Leo Brouwer's Hika: in Memoriam Toru Takemitsu

LEO BROUWER’S HIKA: IN MEMORIAM TORU TAKEMITSU: AN ANALYSIS OF MUSICAL CONTOUR AND FORM

by Julio Orlando Quimbayo Bolaños March 25th, 2019

A dissertation submitted to the Faculty of the Graduate School of Soochow University in partial fulfillment of the requirements for the degree of

Master of Arts

School of Music

ABSTRACT Hika: In Memoriam Toru Takemitsu (1996) is a guitar work by the Cuban composer Leo Brouwer. This piece is partially built on the previous compositional material by the composer himself (specifically on Tres Apuntes: Sobre un Canto de Bulgaria [1959]), and references to Toru Takemitsu’s Rain Tree Sketch II (1992). Besides the pedagogical and philosophical literature1, Hika has not been a focus of study in the field of music analysis. Although, we can find some analytical studies of Brouwer’s compositions, they mainly concentrate on his other major works.2 In addition, none of these analytical studies ever apply contour theory as their methodological approach. A research based on the categorization of various musical contour patterns in Hika is a fresh start to procedurally examine Brouwer’s compositions. The current research intends to discover Brouwer's unique contour vocabularies that represent his personal compositional features. Prompted by this goal, this thesis aims to suggest that certain contour patterns in Hika may be important signifiers announcing the composer's own, marked style, which underlies the musical characteristics in all his guitar repertoire. The greatest strength of contour theory is its ability to systematically categorize and identify the related musical figures projected by pitch, rhythm, dynamics, harmonic density, and texture in music, even if they are usually transformed and imbedded in a new contour figure in the middle of the piece. As Elizabeth West Marvin and Paul Laprade mention: By extention to a non-tonal context, we may predict that listeners will be more likely to assume that non-equivalent sets belong to the same set-class if their contours are the same. In fact, W. J. Dowling and D. S. Figitani have offered experimental justification for the premise that listeners retain brief non-tonal melodies solely in terms of their contours. Thus we may surmise that given the same of similar rhythmic pattern, listeners are generally able to perceive equivalence or similarity among musical contours more easily than among pitch-class sets in melodic settings.3 The current methodologies of contour theory can be categorized into three trends: integer-based, 4 contour typology,5 and image-based.6 Among these three trends, the

1 To name a couple in the past decade, Harm Du Plessis (2016), and Gloria Ariza Adame (2012). 2 For instance, John Bryan Huston’s analysis of Paisaje Cubano con Rumba, Danza Característica, El Decamerón Negro, and Rito de los Orishas (2006) and Carlos Isaac Castilla Peñaranda’s dissertation (2009) about Brouwer’s first Etudes series are the solidest analytical studies of Brouwer’s repertoire up to date. 3 Marvin and Laprade (1987, 225–26). 4 Michael Friedmann (1985 and 1987), Robert Morris (1987 and 1993), Marvin and Laprade (1987), Marvin (1989, 1991, and 1995), Keith Opren and David Huron (1992), Robert Clifford (1995), Ian Quinn (1997 and 1999), Matthew Santa (1999), Sean Carson (2004–2005), Robert Schultz (2008, 2009, and 2016), Aaron Carter-Ényí (2016), among others. 5 Charles Seeger (1960), Charles Adams (1976), Morris (1993), Yi-Cheng Daniel Wu (2012, 2013, and 2019 [forthcoming]).

I 中文摘要从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 integer-based is the most popular and accessible approach in this field. Therefore, I have decided to use the integer-based methodology proposed by Marvin and Laprade (1987) to analyze Brouwer’s Hika. Apart from studying the local musical organization based on the theme, its developments, transformations, variations, rhythm, and dynamics, my analysis will look into a deeper and a more coherent structure constructed by the above elements via their contour representations. Through this analysis, I will propose a stronger and more intimate relationship between the references to Takemitsu’s music as well as to Brouwer’s and the narrative of the formal architecture in Hika.

Keywords: music contour, form analysis, 20th century guitar music.

Written by: ______Supervised by: ______

6 Mieczyslaw Kolinski (1965), Arnold Schoenberg (1967), and Charles Adams (1976).

II 从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 CONTENTS CONTENTS

ABSTRACT ...... I

中文摘要 (CHINESE ABSTRACT) ...... III

第一章 Leo Brouwer 和 Hika: In Memoriam Toru Takemitsu ...... III 第 1.1 节 Brouwer 的简介 ...... III 第 1.2 节 Brouwer 和他的吉他独奏曲目 ...... IV 第 1.3 节 In Memoriam – Brouwer 和 Toru Takemitsu 的关系 ...... V 第二章 Integer-based Contour Theory: 文献回顾 ...... VI 第 2.1 节 Prime form 算法 ...... VI 第 2.2 节轮廓相似性测验 ...... VI 第三章形式与音乐轮廓分析 ...... VII 第 3.1 节声调轮廓 ...... VII 第 3.2 节节奏和动态轮廓 ...... VIII 第四章 Hika 的音乐轮廓关联性 ...... VIII 结论与展望 ...... IX CHAPTER 1 Leo Brouwer and Hika: In Memoriam Toru Takemitsu ...... 1

1.1 Brouwer: Introduction ...... 1 1.2 Brouwer and his Solo Guitar Repertoire ...... 3 1.3 In Memoriam – The relationship between Brouwer and Toru Takemitsu ...... 5 CHAPTER 2 Integer-based Contour Theory: Literature Review ...... 7

2.1 Prime form algorithm ...... 8 2.2 Contour similarity measurements ...... 9 2.2.1 Contour Similarity (CSIM) ...... 10 2.2.2 Contour embedment (CEMB) ...... 10 2.2.3 Contour mutual embedment (CMEMB) and all contour mutual embedment (ACMEMB) ...... 12 CHAPTER 3 Analysis of Form and Musical Contour ...... 14

3.1 Tonal Contour ...... 14 3.2 Rhythmic and Dynamic Contours ...... 28 CHAPTER 4 Relating Musical Contours in Hika ...... 38

FURTHER DISCUSSION ...... 44

CONTENT 从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 APPENDIXES ...... 45

Appendix I: Prime form algorithms demonstration (Marvin/Laprade and Sampaio) ...... 45 Appendix II: Hika – In Memoriam Toru Takemitsu (Leo Brouwer, 1996) ...... 55 BIBLIOGRAPHY ...... 62

从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 LIST OF EXAMPLES

LIST OF EXAMPLES

CHAPTER 2 2.1 Schoenberg’s Suite Op. 29, Mvt. II (m. 1), Csegs A and B...……………………………...... 7 2.2 CEMB(X, Y) Calculation Process………………………………………..……………...…11

CHAPTER 3 3.1.1 Hika (mm. 1–3), Tonal Cseg Relations...... 15 3.1.2 Hika (mm. 4–9), Tonal Cseg Relations...... 16 3.1.3 Hika (mm. 10–17), Tonal Cseg Relations...... 18 3.1.4 Hika (mm. 18–20), Tonal Cseg Relations...... 19 3.1.5 Hika (mm. 21), Tonal Cseg <1 0 3 2>...... 20 3.1.6 Hika (mm. 27–28), Tonal Cseg Relations...... 20 3.1.7 Hika (mm. 38–39, 54, and 61), Tonal Cseg Relations...... 22 3.1.8 Hika (mm. 71–73), Tonal Cseg Relations...... 24 3.1.9 Hika (mm. 74–78), Tonal Cseg Relations...... 25 3.1.10 Hika (mm. 79–81), Tonal Cseg Relations...... 25 3.1.11 Hika (mm. 86–89), Tonal Cseg Relations...... 26 3.1.12 Hika (mm. 89 and 95), Tonal Cseg Relations and Four-part Texture Derivation…………………………..……….…………..………27 3.1.13 Hika (mm. 96–99), Tonal Cseg Relations...... 28

CHAPTER 4 4.1 Hika, mm. 88–89 (Upper Staff) and Tree Rain Sketch II, . mm. 1–2, Right Hand (Lower Staff), Tonal Cseg Relations………………….….….....43

APPENDIXES A.3 Prime Form Algorithm, Sampaio (2012, English Translation Mine)…………………………………….....……..……...……....47 A.3 Prime Form Algorithm, Sampaio (2012, Original Portuguese Text).………………………………………..…………………....48 A.3 Prime Form Algorithm (Sampaio and Kroger 2016, 83)…………………….…....…...48 A.4 Marvin and Laprade’s Prime Form Algorithm, Walking Example (Sampaio’s Interpretation)……………………………….………....49 A.5 Sampaio’s Prime Form Algorithm, Walking Example…………………………….…...50 A.6 Marvin and Laprade’s Original Prime Form Algorithm, Walking Example…………...51

LIST OF FIGURES 从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 LIST OF FIGURES

CHAPTER 2

2.1 COM-matrices of Csegs AP and BP. Relative Height Diagonals (INTn)……………..…....9

2.2 CSIM(AP, BP) ≈ 0.84……………………………………………………………………..10

2.3 CEMB(C, AP) = 0.25……………………………………………………………………..11

CHAPTER 3 3.1.1 CEMB(P.1, P.2) = 0.25…………………………………………………….……..….....15

3.1.2 CSIM(P.4R, P.7) = 1.00………..……………………………………………………...... 17 3.1.3 CEMB( P.8, P.4) ≈ 0.33……..……………………………………………………….....17

3.1.4 CSIM(P.3, P.10a) ≈ 0,66. INT1 = <– + –>……..………………...…………….…….....18

3.1.5 COM-matrix for P.27–28, P.38–39, P.54, and P.61). INT1 = <+ – +>………………….23

3.1.6 CSIM(P.10a, P.71aR) ≈ 0,66.

INT1 Relation Between P.3, P.10a and P.71aR…………………….…………………....24

3.2.1 CSIM(R.1–3 , R.51–53) ≈ 0.66. INT1 = <– + –>………………..………………….….29

3.2.2 CEMB(R.69–70R, R.18–20) ≈ 0.33……………………….….…………..………..…...30

3.2.3 Hika (D.1–17 and D.1–20), INT1 Descending Tendency...... 33 3.2.4 CSIM(D.1–20, D.23–36) = 1.00…………………………………………………….....35

3.2.5 Hika (Part B, Sections 2 and 3), INT1 Descending Tendency………………………....35 3.2.6 Hika (Interlude B, Part A’, and Interlude C),

… INT1 Ascending Tendency…………………………………………………...…………37

3.2.7 CSIM(D.1–20RI, D.100–103) = 1,00…………………………………………………...37

CHAPTER 4 4.1 Hika, Transformation Graph for Csegclasses c.3–2, rc.4–5/8, and c.4–7….……………40

4.2 COM-matrix for P.27–28, P.2, and R.1–3). INT1 = <+ – +>……………………….……41 4.3 Hika (Part C), Geometric Relations Based on c.3–2…………………………………….42

从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 LIST OF TABLES LIST OF TABLES

CHAPTER 2 2.1 Csegs A and B, Prime Form Algorithm Transformation .Based on Sampaio (2012) and Sampaio/Kroger (2016)…………………………….……..8

2.2 CEMB for csegs AP and C...... 11

2.3 CMEMB3(AP, BP) = 0.75...... 12

2.4 ACMEMB(AP, BP) ≈ 0.75 ...... 13

CHAPTER 3 3.1 Hika Formal Division………………………………………………………………....….14 3.1.1 CEMB for Csegs P.1 and P.10a...... 19 3.1.2 CEMB for Csegs P.4 and P.27–28…………………………………………………..….21 3.2.1 Hika (Prelude, Part A, and Interlude A), Rhythmic Cseg Relations……………….…..28 3.2.2 Hika (Part B), Rhythmic Cseg Relations………………………………………….…...29 3.2.3 Hika (Interlude B, Part A’, and Interlude C), Rhythmic Cseg Relations……………....31 3.2.4 Hika (Part C and Codetta), Rhythmic Cseg Relations...... 31 3.2.5 Hika (Prelude, Part A, and Interlude A), Dynamic Cseg Relations…………………....33 3.2.6 Hika (Part B), Dynamic Cseg Relations...... 34 3.2.7 Hika (Interlude B, Part A’, and Interlude C), Dynamic Cseg Relations…………….....36 3.2.8 Hika (Part C and Codetta), Dynamic Cseg Relations...... 37

CHAPTER 4 4.1 Structural Contour Segments in Hika, Prime Forms and Csegclasses…………………..39

APPENDIXES A.1 Equivalent Class with Two Prime Forms (Sampaio 2012, 105 and Sampaio and Kroger 2016, 84)...... 47 A.2 Marvin/Laprade’s Prime Form Algorithm in Csegclass 5-8 (Sampaio’s Interpretation)………………………………………………………..……...52 A.3 Sampaio’s Prime Form Algorithm in Csegclass 5-8………………………………….....53 A.4 Marvin/Laprade’s Original Prime Form Algorithm in Csegclass 5-8…………………...54

从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 CHAPTER 1

CHAPTER 1 Leo Brouwer and Hika: In Memoriam Toru Takemitsu

1.1 Brouwer: Introduction

Leovigildo Brouwer Mesquida, also known as Leo Brouwer, was born on March 1st, 1939 in La Habana (Cuba). He has been developing his musical career as a guitarist, orchestra director, and pedagogue. In addition, he is acclaimed as one of the most important music composers of the twentieth century, especially for his contribution to the development of the guitar repertoire. As Paul Century affirms: (Leo Brouwer) figures prominently among the most active living Cuban musicians today. Regarded worldwide as one of the foremost living composer/guitarists, Brouwer has contributed an essential component to the guitar's repertoire, with many of his works serving as fundamental pedagogical mainstays of the classical guitarist's curriculum. A prolific composer, Brouwer's compositions, apart from the guitar genre, include orchestral and chamber works, instrumental concerti, ballet and theater scores, and film music in styles representing his native Cuban heritage, avant-garde art music, and popular jazz-rock idioms.21 Brouwer started to learn guitar and music theory at the age of thirteen, with the empirical guidance of his father and aunt. His talent allowed him to become a student of Isaac Nicola—the most important cuban guitarist at that time—at Conservatorio Peyrellade (La Habana). During his time in Nicola’s guitar studio, Brouwer not only acquired the necessary technical and musical background related to the guitar performance, but also discovered the tremendous dimensions of the western art music tradition and explored its possibilities of integrating it with his own cultural context. As a consequence, Brouwer began to compose, and self-taught, for guitar and other instruments during his three years studying at the music conservatory. In 1959, the Cuban government granted him a scholarship to study composition at The Juilliard School of Music (New York, USA) and Hartford University (Connecticut, USA). This outstanding learning experience opened his creative horizons and subsequently allowed him to observe and learn diverse compositional styles from great composers such as Leonard Bernstein, Paul Hindemith, Stefan Wolpe, Jean

21 Century 1987, 151. 1

CHAPTER 1 从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 Morel, and Vincent Persichetti, among others.22 Although his professional musical career initiated as a prominent guitar performer—he has performed in the United States, Canada, Japan and Latin America, aside from recording for Deutsche Grammophon, Erato and RCA and others,23 a right-hand injury during one of his tours forced Brouwer to an early retirement in the late 70’s.24 Despite this terrible incident, his professional career as composer and orchestra conductor remained intact until nowadays. In addition, he has been a renowned public figure with great influence in the Cuban music media, as the head of the Music Division from 1969 to 1977 of Instituto Cubano del Arte e Industria Cinematográfica (ICAIC, Art and Film Industry Cuban Institute), and as an honorary member of the International Music Council (IMC, UNESCO) since 1980.25 His peculiar cultural background, aside from his restless creative endeavors, leads Brouwer to merge Afro-Cuban idiomatic elements with Western art tradition’s form and instrumentation in each of his music works. This eclectic constitution of his musical paradigm is the main reason why Brouwer’s composition style is described by Vladimir Witsuba-Álvarez as a complex mix of Cuban, traditional and experimental characteristics.26 This heterogeneous language is an echo of his own thoughts, as Century affirmed that “Brouwer himself has emphasized, the total world surrounding the composer is essential to a complete appreciation of his music.”27 As a result, Brouwer’s music covers diverse musical genres of sonatas, concertos, suites, written for solo instrument, chamber music, choir, orchestra and film music. And in his compositions, there is always a persistent reference of intrinsic Afro-Cuban elements—from programmatic topics in works like his own Cuban Landscapes collection, to rhythmical patterns that are fundamental part of the syncopated Afro-Caribbean music. Therefore, as Witsuba-Álvarez concludes after his analysis of Brouwer’s guitar music: Finally, it is important to mention that cultural identity’s search and definition is possible in wordless music with implicit features recognized as their own as “marks” of Cubanity that, nevertheless, do not exclude the assimilation of features that come from other

22 Rodríguez and Eli 2009, 27–28. 23 Ibid., 55. 24 Calcines, 1999. 25 Century 1987, 151. 26 Witsuba-Álvarez 1991, 24. 27 Century 1987, 152. 2

从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 CHAPTER 1 different subsystems, constituting a process of transculturation and syncretism.28

1.2 Brouwer and his Solo Guitar Repertoire

Most Brouwer’s works are written for the guitar—over fifty pieces for solo guitar, a number that keeps growing thanks to the composer’s ongoing compositive project Danzas y Rituales Festivas. We can divide the development and growth of Brouwer’s compositional styles into three periods based on his music’s particular features. These stylistic variations are also reflected in Brouwer’s works for solo guitar.29 The first stage (1956–1964) shows a profound understanding of the technical possibilities of the instrument. At the same time, he was eager to nurture himself with enough musical knowledge in order to create guitar masterpieces that, in his opinion, were missing in the entire history of the Western art music up to 1950’s. As Marta Rodríguez and Victoria Eli suggest about Brouwer’s earliest creative attempts: To understand his growth, we must consider both the study of the guitar’s standard repertory from Renascence to early twentieth century along with other instrumental masterpieces. This motivates Brouwer’s initial interest to analyze and understand the music from inside out, which later in a few years leads him to further explore the way to compose and create a type of guitar literature that had not been attempted before.30 As a result, although Brouwer’s first works for guitar are mainly based on the traditional tonality and from the Baroque and Classical such as Suite n° 2 in D (1955) and Fugue n° 1 (1957), they also experiment on contemporary harmonies with dissonant intervals and chromatic colorings, with clear references to Afro-Cuban folklore (such as Danza característica [1956] and Dos aires populares cubanos [1957]). Other guitar works from this period that worth mentioning are Tres apuntes (1959)—an homage to Igor Stravinsky, Claude Debussy, Bela Bartok and Manuel de Falla—and Elogio de la danza (1964), which is for Kronenberg “the final, most mature and acclaimed solo guitar piece of Brouwer's first stylistic phase.”31

28 Witsuba-Álvarez 1991, 37 (my translation). The following is the original Spanish text: “Finalmente cabe señalar que la búsqueda y concretización de la identidad cultural es posible en la música sin palabras, pero con rasgos reconocidos como propios por sus ‘marcas’ de cubanidad implícitas que, sin embargo, no excluyen la asimilación de rasgos provenientes de subsistemas diferentes, constituyendo un proceso de transculturación y sincretismos.” 29 Kronenberg 2008, 36–44. 30 Rodríguez and Eli 2009, 16 (my translation). The following is the original Spanish text: “Para comprender su desarrollo, se debe tener en cuenta que la misma práctica de abordar el estudio de éstas y otras composiciones fueron despertando en Leo Brouwer un interés inicial por analizar y conocer la música desde dentro, lo que en pocos años le conduce a sentir la necesidad de componer y crear un tipo de literatura para la guitarra que antes no había sido abordada.” 31 Kronenberg 2008, 42. 3

CHAPTER 1 从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 Brouwer’s second period (1968–1979) is dramatically influenced by his contact with the European avant-garde tradition at that time, after his attendance to the 1961 Polish Warsaw Autumn Festival. As Kronenberg mentions about the impact of this event in Brouwer’s life as a composer: At the event he gained awareness of some of the most advanced contemporary works on the European continent. Attracting Brouwer's interest were reputedly revolutionary composers-Tadeusz Baird, Kazimierz Serocki, Ernest Block, Luigi Nono and Hans Werner Henze. The première of Penderecki's orchestral masterpiece Threnody in memory of the victims of Hiroshima (1960) counts among the many innovative experiences that won Brouwer's deep interest and admiration.32 Therefore, Brouwer’s subsequent guitar compositions echo the influences that he acquired during the early 1960’s, experimenting with serial, aleatoric and electro-acoustic techniques. Brouwer composed four works for solo guitar: Canticum (1968), La espiral eterna (1970)—which is the most important piece in this period, Per suonare a due (1972)—composed for solo guitar and pre-recorded guitar tape,33 Parabola (1973–74), and Tarantos (1973–74). The current stylistic period in Brouwer’s guitar work (from1980 onwards) is characterized by a reutilization of tonal idioms and Afro-Cuban elements, as he already explored in his early music career, with a limited presence of avant-garde compositive techniques. Nevertheless, his music reflects an evolution in his own personality as an erudite and consolidated composer, while at the same time making references to other cultural contexts of jazz, Asian music and literature, and so on. Is in this period of time where most of his major works for solo guitar have been composed, which coincides with his retirement as a guitar performer. Some of Brouwer’s solo guitar works from this period are El Decameron negro (1981), Variations sur un theme de Reinhardt (1984), Sonata n° 1 (1990), Hika: in memoriam Toru Takemitsu (1996), Rito de los Orishas (1993), La ciudad de las columnas (2006), Sonata del caminante (2007), Sonata del Decameron negro (2012), Sonata del pensador (2013) and Sonata Ars Combinatoria (2012–13).34

32 Ibid., 35. 33 Rodríguez and Eli 2009, 52. 34 Updated information about his current compositions can be found in Brouwer’s music publisher official website, Ediciones Espiral Eterna (http://www.eeebrouwer.com). 4

从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 CHAPTER 1 1.3 In Memoriam – The relationship between Brouwer and Toru Takemitsu

As mentioned in Section 1.2, Hika is part of a broader repertoire for guitar composed in Brouwer’s current stylistic period. This composition is named after the demise of Japanese composer Tōru Takemitsu (1930–1996), one of the leading post-war composers from his nation.35 Brouwer and Takemitsu shared a professional admiration and friendship, as Brouwer himself expressed to Rodolfo Betancourt during an interview in 1998: On the way I composed a work, beloved by me and one of the most mature, which has been played a lot lately and that seems to have motivated some great guitarists: [Hika] In Memoriam Takemitsu. Takemitsu, one of the geniuses of the 20th century, was a great friend whose talent I cannot stop to admire.36 Brouwer and Takemitsu met in Japan in the early 1980’s, during a short encounter after a recording session Brouwer was doing in Tokyo.37 However, the story behind the composition of Hika is intrinsically related with Shin-Ichi Fukuda (1955–), a Japanese guitar virtuoso. Fukuda was invited to give a concert at the Spanish Music Festival in Tokyo (1991), where he premiered Brouwer’s Sonata N° 1 in Japan, and a program that also included Takemitsu’s Folios for guitar. Both composers assisted to the recital, which gave them the opportunity to meet each other again and, after a successful concert, Brouwer offered to Fukuda to write a guitar composition for him. Fukuda remembered this episode during his interview with Thérèse Wassily Saba: I was very nervous playing that in front of two great composers. Fortunately, the concert was very successful, and we went to a party together. It was a wonderful night, and Leo also looked very happy. After that, he decided to write something for me. Finally he wrote something five years later, Hika, but ironically, the composition was triggered by the death of Mr. Takemitsu.’38 Hika, alongside with Concierto da Requiem for solo guitar and orchestra (2008), is a tribute to Takemitsu’s lifework. Brouwer uses musical elements in Hika that echo Takemitsu’s works with his own, such as the usage of Brouwer’s Hungarian folk song theme from Tres Apuntes, Mvt III., and the intrinsic relation of Hika with Takemitsu’s Rain Tree Sketch II: in memoriam Olivier Messiaen (further explored in Chapter 3 and 4 of this

35 Burt 1998, 10. 36 Betancourt 1998. 37 Saba 2008, 37. 38 Ibid., 36. 5

CHAPTER 1 从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 thesis, respectively). These musical references, in addition to other compositional techniques that Brouwer adopted from Takemitsu’s works for guitar (such as the unusual scordatura or the occurrence of Lydian sonorities), give to the piece an intellectual significance in Brouwer’s intention to homage the Japanese composer. As Fukuda expressed about this guitar work in the foreword of its first published edition: The concepts Takemitsu expressed during his life of “Dream and Number” and “Sea of Tonality” have been wonderfully re-expressed on the guitar in this new work by Brouwer, who both respected Takemitsu and saw him as his life’s teacher.39

39 Fukuda 1997, 4. 6

从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 CHAPTER 2

CHAPTER 2 Integer-based Contour Theory: Literature Review

Since Michael Friedmann (1985) and Robert Morris’s (1987) initial pioneering works, integer-based contour theory has become an important subject of study for theorists analyzing post-tonal music. Scholars translate pitches in a melody into numerical representations. Based on two sequential dimensions—contour space (from low to high) and sequential time (from the first to the last), they convert pitches of total number n into sequential integers from 0 to n – 1. The resulting numbers are called contour pitches (cps). A set composed by a series of ordered cps is termed as contour segment (cseg). This procedure is known as normalization.40 Example 2.1 uses an excerpt of violin and cello from Schoenberg’s Suite Op. 29, Mvt. II to demonstrate the normalization process. Violin Cello

A = < 3 1 2 0 > B = < 2 1 3 0 >

Example 2.1: Schoenberg’s Suite Op. 29, Mvt. II (m. 1), Csegs A and B

There are four pitches in both contours, where numbers 0 and 3 are assigned to the lowest and highest cps, respectively. As a result, the normalization forms two different csegs, violin cseg A = <3 1 2 0> and cello cseg B = <2 1 3 0>, with the same cardinality n = 4. Both contours describe an overall descending motion, although the melodies possess no pitch in common. Contrarily, both csegs have great commonalities in their rhythm, dynamics, and articulation (forte and pizz). As the music continues throughout the movement, Schoenberg constantly transforms certain csegs projected by various musical elements in order to create an intensive stretto among winds, strings, and the piano. Hence, a detailed contour analysis of the similarities among different csegs becomes a necessary means for the analyst, listener, and performer to understand the deeper textural structure

40 The transformation of a musical contour into “normal form” is proposed by Elizabeth West Marvin and Paul Laprade (1987, 228). 7

CHAPTER 2 从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 underlying the complexity on the musical surface.

2.1 Prime form algorithm

Translating a contour projected by various musical elements into their corresponding numerical csegs is useful to identify certain patterns that could be overlooked at first sight. To achieve this goal, theorists have derived different algorithms based on cp-integers to measure the degree of similarity between any two csegs. Before moving further, we must learn how to identify the prime form XP of a given cseg X. Regarding this task, Marvin and Laprade developed the prime form algorithm,41 which was refined by Marcos da Silva Sampaio and Pedro Kroger.42 For this algorithm’s purpose, there are two geometrical operations43 to perform on any cseg X: inversion (XI, re-write each cp of the cseg as cp – [n – 1]) and retrogradation

(XR, re-write the cseg in reverse order). In consequence, cseg X can be presented in terms of XI, XR, or XRI (the retrograde-inversion of X). The discussion of the algorithm in my thesis is based on Sampaio’s refined method (2012). However, for the sake of the fluidity of my text’s discussion, I will explore the differences between Marvin/Laprade and Sampaio/Kroger algorithms in great details in Apendix 1. Table 2.1 shows the results of applying Sampaio and Kroger’s prime form algorithm to the csegs A and B from Ex. 2.1.

This process assigns either AI or AR, and BR as the prime form of A and B respectively. Workflow Contour Elements cseg X A = <3 1 2 0> B = <2 1 3 0> array arrX arrA arrB Step 1: A is Normal Form? Yes B is Normal Form? Yes Normalize X if necessary → OK → OK arrA[0] = A = <3 1 2 0> arrB[0] = B = <2 1 3 0> Step 2: arrA[1] = A = <0 2 1 3> arrB[1] = B = <1 2 0 3> Operate X, add the four I I arrA[2] = A = <0 2 1 3> arrB[2] = B = <0 3 1 2> versions to arrX R R arrA[3] = ARI = <3 1 2 0> arrB[3] = BRI = <3 0 2 1> arrA[0] = AI = <0 2 1 3> arrB[2] = BR = <0 3 1 2> Step 3: arrA[1] = AR = <0 2 1 3> arrB[1] = BI = <1 2 0 3> Sort arrX arrA[2] = A = <3 1 2 0> arrB[0] = B = <2 1 3 0> arrA[3] = ARI = <3 1 2 0> arrB[3] = BRI = <3 0 2 1> Step 4: AP = <0 2 1 3> BP = <0 3 1 2> XP = arrX[0] Table 2.1: Csegs A and B, Prime Form Algorithm Transformation Based on Sampaio (2012) and Sampaio/Kroger (2016)

41 Marvin and Laprade 1987, 234. 42 Sampaio 2012, 108. 43 I adopt Sampaio’s terms to refer to what we traditionally call Twelve-Tone Operations (TTOs). These terms can be found in a website developed by Sampaio; https://contour.sampaio.me/info (accessed July 30, 2018). 8

从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 CHAPTER 2 2.2 Contour similarity measurements

Once we find a cseg’s prime form, Marvin and Laprade propose four measurements to examine the similarities among different or identical csegs: CSIM (contour similarity),

CEMB (contour embedment), CEMBn (contour mutual embedment), and ACMEMB (all contour mutual embedment).44 All these measurements express the similarity ratio in a decimal number between 0.00 and 1.00, which respectively represents the two most dissimilar and the complete identical csegs. All the operations are realized in a COM-matrix (comparison matrix) developed by Morris.45 A COM-matrix is a structural representation of a cseg, where the contour direction is marked as ‘0’ (static motion), ‘+’ (upwards motion), and ‘–’ (downwards motion). By definition, the COM-matrix’s diagonal is filled with ‘0’, while the rest of the table will reflect the cseg contour nature, in a mirror-shape between the upper and lower triangle of the matrix. Figure 2.1 displays the 46 resulting COM-matrices of csegs AP and BP.

Figure 2.1: COM-matrices of Csegs AP and BP. Relative Height Diagonals (INTn)

COM-matrix also describes a cseg based on the relationship between its consecutive or non-consecutive cps with respect to their relative height. Each of the COM-matrix’s upper-right diagonals, starting with the immediately next to the main zero-diagonal, is 47 labeled as INTn, where n refers to the distance order between two cps. For instance, INT1

44 Marvin and Laprade 1987, 234–46. 45 Morris 1987, 28. 46 Many scholars have been developing their theories based on COM-matrices, such as Marvin (1989, 1991, and 1995) and Ian Quinn (1997). Other theories, while not related to COM-matrices, use cp-integers to design various arithmetic formulas or algorithms to measure the similarity among different contours. The representative studies include: Keith Orpen and David Huron (1992), Morris (1993), Matthew Santa (1999), Sean Carson (2004-2005), Robert Schultz (2008, 2009, and 2016), and Aaron Carter-Enyí (2016). Additionally, there is another trend that classifies all possible contours into several distinctive types. These studies can be found in Charles Seeger (1960), Mieczyslaw Kolinski (1965), Arnold Schoenberg (1967), Charles Adams (1976), Morris (1993), and Yi-Cheng Daniel Wu (2013 and 2019 [forthcoming]). 47 Marvin and Laprade 1987, 231. 9

CHAPTER 2 从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 depicts the relative height of consecutive cps, while INT2 retrieves the height information of cps located two positions apart in the cseg (see Figure 2.1). 2.2.1 Contour Similarity (CSIM) CSIM(X, Y) detects the similarities between two csegs with the same cardinality. It compares each identical position in the upper right-hand triangle of both csegs COM-matrices, counting the matching symbols, and then dividing them by the total number of cells in the COM-Matrix’s upper right-hand triangle. Thus, CSIM(AP, BP) is approximately equal to 0.84. Figure 2.2 highlights the contour similarities between AP and

BP, demonstrating that those csegs only differ between cps 2 and 4.

Figure 2.2: CSIM(AP, BP) ≈ 0.84

2.2.2 Contour embedment (CEMB) CEMB(X, Y) calculates the embedding ratio between two csegs with different cardinalities. First, it translates a given contour subsegment (csubseg) X. The translation process renumbers a csubseg of m distinct cps, from 0 for the lowest cp to m – 1 for the highest cp.48 Then we count the number of times X is embedded into a bigger cseg Y. Example 2.3 lists the detailed procedure of examining a CEMB.

Figure 2.3 highlights CEMB(C , AP) in the COM-matrix of AP, where the embedded csubseg C = <0, 3, 1> contains the first cps of BP. In addition, Table 2.2 summarizes the

CEMB-calculation process between AP and C. The result of the embedding ratio between both csegs is 0.25 (1 out of 4 successful embedment).

48 Marvin and Laprade 1987, 228. 10

从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 CHAPTER 2 Let the array of csubsegs named subY. Step 1. Translate X. Step 2. Extract the csubsegs of Y with the same cardinality of csubseg X. Add them to subY. Step 3. Translate each csubseg in subY. Step 4. Count the identical csubsegs between X and subY. Step 5. CEMB(X, Y) is equal to divide the count result by the total number of csubseg in subY.

Example 2.2: CEMB(X, Y) Calculation Process.

Figure 2.3: CEMB(C, AP) = 0.25

Workflow Contour Elements csubseg X C = <0 3 1> cseg Y AP = <0 2 1 3> csubsegs array: subY subAP Step 1: C = <0 2 1> Translate X Step 2: subA [0] = <0 2 1> subA [2] = <0 1 3> Extract csubsegs and add P P subA [1] = <0 2 3> subA [3] = <2 1 3> them to subY P P Step 3: subA [0] = <0 2 1> subA [2] = <0 1 2> Translate each csubseg in P P subA [1] = <0 1 2> subA [3] = <1 0 2> subY P P subAP [0] = <0 2 1> subA [2] = <0 1 2> Step 4: P subA [1] = <0 1 2> subA [3] = <1 0 2> Count identical with X P P # of matches with C = 1 Step 5: # 표푓 푚푎푡푐ℎ푒푠 CEMB(C, A ) = 1 / 4 = 0.25 CEMB = P 푙푒푛푔푡ℎ 표푓 푠푢푏푋

Table 2.2: CEMB for csegs AP and C 11

CHAPTER 2 从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 2.2.3 Contour mutual embedment (CMEMB) and all contour mutual embedment (ACMEMB) CMEMBn(X, Y) calculates the percentage of csubsegs with cardinality n mutually embedded in csegs X and Y. Note that csegs X and Y here need not to be the same cardinality. CMEMB first translates all the possible n-length csubsegs derived from A and B, and then counts the occurrence of identical contour segments. Finally, CMEMB divides the matches count by the total analyzed csubsegs. The last similarity measurement is ACMEMB(X, Y), which computes the mutually embedding ratio between csegs X and Y, evaluating all the csubsegs from cardinality two to the cardinality of the smaller cseg.49 Tables 2.3 and 2.4 examine the process of CMEMB3(Ap, Bp) and ACMEMB(Ap, Bp), respectively.

csubsegs cardinality n = 3 AP = <0 2 1 3> BP = <0 3 1 2> csubsegs array: subAP csubsegs array: subBP subAP[0] = <0 2 1> subBP[0] = <0 2 1> subAP[1] = <0 1 2> subBP[1] = <0 2 1> subAP[2] = <0 1 2> subBP[2] = <0 1 2> subAP[3] = <1 0 2> subBP[3] = <2 0 1> CMEMB3(AP, BP) = 6 / 8 = 0.75

Table 2.3: CMEMB3(AP, BP) = 0.75

49 CMEMBn(X, A, B) and ACMEMB(A, B) are Marvin and Laprade’s original nomenclatures. This thesis, however, will use a more consistent representation to avoid any confusion between the generic variables (represented with X and Y) and the examples used in Chapter 2 (represented with A and B). In addition, I purposefully omit the first input in CMEMBn(X, Y), which is quite different from Marvin and Laprade’s CMEMBn(X, A, B). This is because their csubseg X is only effective as an internal variable during the calculation process, instead of a formula’s input value. 12

从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 CHAPTER 2

A = <0 2 1 3> B = <0 3 1 2> Cardinality P P csubsegs array: subAP csubsegs array: subBP subAP[0] = <0 1> subBP[0] = <0 1> subAP[1] = <0 1> subBP[1] = <0 1> subA [2] = <0 1> subB [2] = <0 1> n = 2 P P subAP[3] = <1 0> subBP[3] = <1 0> subAP[4] = <0 1> subBP[4] = <1 0> subAP[5] = <0 1> subBP[5] = <0 1> subAP[6] = <0 2 1> subBP[6] = <0 2 1> subA [7] = <0 1 2> subB [7] = <0 2 1> n = 3 P P subAP[8] = <0 1 2> subBP[8] = <0 1 2> subAP[9] = <1 0 2> subBP[9] = <2 0 1> n = 4 subAP[10] = <0 2 1 3> subBP[10] = <0 3 1 2> ACMEMB3(AP, BP) = 18 / 22 ≈ 0.82

Table 2.4: ACMEMB(AP, BP) ≈ 0.75

CHAPTER 3 从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构

CHAPTER 3 Analysis of Form and Musical Contour

The notation used in my discussion to mark each cseg in Chapters 3 and 4 will be {[contour element].[# of measure(s)][a–z]}. The elements considered in this thesis to form a musical contour include pitch (P), rhythm (R), and dynamics (D).50 If the contour lasts for a period of time, the numbers of measure will be divided by an N-dash (–). For example, P.4–5 is a tonal contour that lasts from m.4 to m.5.51 In addition, Table 3.1 details a formal division of Hika based on the following musical contour analysis.

Formal Part Measures Prelude mm. 1–3 Formal Part Measures Part A mm. 4–17 Interlude B mm. 69–70 Interlude A mm. 18–20 Part A’ mm. 71 –82 Introduction mm. 21–22 Interlude C mm. 83–85 Section 1 mm. 23–37 Part C mm. 86–99 Part B Section 2 mm. 38–53 Codetta mm. 100–103 Section 3 mm. 54–68

Table 3.1: Hika Formal Division

3.1 Tonal Contour

Hika starts with a brief three-measure Prelude, grouped together by a tempo mark of Tempo Libero. Measure 1 presents an arpeggiated harmonic segment crossing in a wide pitch space of two octaves (from E4 to E6). The cseg that describes this unfolded arpeggiation is P.1 <0 2 1>, and it will be intertwined throughout the rest of the piece in different musical moments. For instance, m. 2 repeats and doubles the pitch contour in m. 1, generating a new cseg of P.2 <0 2 0 1>. Since P.1 is entirely embedded within P.2, their CEMB(P.1, P.2) equals 0.25. It means, both csegs sketch an identical melodic contour if we compare P.1 with the lowest, higher, and final cps of P.2. This comparation is detailed in Example 3.1.1 and Figure 3.1.1.

50 The lower-case letters are the indicators for different voices. For instance, P.10a in Example 3.1.3 means a pitch contour of the upper voice in m. 10, while P.10b represents the contour of the lower voice in that measure. Meanwhile, to specify the contours associated by the geometrical operations XP, XI, XR, and XRI, I will add these capital letters as a sub-index after a cseg. For instance, P.4R in Table 3.1.2. 51 Note that the term “tonal” in my thesis does not imply the traditional notion of tonality. It simply refers to a cseg produced by the elements of tones (e.g. pitches). 14

从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 CHAPTER 3 Measure 3 differs significantly from the preceding two measures, accentuated by its rhythmical variation and a clearer melodic motion. Nevertheless, P.3 <1 0 2 0> is related with P.2 in its retrograde version. Therefore, P.3R and P.2 are equivalent and, hence, intimately related with P.1. This relation, described in Example 3.1.1, justifies the coexistence of the initial three measures as the preface to Hika, and Brouwer utilizes these three pitch csegs and their associated variations to divide the piece into four formal parts—which are articulated by the occurrence of interludes whose musical materials are mainly based on Hika’s Prelude.

Example 3.1.1: Hika (mm. 1–3), Tonal Cseg Relations

Figure 3.1.1: CEMB(P.1, P.2) = 0.25

A new part of Hika starts from m. 4. Measures 4–9—besides the apparent contextual differences of a more flowing rhythmical momentum superimposed on lower pitch registers—are also built on the same contours P.1 and P.2 as shown in Example 3.1.2. For instance, P.4 and P.5 equal to P.1 <0 2 1>, and the contour described by P.4–5 delineates the same one as P.2 <0 2 0 1>. Since mm. 4–5 are the only moments played without interruption during this section, I must emphasize that these two measures are the synthesis

CHAPTER 3 从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 and integration of the previous P.1 and P.2, creating a temporary unity up to this point. The remaining part of the music (mm. 6–9) presents several distinct fragments based on the material in m. 4. First, m. 6 literally repeats the content of m. 4, consequently creating the same contour (P.6 = P.4 = P.1). Then, m. 7 starts on the second beat of the material used in m. 6, starting from the middle of the ascending motion up to G5. Due to this fragmentation, the contour P.7 equals to <1 2 0>. At the same time, P.7 is completely identical with the retrograde of P.4. This circumstance explains why CSIM(P.4R, P.7) equals to 1.00. The last two fragments in m. 8 and m. 9 are based on the second half of m. 4, where the descending motion from G5 to G4 occurs. Therefore, P.8 and P.9, both being equal to <1 0>, repeat the last two cps of P.4. Measuring CEMB(P.8, P.4) and CEMB(P.9, P.4), the results are approximately 0.33, which confirms P.8 and P.9 embedding in P.4’s last csubseg (see Example 3.1.2, Figures 3.1.2, and 3.1.3).

Example 3.1.2: Hika (mm. 4–9), Tonal Cseg Relations

从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 CHAPTER 3

Figure 3.1.2: CSIM(P.4R, P.7) = 1.00

Figure 3.1.3: CEMB( P.8, P.4) ≈ 0.33

The musical texture of this part starts to densify in m. 10, detailed in Example 3.1.3. The bass line P.10b is still built over P.1, while a new contour appears in the soprano voice, P.10a = <1 0 3 2>. Therefore, this new voice leading preserves the discursive coherence that have been prevailing from the beginning of Hika. It is important to point out that P.3 and P.10a are the only contours with cardinality of four up to this point. Table 3.1.4 examines CSIM(P.3, P.10a), and the result is approximately 0.66. Additionally, their COM-matrix’s INT1 reveal an important piece of information that their contours share the same motion pattern between adjacent cps, <– + –>. Even though P.10a expands the pitch contour variety of this section, its contour shape is closely related with P.3, which was already introduced at the end of Tempo Libero. Another fragmentation process starts from m. 11 to m. 17, which is similar to mm. 4– 9. In order to more clearly associate the contour relationships between P.4 and P.10a, I will use two different colors to differentiate them from one another. Example 3.1.3 highlights the pitch contour related with P.4 in blue color, while contours derived from P.10a are marked in red. With 9 out of 12 blue contours, we can conclude that the main pitch contour of this section springs from P.4. This predominance in the fragmentation process is a consequence of the high embedding relationship between P.4 and the last two cps <3 2> in combination with either one of the first two cps of P.10a, 1 or 0 (see Table 3.1.5). Therefore,

CHAPTER 3 从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 my pitch contour analysis from m. 4 to m. 17 reveals that P.1, expanded in a longer span of time by P.4 and P.10b, is in fact intimately related with the contour development of this part.

Example 3.1.3: Hika (mm. 10–17), Tonal Cseg Relations

Figure 3.1.4: CSIM(P.3, P.10a) ≈ 0,66. INT1 = <– + –>

从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 CHAPTER 3 Workflow Contour Elements csubseg X P.1 = <0 2 1> cseg Y P.10a = <1 0 3 2> csubsegs array: subY subP.10a Step 1: P.1 = <0 2 1> Translate X Step 2: sub P.10a [0] = <1 0 3> sub P.10a [2] = <1 3 2> Extract csubsegs and add them sub P.10a [1] = <1 0 2> sub P.10a [3] = <0 3 2> to subY Step 3: sub P.10a [0] = <1 0 2> sub P.10a [2] = <0 2 1> Translate each csubseg in subY sub P.10a [1] = <1 0 2> sub P.10a [3] = <0 2 1> sub P.10a [0] = <1 0 2> sub P.10a [2] = <0 2 1> Step 4: sub P.10a [1] = <1 0 2> sub P.10a [3] = <0 2 1> Count identical with X # of matches with P.1 = 2 Step 5: # 표푓 푚푎푡푐ℎ푒푠 CEMB(P.1, P.10a) = 2 / 4 = 0.5 CEMB = 푙푒푛푔푡ℎ 표푓 푠푢푏푋

Table 3.1.1: CEMB for Csegs P.1 and P.10a

Example 3.1.4: Hika (mm. 18–20), Tonal Cseg Relations

The first part of Hika—which I refer to as Part A—is followed by Interlude A. This formal articulator is built on variations of Tempo Libero’s material in mm. 18–20 (similar to Interlude B, mm. 69–70). While P.18 and P.19 describe the same pitch cseg as those in mm. 1–2, P.20 depicts a cseg related with P.10a in its retrograde presentation (see Example 3.1.4). This contour variation summarizes the coexistence of P.1 and P.10a at this particular moment, with the extension of the range of pitch contours based on their transformation. The main formal part of Hika, Part B, starts at m. 21 articulated by a velocissimo tempo mark, and stops at m. 68. Since its brilliant compositional style and its rhythmic and 19

CHAPTER 3 从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 dynamic features (these will be discussed fully in Section 3.2), it is undeniable to recognize Part B’s uniqueness in the piece. Despite the contrasting nature of Part B, there are still some reminiscences of tonal csegs from Prelude and Part A. Example 3.1.5 examines the tonal contour in Part B’s Introduction (m. 21), which displays a cseg P.21 <1 0 3 2 4 3>. Analyzing its boundary cps, P.21 can be further translated into <1 0 3 2>, which is identical to P.10a.52 It is important to point out that mm. 21–22 provide an important musical material that will support the articulation of the piece in Interlude C (mm. 83–86).

Example 3.1.5: Hika (mm. 21), Tonal Cseg <1 0 3 2>

Example 3.1.6: Hika (mm. 27–28), Tonal Cseg Relations

52 I follow the methodology proposed by Charles Adams (1979, 196) in order to identify the boundary cps of any given cseg: initial pitch (I), higher pitch (H), lower pitch (L), and final pitch (F). 20

从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 CHAPTER 3 Workflow Contour Elements csubseg X P.4 = <0 2 1> cseg Y P.27–28 = <0 3 1 2> csubsegs array: subY subP.10a Step 1: P.4 = <0 2 1> Translate X Step 2: subP.27–28 [0] = <0 3 1> subP.27–28 = <0 1 2> Extract csubsegs and add subP.27–28 [1] = <0 3 2> subP.27–28 = <3 1 2> them to subY Step 3: subP.27–28 [0] = <0 2 1> subP.27–28 [2] = <0 1 2> Translate each csubseg in subP.27–28 [1] = <0 2 1> subP.27–28 [3] = <2 0 1> subY subP.27–28 [0] = <0 2 1> subP.27–28 [2] = <0 1 2> Step 4: subP.27–28 [1] = <0 2 1> subP.27–28 [3] = <2 0 1> Count identical with X # of matches with P.4 = 2 Step 5: # 표푓 푚푎푡푐ℎ푒푠 CEMB(P.4, P.27–28) = 2 / 4 = 0.5 CEMB = 푙푒푛푔푡ℎ 표푓 푠푢푏푋

Table 3.1.2: CEMB for Csegs P.4 and P.27–28

The core of Part B’s musical material is introduced in its Section 1, in the upper voice of mm. 27–28. Here, Brouwer introduces a theme that was already explored in one of his previous work for solo guitar, Tres Apuntes, Mvt III.53 Example 3.1.6 examines the contour P.27–28 <0 3 1 2>, and how this contour starts to replicate in the lower voice of m. 28 as an immediately answer to the initial Hungarian song’s entrance. In addition, Table 3.1.6 examines CEMB(P.4, P. 27–28) = 0.5. This resultant CEMB is relevant because P.4 is highly embedded within P.27–28 when we consider the first two cps <0 3> in combination with either one of the first two cps of P. 27–28, 1 or 2. This discovery confirms the importance of P.4 as a transverse cseg that supports the tonal contour development in Hika. Contour P.27–28 is modified throughout the rest of Part B to emphasize musical elements that articulate the structure of this formal part. For example, Section 2 (mm. 37– 53) is dominated by a rhythmic motif derived from the first four sixteenth notes of m. 27. This compositional strategy becomes clearer in mm. 38–39, where the motif adopts the form of consecutive sixteenth notes at the beginning of m. 38, and a mirror sequence with pitches (see Example 3.1.7). Considering the accented noes as boundary cps in mm. 38–39, the tonal cseg P.38–39 <1 2 0 1> shares the same INT1

53 The original title in Spanish of this movement is “Sobre un Canto de Bulgaria” (“On a Bulgarian Song”). The musical theme elaborated in this work is also used by Brouwer in Etude VII, from his first Etude Simples. 21

CHAPTER 3 从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 diagonal as P.27–28 (see Figure 3.1.5), which confirms the similarity in melodic motion of both contours. A similar event occurs during Section 3, where the tonal contour P.54 <0 2 1 2> and P.61 <1 3 0 2>—created by the higher pitches of each slur group in m.54 and m.61—describe exactly the same INT1 relationship with P.27–28 (see also Example 3.1.7 and Table 3.1.7).

Example 3.1.7: Hika (mm. 38–39, 54, and 61), Tonal Cseg Relations

从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 CHAPTER 3

Figure 3.1.5: COM-matrix for P.27–28, P.38–39, P.54, and P.61). INT1 = <+ – +>

Another case of pitch contour variation is highlighted in Example 3.1.8, at the beginning of the third part of Hika. Here, P.71a and PC.72b share two different retrograde relationships with P.10a and P.10b respectively. While P.10b and P.72bR project the same prime form, the CSIM between PC.10a and PC.71aR is approximately equal to 0.66, and according to their COM-matrices, they feature the same inner structure of INT1 with that of P.3 (see Figure 3.1.6). These retrograde-related transformations help to postpone the final recapitulation of P.4 in m. 73, which gives an aural flavor of reprise—even though m. 73 chromatically alters many pitches from m.4, their overall contours preserve the same shape <0 2 1>. Example 3.1.9 marks these chromatic variations between P.4 and P.73a in green color.

CHAPTER 3 从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构

Example 3.1.8: Hika (mm. 71–73), Tonal Cseg Relations

Figure 3.1.6: CSIM(P.10a, P.71aR) ≈ 0,66. INT1 Relation Between P.3, P.10a and P.71aR

从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 CHAPTER 3

Example 3.1.9: Hika (mm. 74–78), Tonal Cseg Relations

Example 3.1.10: Hika (mm. 79–81), Tonal Cseg Relations

CHAPTER 3 从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构

Example 3.1.11: Hika (mm. 86–89), Tonal Cseg Relations

After the embellishment of P.73a in m. 74, the music starts to reiterate the contour of P.4 in the upper voice. The constant recurrence of this tonal contour in mm. 75–78 reinforces the idea of recapitalization already introduced in m. 73 (see Example 3.1.9). The local climax is built over more chromatism and a denser polyphonic texture in mm. 79–81. Again, the pitch contours of these measures are deeply connected with P.4. Example 3.1.10 analyzes how P.4 is heard in mm. 79–81 in both prime and retrograde forms. Although Brouwer writes voices separated by significantly wider and more expanded pitch intervals in mm. 71–81, the consistent use of P.4 confirms the relationship between the second and third parts of Hika. Based on the above analysis, I define these two parts as A and A’. Part C (mm. 86–99) starts with a two-voice polyphonic design similar to mm. 80–81. Nevertheless, the main feature of this part is the usage of various geometric transformations of P.2 and P.4 to build its tonal contours. For instance, Example 3.1.11 examines the tonal contours located in mm. 86–89. The first contour of the upper voice in m. 86 share an inversional relation with P.4, and will be successively prolonged in mm. 87–88 over another unfolding of P.4 <0 2 1> in the lower voice. Measure 89 encloses the expansion process of m.87 with a juxtaposition of two contours in the last two beats of m. 89 (marked as P.89c and P.89d), which are retrograde-related with both P.2 and P.4 respectively.

从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 CHAPTER 3 As shown in Example 3.1.12, these last two beats of m. 89 are transformed in mm.

95–96, where the polyphonic texture starts to densify—Brouwer places a harmonic B3 that acts as a pedal over two highly symmetrical contours in the remaining voices, P.96a <0 1 0> (which is a melodic unfolding of two homophonic lines) and P.96b <0 1 2 1 0>. The last three measures of Part C follow a four-part homophonic layout that share an inversional relation with m. 96 not only in their contours but also in the voices’ placement. This compositional strategy helps Brouwer to stabilize the voice leading and resolve the harmonic tension of Part C into a perfect fifth in m. 99 (as detailed in Example 3.1.13).

Example 3.1.12: Hika (mm. 89 and 95), Tonal Cseg Relations and Four-part Texture Derivation

Example 3.1.13: Hika (mm. 96–99), Tonal Cseg Relations

CHAPTER 3 从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 3.2 Rhythmic and Dynamic Contours

Rhythmic and dynamic contours in Hika are relatively less complex than the tonal explored in Section 3.1. Thus, this thesis will analyze rhythmic and dynamic contours from a general and schematic perspective, using the formal division I propose for Hika to group their features. Each formal part contains a certain unique rhythmic pattern. The prelude begins with a pair of ascending arpeggiations with different time-lapse, followed by two pairs of a sixteenth note plus a dotted eighth (see Table 3.2.1). Thus, the Prelude overall describes a rhythmic contour R.1–3 <2 3 0 1>. To proceed, Part A repeats a sequence of static eighth notes, followed by the reappearance of R.1–3 in Hika’s Interlude A (mm.18–20). Table 3.2.1 layouts the rhythmical contours in Prelude, Part A, and Interlude A.

Table 3.2.1: Hika (Prelude, Part A, and Interlude A), Rhythmic Cseg Relations

从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 CHAPTER 3

Table 3.2.2: Hika (Part B), Rhythmic Cseg Relations

Figure 3.2.1: CSIM(R.1–3 , R.51–53) ≈ 0.66. INT1 = <– + –>

CHAPTER 3 从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 Part B is the longest formal section in Hika, lasting for 47 measures in total. Its drive is the fastest in the piece due to its figures composed by small rhythmic values and its tempi velocissimo and vivace. All these factors encourage us to divide Part B from the rest of the piece. Table 3.2.2 describes the rhythmic contour developments of Part B. Section 1 starts with a two-measure introduction that abruptly breaks the rhythmical monotony of the piece. The resulting cseg R.21– 22 <0 2 1 0 2> will reappear later in the piece as part of Section 1’s repetition and as a connector with Part C. Sections 1 and 2 are consistently woven by a stable sixteenth notes rhythmic contour. The transition to Section 3 occurs on mm. 51–53, with a gradual deacceleration that arrives to a half-note with fermata. This compositional strategy produces a cseg R.51–53 <0 3 1

2>, closely related with R.1–3 when we compare their COM-matrix diagonals INT1 as shown in Figure 3.2.1. Since Brouwer uses R.1–3 as a crucial element to divide formal parts in Hika, the fact that R.51–53 displays a similar rhythmical shape reinforces the role of mm. 51–53 as a structural connector between Sections 2 and 3. Finally, Section 3 introduces a new rhythmical pattern that begins with sixteenth triplets, followed by a group of sixteenth notes, a significantly long-tied rhythmic duration, and then concluded by another group of sixteenth notes. Thus, Section 3’s rhythmical outline describes a cseg <1 0 2 0>.

Figure 3.2.2: CEMB(R.69–70R, R.18–20) ≈ 0.33

从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 CHAPTER 3

Table 3.2.3: Hika (Interlude B, Part A’, and Interlude C), Rhythmic Cseg Relations

Table 3.2.4: Hika (Part C and Codetta), Rhythmic Cseg Relations

CHAPTER 3 从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 Table 3.2.3 shows the rhythmical structure of interludes B and C, and Part A’. Interlude B comes immediately after Part B. The rhythmic contour of this interlude shares the same shape with those in Interlude A’s first two cps but in retrograde form, see

CEMB(R.69–70R, R.18–20) in Table 3.2.5. The recycled materials from Prelude and Interlude A subtly announce and hints the reprise in Part A’, with the same eighth-note rhythmical ostinato that also prevailed in Part A. Part A’ ends with an accelerated rhythmic contour at m.82 that prepares the appearance of Interlude C—a mixture of rhythmical elements previously introduced in Hika’s prelude and part B’s introduction. The importance of Part C in Toru Takemitsu’s homage by Brouwer is explicitly marked in Hika’s commentary number 7 (see Appendix II), a feature that will be explored thoroughly in Chapter 4. In terms of rhythm, this part is clearly divided from the rest of the piece, and consists in a gradual dissolution of the rhythmical intensity (see Table 3.2.4). All the phrases in Part C start moving in eighth notes and end their drive with long figures, producing successive expansions and abbreviations of the musical material. Therefore, the resulting cseg R.86–99 <0 3 0 1 0 2>—a contour that may be further translated into as <0 1>— reflects Part C’s rhythmical instability from a general perspective. After the perfect fifth in m. 99, Brouwer introduces a codetta that reuses musical contours from Part A and Interlude A. This reprise of the primal elements in Hika is reinforced by the codetta’s rhythmical contour R.100–103 <0 2 1>, a fact that will be discussed in Chapter 4. It is also possible to group dynamics in Hika following the formal division I propose. It is crucial to point out that Brouwer abundantly uses crescendo and diminuendo marks with no clear dynamic target (specially in Part C). Based on my personal experience of performing Hika, I will mark down the possible dynamics indicated by parenthesis in order to ease the dynamic contour analysis in specific moments of the piece. As shown in Table 3.2.5, Prelude, Part A, and Interlude A describe binary descending contours by themselves in D.1–3, D.4–17, and D.18–20 respectively. When these csegs are combined and simplified by their boundary cps, we obtain two new contours D.1–17 <2 1 0> and D.1–20 <2 0 2 1>. Overall, Prelude, Part A, and Interlude A depict a large-scale decrescendo supported by my analysis of diagonal INT1 when COM-matrices D.1–17 and D.1–20 are examined (see Figure 3.2.3).

从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 CHAPTER 3

Table 3.2.5: Hika (Prelude, Part A, and Interlude A), Dynamic Cseg Relations

Figure 3.2.3: Hika (D.1–17 and D.1–20), INT1 Descending Tendency

CHAPTER 3 从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构

Table 3.2.6: Hika (Part B), Dynamic Cseg Relations

从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 CHAPTER 3

Figure 3.2.4: CSIM(D.1–20, D.23–36) = 1.00

Figure 3.2.5: Hika (Part B, Sections 2 and 3), INT1 Descending Tendency

Table 3.2.6 summarizes the dynamic contour design in Part B. Section 1’s Introduction intensifies the dynamic landscape with a forte–fortissimo crescendo—these dynamics are used in Hika for the first time. Disregarding the difference in their dynamic intensity level, the remaining measures of Section 1 describe a cseg D.23–36 <2 0 2 1>, which is completely identical to D.1–20 (see Figure 3.2.4). The adaptation of similar dynamic contours brings discursive coherence to Part A and the beginning of Part B. Section 1’s dynamic contour can be simplified using its boundary cps. Thus, the resulting

CHAPTER 3 从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 cseg D.21–36 <2 3 0 1> together with D.1–20 establish structural connections between parts A and B (as further examined in Chapter 4) of this thesis. In addition, Sections 2 and 3 share a mainly descending tendency in their phrases’ dynamic layout, as also occurred in Prelude, Part A and Interlude A (see Figure 3.2.5). Table 3.2.7 describe the dynamic contour of Interlude B, Part A’, and Interlude C. In term of dynamics, these three formal parts differ from the large-scale decrescendo that we have been analyzing so far in Hika. This fact can be demonstrated when we examine the diagonal INT1 of the COM-matrices D.69–70, D.71–82, and D.83–85, where the ascending sign ‘+’ dominates the matrices (see Figure 3.2.6). Finally, Table 3.2.8 details the dynamic contour of Part C and Codetta. Part C only uses crescendo and diminuendo marks in order to drive its dynamic plan. Considering that Part C’s phrases always end in a relatively long note (as examined in the rhythmic contour analysis) and the guitar is unable to hold a single note with the same dynamic for an extended span-time, it is convincing to conclude that all the hairpins will produce a dynamic cseg <0 1 0>. Finally, the last dynamic cseg of the piece is depicted in its Codetta, D.100–103 <1 0 2 0>, which shares a retrograde-inversional relationship with D.1–20 (see Figure 3.2.7).

Table 3.2.7: Hika (Interlude B, Part A’, and Interlude C), Dynamic Cseg Relations

从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 CHAPTER 3

Figure 3.2.6: Hika (Interlude B, Part A’, and Interlude C), INT1 Ascending Tendency

Table 3.2.8: Hika (Part C and Codetta), Dynamic Cseg Relations

Figure 3.2.7: CSIM(D.1–20RI, D.100–103) = 1,00 37

CHAPTER 4 从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构

CHAPTER 4 Relating Musical Contours in Hika

As studied in Chapter 3, the musical contour landscape depicted in Hika is based on very specific csegs appearing at structural moments throughout the piece. More importantly, each contour segment continuously builds a network of similar csegs based on their geometric transformations in each musical dimension. Thus, the architectural platform of this piece—crossing the elements of pitch, rhythm, and dynamics—is well balanced and structured. Table 4.1 summarizes the musical contour segments presented in Hika on each formal part, specifying their respective prime form and csegclass. Note that the (R), (I), and (RI) next to the prime form indicate the transformational relationship between a given cseg and its associated prime. If cseg contains nonconsecutive cps, I follow Marvin and Laprade 1987’s method to identify this type of contours—the combination of the two csegclasses separated by a slash sign, where “rc” stands for “repeated-pitch csegclass”.54 For instance, rc.4–5/8 is the combination between c.4–5 and 4–8 identifying the tonal cseg <0 2 0 1> in Prelude (see Table 4.1). This means <0 2 0 1> shares great resemblance of both c.4–5 and c.4–8 based on their COM-matrices.55

54 Marvin and Laprade 1987, 245–48. 55 In a later 1995 article Marvin also mentions about this issue of the nonconsecutive repeated cps (1995, 168). 38

从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 CHAPTER 4

, Prime Forms and Csegclasses Forms Prime , and

Hika

Contourin Segments

Table 4.1: Structural Structural 4.1: Table

CHAPTER 4 从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构

Figure 4.1:

Hika

TransformationGraph Csegclasses c.3 for

–

2, rc.4 2,

–

5/8, and c.4 and 5/8,

–

从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 CHAPTER 4 As shown in Figure 4.1, the predominance of contours from csegclasses c.3–2, c.4– 5/8, and c.4–7 throughout Hika in its various musical dimensions should not be disregarded as an important piece of information to understand the global architecture of this piece. Previously in Chapter 3, I have observed that in Hika’s Prelude all the csegclass representatives that will support the contour diversity and formal division are introduced. These fundamental csegs are P.1 <0 2 1> (c.3–2), P.2 <0 2 0 1> and D.1–20 <2 0 2 1> (rc.4–5/8), and R.1–3 <2 3 0 1> (c.4–7 in retrograde form). Thus, Brouwer uses these specific csegs to articulate form, strategically placing their TTO’s transformations on structural points—a compositional design that explains the musical contour evolution. For instance, the contour design of the formal group Prelude/Part A/Interlude A is relatively simple, based on a contour background over c.3–2 and rc.4–5/8 in their respective tonal and dynamic csegs, while c.4–7 is successively retrograded and transferred from the rhythmical to the tonal dimension. This simplicity prevailing in the first instances of Hika is dramatically modified in Part B’s introduction: c.3–2 is transferred to the rhythmical dimension via retrograde-inversion, and c.4–7 takes places in both tonal and dynamic areas, while leaving rc.4–5/8 in standby. This gesture hints the contrasting complexity that characterizes Part B.

Figure 4.2: COM-matrix for P.27–28, P.2, and R.1–3. INT1 = <+ – +>

CHAPTER 4 从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 As mentioned in Chapter 3, Part B’s tonal material is based on Brouwer’s Hungarian song theme—described by P.27–28 <0 3 1 2>, a representative of c.4–5—and its contour variations. At first glance, the usage of this new musical material might be seen as a mere intertext from Brouwer’s previous works. Nevertheless, the INT1 similarity between P.27– 28, P.2, and R.1–3 (see Figure 4.2), in addition to the embedment connection between P.4 and P.27–28 (already detailed in Chapter 3, section 3.1), demonstrates that the fundamental csegs of Hika remain implicit in Part B’s contour development. Even though, c.4–7 and rc.4–5/8 remain as structural contours on certain moments of part B’s sections in either rhythmic or dynamic dimensions. Interlude B reintroduces the prime form of c.3–2 <0 2 1> on Hika’s tonal dimension. The explicit reprise of c.3–2 brings back stability to the musical contour design and prepares the atmosphere of recapitulation in Part A’. The remaining structural c.4–7 and rc.4–5/8 also reappear, but they only occur momentarily in Interlude C and Part C respectively. Thus, their prominence is reduced by c.3–2’s influence. Starting from Interlude B until the end of the piece, c.3–2’s dominance and expansion throughout all the musical dimensions is evident. It is important to give special attention to Part C, where Brouwer introduces a complex contour design based on c.3–2 with constant transformations via geometric TTO’s operations, while placing the resulting contour segments on Part C’s tonal and dynamic dimensions. As shown in Figure 4.3, the tonal layout of Part C is based on <0 2 1>, which is transformed via inversion and retrograde-inversion into <2 0 1> and <1 0 2> respectively, and both of them related by retrograde. At a same time, cseg <0 2 1> acts as a reference to outline two contour segments with repeated cps. These inversionally related csegs are <0 1 0> and <1 0 1> located in Part C’s dynamic and tonal dimensions respectively.

Figure 4.3: Hika (Part C), Geometric Relations Based on c.3–2 42

从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 CHAPTER 4

Example 4.1: Hika, mm. 88–89 (Upper Staff) and Tree Rain Sketch II, mm. 1–2, Right Hand (Lower Staff), Tonal Cseg Relations

It is crucial to point out that mm. 88–89 in Part C also shares geometric relations in both prime and inversion forms with diverse csegs located in the right hand of Takemitsu’s Tree Rain Sketch II for piano, mm.1–2 (see Example 4.1). Therefore, Part C’s musical contour development reinforces Brouwer’s intention to cite this specific work written by the Japanese composer, encouraging Part C to become an echo of Takemitsu’s death tribute. The intricated symmetry of Part C increases the dramatic tension on this late stage in

Hika—a tension finally discharged into an also symmetric perfect fifth A2-E3-A3 that leads to the epilogue of the piece. The stability of Hika’s contour design is restored due to the final recapitulation of musical materials based on P.4 and P.18 <0 2 1> (csegclass c.3–2) in addition to the dynamic contour background that follows the silhouette of the prime form cseg <1 0 3 2> (c.4–5/8). Since c.3–2 and c.4–5/8 are also present at the very first tonal csegs in P.1 and P.2, the fact that these two csegs also prevail in Hika’s Codetta strengthens the general perception of symmetry that dwells on this piece—starting and finishing with the same musical contour segments.

FURTHER DISCUSSION 从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构

FURTHER DISCUSSION

After a thorough analysis of Hika in terms of tonal, rhythmical, and dynamic contours, multiple strengths and consistencies in its composition are distinguishable—the development of its thematic elements, its formal division and musical narratives of the motivic development, and the apparent reference to other musical intertexts by Brouwer and Takemitsu. For readers who want to extend the current research, I suggest studying how this contour analysis interacts with the analyses of the set classes and the voice leading. Do they support one another? If not, how are we going to interpret the relationships among contour, set classes, and voice leading? As mentioned in Chapter 2, many scholars assert that Brouwer’s works present a series of unique characteristics that feature his own compositional style. These are attributes closely related with his Afro-Cuban cultural background and musical genius. In addition, this thesis sets a precedent by analyzing Brouwer’s repertoire from a musical contour angle. Since contour theory has been an important analytical tool for the fields of ethnomusicology, music theory, and music analysis, the opportunity to systematically identify and relate contour patterns and gestures that support the idea of “uniqueness” throughout Brouwer’s repertoire remains open.

BIBLIOGRAPHY 从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构

BIBLIOGRAPHY

Adams, Charles. 1976. “Melodic Contour Typology.” Ethnomusicology 20 (2): 179– 215. Ariza Adame, Gloria. 2012. “El Método “Brouweriano”: los Estudios como Preparación de sus Obras.” Revista Sexto Orden 1: 21–26. Betancourt, Rodolfo. 1998. “A Close Encounter with Leo Brouwer.” Guitar Review 113: 8–14. Accessed February 20, 2019, http://www.musicweb-international.com/brouwer/rodolfo.htm. Burt, Peter. 1998. “The Music of Toru Takemitsu: Influences, Confluences and Status.” Ph.D. diss., Durham University. Calcines, Argel. 1999. “La música, el infinito y Leo Brouwer.” Accessed February 20, 2019. http://www.opushabana.cu/index.php/entrevistas/59-entrevista/776- Carson, Sean.2004-2005. “The Trace, its Relation to Contour Theory, and an Application to carter’s String Quartet No. 2.” Intégral 18/19: 113–49. Carter-Ényí, Aaron. 2016. “Contour Recursion and Auto-Segmentation.” Music Theory Online 22 (1). Castilla Peñaranda, Carlos Isaac. 2009. “Leo Brouwer’s Estudios Sencillos for Guitar: Afro-cuban Elements and Pedagogical Devices.” Ph.D. diss., University of Southern Mississipi. Castro Pantoja, Daniel Fernando. 2014. “Aguacero: A Semiotic Analysis of Paisaje Cubano con Lluvia by Leo Brouwer.” Trans - Revista Transcultural de Música 14. Century, Paul. 1987. “Leo Brouwer: A Portrait of the Artist in Socialist Cuba.” Latin American Music Review / Revista de Música Latinoamericana 8 (2): 151–171. Clifford, Robert John. 1995. “Contour as a Structural Element in Selected Pre-serial Works by Anton Webern.” Ph.D. diss., University of Wisconsin. Du Plessis, Harm. 2016. “Lyotard’s Sublime: Its Manifestation in the Musical Aesthetic of Toru Takemitsu and Leo Brouwer” Master diss., Capo Town University. Fukuda, Shin-Ichi. 1997. “Foreword”. Hika: In Memoriam Toru Takemitsu, 3–4. Tokyo: Gendai Guitar. Friedmann, Michael. 1985. “A Methodology for the Discussion of Contour: its Application to Schoenberg’s Music.” Journal of Music Theory 29 (2): 223–48. 62

从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 BIBLIOGRAPHY . 1987. “My Contour, Their Contour.” Journal of Music Theory 31 (2): 268– 74. Huston, John Bryan. 2006. “The Afro-cuban and the Avant-garde: Unification of Style and Gesture in the Guitar Music of Leo Brouwer.” Ph.D. diss., University of Georgia. Kolinski, Mieczyslaw. 1965. “The Structure of Melodic Movement: A New Method of Analysis.” Studies in Ethnomusicology 2: 96–120. Kronenberg, Clive. 2008. “Guitar Composer Leo Brouwer: The Concept of a ‘Universal Language’.” Tempo 62 (245): 30–46. Marvin, Elizabeth West and Paul Laprade. 1987. “Relating Musical Contours: Extensions of a Theory for Contour.” Journal of Music Theory 31 (2): 268–74. . 1989. “A Generalized Theory of Musical Contour: Its Application to Melodic and Rhythmic Analysis of Non-Tonal Music and Its Perceptual and Pedagogical Implications.” Ph.D. diss., Eastman School of Music. . 1991. “The Perception of Rhythm in Non-Tonal Music: Rhythmic Contour in the Music of Edgard Varèse.” Music Theory Spectrum 13 (1): 61–78. . 1995. “A Generalization of Contour Theory to Diverse Musical Spaces: Analytical Applications to the Music of Dallapiccola and Stockhausen.” In Concert Music, Rock, and Jazz since 1945 Essays and Analytical Studies. Ed. Elizabeth West Marvin and Richard Hermann, 135–71. Rochester, NY: University of Rochester Press. McKenna, Constance. 1988. “An Interview with Leo Brouwer.” Guitar Review 72: 2– 5. Accessed February 20, 2019, http://www.angelfire.com/in/eimaj/interviews/leo.brouwer.html. Meek, Scott. 2012. “Traditional Japanese Aesthetics within a Modern Frame: Japanese Literary Sources in relation to Toru Takemitsu’s Rain Tree Sketches.” Ph.D. diss., Indiana University. Morris, Robert. 1987. Composition with Pitch-Class: A Theory of Compositional Design. New Haven, CT: Yale University Press. . 1993. “New Directions in the Theory and Analysis of Musical Contour.” Music Theory Spectrum 15 (2): 205–28. Orpen, Keith and David Huron. 1992. “Measurement of Similarity in Music: A Quantitative Approach for Non-parametric Representations.” Computers in Music Research 4: 1–44.

BIBLIOGRAPHY 从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 Quinn, Ian. 1997. “Fuzzy Extensions to the Theory of Contour.” Music Theory Spectrum 19 (2): 232–63. . 1999. “The Combinatorial Model of Pitch Contour.” Music Perception 16 (4): 439–56. Rodríguez Cuervo, Marta and Victoria Eli Rodríguez. 2009. Leo Brouwer, caminos de la creación. Madrid: Ediciones y Publicaciones Autor, S.L.R. Sampaio, Marcos da Silva. 2012. “A Teoria de Relações de Contornos Musicais: Inconsistências, Soluções e Ferramentas” Ph.D. diss., Universidade Federal Da Bahia. . 2018. “Contour Metrics.” Accessed July 30, 2018, https://contour.sampaio.me/info. and Pedro Kroger. 2016. “Contour Algorithms Review.” MusMat: Brazilian Journal of Music and Mathematics 1 (1): 72–85. Santa, Matthew. 1999. “Defining Modular Transformations.” Music Theory Spectrum 21 (2): 200–29. Seeger, Charles. 1960. “On the Moods of a Music-Logic.” Journal of the American Musicological Society 13 (1–3): 224–61. Sakamoto, Haruyo. 2003. “Toru Takemitsu: the roots of his creation.” Ph.D. diss., Florida State University. Schultz, Robert. 2008. “Melodic Contour and Nonretrogradable Structure in the Birdsong of Oliver Messiaen.” Music Theory Spectrum 30 (1): 89–137. . 2009. “A Diachronic-Transformational Theory of Musical Contour Relations.” Ph.D. diss., University of Washington. . 2016. “Normalizing Musical Contour Theory.” Journal of Music Theory 60 (1): 23–50. Schoenberg, Arnold. 1967. Fundamentals of Musical Composition. Ed. Gerald Strang and Leonard Stein. London: Faber and Faber Ltd. Wassily Saba, Thérèse. 2008. “Leo Brouwer’s New Guitar Concerto: An Interview with Leo Brouwer and Shin-Ichi Fukuda.” Classical Guitar 26 (8): 36-40. Witsuba-Álvarez, Vladimir. “Música Guitarrística de Leo Brouwer: una Concreción de Identidad cultural en el Repertorio de la Música Académica Contemporánea.” Revista Musical Chilena 175. 19–41. Wu, Yi-Cheng Daniel. 2013. “A New Similarity Measurement of Pitch Contour for

从 Leo Brouwer 的吉他作品 Hika: In Memoriam Toru Takemitsu 分析音乐轮廓与曲式结构 BIBLIOGRAPHY Analyzing 20th- and 21st-Century Music: The Minimally Divergent Contour Network.” Indiana Theory Review 31 (1/2): 5–51. . 2019 (forthcoming). “Extensions of the Minimally Divergent Contour Network: Considering the Nonconsecutive Repeated Contour Pitches.” Music Theory Spectrum 41 (2).