.

�34 � �

��� �

����

�� �

�� �

��� �

��� �

����

�� �

�� �

��� �� �� � �� �� � �� �� � �� �� � �� ��

�34 �

�� �� ��� �� �

�� �� � �� �� ��� �� � � � � �� �� �

�� ��

� � � � � �� � � � � �� ��

6 5 4 3 6 10 8 6 (d)5 3

� � � � � � � � �

� � � � � �6 6 10 10

� � � � � �

(a) piano reduction

IN IN

C: V6ii6 V; IV6 V6 I(P)

(b) middleground reduction ( = 1 measure of )� 34

� –2

+1

V6ii6 V; 7 I

(c) background reduction–2

+1

7

example 6. Johann Strauss, Jr., Emperor Waltz, Op. 437, mm. 231–38

25 The Beethoven excerpt is also chronologically the earliest under discussion, so it is unsurprising that it fits only nominally into this study. As with other innovations in sequence practice—among which were the gradual length-ening of the pattern and the rise of motivic alterations to patterns (on this see Bass [1996])—NCSs (other than sequential settings of the double voice exchange) seem to have originated in the nineteenth century.

MTS3302_02.indd 130 9/20/11 5:39 PM

This content downloaded from 156.143.240.16 on Thu, 13 Jun 2013 11:43:36 AMAll use subject to JSTOR Terms and Conditions

Page 9: Ricci Non Coinciding Sequences (2011)

non-coinciding sequences 131

<1,3|3,5>.26 Each row contains the realizations that pair the same intervals of transposition; for example, row 1 contains upper-voice patterns joined by t1 and lower-voice patterns joined by t6.27 Each column contains realizations having the same melodic intervals within the pattern.28 Realizations of the other configurations in the same configuration class can be understood from this example by reading the given realizations:

(a) in reverse (=<5,3|3,1>), (b) beginning with the second measure and wrapping around to the first (=<3,5|1,3>), and (c) in reverse beginning with the first measure and wrapping around to the second (=<3,1|5,3>). Example references between the staves are keyed to excerpts discussed thus far, as well as those to be investi-gated. Example 1(e), for instance, corresponds to Realization 5, and Example 4(c) (bass plus alto) corresponds to Realization 10. As the example shows, the twenty-five realizations group into twelve pairs related by retrogression, transposition, and exchange of upper and lower voices, plus one singleton that is invariant (Realization 1).29

26 For convenience, all realizations are represented without accidentals and starting on C in both voices. Harmonic and vertical tritones should thus not be taken at face value; realizations can of course be transported to dif-ferent locations within the scale.

27 Nota bene: tn indicates pitch-class transposition within the diatonic scale; in the realizations in Row 3, for example, the upper voices are pitch-trans-posed up by three steps or down by four—in both cases corresponding to ordered pitch-class interval 3, mod 7.

28 Since melodic unisons within the pattern imply second species, realizations incorporating them are omitted from the table; a melodic unison in one voice corresponds to a non-unison in the other, so there are five realizations

in each row rather than seven. And since the two voices of an NCS are transposed by different intervals, two independent cases of unison transpo-sition of the pattern in one voice are excluded; thus there are five rows in-stead of seven. Special thanks to Julian Hook for suggesting a re-ordering of the columns for clarity.

29 Appendix A specifies the relationships between different realizations of the same configuration with pattern cardinality 2.

� � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � �

� � � � � � �� � � � � � � � � � � � � �

� � � � � � � � � � �� � � � � �

� � � �

� � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � �� � � � � � � �

� � � � � � � � �� � � � � � � � � �

� �

� � � � � � � � � � � � �� �

� � � � � �

� � � � � � � � �� � � � � � � � � � � �

1. 2. 3. 4. 5.

6. 7. 8. 9. 10.

11. 12. 13. 14. 15.

16. 17. 18. 19. 20.

21. 22. 23. 24. 25.

invariant Example 1(e)

Example 3(a) Examples 12 & 13 Examples 4c & 3b

t1

t6

t6

t4

t3

t1

t3

t5

Example 3(c)

t4

t2

example 7. The twenty-five pairs of melodic sequences that realize the configuration <1,3|3,5> Transformations to the left indicate the (diatonic) interval of transposition for each staff

Double-sided arrows join non-coinciding sequences that are related by retrogression, transposition, and exchange of upper and lower voices

MTS3302_02.indd 131 9/20/11 5:39 PM

This content downloaded from 156.143.240.16 on Thu, 13 Jun 2013 11:43:36 AMAll use subject to JSTOR Terms and Conditions

Page 10: Ricci Non Coinciding Sequences (2011)

132 music theory spectrum 33 (2011)

configurations containing ninths: a quartet of examples from popular music

A common configuration class in popular music is [3,7|5,9], shared by three of the excerpts to be examined here.30 The cho-rus of the song “Dilemma” (2002) by the rapper Nelly (featuring Kelly Rowland) consists of the NCS shown in Example 8(a). Its configuration, <9,5|7,3>, contains two dissonant intervals; its first-species model is given in Example 8(b). Both dissonances resolve normatively, with the upper voice resolving down by step within each pattern.31 The lower-voice sequence continues throughout the song: its two-measure pattern occurs a total of fifty times. The chorus occurs five times, resulting in twenty it-erations of the NCS.

The seemingly endless repetition of the NCS befits the nar-rator’s obsession with the beloved. Just as it is difficult to evalu-ate the beloved objectively when one is in love, it is perceptually difficult to orient oneself within the music, because of the four-fold repetition of the NCS within the chorus. This recurrence allows one to hear both ascending and descending transposi-tions in both voices. Tonal ambiguity contributes to the effect: the song may be interpreted in F major or D minor. In the first interpretation, the song lacks a tonic triad, but the melody closes on tonic at the end of each iteration of the NCS; in the second, the tonic triad arrives at the end of each sequence. The music lacks any cues to tilt the balance toward one possible tonic or

the other and the track fades out at the end, without resolving to any defining chord.32

The same first-species realization of <9,5|7,3> is found in Rick Astley’s “Never Gonna Give You Up” (1987); because it employs the same scalar collection relative to the NCS, the same dual-tonal interpretation applies. In Astley’s song, however, the NCS combines with a CS (see the transcription in Example 9[a]). In the first half of the chorus, the strings and bass articu-late a <7,5|7,5> CS that ascends by step, while the voice sings a sequence that descends by step. I view the NCS between voice and bass as primary and therefore include only those parts in the first-species model in Example 9(b). The first half of the chorus realizes configuration <9,5|7,3>; the bass’s sequence is repeated in the second half of the chorus, while the vocal line descends by step once more. The expected continued descent—shown in the ossia staff—is interrupted in the final measure of the chorus, resulting in a CS in the third and fourth measures. This passage thus suggests the possibility of an NCS containing three patterns: <9,5|7,3|5,1>.33 Such a continuation, with its melodic resolution to B b, would have tilted the tonal interpreta-tion toward B b minor. Unusually, the CS acts as an alteration to the NCS rather than vice versa.34 Such an alteration suits the

30 I use “9” in the prime form because reducing it to “2” would obscure the constant difference between harmonic intervals in adjacent patterns. Of course, there is also substantial historical justification in the figured-bass tradition for maintaining the distinction between “2” and “9.”

31 One might understand the upper voice of the second pattern as arising from an inner voice of the first.

32 The lack of a leading tone is characteristic of many popular-music styles. See Everett (2008, 156–60) for a sampling of pop-rock songs in various modes. The song on which the chorus of “Dilemma” is based, Patti La-Belle’s “Love, Need and Want You” (1983), lacks the lower-voice melodic sequence; in its place is a pedal point on the dominant that (in combination with a cadence elsewhere in the song) clearly establishes major mode.

33 Due to the repetition of the same pitches in the bass line, a realization of the ossia staff would have produced two NCSs: <9,5|7,3> followed by <7,3|5,1>.

34 Examples 12 and 13 feature NCSs as adjustments to CSs.

� � � � � � � � � � � � � � � � � � � � � � � �

No mat - ter what I do, All I think a - bout is you.E - ven when I’m with my boo, Boy ya know I’m cra - zy o - ver you.

� � � � � � � � �

� � � � � � �9 5 7 3

� � � � � � �

(a) transcription

(b) first-species model

1., 3.2., 4.

4x

F: ii9 V iii7 vi

d: iv9 VII v7 i

+1

–1

Voice

Bass

example 8. Nelly (featuring Kelly Rowland), “Dilemma” (2002), chorus

MTS3302_02.indd 132 9/20/11 5:39 PM

This content downloaded from 156.143.240.16 on Thu, 13 Jun 2013 11:43:36 AMAll use subject to JSTOR Terms and Conditions

Page 11: Ricci Non Coinciding Sequences (2011)

non-coinciding sequences 133

narrator’s purpose: the ascending melodic sequence in the sec-ond half of the chorus better portrays his insistence on his com-mitment to his beloved than does the descending sequence in the ossia part.35

Example 10 presents an excerpt from Billy Joel’s “James” (1976) which, like “Never Gonna Give You Up,” suggests but does not quite realize a three-pattern NCS.36 Walter Everett summarizes the song’s narrative as follows:

James, a friend of the singer, is a sensitive soul who “pursued an education” (apparently in creative writing) while the unschooled singer “went on the road” to become a practicing musician. The singer finds James frustrated, working hard, well behaved, living up to others’ expectations, and advises him to be true to himself in order to produce his masterpiece, which has yet to appear.37

The third and fourth measures of Example 10(a), constituting a transition from verse to chorus, are overtly sequential, and share

motivic and rhythmic content as well. These measures are ton-ally salient because of their chromaticism and pitch content: while much of the song reiterates the diatonic descending-fifths progression d–g–C–F, these measures outline a chromatic descending-fifths progression, F–B b –E–A, which contains the aggregate-completing G #.38 As Example 10(b) shows, these measures articulate configuration <7,3|9,5>, retrograding the patterns of the configuration found in the two previous songs. The vertical arrows indicate the way in which the vocal part out-lines a compound melody moving from an outer to inner voice and back within each of these patterns; the motion from B b to D is completed by an ascending stepwise line in the Fender Rhodes part (shown in small noteheads in Example 10[a]).39

The end of the verse sows the seeds for the NCS; the ca-dence contains the previous pattern in the bass (G–C) and the second pitch of the upper voice’s pattern (C). The upper-voice

35 Since the same music—without the voice part—occurs as the song’s intro-duction, the voice part in the chorus in this larger context is grafted onto an underlying CS.

36 The transcription notates the rhythm of the first verse, which is varied slightly in subsequent verses; the pitch material remains the same but for the substitution of G for Bn in m. 1, beat 4, in some verses.

37 Everett (2000, 119).

38 See Baker (1993) and Burnett and O’Donnell (1996) regarding the struc-tural role of aggregate completion in tonal music.

39 Everett (2000, 120–21) interprets the passage differently as a <7,8|9,10> succession which, from the standpoint of my interpretation, conflates two different voices. He also views it as “structurally 7–8, 7–8, but varied by the raising of the vocal part into a temporary descant role for an emphatic 9–10.”

� �� � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � �

Nev-er gon-na give you up, Nev-er gon-na let you down, Nev-er gon-na run a - round and de - sert you.Nev-er gon-na make you cry, Nev-er gon-na say good - bye, Nev-er gon-na tell a lie and hurt you.

� � � � � � � � � � �� � � � � �� � � � � � � � � �� � � � �

� � � � � � � �� � � � � � � � �

Voice

Strings

Bass

(a) transcription

(b) first-species model

: ii9 V iii7 vi

: iv9 VII v7 i

� �

� � � � � � � � � � � � � � �

9 5 7 3 7 3 7 3

5 1

� � � � � � �� � � � � � � �

Voice8

(reduced)

2nd time:

1.2.

D �b�

ii7

iv7

V

VII

iii7

v7

vi7

i7

Bass

+1

–1 –1

–1

0

8

8

7 5 7 5 7 5

example 9. Rick Astley, “Never Gonna Give You Up” (1987), chorus

MTS3302_02.indd 133 9/20/11 5:39 PM

This content downloaded from 156.143.240.16 on Thu, 13 Jun 2013 11:43:36 AMAll use subject to JSTOR Terms and Conditions

Page 12: Ricci Non Coinciding Sequences (2011)

134 music theory spectrum 33 (2011)

D in the first measure of the model in Example 10(b) is merely implied.40 Because of the different rhythmic and motivic sur-roundings, the <5,1> harmonic interval succession does not form part of the NCS proper.41 Joining the <5,1> pattern to the subsequent measures makes sense for textual reasons, since the words of the overt sequence twice respond to those of the end of the verse. The first time, the latter text specifies the different directions taken by the narrator and James in their lives; the third time, the latter text is an admonishment, the only one in the song.42 The line “Do what’s good for you, or you’re not good for anybody,” is the narrator’s most impassioned plea urging James to find and follow his true calling; its setting as a chro-matic NCS thus seems especially appropriate.

A different context for a dissonant ninth is found in an NCS from Gwen Stefani’s “Sweet Escape” (2006), whose opening guitar vamp and subsequent NCS are transcribed in Example 11. The passage articulates configuration <10,6|9,5> with the two voices rhythmically displaced. The vamp establishes D b and F as pedal tones that persist through the NCS. The F and D b in the vocal part of the NCS thus can be heard as the pedal tones’ continuation. In the ninth between the bass and voice parts, then, the bass E b is the source of the dissonance. The bass line, which harmonizes the descending line B b–A b–G n–G b in the guitar part, lies closer to the foreground than the voice’s tonic arpeggiation.

a quartet of examples from art music

Like the popular-music examples, the excerpted selections below are interrelated in various ways. The first pair embed their NCSs within a CS; such NCSs can be profitably under-stood as a category of alterations to the CS of which they form a part.43 Such NCSs constitute a particularly effective altera-tion to a CS by preserving the momentum of sequential con-tinuation in individual voices. The second pair of examples do the converse: each embeds a CS within an NCS whose pattern is longer. Interestingly, three of the four excerpts share a me-lodic sequence that alternates descending fourths and ascend-ing thirds.

40 As Everett (2000) argues, “James” exemplifies Billy Joel’s “learned” style and, as such, we should not be surprised to find occasional implied tones in this piece. Interestingly, in a promotional video for the song, Joel sings the D during the final verse (at 3:05); what previously was only implied becomes overt. See http://www.youtube.com/watch?v=sOwtkQWyoT8 (accessed 28 December 2009). I am not claiming that Joel’s performance of this verse justifies my reading of an implied D, but the fact that he sings it here nonetheless argues for its implicitness.

41 Beginning with a pattern whose connection to subsequent patterns is sub-surface offers the reverse of what Forte and Gilbert (1982, 85) observe in connection with linear intervallic patterns: that is, a LIP can continue after the conclusion of a sequence.

42 In the rest of the song, the narrator recounts past events or queries James. This music occurs twice more: in the fourth iteration, a sax solo plays over the verse, followed by the line “I went on the road. . .”; in the fifth iteration, the second question (“Will you ever write your masterpiece?”) is followed by a repeat of the admonishment.

43 Bass (1996) studies some ways in which alterations to harmonic sequences can be motivic.

� � 44 24 44� � � � �� � � � � � � � � � � �� � � � � � �� � � � ��� � � � � � � � � � � � � � �

1. And we had to go our sep’ - rate ways. 1., 4. I went on the road, You pur-sued an e - du - ca - tion.2., 5. Will you ev - er write your mas - ter - piece? 2. Are you still in school, Li - ving up to ex - pec - ta - tions?

3. Some-one el - se's dream of who you are? 3., 5. Do what’s good for you Or you're not good for a - ny - bo - dy.

� � 44 24 44� � �� � � � � � � � �� �

�Bass

� � � � � �� � � ���� � � �� � � �

5 1 7 3 9 5

� � � � � � � �� �� � � � � �

Voice

(a) transcription

(b) first-species model

F: P 6 V IV

d: V 9VI

ii9I 7 V V7

V

( )

–1

+1 +1

–1

8

8

[Fender Rhodes]

example 10. Billy Joel, “James” (1976), end of verse plus transition

MTS3302_02.indd 134 9/20/11 5:39 PM

This content downloaded from 156.143.240.16 on Thu, 13 Jun 2013 11:43:36 AMAll use subject to JSTOR Terms and Conditions

Page 13: Ricci Non Coinciding Sequences (2011)

non-coinciding sequences 135

Example 12 presents my analysis of a sentential structure from the Prelude to Wagner’s Parsifal featuring multiple trans-positions of the “Faith” motive. Example 12(a) reproduces a du-rational reduction and harmonic analysis of mm. 45–55 by David Lewin.44 In its entirety this passage modulates from A b major to E b minor by way of a CS with alterations: its first pat-tern (mm. 45–47) moves from I to V in A b major, while its second pattern (mm. 48–50) transposes the first up a minor third to C b major; its third pattern (mm. 51–55) begins as a minor-third transposition of its second pattern to E b b major. Lewin’s analysis, which appears within the context of a study of the incommensurability of Stufen-space and Riemann-space, shows that there is no structural enharmonicism in this passage: he represents the third pattern as modulating from E bb major to E b major, arguing that Wagner’s notation of E b b major as D major is largely a matter of notational convenience.45 Example 12(b), my reduction and analysis of the third pattern, employs the notationally more-expedient D major and continues in D # minor so as not to imply an enharmonic modulation. This phrase, itself a sentence structure, begins with a CS, but contin-ues with two overlapping NCSs, the models of which are no-tated on the ossia staves.46 The phrase as a whole accomplishes a modulation from D major to D # minor in which both outer voices descend by a major seventh. At this Prelude’s tempo, a CS traversing a seventh would likely be tedious; by enabling each voice to take a shortcut through the seventh, the altera-tions help to avoid monotony. The first alteration of the smaller CS occurs in mm. 52–53, when the upper voice continues to descend by step while the lower voice is transposed down three steps, producing the NCS represented on the upper ossia staff.47

44 Lewin (1984, 348). 45 See also Harrison (2002) for more on notational versus structural enhar-

monicism. 46 The NCSs here are thus alterations on two levels: both to the smaller-scale

CS beginning in m. 51 and the larger-scale one beginning in m. 45. 47 Murphy (2001) discusses this NCS in terms of a divergence between the up-

per-voice melodic sequence and Neo-Riemannian transformations relating the underlying harmonies. This NCS corresponds to Realization 9 in Example 7.

The bass line’s downbeat pitches in mm. 52–53 imitate the so-prano’s C #–G # (m. 52) in augmentation. In m. 53, the pattern is expanded by the interpolation of a new chord, one that influ-ences the continuation in the lower voice but not that in the upper voice. The lower voice’s pattern in m. 54 deletes the first pitch of the previous pattern, thus preserving only the melodic ascending fourth from m. 53, an inversion of the upper voice’s intra-pattern interval; the upper voice in m. 54 omits the middle pitch from m. 53’s pattern, thereby returning to the pattern em-ployed in mm. 51–52.48 Combining these two elements of the two voices produces the NCS given on the lower ossia staff. The unusual and dissonant 11–5 voice exchange in its first pattern is transformed by displacement on the musical surface; the disso-nant fourth in the second pattern remains, the upper voice func-tioning as a non-chord tone.

Like the Parsifal passage, the excerpt below from Chopin’s Trois Nouvelles Études, No. 2, also features an NCS within a CS. The connection between the two excerpts is even closer: the NCS, which occupies mm. 33–34, is T4-related to mm. 52–53 of the Parsifal prelude.49 (Example 13 supplies a reduction of the passage.) As in the latter, the phrase of which the NCS is a part—mm. 33–36—constitutes a pattern in a larger-scale CS. But unlike the Parsifal passage, there is no smaller-scale CS subjected to alterations. The irregular hexagon in the second line marks the alterations to the CS; the third line notates the pitches of an unaltered transposition of mm. 29–30. The NCS, which articulates the configuration <8,10|10,12>, replaces the tenth and twelfth of m. 29 with an octave and tenth. Relative to a -transformation of m. 29, the upper voice in m. 33 is

� � � � � � 44 � � � � �� � � � � �� � �

Woo - hoo yee - hoo

10 6 9 5

� � � � � � 44� ��� ��� �� � ��� ��� �� � ��� ��� �� � ��� ��� ��� � �� � �� � ��� � � � �� � �� � ��� �

� � � � � �44 � � � � �� � � � ��� �

� �

Voice

Bass

Guitar

4x4x� � �= ��

: IV(9)b� IIIi VI(7)�

+2

+3

example 11. Gwen Stefani (featuring Akon), “The Sweet Escape” (2006), opening vamp and non-coinciding sequence

48 One might interpret the lower voice in mm. 53–54 as containing a se-quence by descending third (i.e., G # F # | E #), with the alteration occurring in m. 54. But this interpretation is insensitive to the strong expectation created by the half-diminished seventh chord on the downbeat: in other words, since the A # dominant is so strongly suggested, it is not likely to be heard as an alteration. (The seventh of the iiØ7 is present in the score but not in my reduction.) The bass’s stepwise line (G #–F #–E #) continues to D # in the following measure.

49 Thus, it also corresponds to Realization 9 in Example 7.

MTS3302_02.indd 135 9/20/11 5:39 PM

This content downloaded from 156.143.240.16 on Thu, 13 Jun 2013 11:43:36 AMAll use subject to JSTOR Terms and Conditions

Page 14: Ricci Non Coinciding Sequences (2011)

136 music theory spectrum 33 (2011)

transposed up a half-step, while the lower voice is transposed down eight half-steps, resulting in the changing harmonic in-tervals from m. 33 to m. 34.50 Like the upper voice in m. 33, the first pitch of the lower voice in m. 34 is also transposed up a half-step relative to a -transformation. The substitution of C minor in m. 34 for the expected C b major makes a harmonic association between m. 34 and mm. 28–29, rendering m. 29 and m. 34 virtually identical51 and thereby creating the impression that mm. 33–36 retrace part of the harmonic path of mm. 29–32. Since the structural harmony of m. 28 (not shown in the example) is Cb major, C minor standing in for Cb major in m. 34 telescopes the former succession of C b major by C minor in

mm. 28–29. In addition to recomposing mm. 28–32, these al-terations intensify the rhetoric of the second pattern by expand-ing the registral space between the outer voices; at the same time, however, the alterations increase the harmonic stability of mm. 33–36. Measures 33–40 as a whole articulate a progression in F minor—i (mm. 33–35)–iv (m. 36)–V (mm. 37–40)—that enables a smooth retransition to the opening material in m. 41 in the home key of A b major.

Both of the following excerpts feature a CS nested within an NCS. They also employ the same sequential progression—root motion by ascending (minor) third alternating with root motion by ascending (perfect) fifth—as harmonic substrate. Example 14(a) supplies a durational reduction of a phrase from Dvořák’s concert overture Othello.52 The NCS, which occurs in mm. 261–66, interrupts the phrase’s functional motion. The first three hypermeasures expand the tonic via two neighbors, 6 and #2.53

50 The harmonic intervals are measured in diatonic space in the example. In chromatic space, the (simple counterpart of the) first harmonic interval in m. 29 is three half-steps; and the difference between the respective intervals of transposition of the upper and lower voices is (1 – (– 8) =) 9 half-steps; thus, the first harmonic interval in m. 33 is (3 + 9 =) 12 half-steps, or an octave; likewise, the second harmonic interval of m. 33 is (7 + 9 =) 16 half-steps, or a major tenth.

51 While they are identical in my reduction, there are slight differences at the musical surface.

52 I have enharmonically respelled some pitches to better reflect underlying voice leading.

53 One might view the chord in the second hypermeasure as a common-tone half-diminished seventh chord (D–F–A–B) spelled enharmonically. (A is present in the score but not in my reduction.)

� � � � � �� �� � �� �� �� � �� ��� �� ��� ���� ���� �� �� ���� �� � �� ����I

45

IIII V III v ii IV vi iii iii (I

I iii/ iv i ii V i (VI

(David Lewin, “Amfortas’s Prayer to Titurel and the Role of D in Parsifal:The Tonal Spaces of the Drama and the Enharmonic C � /B.” 19th-CenturyMusic 7 (3): 336-49. ©1984, The Regents of the University of California.

Used by permission. All rights reserved.)

(a) Lewin (1984), Figure 4 (with measure numbers added): = 1 measure of� 64

� � � � � � � � � � � � �� � � � � �� � � � � �

� � � � � 64 44�� � �� � �� � �� � �� � �� � �� � �� � �� � � �� ��� � � � � � � � � � � � � � � � � � � � � � �8 10 8 10 10 (10) 12 10 11 10

� � � � � 64 44�� � �� � �� � �� � �� � �� � �� � �� � � � �

���

�� � � � � �� � � � � �� � �� � �� � �� � �� �

�� � � � � � � � � � � �� � � � � � � � � � � �

(b) Reduction of mm. 51–55

�48 51 53 55

:A �:C�

:E��

�:e

8 3 10 5

11 5 10 4

V77�:d� ii iVIi6iviiiID:

–1 –1 –2

–1 –3–1

V

example 12. Wagner, Prelude to Parsifal

MTS3302_02.indd 136 9/20/11 5:39 PM

This content downloaded from 156.143.240.16 on Thu, 13 Jun 2013 11:43:36 AMAll use subject to JSTOR Terms and Conditions

Page 15: Ricci Non Coinciding Sequences (2011)

non-coinciding sequences 137

two root motions (an ascending minor third and ascending perfect fifth) governs the chords of the NCS, chords the func-tional meaning of which is unclear. Moving against the bass’s melodic sequence, the sequential progression creates a hemiola. The acceleration of harmonic rhythm, interruption of func-tional momentum, and rhythmic dissonance between sequential progression and melodic sequence combine to produce a daz-zling, disorienting effect.

The NCS has a programmatic function here: Dvořák anno-tated the manuscript with notes providing a glimpse into his realization of Shakespeare’s narrative. His gloss above m. 246 indicates that here Othello and Desdemona “embrace in silent ecstasy.” The musical disorientation intimates the intoxicating effect of their lovemaking. Shortly before this music returns, transposed, in the recapitulation, Dvořák writes that Othello, having now murdered Desdemona, “kisses her for the last time”; more softly orchestrated, this music corresponds to Othello’s reminiscence.55 After two patterns of non-coincidence, the lower voice follows the interval of transposition of the upper voice, producing a CS in mm. 264–68. Predominant function is restored by the Neapolitan, which substitutes for the C b-major triad that would be produced by a continuation of the sequential progression. In terms of root motions, the substitution of Eb major in m. 267 for the expected C b major inverts the substitu-tion of F major for A major in m. 261; in this sense the har-monic entrance into and exit from the NCS counterbalance each other.

Without the bass line, the music of mm. 261–66 would ar-ticulate a harmonic sequence. Example 14(c) suggests another way of understanding the NCS: it interprets the bass’s noncon-formance as a departure from an underlying CS. The middle four chords constitute a CS with an interval of transposition down by whole step. The first and last bass pitches can be ob-tained through a projection backward and forward of the me-lodic ascending third inherent in the pattern. Thus, there is a sense in which mm. 261–66 can be viewed as a CS embedded in an NCS.56

55 The resemblance between this passage and Wagner’s “Sleep” motive from Die Walküre has been noted; see, for example, Clapham (1979, 111) and Hurwitz (2005, 164). The “Sleep” motive also contains a chromatically de-scending upper voice and a bass melodic sequence containing ascending minor thirds, but Wagner’s pattern is four chords long and is part of a CS.

56 The NCS is part of a much longer cycle of some theoretical interest. The cycle is thirty-six chords long, consisting of a twelve-chord harmonic cycle (six two-chord patterns) repeated thrice and a thirty-six-pitch melodic se-quence in the bass (twelve three-pitch patterns). The figured-bass cycle is of length 18: relative to the harmonic sequence’s triads, the bass is either the chordal third, the root, or a half-step above the chordal fifth (in which case the “chord” is an instance of set-class [0148]). One can proceed through up to ten chords of the thirty-six-chord cycle without encounter-ing any dissonant [0148]s, a six-chord segment of which is employed by Dvořák. While the part of the cycle that includes [0148] may not be at home within his style, one can imagine Liszt and other more harmonically progressive late-nineteenth-century composers employing it. For a classic example of [0148] in Liszt’s oeuvre, see his “Nuages gris,” discussed by Forte (1987).

� � � � � 24 � � �� �� � �� �

10 12 10 12 6 5 104 3

� � � � � 24 � � � � � �� ��

� � � � � 24� � � � � � � �

8 10 10 12 6 5 10

� � � � � 24 � � � � � � �

� � � � � 24�� ��

10 5

� � � � � 24 � �� ��

29

33 37–40

:A� iii a: iv V ic d E ae� B�

T p1

if: iv V ivf CFe� b �

Vc

–1

+4

with exactsequence

d � f � C �

4 3

example 13. Chopin, Trois Nouvelles Études, No. 2 in A b Major, mm. 29–36 (reduced); non-coinciding sequence occurs

in mm. 33–34

The fourth hypermeasure varies the second, replacing 6 with b6 and #2 with 4, and harmonizing them with iv. The fifth hyper-measure could have continued either with a return to tonic, as in the third hypermeasure, thereby making the iv chord into a functional subdominant, or with a continuation to the domi-nant, making the iv chord into a functional predominant. The arrival of the bass on 5 in m. 261 supports the latter option, but the inner voices (C n and F n) undermine the dominant: a first-inversion F-major triad here substitutes for an A-major triad. Measure 261 also initiates a much more rapid harmonic rhythm (one chord per q) and a melodic sequence in the bass the pattern of which contains three pitches and is transposed up by half-step, as shown in Example 14(b). (This grouping into threes motivates the notated change of meter in the durational reduc-tion.) The chromatic descending line in the upper voice is grouped into threes through its pairing with the lower voice, thereby producing an NCS.54 A sequential progression containing

54 For another unusual harmonization of a descending chromatic scale (into groups of four pitches), see mm. 85–88 of Chopin’s Mazurka in A b Major, Op. 59, No. 2. The most common tonal setting of a descending chromatic line is by a sequential progression consisting exclusively of root motion by descending (perfect) fifth (see also Note 7). In the diatonic realm, the so-called “Pachelbel sequence” features a descending stepwise line in the so-prano voice that is perceptually grouped into twos by the alternately leaping and stepping bass line.

MTS3302_02.indd 137 9/20/11 5:39 PM

This content downloaded from 156.143.240.16 on Thu, 13 Jun 2013 11:43:36 AMAll use subject to JSTOR Terms and Conditions

Page 16: Ricci Non Coinciding Sequences (2011)

138 music theory spectrum 33 (2011)

My analysis of an excerpt from the first movement of Brahms’s String Quintet in G Major, Op. 111, that similarly embeds a CS in an NCS, is given as Example 15. As in the Dvořák example, the NCS arrives in the midst of a predomi-nant expansion—here a harmonic motion from the diatonic submediant (mm. 6 – 7) to the chromatic submediant (m. 11)—and introduces a much more rapid harmonic rhythm of one chord per q that produces a hemiola.57 Example 15(a) furnishes an annotated reduction of mm. 7–12, bracketing two similar segments for closer study. Example 15(b) transcribes Segment 2, highlighting perhaps the most salient aspect of this passage, the harmonic sequence by descending major second, which ar-ticulates tonic-to-dominant progressions in G major and F major, concluding on E b major.58 In the Segment-1 sequence, shown in Example 15(c), the bass line is metrically displaced relative to Segment 2: the melodic descending fourths in the bass begin on beat 2 rather than beat 1. In addition, two of the triads are minor rather than major, lessening the impression of

tonic-dominant chord pairs. While Segment 2 is progressional, Segment 1 is prolongational, serving to expand B major, which functions as E minor’s dominant.59 Segment 2 is thus a neces-sary successor to Segment 1: in terms of the passage’s overall function, Segment 1 fails, while Segment 2 succeeds.

Example 15(d) displays the NCS, which joins the last four chords of Example 15(c) with the first four chords of Example 15(b). The upper voice is transposed down by step, while the lower is transposed up by step; the harmonic intervals of the second pat-tern (3,8,3,8) are thus two steps smaller than those of the first pattern (5,10,5,10).60 The metric and harmonic context of mm. 7–12 as a whole forges a strong connection between the respective beginnings of each segment; as shown in Example 15(a), the two

57 The more-rapid harmonic rhythm is prepared in mm. 6 – 7 by tonic-dom-inant motion over a pedal 6.

58 The upper voice of mm. 10 – 11 is transposed down an octave for convenience in this and subsequent examples. Other well-known instances of this sequence can be found in the opening phrases of Beethoven’s “Waldstein” Sonata, Op. 53, and Schubert’s String Quartet in G Major, D. 887.

59 Schoenberg ([1954] 1969, 81–84) also identifies B major as in effect in mm. 7–9; he singles out the opening period (mm. 1–16) as a model of the exploration of many harmonic regions within a short span. Had Segment 1 begun with a D #-minor triad, the sequential progression <ascending minor third, ascending perfect fifth> would have been intact throughout Segment 1 as it is in Segment 2. Beginning instead with a B-major triad strengthens the prolongational function of Segment 1.

60 Thus, as with some earlier examples, the upper voice of the second pattern may be viewed as beginning on a “lower” chord member than in the first pattern. Here, there is no physically sounding voice in the first pattern tracing the chord members of the upper voice’s second pattern.

�� � 44 34 24 44� � � � � � � � � � �� � �� � �� �� �� �� �� � � � �

�� � 44 34 24 44�� � �� �� �� �� �� �� �� �� �� �� ��� ���� � �� � � �� � � � � � � ��

�� � 44 34 24 44� � � � � �� �� �� �� ��

�� �� � � �

�� � 34 24

� �� � �� �� �� �� ��

12 8 4 8 4 0 8

+3

�� � 34 24� �� �� �� �� ��

�� ��

�� � � �� � �� �� ��

�� � � �� �� �� �� ��

= 1 measure of� 34

(a) durational reduction of mm. 245–80

(b) mm. 261–68 as a non-coinciding sequence becoming a coinciding sequence

(c) mm. 261–66 as a coinciding sequence within a non-coinciding sequence

267253 273

D: I53

62� 5

3 iv [non-coinciding sequence] 6II� Gr+6 V64

6� 5 I3

+3 +3 +3

T p–3 T p

–3

T p+1 T p

–3 mel. intervals:

E� G� D � F �A�F E� B�666 6

E� G� D � F �A�F666

T10 T10

4

T p–2

example 14. Dvořák, Othello, Op. 93, mm. 245–80

MTS3302_02.indd 138 9/20/11 5:39 PM

This content downloaded from 156.143.240.16 on Thu, 13 Jun 2013 11:43:36 AMAll use subject to JSTOR Terms and Conditions

Page 17: Ricci Non Coinciding Sequences (2011)

non-coinciding sequences 139

example 15. Brahms, String Quintet in G Major, Op. 111, I, mm. 7–12

�� 98 � �� � � � � �� � � � � � �� � � �� � � �� � �� � �� � � � �� ��� � �� �� � � � � � � � �� � �� � � � � �� � � � � � � � � � �� � � � � ��� �� � ���

�� 98

� �� � �� �� �� � � �� � � � � � � � ���

� �

� �� � �� � �� � �� � �� ���

�� �� � �� � �� �

�� 98 � � �� � � � �� � � �� � � � �� � � � �� �

G D F C E

�� 98 � � � � �� � �� � �� �

�� 98

� � �� � � � � � �� �� � � � � �� � �� � � �B F c e B

�� 98 �� � � � �� � � � � �

7

(a) reduction of mm. 7–12

(b) Segment 2 (c) Segment 1

B: 64

53 I

vcl.

vla. II

vla. II

vcl.

7�ii Viv iv 7�ii

G: 64

53

7�vii Vvi �vii 43

65VVI�

vla.

Segment 1 Segment 2

B: 6I V ii(?) iv IG: I V I IV

I VII� VI�

� � �

�� �� � � � � � �� � � � �� � � � �� �

5 10 5 10 3 8 3 8

��

� � �� � � � � � � � � � �� � �� �

�� � �� � � �� � �� � �� �

10 5 10 5 10 3 8 3 8 3

��

�� � �� � � � �� �� �� ��

(d) Non-coinciding sequence (e) Double counterpoint

–1

+1

Segment 1 Segment 2 Segment 1 Segment 2

+5

–6

(f ) coinciding sequences within non-coinciding sequence within double counterpoint

Mel. intervals:

F�5 C�5 E5 D�5B4 A4 C5B4 D5 G4

–3 –3+2 +2 +2 +2–3 –3

D�4 F�4 E4 F4 C4G4 D4

Mel. intervals: 10 105 5 3 810 3 8 3

C�4 B3 E �4

Mel. intervals: +2 +2–3 –3 –3 –3+2 +2

–1

+1

+5

–6

MTS3302_02.indd 139 9/20/11 5:39 PM

This content downloaded from 156.143.240.16 on Thu, 13 Jun 2013 11:43:36 AMAll use subject to JSTOR Terms and Conditions

Page 18: Ricci Non Coinciding Sequences (2011)

140 music theory spectrum 33 (2011)

chords preceding the downbeats of mm. 8 and 10 are the same, and the first two beats of each segment articulate a V 6-5 motion.61

Harmonic motion within each segment operates on a differ-ent level, however. Within mm. 7 – 12 as a whole, the second viola (doubled part of the time by the first viola) functions as the bass, while the cello has the primary line in the middle of the texture. Immediately after the downbeats of mm. 8 and 10, both viola parts either drop out or move to a higher register and the cello becomes the lowest sounding part. These junctures thus effectuate harmonic elisions, with each segment interrupt-ing the functional progression leading into it. The sequences emerge and are somewhat separate from the larger harmonic context, with the downbeats of mm. 8 and 10 serving as har-monic pivots. The downbeat of m. 8 is therefore both a caden-tial six-four chord and a first-inversion tonic triad (in B major), and the downbeat of m. 10 is at once a cadential six-four and a root-position tonic triad (in G major).

Within the two segments, the outer voices contain only two melodic intervals, descending fourths and ascending thirds, in-tervals that occur in strict alternation; a descending fourth in one voice accompanies an ascending third in the other. Example 15(e) displays the pitches of the outer voices, showing that there are only two discrete melodic strands, beginning with a de-scending fourth and an ascending third, respectively.62 The outer

voices trade strands, producing double counterpoint at the twelfth: the upper voice in Segment 1 is transposed down a major seventh to become the lower voice of Segment 2, and the lower voice of Segment 1 is transposed up a minor sixth to be-come the upper voice of Segment 2. Tenths invert into thirds and fifths invert into octaves in this type of double counter-point; they are the only four intervals that remain consonant in this type. The passage is very similar to an example from Fux’s Gradus reproduced as Example 16 that—strikingly—serves to illustrate double counterpoint at the twelfth.63 I have aligned mm. 8–9 of the Brahms with the first part of the Fux to show the correspondence. In the Fux passage, passing tones fill in the melodic ascending third in the upper voice and the melodic de-scending fourth in the lower voice; the Brahms passage uses Fux’s upper voice as the pattern for both voices.64

Example 15(f ) combines aspects of Examples 15(b), (c), (d), and (e), employing a format similar to that of Example 5(b).

61 Omitted from Example 15(a) is an indication of the larger-scale prolonga-tion of E minor; in terms of this prolongation, the two E-minor triads that directly precede each segment may both be viewed as local pivots—from E minor to B major and E minor to G major, respectively.

62 These melodic strands are also paired in the outer voices of the Wagner and Chopin excerpts. Both mm. 51–53 of the Parsifal prelude and mm. 29–30

of the Chopin etude contain five pitches of this sequence in the upper voice and four in the lower; the former produces the linear intervallic pattern of Segment 2 (starting with the octave), and the latter produces the linear intervallic pattern of Segment 1. A similar passage occurs in mm. 31–32 and 46 – 49 of the third movement of Brahms’s Triumphlied, Op. 55.

63 Mann ([1958] 1987, 126). 64 Brahms’s interest in counterpoint is well documented: see, for example,

Brodbeck (1994). According to Geiringer (1982, 338), Brahms “possessed the works of the most important musical theorists, beginning with Fux, Forkel, and Mattheson, down to the end of the nineteenth century.” Brahms uses the same melodic sequence (with a passing tone filling in the melodic third) in the opening of the subject to the first fugue (“Verwirf mich nicht von deinem Angesicht”) of his Motet, Op. 29, No. 2, which sets the text of Psalm 51.

� � � � � � � � � � � � � �

5 6 7 8 10 5 6 7 8 10 5 6 7 8 10 5 6 7 8 10

� � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � �8 7 6 5 3 8 7 6 5 3 8 7 6 5 3 8 7 6 5 3

� � � � �� � � � � � � � � �

�� 98

�� � � �� � � � � � � �� �� ��

10 9 8 5 6 10 9 5 6 10

�� 98 �� � �� �� � �� � �� � � � � �

Brahms, mm. 8–9:Violin I and Cello

Transposition to the lower twelfth

example 16. Example of double counterpoint at the twelfth from Fux, Gradus ad Parnassum (Mann [{1958} 1987, 126], used with permission by Dover Publications, Inc.) and Brahms’s mm. 8–9 (brackets and harmonic interval labels added)

4-3

MTS3302_02.indd 140 9/20/11 5:39 PM

This content downloaded from 156.143.240.16 on Thu, 13 Jun 2013 11:43:36 AMAll use subject to JSTOR Terms and Conditions

Page 19: Ricci Non Coinciding Sequences (2011)

non-coinciding sequences 141

In that the upper and lower voices are transpositionally related, Segments 1 and 2 of the Brahms excerpt are similar to the Beethoven violin sonata excerpt examined earlier. Whereas in the latter the two voices move in parallel motion, producing only one type of harmonic interval within each segment, in the former, the voices move in contrary motion, producing two dif-ferent harmonic intervals within each segment.65 And in the Beethoven excerpt there is only one melodic interval in each voice, while here there are two. Thus the two voices are de facto canonically related at the distance of one beat. This richly com-plex interaction between double counterpoint, canon, NCS, and CS seems characteristically Brahmsian.66

conclusions and generalizations

The musical excerpts examined in this essay suggest ways in which NCSs serve certain common functions. One of these is disorientation: the chorus of “Dilemma” disorients because its repetition of an NCS confuses the listener’s perception of the

direction of each voice, an effect that reinforces the meaning of the lyrics. Another related function is disruption: the excerpts from Brahms’s Op. 111 and Dvořák’s Othello are both metrically and tonally disruptive, and in the latter the disruption is inti-mately tied to programmatic meaning. Other NCSs serve mod-ulatory purposes: the excerpt from the Prelude to Parsifal smoothly modulates from D major to D # minor by way of two overlapping NCSs; the Chopin Prelude passage becomes more tonally settled by way of an NCS; and the NCS in Billy Joel’s “James” modulates from (the dominant of ) F major to (the dominant of ) D minor. And some NCSs serve rhetorical pur-poses: in the Chopin Prelude, the NCS expands the registral space at the opening of the CS’s second pattern, while in the chorus from “Never Gonna Give You Up,” the incipient closure of the NCS is interrupted by a CS in the narrator’s attempt to convince the beloved of his sincerity.

What overall picture is painted by the excerpts examined here? Example 17 lists the passages I have analyzed, organized by configuration class, and then by configuration. Eight differ-ent configuration classes were instantiated by excerpts, [1,3|3,5], [3,3|6,6] and [1,5|3,7|5,9] by multiple excerpts.67 As discussed above, realizations of configurations containing dissonant inter-vals are constrained by rules of dissonance treatment, so configu-rations containing only consonances are especially privileged. As such, configurations [1,3|3,5], [1,3|6,8], [3,3|6,6], [3,5|6,8], and [3,6|5,8] are special; and of these, [1,3|3,5] is singular in that it can support a harmonic sequence with all chords in root position. As shown in Example 7, of the twenty-five first-species realizations of configuration class [1,3|3,5], four were instantiated

67 I omit the paradigmatic NCSs given in Example 3 (other than those in-stantiated in Example 4), although presumably there are many extant examples of these.

65 In general, when an NCS is embedded within a double-counterpoint seg-ment, the distance between identical harmonic intervals is twice the differ-ence between the respective cardinalities of the double-counterpoint segment and the NCS pattern. In the Beethoven excerpt, the cardinalities are the same, meaning that there is no room for harmonic variety; but in the Brahms, the double-counterpoint segment is five pitches long and the NCS pattern is four pitches long, and since twice the difference is (2(5 – 4) =) 2, every other harmonic interval must be identical, meaning that only two distinct harmonic intervals are possible in each segment. It follows that there can be only two melodic intervals. Appendix B supplies a proof.

66 Perhaps surprisingly, the passage occurs in this form only once. It is simpli-fied in the recapitulation: Segment 1 occurs over a dominant pedal and Segment 2 over a pedal 4; in the place of two canonically related voices in contrary motion are parallel thirds and parallel sixths, respectively.

example 17. List of configurations keyed to excerpts

MTS3302_02.indd 141 9/20/11 5:39 PM

This content downloaded from 156.143.240.16 on Thu, 13 Jun 2013 11:43:36 AMAll use subject to JSTOR Terms and Conditions

Page 20: Ricci Non Coinciding Sequences (2011)

142 music theory spectrum 33 (2011)

by excerpts from the literature (Examples 1[e], 4[c], 12, and 13).68 Are these exemplars representative, and if so, why are certain realizations of this configuration class more common than others? One hypothesis is that smaller intervals of transpo-sition are more common than larger ones for reasons of registral continuity. Indeed, common harmonic sequences feature inter-vals of transposition by second or third. Only Realizations 1 through 5 exclude intervals of transposition larger than a third, and these realizations are especially privileged because one voice is transposed up by step and one down by step. However, since NCSs usually contain only two patterns, they typically do not threaten registral continuity. (The first row of realizations does not include Examples 12 and 13, the NCSs of which contain only two patterns.) Moreover, certain realizations may be faulted for hidden fifths (e.g., m. 2 of Realizations 8 and 13).

While NCSs in tonal music are somewhat constrained by norms of dissonance treatment, those in non-tonal contexts would logically be less restricted, suggesting another avenue of inquiry. And what types of NCSs might be employed in scalar universes other than the diatonic and chromatic? Extensions to common subsets of the chromatic (hexatonic, whole-tone, octatonic) and to microtonal universes may prove useful. While NCSs may be rare relative to their coinciding brethren, they hint at a substantial richness of sequence practice that remains to be fully explored.

appendix a

relationships between realizations of a particular configuration with pattern cardinality 2

The twenty-five first-species realizations of configuration <1,3|3,5> in Example 7 pair a melodic sequence in the upper voice with a particular melodic sequence in the lower voice. Transposing the upper and/or lower voice in each realization changes the harmonic intervals between the voices, but does not alter the differences between successive harmonic intervals, or harmonic-interval-difference series. The difference series for <1,3|3,5> is <(3–1),(3–3),(5-3)> = <2,0,2>. (Consider again Example 2 in this context: the two transposition graphs are in a sense independent of one another.) In general, the realiza-tions that exemplify a particular harmonic-interval-difference series group into pairs that are related by retrogression, trans-position, and exchange of upper and lower voices; these pairs are indicated by double-sided arrows in Example 7. This fea-ture depends upon the fact that the melodic-interval series for any p = 2 melodic sequence takes the form <x, q, x>. Retrograding the pitch classes of a series retrogrades and in-verts the ordered pitch-class intervals of the series, but since the interval series is R-symmetric, the net effect is to invert the intervals. Let the upper-voice melodic sequence be <x, q, x>

68 The Brahms string quintet passage (Example 15) might be included here: it corresponds to Realization 5 in retrograde, except that the pattern is doubled in length. From this perspective, the CS thus constitutes a kind of composing-out of the NCS via transposed repetition.

and the lower-voice melodic sequence <y, r, y>. The harmonic-interval-difference series is thus <x–y, q–r, x–y>. Retrograding the realization and exchanging upper and lower voices pro-duces the melodic sequence <–y, –r, –y> in the upper voice and <–x, –q, –x> in the lower, resulting in the same harmonic-interval-difference series.

If the initial and final harmonic intervals in a configura-tion are different, then maintaining the same configuration also requires that one or both voices be transposed to “reset” the opening harmonic interval. For configuration <1,3|3,5>, the difference between the final and initial harmonic intervals is (5 – 1 =) 4. Retrogression results in configuration <5,3|3,1>, and exchanging the voices produces its double-counterpoint-at-the-octave partner, <4,6|6,8>. The resultant harmonic in-tervals must be increased by four steps (or decreased by three) to produce configuration <1,3|3,5>; thus, the difference be-tween the transposition of the lower voice (which becomes the new upper voice) and that of the upper voice (which be-comes the new lower voice) must be 4. Transforming Realization 6 into Realization 11 involves t4 of the lower voice (= new upper voice) and t0 of the upper voice (= new lower voice): 4 – 0 = 4; transforming Realization 8 into Realization 12 involves t2 of the lower voice and t5 of the upper voice: 2 – 5 = 4; and Realization 1 is transformed into itself by the same transpositions.

Realization 1 is invariant since the melodic sequences in its upper and lower voices are inversionally related.69 Let the melodic intervals in the upper voice equal x and y—thus, the melodic intervals in the lower voice are –x and –y—and the harmonic intervals be f, g, h, and i. Thus, the first harmonic-interval differ-ence is x – (–x) = 2x = g–f, and the second harmonic-interval difference is 2y = h–g. (The third harmonic-interval difference is equivalent to the first.) The number of invariant realizations of a given configuration is the product of the number of solutions to these two equations; the number of solutions to each equation depends upon scale cardinality s. By elementary number theory, if d = gcd(2,s), then there are d solutions to 2x = g–f (mod s) if and only if d divides g–f and no solutions if d does not divide g–f .70 The number of invariant NCSs differs depending only upon whether s is odd or even.

If s is odd, then d = 1; and since 1 divides all integers, there is exactly one solution to each equation and (1×1 =) 1 invari-ant realization. In the case at hand (s = 7), d = gcd(2,7) = 1 and 1 divides both g–f (= 2) and h–g (= 0). Thus, there is exactly one invariant NCS of pattern cardinality 2 in a scale of seven tones.

If s is even, then d = 2. Since the two voices of an invariant realization are inversionally related and any integer multiplied by 2 is even, the harmonic-interval differences are always even. Since 2 divides any even number, there are four (2×2) invariant

69 It is also degenerate, however; since both voices have melodic unisons in their interior, the pattern boundary is not well articulated. (Of course, mov-ing inner voices could help to differentiate the two B–D tenths.)

70 Jones and Jones (1998, 46 – 47).

MTS3302_02.indd 142 9/20/11 5:39 PM

This content downloaded from 156.143.240.16 on Thu, 13 Jun 2013 11:43:36 AMAll use subject to JSTOR Terms and Conditions

Page 21: Ricci Non Coinciding Sequences (2011)

non-coinciding sequences 143

Since xi and yi are constants, the harmonic intervals within each double-counterpoint segment are all identical, entailing parallel motion between upper and lower voices. In the case of m. 189 of the Beethoven, p = 14 (double counterpoint at the double octave) and q = (10 – 6 =) 4. So yi = 9 and xi = 5, corresponding to tenths and sixths, respectively.

2. d > n

I first prove that for an NCS pattern embedded within a dou-ble-counterpoint segment, the distance between equivalent har-monic intervals is equal to 2(d – n).71 Second, I prove that the distance between equivalent melodic intervals is also equal to 2(d – n). Third, I list the different possible realizations of the harmonic-interval-difference series in the Brahms.

a. Harmonic intervals

The Brahms passage may be represented as in Example 19(a):

example 19a. Cardinality of double-counterpoint segment,cardinality of NCS pattern, and harmonic intervals in

the Brahms passage

Here, p = 10 + 3 = 5 + 8 = 13 and q = 3 – 5 = 8 – 10 = –2 ≡ 5 mod 7.

The general situation may be represented as in Example 19(b):

x1, x2, …, xd – n + 1, xd – n + 2, …,xd y1, y2, …, yn, …, yd

length n

length d

length n

length d

example 19b. A generalization of the interrelationships between double-counterpoint segment, NCS pattern, and harmonic

intervals in the Brahms passage

Thus, q = y1 – xd – n + 1 = y2 – xd – n + 2 = … = yn – xd. In general, q = yj – xd – n + j for 1 ≤ j ≤ n.

� � �� � �� � � � � � � �� �� � �� �� �

0 2 4 6 0 2 4 6 0 2 4 6 0 2 4 6

�� �� �� � � � � �� � � �� �� � �� � ��

realizations. Example 18 displays the four invariant realizations of <0,2|4,6> mod 12. (The first realization would need to be elaborated in some fashion to articulate a p = 2 pattern.)

appendix b

inherent relationships that obtain when the upper and lower voices of an ncs contain the same

melodic-interval content

The NCSs from Beethoven’s Violin Sonata, Op. 23, and Brahms’s String Quintet, Op. 111, both feature upper- and lower-voice melodic sequences that are transpositionally related. The following proves the relationships that obtain in this spe-cial case; I use “double-counterpoint segment” here in the sense in which I used it in my analysis of the Brahms passage.

Let d equal the length of the double-counterpoint segment and n equal the cardinality of the NCS pattern.

Let xi represent the harmonic intervals of the first double-counterpoint segment and yi the harmonic intervals of the sec-ond double-counterpoint segment: x1, x2, …, xd | y1, y2, …, yd for 1 ≤ i ≤ d.

Let p equal the double-counterpoint interval. In general, xi + yi = p for 1 ≤ i ≤ d.

Let q equal the difference between corresponding harmonic intervals in successive patterns.

I first prove the case in which the double-counterpoint seg-ment and NCS pattern have the same cardinality (correspond-ing to the Beethoven), then the case in which they have different cardinalities (corresponding to the Brahms).

1. d = n

The harmonic intervals are given by the set of integersx1, x2, … , xd, y1, y2, … , yd

that satisfyxi + yi = p, for 1 ≤ i ≤ d

andyi – xi = q.

Thus, xi = p – yi and xi = yi – q, so p – yi = yi – q.2yi = p + q

example 18. The four invariant realizations of <0,2|4,6> mod 12

71 I thank Paul Duvall for assistance with this proof.

MTS3302_02.indd 143 9/20/11 5:39 PM

This content downloaded from 156.143.240.16 on Thu, 13 Jun 2013 11:43:36 AMAll use subject to JSTOR Terms and Conditions

Page 22: Ricci Non Coinciding Sequences (2011)

144 music theory spectrum 33 (2011)

The harmonic intervals are given by the set of integersx1, x2, … , xd, y1, y2, … , yd

that satisfyxi + yi = p, for 1 ≤ i ≤ d

andyj – xd – n + j = q for 1 ≤ j ≤ n.

Now suppose that 1 ≤ k ≤ 2n – d.72

Then since yk – xd – n + k = q, or yk = q + xd – n + k, and xk = p – yk, we havexk = p – yk = p – q – xd – n + k. [1]

Applying this equation with k replaced by d – n + k, we obtainxd – n + k = p – q – x2(d – n) + k. [2]

Combining [1] and [2]:xk = p – q – xd – n + k = p – q – (p – q – x2(d – n) + k) = x2(d – n) + k.

Thus, xis that are 2(d – n) apart are equal. Since xi + yi = xj + yj = p, xi – xj = yj – yi. Thus, if xi = xj, then yi = yj and so yis that are 2(d – n) apart are also equal.

In the case of the Brahms, d = 5 and n = 4. If d = n + 1, har-monic intervals that are 2(1) = 2 apart are equal, meaning that the harmonic intervals within each double-counterpoint seg-ment are either identical or strictly alternate between two val-ues. As the next part of the proof shows, the former possibility necessitates identical melodic intervals as well, which entails the case where d = n; the latter is what occurs in the Brahms pas-sage, as shown in Examples 15(e) and 15(f ).

b. Melodic intervals

There are a total of d – 1 melodic intervals in each voice. Because the two voices are related by double counterpoint, the melodic intervals of the upper voice in the first segment are identical to the melodic intervals of the lower voice in the sec-ond; likewise, the melodic intervals of the lower voice in the first segment are identical to the melodic intervals of the upper voice in the second. The general situation may be represented as in Example 19(c):

Thus, ud – n + 1 = l1, ud – n + 2 = l2, …, ud – 1 = ln – 1 and ld – n + 1 = u1, ld – n + 2 = u2, …, ld – 1 = un – 1. In general, ud – n + j = lj and ld – n + j = uj, for 1 ≤ j ≤ n – 1.

Now suppose that 1 ≤ k ≤ 2n – d.

Then we have ud – n + k = lk [1] and ld – n + k = uk [2].

Applying equation [2] with k replaced by d – n + k, we obtain

l2(d – n) + k = ud – n + k. [3]

Substituting in equation [3] via equation [1], we obtain

lk = l2(d – n) + k.

Thus, lk’s that are 2(d – n) apart are equal; similarly, uk’s that are 2(d – n) are equal.

In the case of the Brahms passage, d = 5 and n = 4. If d = n + 1, then melodic intervals that are 2 apart are equal, so either the melodic intervals are all identical (in which case parallel motion results, entailing the case in which d = n) or the melodic inter-vals within each voice strictly alternate between two values. The latter is what occurs in the Brahms, as shown in Examples 15(e) and 15(f ).

c. The number of realizations of the harmonic-interval-difference series of the Brahms passage

The case where d = n + 1 entails an upper-voice melodic se-quence that can be described by <x, y, x, y, . . . x>. The lower-voice melodic sequence must then be <y, x, y, x, … y>. Only one melodic interval is freely chosen, since the difference between x and y is given by the difference between the two harmonic in-tervals. The harmonic-interval difference between 5 and 10 (= 3 mod 7) is 2, a condition that is fulfilled by x = 4 and y = 2, Brahms’s solution. Disallowing melodic unisons, there are three remaining solutions: x = 3, y = 1; x = 5, y = 3; and x = 6, y = 4.73 Retrograding the passage results in melodic sequences <35> and <53> in upper and lower voices, respectively. The two remaining solutions—<31>, <13> and <64>, <46>—are also R-related; these sequences have intervals of transposition of 3 or 4, making them less viable because such intervals of transposition threaten registral continuity.

works cited

Aldwell, Edward, and Carl Schachter. 2003. Harmony and Voice Leading. 3rd ed. Belmont [CA]: Thomson/Schirmer.

Astley, Rick. 1987. “Never Gonna Give You Up.” Written by Matt Aitken, Mike Stock, and Pete Waterman. Whenever You Need Somebody. RCA 75150.

Baker, James. 1993. “Chromaticism in Classical Music.” In Music Theory and the Exploration of the Past. Ed. Christopher Hatch and David W. Bernstein. 233–307. Chicago: The University of Chicago Press.

72 This ensures that 2(d – n) + k (in equation [2] below) will be no greater than d. If d is greater than or equal to 2n, then there is no repetition of harmonic intervals within the pattern.

u1, u2, …, ud – n + 1, ud – n + 2, …, ud – 1 l1, l2, …, ln – 1, …, ld – 1

l1, l2, …, ld – n + 1, ld – n + 2, …, ld – 1 u1, u2, …, un – 1, …, ud – 1

length d – 1

length n – 1 length n – 1

length d – 1

example 19c. A generalization of the interrelationships between double-counterpoint segment, NCS pattern, and melodic

intervals in the Brahms passage

73 The solution x = 1, y = 6 results in a unison interval of transposition.

MTS3302_02.indd 144 9/20/11 5:39 PM

This content downloaded from 156.143.240.16 on Thu, 13 Jun 2013 11:43:36 AMAll use subject to JSTOR Terms and Conditions

Page 23: Ricci Non Coinciding Sequences (2011)

non-coinciding sequences 145

Bass, Richard. 1996. “From Gretchen to Tristan: The Changing Role of Harmonic Sequences in the Nineteenth Century.” 19th-Century Music 19 (3): 263–85.

Brodbeck, David. 1994. “The Brahms-Joachim Counterpoint Exchange; or, Robert, Clara, and ‘the Best Harmony between Jos. and Joh.’ ” In Brahms Studies. Vol. 1. Ed. David Brodbeck. 30 – 80. Lincoln: University of Nebraska Press.

Buchler, Michael. 2007. “Reconsidering Klumpenhouwer Networks.” Music Theory Online 13 (2). http://www.mtosmt.org/issues/mto.07.13.2/mto.07.13.2.buchler.html (accessed 8 May 2011).

Burnett, Henry, and Shaugn O’Donnell. 1996. “Linear Ordering of the Chromatic Aggregate in Classical Symphonic Music.” Music Theory Spectrum 18 (1): 22–50.

Clapham, John. 1979. Dvořák. New York: W. W. Norton.Everett, Walter. 2000. “The Learned vs. the Vernacular in the

Songs of Billy Joel.” Contemporary Music Review 18 (4): 105–29.

———. 2008. “Pitch Down the Middle.” In Expression in Pop-Rock Music: Critical and Analytical Essays. 2nd ed. Ed. Walter Everett. 111–74. New York: Routledge.

Forte, Allen. 1974. Tonal Harmony in Concept and Practice. 2nd ed. New York: Holt, Rinehart and Winston.

———. 1987. “Liszt’s Experimental Idiom and Music of the Early Twentieth Century.” 19th-Century Music 10 (3): 209–28.

Forte, Allen, and Steven E. Gilbert. 1982. Introduction to Schenkerian Analysis: Form and Content in Tonal Music. New York: W. W. Norton.

Geiringer, Karl, in collaboration with Irene Geiringer. 1982. Brahms, His Life and Work. 3rd ed. New York: Da Capo Press.

Gjerdingen, Robert O. 1986. “The Formation and Deformation of Classic/Romantic Phrase Schemata: A Theoretical Model and Historical Study.” Music Theory Spectrum 8: 25 – 43.

Harrison, Daniel. 2002. “Nonconformist Notions of Nineteenth-Century Enharmonicism.” Music Analysis 21 (2): 115–60.

———. 2003. “Rosalia, Aloysius, and Arcangelo: A Genealogy of the Sequence.” Journal of Music Theory 47 (2): 225 – 72.

Hook, Julian. 2008. “Signature Transformations.” In Music Theory and Mathematics: Chords, Collections, and Transformations. Ed. Jack Douthett, Martha M. Hyde, and Charles J. Smith. 137–60. Rochester: University of Rochester Press.

Hurwitz, David. 2005. Dvořák: Romantic Music’s Most Versatile Genius. Pompton Plains [NJ]: Amadeus Press.

Joel, Billy. 1976. “James.” Written by Billy Joel. Turnstiles. Columbia PCQ-33848.

Jones, Gareth A., and J. Mary Jones. 1998. Elementary Number Theory. New York: Springer.

LaBelle, Patti. 1983. “Love, Need and Want You.” Written by Jimmy Davis, Roger “Ram” Ramirez, and Jimmy Sherman. I’m in Love Again. Philadelphia International 25398.

Laitz, Steven G. 2008. The Complete Musician: An Integrated Approach to Tonal Theory, Analysis, and Listening. 2nd ed. New York: Oxford University Press.

Lewin, David. 1984. “Amfortas’s Prayer to Titurel and the Role of D in Parsifal: The Tonal Spaces of the Drama and the Enharmonic C b/B.” 19th-Century Music 7 (3): 336 – 49.

Mann, Alfred. [1958] 1987. The Study of Fugue. New Brunswick: Rutgers University. Repr. Mineola [NY]: Dover Publications.

Murphy, Scott. 2001. “Wayward Faith: Divergence and Reconciliation of Melodic Sequence and Harmonic Cycle in Some Measures from the Prelude to Wagner’s Parsifal.” Paper presented at the Annual Meeting of the Society for Music Theory, 3 November, Philadelphia.

Nelly, and Kelly Rowland. 2002. “Dilemma.” Written by Bam Bam [Antoine Macon] & Pebbles, Ryan Bowser, Kenny Gamble [Kenneth Gamble], Nelly [Cornell Haynes, Jr.], and Bunny Sigler. Nellyville. Universal 017747.

O’Donnell, Shaugn. 1998. “Klumpenhouwer Networks, Isography, and the Molecular Metaphor.” Intégral: The Journal of Applied Musical Thought 12: 53–80.

Piston, Walter. 1978. Harmony. 4th ed. Rev. and expanded by Mark DeVoto. New York: W. W. Norton.

———. 1987. Harmony. 5th ed. Rev. and expanded by Mark DeVoto. New York: W. W. Norton.

Rahn, John. 1980. Basic Atonal Theory. New York: Schirmer Books.

Ricci, Adam. 2002. “A Classification Scheme for Harmonic Sequences.” Theory and Practice 27: 1 – 36.

———. 2004. “A Theory of the Harmonic Sequence.” Ph.D. diss., University of Rochester.

Roeder, John. 1989. “Harmonic Implications of Schoenberg’s Observations of Atonal Voice Leading.” Journal of Music Theory 33 (1): 27 – 62.

Schoenberg, Arnold. 1978. Theory of Harmony. Trans. Roy E. Carter. Berkeley: University of California Press.

———. [1954] 1969. Structural Functions of Harmony. Rev. ed. Ed. Leonard Stein. New York: W. W. Norton.

Stefani, Gwen, with Aliaune “Akon” Thiam. 2006. “The Sweet Escape.” Written by Gwen Stefani, Aliaune “Akon” Thiam, and Giorgio Tuinfort. The Sweet Escape. Interscope 0008099.

Taneyev, Sergey Ivanovich. 1962. Convertible Counterpoint in the Strict Style. Trans. G. Ackley Brower. Boston: B. Humphries.

Wagner, Naphtali. 1995. “No Crossing Branches? The Overlapping Technique in Schenkerian Analysis.” Theory and Practice 20: 149–76.

Music Theory Spectrum, Vol. 33, Issue 2, pp. 124–145, ISSN 0195-6167, electronic ISSN 1533-8339. © 2011 by The Society for Music Theory. All rights reserved. Please direct all requests for permission to photocopy or reproduce article content through the University of California Press’s Rights and Permissions website, at http://www.ucpressjournals.com/ reprintinfo.asp. DOI: 10.1525/mts.2011.33.2.124

MTS3302_02.indd 145 9/20/11 5:39 PM

This content downloaded from 156.143.240.16 on Thu, 13 Jun 2013 11:43:36 AMAll use subject to JSTOR Terms and Conditions