Measuring the Complexity of Musical Rhythm

This thesis studies measures of musical rhythm complexity. Informally, rhythm complexity may be thought of as the difficulty humans have performing a rhythm, listening to a rhythm, or recognizing its structure. The problem of understanding rhythm complexity has been studied in musicology and psychology, but there are approaches for its measurement from a variety of domains. This thesis aims to evaluate rhythm complexity measures based on how accurately they reflect human-based measures. Also, it aims to compare their performance using rhythms from Africa, India, and rhythms generated randomly. The results suggest that none of the measures accurately reflect the difficulty humans have performing or listening to rhythm; however, the measures do accurately reflect how humans recognize a rhythm’s metrical structure. Additionally, the results suggest a need for normalization of the measures to account for variety among cultural rhythms.

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

[2]  L. Guibas,et al.  Finding color and shape patterns in images , 1999 .

[3]  Peter Lawrence,et al.  Transcribe: A Comprehensive Autotranscription Program , 1993, ICMC.

[4]  Remco C. Veltkamp,et al.  Using transportation distances for measuring melodic similarity , 2003, ISMIR.

[5]  L. Lefkovitch.  MATRICES, A GENERAL MEASURE OF BIOGEOGRAPHICAL DISTANCE, AND THEIR APPLICATION , 1984 .

[6]  Graham K. Rand,et al.  Quantitative Applications in the Social Sciences , 1983 .

[7]  D. Wolfe,et al.  Nonparametric Statistical Methods. , 1974 .

[8]  C. Spearman.  The proof and measurement of association between two things. , 2015, International journal of epidemiology.

[9]  L. Cronbach.  Coefficient alpha and the internal structure of tests , 1951 .

[10]  M. J. O’Brien,et al.  Cladistics Is Useful for Reconstructing Archaeological Phylogenies: Palaeoindian Points from the Southeastern United States , 2001 .

[11]  Charlotte Truchet,et al.  Computation of words satisfying the "rhythmic oddity property" (after Simha Arom's works) , 2003, Inf. Process. Lett..

[12]  Allen Forte.  The Domain and Relations of Set-Complex Theory , 1965 .

[13]  Horst Bunke,et al.  Non-Euclidean or Non-metric Measures Can Be Informative , 2006, SSPR/SPR.

[14]  C. Krumhansl,et al.  Mental representations for musical meter. , 1990, Journal of experimental psychology. Human perception and performance.

[15]  Syncopation: A Study of Musical Rhythms , 1939 .

[16]  Milton Friedman,et al.  A Correction: The Use of Ranks to Avoid the Assumption of Normality Implicit in the Analysis of Variance , 1939 .

[17]  Mark Goldstein,et al.  On Finding Rhythmic Patterns in Musical Lines , 1985, ICMC.

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

[19]  Phylogenetic Tree (Phylogenetic Reconstruction, Phylogeny, Phylogeny Reconstruction) , 2004 .

[20]  Dirk-Jan Povel,et al.  Rhythm complexity measures for music pattern recognition , 1998, 1998 IEEE Second Workshop on Multimedia Signal Processing (Cat. No.98EX175).

[21]  P. Essens,et al.  Structuring temporal sequences: Comparison of models and factors of complexity , 1995, Perception & psychophysics.

[22]  Abraham Lempel,et al.  Compression of individual sequences via variable-rate coding , 1978, IEEE Trans. Inf. Theory.

[23]  Richard D. McKirahan,et al.  Pythagoras: His Life, Teaching, and Influence , 2005 .

[24]  William H. Press,et al.  Numerical Recipes in FORTRAN - The Art of Scientific Computing, 2nd Edition , 1987 .

[25]  Ehrhard Behrends,et al.  Music and mathematics , 2006 .

[26]  Godfried T. Toussaint,et al.  C G ] 5 J ul 2 00 5 An O ( n log n )-Time Algorithm for the Restricted Scaffold Assignment , 2009 .

[27]  David G. Stork,et al.  Pattern classification, 2nd Edition , 2000 .

[28]  N. Saitou,et al.  The neighbor-joining method: a new method for reconstructing phylogenetic trees. , 1987, Molecular biology and evolution.

[29]  B. Bronson.  Mechanical Help in the Study of Folk Song , 1949 .

[30]  J. Felsenstein.  Evolutionary trees from DNA sequences: A maximum likelihood approach , 2005, Journal of Molecular Evolution.

[31]  Donald Byrd,et al.  Problems of music information retrieval in the real world , 2002, Inf. Process. Manag..

[32]  D. Huson,et al.  Application of phylogenetic networks in evolutionary studies. , 2006, Molecular biology and evolution.

[33]  R. Parncutt.  Systematic Musicology and the History and Future of Western Musical Scholarship , 2007 .

[34]  Dharmesh Patel.  Rhythm , 1919, The Craft of Poetry.

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

[36]  Eric Thul and Godfried T. Toussaint.  A Comparative Phylogenetic-Tree Analysis of African Timelines and North Indian Talas , 2008 .

[37]  David Lewin,et al.  Forte's "Interval Vector, My Interval Function", and Regener's "Common-Note Function" , 1977 .

[38]  D J Povel,et al.  Internal representation of simple temporal patterns. , 1981, Journal of experimental psychology. Human perception and performance.

[39]  Ichiro Fujinaga,et al.  Implementation of exemplar-based learning model for music cognition , 1998 .

[40]  Brian A. Davey,et al.  Introduction to Lattices and Order: Frontmatter , 2002 .

[41]  J. S. Tanaka,et al.  A fit index for covariance structure models under arbitrary GLS estimation , 1985 .

[42]  Peter Y. Chen,et al.  Correlation: Parametric and Nonparametric Measures , 2002 .

[43]  C. Spearman.  ‘FOOTRULE’ FOR MEASURING CORRELATION , 1906 .

[44]  M. Friedman.  The Use of Ranks to Avoid the Assumption of Normality Implicit in the Analysis of Variance , 1937 .

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

[46]  I. Shmulevich,et al.  A system for machine recognition of music patterns , 1998, Proceedings of the 1998 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP '98 (Cat. No.98CH36181).

[47]  Jacob Cohen,et al.  Applied multiple regression/correlation analysis for the behavioral sciences , 1979 .

[48]  M. Kassler.  Toward Musical Information Retrieval , 1966 .

[49]  P. Hogeweg,et al.  The alignment of sets of sequences and the construction of phyletic trees: An integrated method , 2005, Journal of Molecular Evolution.

[50]  Guido van Rossum,et al.  Python Programming Language , 2007, USENIX Annual Technical Conference.

[51]  Charles Seeger,et al.  Rhythm and Tempo; A Study in Music History , 1953 .

[52]  I. Shmulevich,et al.  Measures of Temporal Pattern Complexity , 2000 .

[53]  Brian A. Davey,et al.  An Introduction to Lattices and Order , 1989 .

[54]  Charles Seeger.  AN INSTANTANEOUS MUSIC NOTATOR , 1951 .

[55]  N. Rashevsky,et al.  The mathematical basis of the arts , 1949 .

[56]  Shmuel Tomi Klein,et al.  Fuzzy Hamming Distance: A New Dissimilarity Measure , 2001, CPM.

[57]  J. Sylvester,et al.  Music and Mathematics , 1886, Nature.

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

[59]  Joseph Schillinger,et al.  The Schillinger System of Musical Composition , 1945 .

[60]  R. H. Stetson,et al.  A Motor Theory of Rhythm and Discrete Succession. , 1905 .

[61]  Arbee L. P. Chen,et al.  Query by rhythm: an approach for song retrieval in music databases , 1998, Proceedings Eighth International Workshop on Research Issues in Data Engineering. Continuous-Media Databases and Applications.

[62]  Andranick Tanguiane.  Artificial Perception and Music Recognition , 1993, Lecture Notes in Computer Science.

[63]  Jack Douthett,et al.  Vector Products and Intervallic Weighting , 1994 .

[64]  Jan La Rue,et al.  On Style Analysis , 1962 .

[65]  Pierre L'Ecuyer,et al.  Random Number Generators: Selection Criteria and Testing , 1998 .

[66]  Joseph Felsenstein,et al.  Maximum Likelihood and Minimum-Steps Methods for Estimating Evolutionary Trees from Data on Discrete Characters , 1973 .

[67]  C. D. Kemp,et al.  Density Estimation for Statistics and Data Analysis , 1987 .

[68]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[69]  William H. Press,et al.  Numerical recipes in C. The art of scientific computing , 1987 .

[70]  Ian H. Witten,et al.  Towards a digital library of popular music , 1999, DL '99.

[71]  D J Povel,et al.  A theoretical framework for rhythm perception , 1984, Psychological research.

[72]  Theodore F. Normann.  Rhythm and Tempo: A Study in Music History , 1953 .

[73]  Ajay Kapur,et al.  Computational Ethnomusicology Hesaplamalı Etnomüzikoloji , 2007 .

[74]  Douglas A. Wolfe,et al.  Nonparametric Statistical Methods , 1973 .

[75]  Edward J. Coyle,et al.  Perceptual Issues in Music Pattern Recognition: Complexity of Rhythm and Key Finding , 2001, Comput. Humanit..

[76]  E. Dietz.  Permutation Tests for Association Between Two Distance Matrices , 1983 .

[77]  Brian J. McCartin.  Prelude to Musical Geometry , 1998 .

[78]  Pierre L’Ecuyer,et al.  Random Number Generation , 2008, Encyclopedia of Algorithms.

[79]  Ichiro Fujinaga,et al.  Feature Selection Pitfalls and Music Classification , 2006, ISMIR.

[80]  Jay Rahn,et al.  Turning the Analysis around: Africa-Derived Rhythms and Europe-Derived Music Theory , 1996 .

[81]  P. L'Ecuyer,et al.  Uniform random number generators , 1998, 1998 Winter Simulation Conference. Proceedings (Cat. No.98CH36274).

[82]  A. S. Tanguiane,et al.  A Principle of Correlativity of Perception and Its Application to Music Recognition , 1994 .

[83]  Andrew J. Rosenfeld,et al.  Perception and Production of Syncopated Rhythms , 2007 .

[84]  Milton Babbitt.  The Use of Computers in Musicological Research , 1965 .

[85]  Satoru Miyano,et al.  Open source clustering software , 2004 .

[86]  W. Kruskal.  Ordinal Measures of Association , 1958 .

[87]  Dirk-Jan Povel.  Internal representation of simple temporal patterns. , 1981 .

[88]  R. Sokal,et al.  A METHOD FOR DEDUCING BRANCHING SEQUENCES IN PHYLOGENY , 1965 .

[89]  Bjarne Stroustrup,et al.  The C++ Programming Language, Second Edition , 1991 .

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

[91]  Tobias Scherbaum.  Gentoo Linux , 2008 .

[92]  Karl Pearson.  II. On an Extension of the Method of Correlation by Grades or Ranks , 1914 .

[93]  Burton R. Wolin,et al.  Informational models and their uses , 1964 .

[94]  Ronald L. Rivest,et al.  Introduction to Algorithms, Second Edition , 2001 .

[95]  K. Pearson.  Mathematical contributions to the theory of evolution.—On the law of reversion , 1900, Proceedings of the Royal Society of London.

[96]  Abraham Lempel,et al.  On the Complexity of Finite Sequences , 1976, IEEE Trans. Inf. Theory.

[97]  Paul Nauert.  Theory and Practice in Porgy and Bess: The Gershwin-Schillinger Connection , 1994 .

[98]  R. H. Stetson.  A motor theory of rhythm and discrete succession: II. , 1905 .

[99]  Jonathan Katz.  Lecture 4 , 2004 .

[100]  Olivier Messiaen.  Technique de mon langage musical , 1944 .

[101]  L. Cavalli-Sforza,et al.  PHYLOGENETIC ANALYSIS: MODELS AND ESTIMATION PROCEDURES , 1967, Evolution; international journal of organic evolution.

[102]  Ronald L. Rivest,et al.  Introduction to Algorithms , 1990 .

[103]  Steve Reich.  Writings about music , 1974 .

[104]  W..  Relative Neighborhood Graphs and Their Relatives , 2004 .

[105]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[106]  Peter Essens,et al.  Perception of Temporal Patterns , 1985 .

[107]  Harold Hotelling,et al.  Rank Correlation and Tests of Significance Involving No Assumption of Normality , 1936 .

[108]  J. Carroll.  The nature of the data, or how to choose a correlation coefficient , 1961 .

[109]  G. Toussaint,et al.  Geometric properties of musical rhythms , 2006 .

[110]  Kofi Agawu,et al.  Structural Analysis or Cultural Analysis? Competing Perspectives on the "Standard Pattern" of West African Rhythm , 2006 .

[111]  John M. Hancock,et al.  Dictionary of bioinformatics and computational biology , 2004, Choice Reviews Online.

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

[113]  Richard M. Stallman,et al.  Using the GNU Compiler Collection , 2010 .

[114]  Godfried T. Toussaint,et al.  An algorithm for computing the restriction scaffold assignment problem in computational biology , 2005, Inf. Process. Lett..

[115]  M. Collard,et al.  Investigating cultural evolution through biological phylogenetic analyses of Turkmen textiles , 2002 .

[116]  Martin Clayton,et al.  Time in Indian Music , 2000 .

[117]  John Clough.  Aspects of Diatonic Sets , 1979 .

[118]  W. Fitch,et al.  Construction of phylogenetic trees. , 1967, Science.

[119]  Dirk-Jan Povel,et al.  Complexity Measures of Musical Rhythms , 1998 .

[120]  Gerald Myerson,et al.  Variety and Multiplicity in Diatonic Systems , 1985 .

[121]  Brian Everitt,et al.  Cluster analysis , 1974 .

[122]  Grosvenor W. Cooper,et al.  The Rhythmic Structure of Music , 1971 .

[123]  O Gascuel,et al.  BIONJ: an improved version of the NJ algorithm based on a simple model of sequence data. , 1997, Molecular biology and evolution.

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

[125]  Alexander V. Arhangel'skii,et al.  General topology I: Basic concepts and constructions dimension theory , 1990 .

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

[127]  Paul C. Vitz,et al.  A coded element model of the perceptual processing of sequential stimuli. , 1969 .

[128]  Yang Liu,et al.  A Tessellation-Transformation Method for Categorizing and Generating Geometric Textile Design Patterns , 2008 .

[129]  S. Lloyd,et al.  Measures of complexity: a nonexhaustive list , 2001 .

[130]  Godfried T. Toussaint,et al.  Analysis of musical rhythm complexity measures in a cultural context , 2008, C3S2E '08.

[131]  Daniel H. Huson,et al.  Analyzing and Visualizing Sequence and Distance Data Using SplitsTree , 1996, Discret. Appl. Math..

[132]  S. Lloyd.  Programming the Universe: A Quantum Computer Scientist Takes on the Cosmos , 2006 .

[133]  Godfried T. Toussaint,et al.  Analysis of Emergent Beat-Class Sets in Steve Reich's "Clapping Music" and the Yoruba Bell Timeline , 2021 .

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

[135]  Jerome L. Myers,et al.  Research Design and Statistical Analysis , 1991 .

[136]  Susan A. Murphy,et al.  Monographs on statistics and applied probability , 1990 .

[137]  Raymond Boyd,et al.  African polyphony and polyrhythm: Map , 1991 .

[138]  M. Jones,et al.  African Rhythm , 1954, Africa.

[139]  Pierre L'Ecuyer,et al.  Uniform random number generation , 1994, Ann. Oper. Res..

[140]  Vincent Moulton,et al.  Biogeographic interpretation of splits graphs: least squares optimization of branch lengths. , 2005, Systematic biology.

[141]  John W. Tukey,et al.  Exploratory Data Analysis. , 1979 .

[142]  Joab R Winkler,et al.  Numerical recipes in C: The art of scientific computing, second edition , 1993 .

[143]  Abraham Lempel,et al.  A universal algorithm for sequential data compression , 1977, IEEE Trans. Inf. Theory.

[144]  George Marsaglia,et al.  Uniform Random Number Generators , 1965, JACM.

[145]  W. Anku,et al.  Circles and Time: A Theory of Structural Organization of Rhythm in African Music , 2000 .

[146]  J. Stephen Downie,et al.  The Scientific Evaluation of Music Information Retrieval Systems: Foundations and Future , 2004, Computer Music Journal.

[147]  D. Temperley.  Syncopation in rock: a perceptual perspective , 1999, Popular Music.

[148]  M. Kendall.  A NEW MEASURE OF RANK CORRELATION , 1938 .

[149]  T. Eerola,et al.  Perceived complexity of western and African folk melodies by western and African listeners , 2006 .

[150]  A W EDWARDS,et al.  A METHOD FOR CLUSTER ANALYSIS. , 1965, Biometrics.

[151]  David Huron,et al.  An Empirical Study of Syncopation in American Popular Music, 1890–1939 , 2006 .

[152]  Jeff Pressing,et al.  Cognitive complexity and the structure of musical patterns , 1999 .

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

[154]  Eric Regener,et al.  ON ALLEN FORTE'S THEORY OF CHORDS , 1974 .

[155]  Ichiro Fujinaga,et al.  Style-Independent Computer-Assisted Exploratory Analysis of Large Music Collections Büyük Müzik Koleksiyonlarinin Biçemden Baimsiz Bilgisayar Destekli Keif Niteliinde Çözümlenmesi , 2007 .

[156]  Bjarne Stroustrup,et al.  C++ Programming Language , 1986, IEEE Softw..

[157]  David Liptak,et al.  The Stratification of Musical Rhythm , 1978 .

[158]  R. Forthofer,et al.  Rank Correlation Methods , 1981 .

[159]  N. G. de Bruijn.  Sorting by means of swappings , 1974, Discret. Math..

[160]  L. Lefkovitch,et al.  Further nonparametric tests for comparing dissimilarity matrices based on the relative neighborhood graph , 1985 .

[161]  P. Vitz.  Information, run structure and binary pattern complexity , 1968 .

[162]  P. Haggett,et al.  The application of multidimensional scaling methods to epidemiological data , 1995, Statistical methods in medical research.

[163]  Godfried T. Toussaint,et al.  The Geometry of Musical Rhythm , 2013, JCDCG.

[164]  E. V. Hornbostel,et al.  African Negro Music , 1928, Africa.

[165]  Hokky Situngkir.  Towards Complexity Studies of Indonesian Songs , 2007 .

[166]  Godfried T. Toussaint,et al.  An Experimental Comparison of Formal Measures of rhythmic Syncopation , 2007, ICMC.

[167]  W. R. Garner.  Uncertainty and structure as psychological concepts , 1975 .

[168]  Ichiro Fujinaga,et al.  On-demand metadata extraction network (OMEN) , 2006, Proceedings of the 6th ACM/IEEE-CS Joint Conference on Digital Libraries (JCDL '06).