When Quality Reaches Its Limit: A New Interpretative Approach to Understand Post-tonal Harmony

Abstract

To make post-tonal harmony more accessible, many music scholars derive various theories, which rely on the traditional notions of consonance and dissonance, to describe the quality of a chord. Most of these theories use the quality of an interval as the criterion to measure the proportion between the consonances and dissonances within a chord. All intervals can be categorized as either consonances or dissonances except for one—the tritone. Although traditionally in tonal music, the tritone was perceived as a dissonance; its quality in twentieth-century music, however, has been challenged by many scholars. It may sound like a dissonant (Paul Hindemith), a consonant (Joseph Straus), or a neutral interval (Charles Seeger). If different viewpoints abound in the tritone’s quality, do we hear it as adding the consonant or dissonant sound to a chord? Confronted with this issue, this article suggests an alternative way of perceiving all intervals—their space—and further proposes a method that assists us in hearing a chord in terms of its degree of compactness. Based on this method, we can experience the dynamics created by the confrontation between spatial and compact chords, and find post-tonal harmony more accessible.

Notes

¹ In comparison, based on the size of an interval and the number of semitones therein, we can categorize all intervals into five different qualities: major, minor, perfect, augmented, and diminished. Since this categorization contains more categories than that of consonance and dissonance, I refer to the latter as the minimal number of categories in the text.

² Charles Seeger, ‘On Dissonant Counterpoint’, Modern Music 7 (1930), 25.

³ Initially, the only consonant intervals were perfect ones: fourth, fifth, and octave. By the end of the fourteenth century, the category of consonances included thirds and the major six. Finally, the dominant seventh joined the consonant family in the mid-nineteenth century. For more details, see Charles Seeger, ‘Dissonance and the Devil: An Interesting Passage in Bach Cantata’, The Baton 9 (1930), 7–8.

⁴ Seeger, ‘On Dissonant Counterpoint’, 25.

⁵ An exhaustive illustration of his theoretical principles can be found in one of Seeger’s articles, ‘Manual of Dissonant Counterpoint’, which appears in his previously unpublished book manuscript, Tradition and Experiment in (the New) Music. Fifteen years later, after Seeger’s death in 1979, Ann M. Pescatello edited and published Seeger’s Studies in Musicology II: 1929–1979, in which Pescatello also includes Tradition and Experiment in (the New) Music as the first section of this book. For more information, see Charles Seeger, ‘Tradition and Experiment’, in Studies in Musicology II: 1929–1979, ed. Ann M. Pescatello (Berkeley and Los Angeles: University of California Press, 1994), 17–273. Additionally, during Seeger’s invention of dissonant counterpoint (1913), there also appeared other harmonic treatises that studied the possibilities of forming harmonies other than the traditional triads or seventh chords. According to Jonathan Bernard’s discussion , these treatises include Bernard Ziehn’s Manual of Harmony (1907) and Five- and Six-Part Harmonies (1911), Arnold Schönberg’s Harmonielehre (1911), René Lenormand’s Etude sur l’harmonie moderne (1912), and Arthur Eaglefield Hull’s Modern Harmony (1914) (Jonathon Bernard, 1997., 13–21). See Jonathan Bernard, ‘Chord, Collection, and Set in Twentieth-century Theory’, in Music Theory in Concept and Practice, ed. James Baker, David Beach, and Jonathan Bernard (Rochester: University of Rochester Press, 1997), 11–51. Each of these four scholars spends one or more chapters explaining the derivation of the new chords, which are found in some particular compositions written in the first decade of the twentieth century. For instance, Hull provides several harmonic examples from Schönberg’s Kammersymphonie (1906), Drei Klavierstücke (1909), and Fünf Orchesterstücke (1909). Meanwhile, the last chapter from Schönberg’s Harmonielehre discusses chords that contain six or more tones, which are accompanied by passages from Anton Webern’s Fünf Sätze für Streichquartett (1909), Alban Berg’s Vier Lieder (1909–10), and Schönberg’s Erwartung (1909). In these compositions, Ziehn, Schönberg, Lenormand, and Hull have discovered the multitudinous possibilities of creating new harmonies. However, their perception of these harmonies still remains in the framework of the ‘nineteenth-century ideas of tonal harmony as augmented by chromaticism’ (Jonathon Bernard, 1997., 21). This judgment, in fact, is not unlike that of Seeger’s, which similarly regards this type of composition as the ‘elaboration and extension of the old diatonic and chromatic harmony.’ Accordingly, although Seeger did not comment on any of the foregoing harmonic treatises, he would certainly agree with Bernard’s point of view— Ziehn, Schönberg, Lenormand, and Hull’s treatments of the new harmonies are still ‘conceived as supplements of the older harmonic theory,’ and ‘[do] not provide fundamentally new ideas about pitch organization’ (Jonathon Bernard, 1997., 13).

⁶ Seeger’s dissonant counterpoint appears eight years ahead of Schönberg’s invention of serialism (1921). Although Seeger introduces his theory in the 1930 article ‘On Dissonant Counterpoint,’ we can trace the development of dissonant counterpoint back to as early as 1913, during the period (1912–19) when Seeger was teaching at the University of California, Berkeley. For more information, see Ann M. Pescatello, Charles Seeger: A Life in American Music (Pittsburg: University of Pittsburg Press, 1992), 93. In one sense, Seeger’s technique represents the pioneering compositional method that systematically organizes the sound in post-tonal music. However, his dissonant counterpoint is less familiar to musicians than Schönberg’s serialism. From my view, this is because there still exists hierarchical levels of intervals in Seeger’s method. The preference for dissonances emancipates all the dissonant intervals, but the appearance of consonances, contrarily, is restricted. Thus, consonant intervals do not have equal structural importance as dissonant ones; they only appear to embellish dissonance. This type of hierarchy disappears in Schönberg’s serialism, for all intervals have equal weight. In a series, twelve pitches are ordered according to a fixed pattern of eleven intervals, which are carefully designed by the composer. The goal for these intervals progressing from one to another is to produce a unique ordering of twelve pitches, not to prepare or resolve embellishing intervals. Additionally, this feature also allows composers in contemporary and later generations to develop other serial techniques. For example, Webern’s derived row and Stravinsky’s rotational row explore the interesting combination of intervals, and the integral serialism of Boulez, Messiaen, Stockhausen, and Babbitt integrates domains other than pitch elements into a row (such as dynamics and durations).

⁷ Hindemith published his treatise in German in 1937: Paul Hindemith, Unterweisung im Tonsatz: Theoretischer Teil (Mainz: B. Schott’s Söhne, 1937). To provide a more accessible discussion, this article uses the English version of Hindemith’s treatise translated by Arthur Mendel: The Craft of Musical Composition, Book I: Theory (New York: Shott Music, 1984).

⁸ Hindemith, The Craft, 57–8. Additionally, his series 1 refers to the order of the twelve chromatic pitches derived from the overtone series.

⁹ Ibid., 64 and 81.

¹⁰ Ibid, 85. For those interested in reading this quotation in the original German text, the following is Hindemith’s statement from his Unterweisung im Tonsatz: ‘Die konsonanten Klänge wären demnach auf der linkem Seite der Reihe 2 beheimatet, die dissonanten rechts’ (Hindemith, Unterweisung, 101).

¹¹ Hindemith, The Craft, 85. Also, here I provide Hindemith’s original German text: ‘Zwischen der Oktave als dem vollkommensten Klang und der großen Septime als dem unvollkommensten ist eine Reihe von Intervallpaaren angeordnet, deren Wohklang in dem Maße abnimmt, wie sie sich von der Oktave in Richtung auf die große Septime entfernen. Der Tritonous kann weder in die Region des Wohlklanges eingeordenet noch als Mißklang angesehen werden; er steht als das eigenartigste Intervall auch hier wieder abseits’ (Hindemith, Unterweisung, 101).

¹² Joseph Straus, ‘Voice Leading in Set-class Space’, Journal of Music Theory 49 (2005), 45–108. Besides Straus’s 2005 article, his fuzzy transformational voice leading also appears in his other two publications: Joseph Straus, ‘Voice Leading in Atonal Music’, in Music Theory in Concept and Practice ed. James Baker, David Beach, and Jonathan Bernard (Rochester: University of Rochester Press, 1997), 237–74; and Joseph Straus, ‘Uniformity, Balance and Smoothness in Atonal Voice Leading’, Music Theory Spectrum 25 (2003), 305–52.

¹³ Straus, ‘Voice Leading in Set-class Space’, 73.

¹⁴ In comparison, if we consider the eighty-eight keys on the piano keyboard, measuring the number of semitones between any two keys derives an (un)ordered pitch interval, which may range from zero to 87. The difference between ordered and unordered pitch interval is that the former contains an additional plus or minus sign, which indicates the ascending or descending direction of a melodic interval. An ordered pitch-class interval measures the number of semitones in an ascending numerical order between two pcs. That is, we always count the number of semitones from the first to the second pitch class (pc) in the order from pcs 0–11. Thus, an ordered pc interval may range from zero to eleven. These three types of intervals all have significantly more members than those of the unordered pc interval. Thus, I refer to the seven ics as the minimal number of categories that define the space of all intervals.

¹⁵ The same idea has been used to describe the Z-related pcsets. Two pcsets, despite belonging to two different scs, can still be considered as producing a similar sound based on their identical ic vector. We call the relationship between these two scs a Z-relation. For instance, sc 4-Z29 and sc 4-Z15 are Z-related by sharing the same ic vector [111111].

¹⁶ For the detailed summary and comparison of different sc similarity measurements, see Eric Isaacson, ‘Similarity of Interval-class Content between Pitch-class Sets: The IcVSIM Relation’, Journal of Music Theory 34 (1990), 1–29; Richard Hermann, ‘A General Measurement for Similarity Relations: A Heuristic for Constructing or Evaluating Aspects of Possible Musical Grammars’ (PhD thesis, University of Rochester, 1993); Marcus Castrén, ‘RECREL: A Similarity Measure for Set-classes’ (PhD thesis, Sibelius Academy, 1994); Michael Buchler, ‘Relative Saturation of Interval and Set Classes: A New Model for Understanding Pcset Complementation and Resemblance’, Journal of Music Theory 45 (2001), 263–343; and Tuire Kuusi, ‘Chord Span and Other Chordal Characteristics Affecting Connections between Perceived Closeness and Set-class Similarity’, Journal of New Music Research 34 (2005), 259–71.

¹⁷ This is the first tetrachord in this movement, in which each string instrument plays a different pitch. This chord stops at the third beat of b. 21, because the first violin moves the pitch D₄ to F_#4.

¹⁸ All of the theoretical terms and their definitions are summarized in Appendix 1. It is advisable to read the text along with this glossary.

¹⁹ In my discussion, the elements in a VPicset, reading from left to right, always correspond to the lowest to the highest VPics. Additionally, I must point out that because this harmonic perception fully relies on the different instrumental timbres, the relationship between the immediately adjacent string instruments becomes far more intimate and direct than that between the non-adjacent ones. Hence, I only consider the adjacent instruments and exclude the non-adjacent ones. Since there are four string instruments in Figure 6, the VPicset includes three VPics, each of which, from low to high, appears between the adjacent voices of [Vc–Vla], [Vla–Vln II], and [Vln II–Vln I]. Significantly, the same harmonic perception explains how I understand and hear a chord latter in the analyses of Stravinsky and Kurtág’s music at the end of this article.

²⁰ Voice crossings constantly appear throughout this movement. (Voice crossing here refers to a pair of the adjacent string instruments whose registral orderings do not follow the traditional, standard string quartet setting, in which Vln I, Vln II, Vla, and Vc, respectively, project the soprano, alto, tenor, and bass voices.) Ellie Hisama uses the term ‘degree of twist’ to define the number of voice crossings within a chord. Ellie Hisama, ‘The Question of Climax in Ruth Crawford’s String Quartet, Mvt. 3’, in Concert Music, Rock, and Jazz since 1945, ed. Elizabeth West Marvin and Richard Hermann (Rochester: University of Rochester Press, 1995), 298. For instance, the chord in Figure 6 has a degree of twist 2—Vc over Vla and Vln II over Vln I. Based on her analysis, Hisama finds multiple climaxes achieved by the gradual progressions moving toward and then away from an exceedingly twisted texture, which do not coincide with the registral and dynamic climax at b. 75. For Hisama, the highly twisted texture represents what she believes to be a ‘feminist’ climax (ibid., 305), which is opposed to the register and dynamics representing a more contextual, traditional, and ‘masculine’ form of climax in a composition. Also, Edward Gollin extends Hisama’s study by applying a Cayley graph to analyze how the registral permutations of the four string instruments create a transformational network, and further uses his results to support Hisama’s view of feminist climax. Edward Gollin, ‘Form, Transformation and Climax in Ruth Crawford Seeger’s String Quartet, Mvmt. 3’, Mathematics and Computation in Music 37 (2009), 340–6.

²¹ In his ‘Voice Leading in Atonal Music’ (237–8), Joseph Straus categorizes contemporary atonal voice leading theories into three different models: prolongational, associational, and transformational. The perception of voice leading discussed in the text is akin to Straus’s definition of the associational, which uses contextual means (in Crawford’s case, the timbre) to connect the adjacent pitches between successive chords. The transformational refers to the application and extension of David Lewin’s generalized interval system, which uses transformation to analyze the voice leading. David Lewin, Generalized Musical Intervals and Transformations (New Haven: Yale University Press, 1987). The theories representing this model include Henry Klumpenhouwer’s K-net and Joseph Straus’s fuzzy transformational voice leading. The prolongational model, which ‘has its roots in the theories of Heinrich Schenker’ (ibid., 237), uses graphical notations—such as slurs and stems—to identify structural pitches from the embellishing ones. Scholars whose works feature this model include Roy Travis, James Baker, and Paul Wilson. Besides categorizing various voice leading techniques into different models, Straus also critiques and evaluates these techniques, pointing out their potential weaknesses and practical strengths.

²² Actually this harmonic perception is similar to Allan Chapman’s voice pairs interval set (VP). Allan Chapman, ‘Some Intervallic Aspects of Pitch–Class Set Relations’, Journal of Music Theory 25 (1981), 278. A VP gauges the ordered pc interval between each pair of the adjacent voices in one chord. If chords whose VPs share the same combination but with different orderings of the ordered pc intervals, they create the VP-related sets (ibid., 280). However, unlike the topic of my article, Chapman is concerned more with the scs than the space produced by VPs or VP-related sets, because different orderings of a VP can result in different scs.

²³ For instance, applying my formula to measure the average-VPicset in Figure 6: (1 + 3 + 1) / 3 = 1.67.

²⁴ If a TVPicset is only composed of the most compact TVPic 0, summing up all the TVPics and averaging the total derives the average-TVPicset 0.00, which corresponds to the smallest number and represents the most compact chord. Contrarily, if a TVPicset is only composed of the most spatial TVPic 6, summing them up and averaging the total derives the average-TVPicset 6.00, which corresponds to the largest number and represents the most spatial chord. Since the smallest and the largest average-TVPicsets, respectively, are 0.00 and 6.00, connecting these two numbers with each other then forms a limited range, which covers all of the possible average-TVPicsets derived from the application of my harmonic measurement.

²⁵ This footnote provides the reader with other alternative theories that also use the concept of space to study chords in twentieth-century and twenty-first-century music. Jonathan Bernard gauges the total number of semitones from the lowest to the highest pitches. Jonathan Bernard, ‘Pitch/Register in the Music of Edgard Varèse’, Music Theory Spectrum 3 (1981), 1–25. Based on this intervallic perception, he studies the structural similarity between different chords in terms of the arrangements of their intervals. The analytical techniques he proposes include symmetry, mirror symmetry, parallel symmetry, projection, partial projection, rotation, expansion, and contraction. Also, grounded in the same definition of an interval, Bernard extends his discussion to include the subject of voice leading. See Jonathan Bernard, ‘Voice Leading as a Spatial Function in the Music of Ligeti’, Music Analysis 13 (1994), 227–53. He examines how the formation of the voice leading is reinforced not by ‘literal, registral proximity,’ but by the spatial context of the ‘absolute sizes of intervals’ in Ligeti’s Lux aeterna and his Second String Quartet, Mvt. III (ibid., 230).

²⁶ Straus, ‘Voice Leading in Set-class Space’, 97.

²⁷ This passage is also no exception, which is part of the conclusion of Threni.

²⁸ Straus, ‘Voice Leading in Set-class Space’, 97.

²⁹ Note that the last IR is different from a more conventionally adopted Schoenbergian RI—the retrograde of the inversion. Stravinsky’s use of IR may be influenced by Ernst Krenek. He introduces IR in addition to RI that associates with P, I, and R. Krenek, Studies in Counterpoint Based on the Twelve-tone Technique (New York: G. Shirmer, 1940), 11. For a detailed discussion about the linkage between Stravinsky and Krenek’s twelve-tone rows, see: Joseph Straus, Stravinsky’s Late Music (Cambridge: Cambridge University Press, 2001), 174; Joseph Straus, Twelve-tone Music in America (Cambridge: Cambridge University Press, 2009), 34–40; and Arnold Whitall, The Cambridge Introduction to Serialism (Cambridge: Cambridge University Press, 2008), 138–41.

³⁰ For a more thorough and technical discussion about Stravinsky’s other twelve-tone passages similar to Figure 9, see Joseph Straus, ‘Stravinsky’s “Construction of Twelve Verticals”: An Aspect of Harmony in the Late Music,’ Music Theory Spectrum 21 (1999), 43–73; Straus, Stravinsky’s Late Music, 165–82; Straus, Twelve-tone Music in America, 34–40; and David Smyth, ‘Stravinsky as Serialist: The Sketches for “Threni”’, Music Theory Spectrum 22 (2000), 205–24.

Additionally, in his two other recent articles, Straus also studies Stravinsky’s serial music but from rather different, innovative approaches. Joseph Straus, ‘Contextual-inversion Spaces’, Journal of Music Theory 55 (2011): 43–88; and Joseph Straus, ‘Harmony and Voice Leading in the Music of Stravinsky’, Music Theory Spectrum 36 (2014), 1–33. These include the adoption of David Lewin’s contextual inversion operation and his own bi-quintal structure. See David Lewin, Musical Form and Transformation: Four Analytical Essays (New Haven: Yale University Press, 1993), 7.

³¹ Although Stravinsky does not rearrange the pcs in chord ⑥, he adds another pc 2 in the soprano marked with an asterisk sign in the array.

³² The reason why pc 6 here stands for a G^b₃ instead of its enharmonic F_#3 in the bass is that Stravinsky always realizes pc duplication in his serial music as the octave doubling. Since the tenor sings a G^b₄, its lower octave doubling must be a G^b₃.

³³ Sufficient evidence supports this claim. This chord reappears later in bb. 416–17, and its bass is a G^b.

³⁴ Straus, Stravinsky’s Late Music, 170, footnote 24. A similar notational error also appears in ’Exaudi’ from Stravinsky’s Requiem Canticles (bb. 71–76; see ibid., 172).

³⁵ Straus actually uses the term ‘degree of chromaticness’ to define the space of a chord (‘Voice Leading in Set-class Space’, 95). However, for the purpose of consistency, I substitute his term with my own ‘compactness’ throughout the discussion.

³⁶ Straus, ‘Voice Leading in Set-class Space’, 95.

³⁷ Note that both chords ④ and ⑤ significantly expand their spaces in the contextual realizations (especially chord ④, it changes its average-TVPicset from the smallest 1.67 to the largest 5.33 in the entire Figure 11). This is because Stravinsky moves the duplicated pcs next to each other (see the contextual realizations in Figure 9: the two pcs 8 in chord ④ and the two pcs 1 in chord ⑤), which form an ic 0 representing the most spatial TVPic6 and significantly increasing the space of that chord.

³⁸ Instead of the one on the extreme right, which spans 0.70 degrees.

³⁹ According to her article, Hyde points out that Kurtág’s music captures the essence of the Romantic fragment, because it is both ‘imperfect and yet complete: imperfect in the sense that its meaning expands ever outward, yet complete or self-standing in its form.’ See Martha Hyde, ‘Semiotics and Form: Reading Kurtág’s Kafka Fragments’, Interdisciplinary Studies in Musicology 5 (2005), 186. Note that Hyde uses the slightly different term ‘imperfect’ from my ‘incomplete.’ This term ‘imperfect’ is also used by Charles Rosen in the second chapter ‘Fragments’ of his book. Charles Rosen, The Romantic Generation (Cambridge: Harvard University Press, 1995), 41–115. In Rosen’s discussion, he not only provides many nineteenth-century music examples representing the Romantic style fragments, but also explicitly explains the history and development of the concept of a fragment.

For a Romantic composer, the challenge in creating a fragment is to strive to find a balance between supplying it with many ‘destabilize[d] … codes of tonal syntax’ and yet, at the same time, superimposing it on a ‘complete, formal musical structure’ (Hyde, ‘Semiotics and Form’, 186). However, the challenge in creating a fragment for Kurtág, whose works essentially cross the confines of tonality, is rather different. If his fragments seemingly recapitulate the tradition of the Romantic fragment and correspondingly project complete musical forms, how are they destabilized in the absence of non-tonally syntactic codes?

Confronted with this issue, Hyde selects some fragments from Kurtág’s Kafka Fragments (Op. 24) as her test cases, applying musical topoi (i.e. topics)—defined by Raymond Monelle, The Sense of Music: Semiotic Essays (Princeton: Princeton University Press, 2000)—to analyze her selections. Her results show that Kurtág’s fragments always unfold the particular topoi that ‘historically signify unstable or contradictory meanings, which often have evolved over centuries and span both tonal and pre-tonal styles’ (Hyde,‘Semiotics and Form’, 186). Hence, from Hyde’s analytical perspective, Kurtág’s fragments then carry the sense of imperfect, for they contain the topoi whose meanings are ‘unstable’ and ‘contradictory.’

⁴⁰ Since the soprano is the leading, principal voice among the three imitative voices, I regard it as the primary melody accompanied by two other bass instruments and focus on its melodic contour.

⁴¹ As Rachel Beckles Willson would probably observe, Kurtág’s use of perfect fifths in the soprano’s first phrase invokes the ‘innovation of God,’ which can be regarded as the depiction of the word ‘sunrise’ in the text, while his use of a pitch C^#₄ that is chromatic and dissonant to the whole tone collections {0, 2, 8, T} in the soprano’s second phrase (I define these pitches as all-but-one whole-tone collections) suggests ‘sin’ and ‘death,’ which might judge those who do not get up early. Rachel Beckles Willson, Ligeti, Kurtág, and Hungarian Music During the Cold War (Cambridge: Cambridge University Press, 2007), 107–11. Combining her observation with my analysis, I find that not only is there a phonological change between the two phrases (i.e. pitch contour) but also a semiotic one (i.e. the intervals). But how does my analysis of Figure 22 support this semiotic articulation?

Beckles Willson also mentions Kurtág’s use of chords to project what she refers to as ‘dark, dirty’ and ‘light, clean’ sonorities (ibid., 106). In particular, she uses space to describe each of these two types of sonorities. While the first sounds ‘closed,’ the second sounds ‘open’ (ibid., 106). If we replace Beckles Willson’s terms with my words, the open and closed sonorities would correspond to spatial and compact chords, respectively. Then, the overall harmonic progression from the spatial to the compact in Figure 22 beautifully supports Kurtág’s semiotic design, in that the spatial chord (which signifies the topic ‘light, clean’) along with the perfect fifths delineate the picture of the ‘sunrise,’ and the compact chord (which signifies the topic ‘dark, dirty’) along with the all-but-one whole-tone collections delineate the picture of ‘no one gets up.’

⁴² It is worth mentioning that a few years ago the Italian music theory journal Gruppo di Analisi e Teoria Musicale published a special issue concentrating on Kurtág’s music. This issue contains articles by Friedemann Sallis, Antonio Rostagno, Egidio Pozzi, and Mario Baroni. For more information, see Gruppo di Analisi e Teoria Musicale XVI/1 (2010), 7–122.

Additional information

Notes on contributors

Yi-Cheng Daniel Wu

Yi-Cheng Daniel Wu completed his PhD (2012) in Music Theory at the University at Buffalo (Buffalo NY, USA). His research interests focus on the topics of musical form, harmony, voice leading, and pitch contour in twentieth-century and twenty-first-century music. Prior to coming to Soochow University School of Music (Suzhou, China) in Fall 2013 as the assistant professor of Music Theory, he taught at Wesleyan University (Middletown CT, USA), where he served as the visiting assistant professor of Music. His articles appear in Indiana Theory Review, Music Analysis, and Musicology Australia.