Analysis of Emergent Beat-Class Sets in Steve Reich's "Clapping Music" and the Yoruba Bell Timeline

Steve Reich’s Clapping Music consists of a rhythmic pattern played by two performers each clapping the rhythm with their hands. One performer repeats the pattern unchangingly throughout the piece, while the other shifts the pattern by one unit of time after a certain fixed number of repetitions. This shifting continues until the the performers are once again playing in unison, which signals the end of the piece. Two intriguing questions in the past have been: how did Steve Reich select his pattern in the first place, and what kinds of explanations can be given for its success in what it does. Here we compare the Clapping Music rhythmic pattern to an almost identical Yoruba bell timeline of West Africa, which strongly influenced Reich. Reich added only one note to the Yoruba pattern. The two patterns are compared using two mathematical measures as a function of time as the piece is performed. One measure is a dissimilarity measure between the two patterns as they are being played, and the other is a measure of syncopation computed on both patterns, also as they are played. The analysis reveals that the pattern selected by Reich has greater rhythmic changes and a larger variety of changes as the piece progresses. Furthermore, a phylogenetic graph computed with the dissimilarity matrix yields additional insights into the salience of the pattern selected by Reich.

[1]  E. Marvin,et al.  Relating Musical Contours: Extensions of a Theory for Contour , 1987 .

[2]  Clifton Callender,et al.  Generalized Voice-Leading Spaces , 2008, Science.

[3]  Richard Cohn Transpositional Combination of Beat-Class Sets in Steve Reich's Phase-Shifting Music , 1992 .

[4]  Pedro van der Lee Zarabanda : esquemas rítmicos de acompañamiento en 6/8 , 1995 .

[5]  Eric J. Isaacson Similarity of Interval-Class Content between Pitch-Class Sets: The IcVSIM Relation , 1990 .

[6]  P van Leeuwen,et al.  Musical rhythms in heart period dynamics: a cross-cultural and interdisciplinary approach to cardiac rhythms. , 1999, American journal of physiology. Heart and circulatory physiology.

[7]  Joel K. Haack Clapping Music—a Combinatorial Problem , 1991 .

[8]  Akin Eúbà Yoruba drumming : the dùndún tradition , 1990 .

[9]  Godfried T. Toussaint,et al.  The Euclidean Algorithm Generates Traditional Musical Rhythms , 2005 .

[10]  Henri Pousseur,et al.  The Question of Order in New Music , 1966 .

[11]  Philip N. Johnson-Laird,et al.  Rhythm and meter: A theory at the computational level. , 1991 .

[12]  Mark Lindley,et al.  The New Grove Dictionary of Music and Musicians , 2001 .

[13]  Steve Reich Writings about music , 1974 .

[14]  Steve Reich,et al.  Writings on Music, 1965-2000 , 2002 .

[15]  柴田 南雄,et al.  ニューグローヴ世界音楽大事典 = The new Grove dictionary of music and musicians , 1879 .

[16]  R. Jackendoff,et al.  A Generative Theory of Tonal Music , 1985 .

[17]  Daniel H. Huson,et al.  SplitsTree: analyzing and visualizing evolutionary data , 1998, Bioinform..

[18]  Godfried T. Toussaint,et al.  A Mathematical Analysis of African, Brazilian, and Cuban Clave Rhythms , 2002 .

[19]  Timothy A. Johnson Minimalism: Aesthetic, Style, or Technique? , 1994 .

[20]  Godfried T. Toussaint,et al.  Classification and Phylogenetic Analysis of African Ternary Rhythm Timelines , 2003 .

[21]  M. Asch,et al.  Social Context and the Musical Analysis of Slavey Drum Dance Songs , 1975 .

[22]  Ian Quinn,et al.  Minimal Challenges: Process Music and the Uses of Formalist Analysis , 2006 .

[23]  Robert Morris A Similarity Index for Pitch-Class Sets , 1979 .

[24]  David W. Rogers,et al.  A Geometric Approach to PCSet Similarity , 1999 .

[25]  Godfried T. Toussaint,et al.  A Comparison of Rhythmic Similarity Measures , 2004, ISMIR.

[26]  Godfried T. Toussaint,et al.  Mathematical Features for Recognizing Preference in Sub-saharan African Traditional Rhythm Timelines , 2005, ICAPR.

[27]  H. C. Longuet-Higgins,et al.  The Rhythmic Interpretation of Monophonic Music , 1984 .

[28]  Martha Ellen Davis,et al.  Instrumentos de la musica folklorico-popular de Cuba , 2003 .

[29]  Godfried T. Toussaint,et al.  Mathematical Measures of Syncopation , 2005 .

[30]  Ian Quinn,et al.  Fuzzy Extensions to the Theory of Contour , 1997 .

[31]  John Roeder Beat-Class Modulation in Steve Reich's Music , 2003 .

[32]  Joseph N. Straus Uniformity, Balance, and Smoothness in Atonal Voice Leading , 2003 .

[33]  Eric J. Isaacson,et al.  THE INTERVAL ANGLE: A SIMILARITY MEASURE FOR PITCH-CLASS SETS , 1998 .

[34]  Godfried T. Toussaint,et al.  El Compás Flamenco: A Phylogenetic Analysis , 2004 .

[35]  VOICE LEADINGS AS GENERALIZED KEY SIGNATURES A rational reconstruction of Western musical notation , 2005 .

[36]  Michael Keith From Polychords to Polya : Adventures in Musical Combinatorics , 1991 .

[37]  Henkjan Honing,et al.  Evaluating and Extending Computational Models of Rhythmic Syncopation in Music , 2006, ICMC.

[38]  R. Morris Equivalence and Similarity in Pitch and Their Interaction with PCSet Theory , 1995 .