Penta ntatonicism tonicism from th the e Eighteenth Century to Debussy Jeremy Day-O’Connell. University of Rochester Press,
2007.
reviewed by NICOLE BIAMONTE
Analytical studies of pentatonic structures in western art music are few, and most have focused on individual works. 1 Some broader considerations of pentatonicism in folk and rock musics have been published, 2 but Jeremy Day-O’Connell is the first scholar to undertake such a study for art music, and the first to do so on such an ambitious scale. Pentatonicism from the Eighteenth Century to Debussy, Debussy , an expansion of the author’s dissertation (Day-O’Connell 2002), is a significant addition to the literature on this topic. The book’s five chapters are grouped into three parts, followed by a brief afterword surveying pentatonic practice after Debussy, and a catalog of over 400 examples. Part I, “Scale,” comprises a single chapter (Ch. 1) focused on the role of 6 in the nineteenth century. Part II, “Signification,” explores topics that have historically been represented through pentatonicism: the “Pastoral-Exotic” (Ch. 2) and the religious (Ch. 3). Part III, “Beyond Signification,” focuses on coloristic rather than symbolic expressions of the pentatonic, in in the form of glissandos (Ch. 4) and in the music of Debussy (Ch. 5). There is no single s ingle thesis statement in the introduction, but Day-O’Connell Day-O’Connell suggests that he will provide an “ideological, analytical, and theoretical” (2) examination of pentatonicism’s history and pre-history in the nineteenth century. (In similar manner, the jacket copy describes a weaving together of “historical “historical commentary with theory and analysis,” although the author may well have had nothing to do with the writing of this text.) This implicit promise is met, but only in part: the book offers extensive historical commentary, hermeneutic interpretation, and an abundance of musical examples, and several of the analyses are insightful and illuminating, but in many other cases they are cursory, and the theoretical grounding for the study could be more thorough. In the introduction, “pentatonic” is defined as the familiar anhemitonic major pentatonic, 1–2–3–5–6 of the diatonic major scale (3–4). The other rotations of this scale are mentioned, but according to the author they have not been explored as a compositionall resource in western art music, with the exception of the minor pentatonic compositiona
See, for example, Kopp 1997 and Day-O’Connell 2009 on Debussy, Beveridge 1977 and Beckerman 1996 on Dvoř ák, ák, Harrison 1997 on Milhaud, and Hefling 2000 and Williamson 2011 on Mahler. 1
On the pentatonic in Anglo-American folk music, see Bevil 1986, which surveys earlier work by Gilchrist, Sharp, Foss, and Bronson; in rock music, see Biamonte 2010, Capuzzo 2009, Doll 2013, and Everett 2004. 2
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in the late nineteenth (5) or the early twentieth century, in the music of Bartók and Stravinsky (n.12).3 Defining and labeling the pentatonic in reference to the diatonic is appropriate in this context, because it reflects the way that pentatonic structures would most likely have been understood by contemporaneous composers, listeners, and critics. Several of the special properties shared by the pentatonic and diatonic scales are briefly listed in the text, but are described only as conferring “certain advantages” (5), without further consideration of their musical consequences. An explanation of the effects of these properties would be helpful for readers not conversant with scale theory, so I have expanded on the author’s summary below. Both collections constitute “systems” (Dahlhaus 1990) 4 or “well-formed” scales (Carey and Clampitt 1989) 5; thus they have a clear distinction between steps and leaps, which is an important point in light of focus of this study, since the pentatonic scale is perceived as gapped when heard against the background of the diatonic. Both are “generated collections” constructed from a cycle of fifths, but both require one non-perfect fifth to complete the cycle: a tritone in the diatonic collection (a perfect fifth diminished by one semitone), and a minor sixth in the pentatonic (a perfect fifth enlarged by a semitone). 6 This single differing interval ensures maximal individuation—each note has a unique relationship to the other notes, allowing for accurate position-finding within the scale. Both collections are maximally even but not perfectly symmetrical, 7 and have two sizes of scale step. This near-symmetry allows for smooth melodic motion and voice leading but also for slight variations in interval size when motives or chords are transposed, creating musical interest.
One example of a different pentatonic rotation is the B section of Liszt’s Klavierstuck no. 1 (mm. 23–30), which features the third rotation of the major pentatonic as a bass ostinato (E–G–A–C–D). 3
In Dahlhaus’s model, the pentatonic and diatonic scales constitute tonal systems because they contain no ambiguities between scale steps and leaps. In contrast, in the harmonic minor scale, the scalar step of an augmented second spans the same number of semitones as a leap of a minor third. Similarly, in the natural hexachord, the distance from A up to C is technically a step, but it is the same size as the minor thirds between D and F or E and G. See Dahlhaus 1990, 172. 4
A well-formed scale is generated from a cycle of a single generic interval (each interval spans the same number of scale steps, but not necessarily the same number of semitones). Well-formed scales have either one size (“degenerate”) or two sizes (Myhill’s property) of scale step. See Carey and Clampitt 1989. 5
Collections generated from only one specific interval size, such as the whole-tone and chromatic scales, are called “degenerate” well-formed scales, and are more symmetrical than other scales. See Carey and Clampitt 1989. 6
In a maximally even collection, notes are as evenly distributed as possible. Maximally even collections have either one size of scale step and thus are perfectly symmetrical, as in the whole-tone scale, or two consecutive sizes, which are either symmetrical, as in the octatonic scale, or near-symmetrical, as in the diatonic and pentatonic scales. See Clough and Douthett 1991. 7
Nicole Biamonte – Review of Pen tatonicism from the Eighteenth Century to Debussy
Both scales possess optimum tonal consonance, 8 and thus have many consonant subsets. The diatonic collection is “deep,” with a unique multiplicity of each intervalclass9; the pentatonic is “near-deep” (my term), because it does not quite have unique multiplicity— there are no semitones and no tritones, so there are two zeroes in its interval-class vector. Nonetheless the pentatonic collection has unique multiplicity for each interval class that it does contain. The varied density of interval classes in both collections establishes a hierarchy of common and rare intervals within each collection, as well as a hierarchy of close and distant modulations because of the varying numbers of common tones. Both the diatonic and pentatonic scales alter by a single semitone when transposed by fifth, 10 permitting maximally smooth modulation by fifth. An additional property common to the pentatonic and diatonic collections but not mentioned in the text is inversional symmetry, which also contributes to efficient modulation. The pentatonic scale shares all of these properties with the diatonic scale because of its close relationship as both complement and subset. More important to the focus of Day-O’Connell’s study, however, are their differences. The author notes (3-4) that the pentatonic scale is a subset of the diatonic scale, and thus it has fewer available pitches. Pentatonic steps are larger than diatonic steps, resulting in a structure often perceived as “gapped,” and one that severely limits the possibilities for triadic harmonization. Because of the different scale-step sizes, the two collections have different characteristic scalar subsets: minor second/major second combinations for the diatonic, major second/minor third for the pentatonic. The pentatonic scale has greater intervallic consonance than the diatonic scale, because of its lack of tritones and semitones. For the same reason, position-finding, including the establishment of a clear tonic, is more difficult in a pentatonic context than in a diatonic one. Of these characteristics, gapped melody, lack of semitones, and restricted harmony seem likely to be the most analytically salient pentatonic features. The examples in the book are indeed focused on instances of pentatonic melody as expressed through characteristic major second/minor third figures. However, the incorporation of these melodic figures within the context of a highly sophisticated system of harmony begs for a consideration of their relationships to their accompaniments, which is missing from this study. Various strategies are manifested in the book’s musical examples; selected examples Alone among the pentachords, the pentatonic scale lacks the two most dissonant interval classes, the semitone and tritone, and the diatonic scale has the fewest of any heptachord (two semitones and one tritone). Both sets have the highest aggregate dyadic consonance for their cardinalities. See Huron 1994. 8
9
10
See Gauldin 1983 and Clough and Myerson 1985. Also called the “F to Fƒ property.” See Zweifel 1996 and Tymoczko 2011, 17–19 and 129–36.
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of each technique are cited in footnotes(example numbers begin either with a chapter number or a P, which refers to the catalogue of pentatonic examples at the end of the book): unison texture, 11 octave doubling, 12 pedal point,13 static harmony,14 I–vi oscillation, 15 incomplete triads,16 planing, 17 and quartal18 or quintal19 harmony (as in much modern jazz). Quartal harmonies can of course be understood as stacks of fourths, but underneath a pentatonic melody it makes more sense to interpret them as pentatonic thirds; i.e., as combinations of major thirds and perfect fourths. 20 Pentatonic melodies can also be treated in a more conventionally tonal manner, with diatonic or chromatic harmonizations, and indeed, most of the examples presented in this study fall into this category. Such harmonizations may be either consonant or dissonant with regard to the melody; a greater tolerance for dissonance may even have been a factor in the increasing use of pentatonicism in the late nineteenth century, along with other melodic structures that are difficult to harmonize diatonically, such as the whole-tone and octatonic scales. An excerpt from Rimsky-Korsakov’s Sadko features a unison pentatonic melody (P252, 357). Unison pentatonic melodies alternate with harmonized versions in the “Promenade” of Mussorgsky’s Pictures at an Exhibition (P266, 364). A unison pentatonic example not noted by the author is the first six bars of the section “Brüder! über’m Sternenzelt” from the finale of Beethoven’s Ninth Symphony (mm. 611–17). 11
Pentatonic melodies are set with octave doublings at the end of Saint-Saëns’s La princesse jaune (P15, 218) and at the beginnings of the Sullivan’s The Mikado overture (P25, 222) and Chopin’s Krakowiak (P242, 348). 12
A pentatonic melody is supported with a bass pedal in an excerpt from Debussy’s Voiles (5.15, 169), and a bass melody is accompanied with an upper-register dominant pedal at the beginning of the overture to Nicolai, Die lustigen Weiber von Windsor (P222, 334). Pentatonic counterpoint is set over a bass pedal in an excerpt from Argento, “To Be Sung upon the Water” (A5, 187). 13
Static tonic harmony supports pentatonic melodies in the openings of Rossini’s William Tell overture (P70, 248), Wagner’s Das Rheingold (P96, 265), and Rachmaninoff’s “The Lilacs” (P214, 330). Two additional examples are the central section of Debussy's "Nuages" (mm. 64-68) and the opening of his “Sirènes” (mm. 1-4, transposed up a minor third in mm. 5-7 and again in mm. 8-11). 14
Oscillations between I and vi occur in several excerpts from Saint-Saëns’s La princesse jaune (P8, 213; P10, 214; P11, 215), an excerpt from Wagner’s Lohengrin (P99, 267), and the opening of Hahn’s “L’heure exquise” (P168, 308). 15
Vogler’s Pente chordium (Ex. P2, 206–7) conforms to the black-key pentatonic collection in both its melody and its accompaniment, which alternates between tonic triads and incomplete dominant triads with no third. 16
The choral entrance at the beginning of Borodin’s Prince Igor features planed triads outlining a pentatonic melody (P243, 349), and quintal harmonies are pentatonically planed in Debussy’s La fille aux cheveux de lin, mm. 24–27, and the opening of his La cathédrale engloutie (these two instances are not included in the book, although the author analyzes La fille aux cheveux de lin; see 160-163 and Day-O’Connell 2009). 17
18
Quartal harmony is set over a tonic pedal in an excerpt from Ravel’s Daphnis et Chloé (5.2, 159).
A pentatonic melody in the bass is accompanied with quintal harmony in an excerpt from Debussy’s Pagodes (5.22, 175). The opening of Hahn’s “D’une Prison” can be interpreted as static tonic harmony with an added sixth, but it is voiced mostly in fifths (the lowest interval is a major sixth). 19
20
The author makes this point in relation to Debussy's Pagodes (175 and 2009, 228). For additional examples from both art music and jazz, see Tymoczko 2011, 128 and 357-60.
Nicole Biamonte – Review of Pen tatonicism from the Eighteenth Century to Debussy
Chapter 1, “The Rise of 6 in the Nineteenth Century” (slightly revised and expanded from Day-O’Connell 2006), explores two contrasting voice-leading treatments of 6 in
theory and practice: “classical” 6 as a tendency tone that descends to 5 versus “nonclassical” 6 as a pentatonic step that ascends to 8. The descending resolution of 6 to 5 is analogous to that of 4 to 3, although Fétis’s term for 6, “superdominant” rather than
“submediant” (17), suggests a closer analogy with the resolution of 2 (“supertonic”) to 1. The author cites theoretical treatises by Rameau, Heinichen, Hauptmann, Louis, Thuille, and Curwen that either implicitly or explicitly posit a break or gap between 6 and 7 in the ascending scale (15-16). One compositional solution to this perceived
difficulty is to leap down a seventh from 6 down to 7 in a lower octave, as in the third movement of Haydn’s String Quartet op. 50 no. 6, m. 8 (ex. 1.11, 25). This gesture has precursors in numerous Baroque fugue subjects that feature a descending diminished-seventh leap from the lowered form of 6 in minor down to the leading tone 21—for that matter, fugue subjects featuring the “classical” gesture 5–6–5 are also very common. Another compositional solution is to treat 6 as a surrogate leading tone and ascend directly to 8, and Day-O'Connell offers a variety of examples supporting the emergence of this gesture in Romantic music. The examples of cadential 6–8 (Table 1.2, 30) are convincing, although it would have been nice to include years of composition for each work to provide additional chronological context; the ending of Mendelssohn’s Midsummer Night’s Dream overture is another instance of this phenomenon. Overall, the content of Ch. 1 is interesting and informative, and as McFarland 2009 points out, it is a worthy successor to the study of the subdominant expansion in the nineteenth century presented in Stein 1983. It is problematic, however, that this exploration of 6 comprises the entirety of Part I, entitled “Scale,” which the reader might reasonably expect to focus on the pentatonic scale. As a way of recontextualizing 6–8 as a step, Day-O’Connell posits a pentatonic pitch space that lies between triadic and diatonic space (Fig. 1.7, 31), in a model indebted to Lerdahl’s Tonal Pitch Space (2005, 161). The author runs into difficulty in his attempts to connect the plagal harmonic
motion supporting most of the examples with pentatonic space. In numerous instances, 4 (which lies outside the pentatonic collection as defined here) is the penultimate bass note supporting the ostensibly pentatonic 6 in the melody. A disclaimer in the introduction notes that most of the examples of pentatonicism pertain to melody alone
(4), yet even considered purely melodically, these examples are not very satisfying. Many of the “non-classical” 6–8 motives are approached from 5, creating the characteristic pentatonic (025) trichord, but most comprise only a few notes, and
Bach’s Ein musikalisches Opfer and Fugue no. 20 in A minor from the Well-Tempered Clavier Bk. II, “And with his stripes” from Handel’s Messiah, the Kyrie from Mozart’s Requiem, and numerous others. 21
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with two exceptions, 22 they do not constitute pentatonic melodies in any larger sense. At best, they could be described as pentatonic fragments or brief inflections, rather than instantiations of pentatonic space. Thus the nature of the pentatonic is here reduced to brief emphases on a single melodic scale degree. These are two related but separate topics; indeed, “The Rise of 6 in the Nineteenth Century” could serve as a title for the first three chapters, since little is said about other degrees of the pentatonic scale. A mere handful of examples demonstrate melodic gaps between 3 and 5. Yet if 6 is a substitute for the “missing” 7, is there no equivalent substitute for the “missing” 4, such as a new tendency to approach a cadential 3 directly from 5, functioning as a descending pentatonic step (as in Chopin’s Nocturne op. 55 no. 1 and Debussy’s “La flûte de Pan” and Danseuses de Delphes)?
The last example in Ch. 1 lists several specimens of “speech thirds,” such as “tattletale,” “airball,” and “left-right-left!” (1.35, 42), which Day-O’Connell conjectures may relate to the pentatonic minor third. Additional instances of “calling thirds” are offered in the next two chapters. These examples resonate strongly with the speculative narrative described in Van der Merwe 2004, in which the major seconds and minor thirds of children’s chants from around the world coalesced into the pentatonic, which became the background structure for diatonic music, which then developed into classical tonality. This evolutionary model of tonality remains controversial, and is at best an oversimplification, at worst a pernicious myth—see Gelbart 2007 for a compelling critique of precisely this construct. Nonetheless, the broad geographical distribution of both calling thirds and pentatonic structures are well documented and suggestive of a potential correlation. 23 Part II of the book, “Signification,” is a historical and hermeneutic investigation of topics associated with the pentatonic scale. The first section of Ch. 2, “The PastoralExotic Pentatonic,” is a detailed survey of the eighteenth- and nineteenth-century European reception of other musics, mostly Chinese music, concluding with a handful of examples of pentatonic exoticism. The second and much larger section explores “domestic” pentatonicism in a wide variety of guises grouped under the aegis of the pastoral: traditional pastorales, horn calls, vocal calls, lullabies, birdsong, and bells. Many of the examples in this section are subsets of the pentatonic and thus could also be interpreted as triadic arpeggiations ornamented with passing ( 2) or neighboring ( 6) tones.
Two exceptions are the repeated opening phrase of Schubert’s “Gute Nacht” from Winterreise (1.22, 33), and the opening (a) and closing (a’) phrases of the A section of Dvo ř ák’s 9th Symphony slow movement (1.32, 39–40). 22
23
See, for example, Gauldin 1983, Burns 1999, Van der Merwe 2004, and McDermott and Hauser 2005.
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A brief discussion of dance highlights the importance of 6 in the waltz and other Viennese dances, especially as an appoggiatura meant to prompt physical motion through its dissonance (80). The chapter concludes with a consideration of Scottish pentatonicism, which is described as presumptively familiar to continental Europeans (84), but might be better placed in the “exotic” section, especially in light of the not-quite-wholesale manufacture of the Scottish national identity in the nineteenth century (see Gelbart 2007). Chapter 3 concerns “The Religious Pentatonic,” beginning with a survey of the nineteenth-century religious revival and concomitant reawakening of interest in plainchant and other older religious music, and ending with a variety of musical examples. In sacred music with pentatonic associations, the characteristic 025 motive frequently occurs as a melodic incipit as well as a cadential figure. A quite different use of the pentatonic is demonstrated with examples of rapid pentatonic arpeggiations that suggest the spiritual or supernatural, as in the magic fire music at the end of Wagner’s Die Walküre (3.29, 138–39). The most substantial example discussed in this chapter is the opening one, “In Paradisum” from Fauré’s Requiem, which features a D-major pentatonic melody and bass—but a hexatonic accompaniment pattern that includes G— for the first fifteen bars (3.1, 99–101). After this point, both the melody and harmony are chromaticized, but Day-O’Connell continues his pentatonic interpretation into this chromatic territory, explaining the progression B7 to D7 in mm. 17–18 as a pentatonic lower-neighbor figure in the bass (B up to D) combined with a chromatic upper neighbor
(D down to D) in the melody (3.2, 101; note that C in m. 18 is missing a natural sign). An advantage of this contrapuntal analysis is that it highlights the contrary-motion approach to the tonic D. More harmonically-oriented interpretations would label this motion as either a common-tone dominant-seventh relationship (preserving F and A as common tones) or as a chromatic-third dominant substitute, which is very characteristic of Fauré’s harmony. 24 The author cites the “Pie Jesu” from this work for its many instances of the 5–6–8 motive, and the choral “Hosanna” for its omission of 4, but his focus on the major pentatonic has led him to omit consideration of the strong minor pentatonic inflections of the opening movement. While the two chapters in Part II treat closely related topics, Part III, “Beyond Signification,” pairs two chapters that are unrelated apart from not fitting comfortably into either of the first two sections. Chapter 4, “The Pentatonic Glissando,” is much
shorter than the others, and is more of an extended footnote than an essential part of A similar chromatic-third dominant substitute, followed by its diatonic-third substitute, is employed at the end of the tenor solo in the “Agnus Dei” movement: the authentic cadence G7–C is echoed by E–C and then Em–C (mm. 39–45). This arrival serves as the point of departure for yet another chromatic-third progression, in which the tonic C becomes the third of A-flat major. 24
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the text. Most of the chapter is taken up with descriptions of nineteenth-century harp technology and repertory. A fuller exploration of the relationship between organology and the pentatonic scale would include earlier harps and their tunings, a more in-depth discussion of keyboard topology and its potential influence (a few black-key passages are cited, including Chopin’s well-known Etude in G-flat, op. 10 no. 5), and some consideration of the guitar, which is tuned in pentatonic thirds—perfect fourths plus one major third—and is notable for the ease with which pentatonic scales can be played on the fretboard in a simple box pattern. The author ascribes the development of additive harmony to added-note harp arpeggios, particularly the major triad with added sixth (147, 156), but a much earlier precedent was set by one of Rameau’s most famous theoretical rationalizations, the chord of the added sixth, which is discussed in treatises from the late seventeenth century and was used in practice even earlier. 25 The final chapter, “Debussy and the Pentatonic Tradition,” presents musical analyses on a larger scale. An example from Ravel’s Daphnis et Chloé is offered near the beginning of the chapter (5.2 ,159) but the remainder examines works of Debussy, with the most attention given to La fille aux cheveux de lin (160–3), La Mer (164–66), and Pagodes (171–81). These analyses appear in Day-O’Connell 2009 with expanded Schenkerian readings, along with other material from earlier in the book. Although readers might disagree with some of the author’s interpretations, and orthodox Schenkerians may likely disapprove of his methodology and the resulting graphs, the analyses are clearly and coherently presented. In this chapter the author is primarily concerned with purely musical relationships; however, Monelle 2009 observes a common thread of topics and characters deriving from ancient mythology that can be traced through several of the examples. The book concludes with a one-page afterword that lists some later art-music composers noted for their pentatonicism and comments on the prevalence of the minor pentatonic in popular music. A brief catalog of twelve interesting and provocative examples drawn from these repertories follows, but, frustratingly, they are not discussed at all. This is also the case with many of the examples from the much larger appendix at the end of the book, comprising 416 music examples from 1700–1926 in a variety of styles and genres, many from little-known works— an impressive compendium, but much longer than necessary. While some examples are excerpted in the text and presented in fuller versions in the appendix, numerous others are reproduced unchanged, for no discernible reason. As noted earlier in this review, many examples are fragmentary (two
See Denis Delair, Traité d’acompagnement pour le théorbe et le clavessin (Paris: Delair, 1690), Charles Masson, Nouveau Traité des regles pour la Composition de la Musique (Paris: Ballard, 1699), and Jean-Philippe Rameau, Traité de l’harmonie (Paris: Ballard, 1722). 25
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or three bars of a large-scale work), and would better have been omitted to allow for more discussion of the longer and structurally significant examples. All of the examples in the catalog are cross-referenced in the main text, but it would also have been helpful to include annotations in the catalog: an ideal caption not only identifies the musical excerpt but also what it is meant to demonstrate. Brief descriptions of the point of each musical example would be especially useful in this case, since not all of the examples in the “Catalog of Pentatonic Examples” are in fact examples of pentatonicism. The text is well-written and eminently readable, but there are occasional problems with terms that are defined inadequately or not at all. For example, the minor third formed by 3–5 is labeled “infra-diatonic” (4), a term that presumably refers to the status of the pentatonic scale as an older subset of the diatonic in the evolutionary model of tonality presented in Yasser 1932, but the term is not defined here and is not in general use. A larger issue with terminology is the definition of “mode” in the musictheoretical sense: in the introduction the author defines the pentatonic scale as a mode, “that is, with tonic 1” (5), which is a better definition of “scale” than “mode.” In the discussion that follows, “mode” is used to refer to the set of rotations of the pentatonic scale, but preserving the scale-degree labeling of the original rotation, so that none of the other putative tonics are labeled 1. At the beginning of Ch. 1, the author defines mode still differently (13–14 and 31), this time as a negotiation between scale and melodic archetype, as in the dichotomy of “particularized scale” versus “generalized tune” posited in the New Grove article on the subject (Powers 2001).
It is always a tricky balancing act to intertwine history and theory, and this book succeeds more on the historical side than on the theoretical, with music analysis as a mediating ground. But for all its weaknesses, Day-O’Connell is to be commended for undertaking such an ambitious project and realizing it to this extent. As noted at the beginning of this review, analytical studies of pentatonicism in the western tradition are surprisingly few, given its prevalence in nonwestern musics and western vernacular musics, and, as the author has demonstrated, its significance to art music from the nineteenth century and later. In his survey of an extraordinary breadth of repertoire, Day-O’Connell has made a significant beginning toward remedying that lack.
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Beckerman, Michael. 1996. “Dvorák’s Pentatonic Landscape: The Suite in A Major,” in Rethinking Dvorák: Views from Five Countries, ed. David Beveridge, 257–63. Oxford: Clarendon Press. Beveridge, David. 1977. “Sophisticated Primitivism: The Significance of Pentatonicism in Dvorák’s American Quartet.” Current Musicology 24: 25–36. ˇ
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Bevil, J. Marshall. 1986. “Scale in Southern Appalachian Folksong: A Reexamination.” College Music Symposium 26: 77–91. Biamonte, Nicole. 2010. “Triadic Modal and Pentatonic Patterns in Rock Music.” Music Theory Spectrum 32: 95–110. Burns, Edward M. 1999. “Intervals, Scales, and Tuning.” In The Psychology of Music , ed. Diana Deutsch, 2nd ed., 215–64. San Diego: Academic Press. Capuzzo, Guy. 2009. “A Pedagogical Approach to Minor Pentatonic Riffs in Rock Music.” Journal of Music Theory Pedagogy 23: 39–55. Carey, Norman, and David Clampitt. 1989. “Aspects of Well-Formed Scales.” Music Theory Spectrum 11: 187–206. Chacko, Rachel. 2011. Review of Pentatonicism from the Eighteenth Century to Debussy by Jeremy Day-O’Connell. Journal of Musicological Research 30: 257–61. Clough, John, and Jack Douthett. 1991. “Maximally Even Sets.” Journal of Music Theory 35: 93– 173. Clough, John, and Gerald Myerson. 1985. “Variety and Multiplicity in Diatonic Systems.” Journal of Music Theory 29: 249–70. Dahlhaus, Carl. 1990. Studies on the Origin of Harmonic Tonality. Trans. Robert O. Gjerdingen. Princeton: Princeton University Press. Day-O’Connell, Jeremy. 2002. “Pentatonicism in Nin eteen th-Century Music.” Ph.D. diss., Cornell University. ———. 2006. “The Rise of 6 in the Nineteenth Century.” Music Theory Spectrum 24: 35–67. ———. 2009. “Debussy, Pentatonicism, and the Tonal Tradition.” Music Theory Spectrum 31: 225–61. Doll, Christopher. “Pentatonic Scale Fragments in Pop-Rock Harmony.” Paper presented at the International Conference on Analysing Popular Music, University of Liverpool, July 2013. Everett, Walter. 2004. “Making Sense of Rock’s Tonal Systems.” Music Theory Online 10.4.
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Stein, Deborah. 1983. “The Expansion of the Subdominant in the Late Nineteenth Century.” Journal of Music Theory 27: 153–80. Tymoczko, Dmitri. 2011. A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice. Oxford: Oxford University Press. Van der Merwe, Peter. 2004. Roots of the Classical: The Popular Origins of Western Music. Oxford: Oxford University Press. Williamson, John. 2011. “Fragments of Old and New in Der Abschied .” Nineteenth-Century Music Review 8:197–217. Yasser, Joseph. 1932. A Theory of Evolving Tonality. New York: American Library of Musicology, rept. Da Capo Press, 1975. Zweifel, Paul. 1996. “Generalized Diatonic and Pentatonic Scales: A Group-Theoretic Approach.” Perspectives of New Music 34: 140–61.
