Computational geometric aspects of rhythm, melody, and voice-leading

Many problems concerning the theory and technology of rhythm, melody, and voice-leading are fundamentally geometric in nature. It is therefore not surprising that the field of computational geometry can contribute greatly to these problems. The interaction between computational geometry and music yields new insights into the theories of rhythm, melody, and voice-leading, as well as new problems for research in several areas, ranging from mathematics and computer science to music theory, music perception, and musicology. Recent results on the geometric and computational aspects of rhythm, melody, and voice-leading are reviewed, connections to established areas of computer science, mathematics, statistics, computational biology, and crystallography are pointed out, and new open problems are proposed.

[1]  Manal Mohamed,et al.  Necklace Swap Problem for Rhythmic Similarity Measures , 2005, SPIRE.

[2]  Peter Gilbert,et al.  There Are No New Homometric Golomb Ruler Pairs with 12 Marks or Less , 1994, Exp. Math..

[3]  S. S. Ravi,et al.  Heuristic and Special Case Algorithms for Dispersion Problems , 1994, Oper. Res..

[4]  Emmanuel Amiot.  David Lewin and maximally even sets , 2007 .

[5]  C. Matthews.  25th Anniversary Issue , 2001 .

[6]  Thomas Sikora,et al.  EFFICIENTLY COMPUTABLE SIMILARITY MEASURES FOR QUERY BY TAPPING SYSTEMS , 2004 .

[7]  Godfried T. Toussaint,et al.  Similaridad y evolución en la rítmica del flamenco: una incursión de la matemática computacional , 2005 .

[8]  Carlton Gamer,et al.  Some Combinational Resources of Equal-Tempered Systems , 1967 .

[9]  A. J. Cohen,et al.  D. Temperley, The Cognition of Basic Musical Structures , 2004 .

[10]  Laz E.N. Ekwueme,et al.  Concepts of African Musical Theory , 1974 .

[11]  Justin London,et al.  音楽のリズム に関する最近の神経科学的研究について : 『Hearing in Time』第二版､ 第三章｢神経生物学とリズムの発達｣より抜粋 , 2012 .

[12]  Frans Wiering.  Transportation Distances and Their Application in Music-Notation Retrieval , 2004 .

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

[14]  Minghui Jiang,et al.  On the sum of distances along a circle , 2008, Discret. Math..

[15]  G. Oddie.  Likeness to Truth , 1986 .

[16]  Gerhard J. Woeginger,et al.  Fast algorithms for the maximum convolution problem , 1994, Oper. Res. Lett..

[17]  Godfried T. Toussaint,et al.  Some properties of Matusita's measure of affinity of several distributions , 1974 .

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

[19]  Gad M. Landau,et al.  Approximation algorithm for the cyclic swap problem , 2005, Stringology.

[20]  Kresimir Josic,et al.  The Mathematics of Musical Instruments , 2001, Am. Math. Mon..

[21]  Erhan Erkut,et al.  The discrete p-Maxian location problem , 1990, Comput. Oper. Res..

[22]  A. Forte.  The Structure of Atonal Music , 1973 .

[23]  F. Pillichshammer.  A note on the sum of distances under a diameter constraint , 2001 .

[24]  Paul Erdös,et al.  Variations on the theme of repeated distances , 1990, Comb..

[25]  David Lewin,et al.  Re: The Intervallic Content of a Collection of Notes, Intervallic Relations between a Collection of Notes and Its Complement: An Application to Schoenberg's Hexachordal Pieces , 1960 .

[26]  Diana Cukierman,et al.  Online Music Search by Tapping , 2006, Ambient Intelligence in Everyday.

[27]  Costas S. Iliopoulos,et al.  Approximate string matching for music analysis , 2004, Soft Comput..

[28]  David Rappaport.  Geometry and Harmony , 2005 .

[29]  Steven Skiena,et al.  A partial digest approach to restriction site mapping , 1993, ISMB.

[30]  Friedrich Pillichshammer.  On the sum of squared distances in the Euclidean plane , 2000 .

[31]  L. Fejes Tóth,et al.  On the sum of distances determined by a pointset , 1956 .

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

[33]  Kai Fikentscher,et al.  Some Observations on the Naming of Musical Instruments and on Rhythm in Oman , 1990 .

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

[35]  Richard M. Karp.  Complexity of Computation , 1974 .

[36]  E. Altman.  On a Problem of P. Erdös , 1963 .

[37]  R. Graham,et al.  Spearman's Footrule as a Measure of Disarray , 1977 .

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

[39]  Refael Hassin,et al.  Approximation algorithms for maximum dispersion , 1997, Oper. Res. Lett..

[40]  Jack Douthett,et al.  Maximally Even Sets , 1991 .

[41]  Alice Singer,et al.  THE METRICAL STRUCTURE OF MACEDONIAN DANCE , 1974 .

[42]  L. Tóth,et al.  Über eine Punktverteilung auf der Kugel , 1959 .

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

[44]  A. M. Jones,et al.  Studies in African Music , 1960 .

[45]  N. Collins.  Uniqueness of Pitch Class Spaces , Minimal Bases and Z Partners , 2022 .

[46]  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 .

[47]  Richard A. Chrisman.  Identification and Correlation of Pitch-Sets , 1971 .

[48]  Godfried T. Toussaint,et al.  Algorithms for Computing Geometric Measures of Melodic Similarity , 2006, Computer Music Journal.

[49]  Luke Windsor,et al.  Make Me a Match: An Evaluation of Different Approaches to ScorePerformance Matching , 2000, Computer Music Journal.

[50]  Owen Wright,et al.  The modal system of Arab and Persian music, A.D. 1250-1300 , 1976 .

[51]  Michael A. Trick,et al.  Comments on the Paper: 'Heuristic and Special Case Algorithms for Dispersion Problems' by S. S. Ravi, D. J. Rosenkrantz and G. K. Tayi , 1998 .

[52]  M. Maes,et al.  On a Cyclic String-To-String Correction Problem , 1990, Inf. Process. Lett..

[53]  D. Temperley.  The Cognition of Basic Musical Structures , 2001 .

[54]  Sung-Hyuk Cha,et al.  On measuring the distance between histograms , 2002, Pattern Recognit..

[55]  Godfried T. Toussaint.  On a Simple Minkowski Metric Classifier , 1970, IEEE Trans. Syst. Sci. Cybern..

[56]  Remco C. Veltkamp,et al.  A Pseudo-Metric for Weighted Point Sets , 2002, ECCV.

[57]  Lenwood S. Heath,et al.  Sorting by Short Swaps , 2003, J. Comput. Biol..

[58]  Friedrich Pillichshammer.  On extremal point distributions in the Euclidean plane , 2003 .

[59]  Dan Gusfield.  Algorithms on Strings, Trees, and Sequences - Computer Science and Computational Biology , 1997 .

[60]  Leon Harkleroad,et al.  The Math Behind the Music , 2006 .

[61]  Douglas Eck.  A Positive-Evidence Model for Classifying Rhythmical Patterns , 2000 .

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

[63]  G. N. Srkzy.  On a problem of P. Erd?s , 1992 .

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

[65]  Sándor P. Fekete,et al.  Maximum dispersion and geometric maximum weight cliques , 2000, APPROX.

[66]  Ilona Palásti.  A distance problem of P. Erdös with some further restrictions , 1989, Discret. Math..

[67]  William A. Sethares,et al.  Rhythm and Transforms , 2007 .

[68]  Robert Morris,et al.  Voice-Leading Spaces , 1998 .

[69]  J. Iglesias.  On Patterson's cyclotomic sets and how to count them , 1981 .

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

[71]  Erik D. Demaine,et al.  The distance geometry of music , 2007, Comput. Geom..

[72]  P. Seymour,et al.  The Structure of Homometric Sets , 1982 .

[73]  Godfried T. Toussaint,et al.  Perception of meter similarity in flamenco music , 2007 .

[74]  Richard M. Karp,et al.  The Restriction Scaffold Problem , 2003, J. Comput. Biol..

[75]  Don B. Gibson.  The Aural Perception of Nontxaditional Chords in Selected Theoretical Relationships: A Computer-Generated Experiment , 1986 .

[76]  Godfried T. Toussaint,et al.  Sharper lower bounds for discrimination information in terms of variation (Corresp.) , 1975, IEEE Trans. Inf. Theory.

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

[78]  Tayuan Huang,et al.  Metrics on Permutations, a Survey , 2004 .

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

[80]  Chung Chieh.  Analysis of cyclotomic sets , 1979 .

[81]  Grzegorz Kondrak,et al.  Phonetic Alignment and Similarity , 2003, Comput. Humanit..

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

[83]  Ronald L. Graham,et al.  The Mathematics of Paul Erdős II , 1997 .

[84]  Anna Lubiw,et al.  Pattern Matching in Polyphonic Music as a Weighted Geometric Translation Problem , 2004, ISMIR.

[85]  Godfried T. Toussaint,et al.  Mathematical Models for Binarization and Ternarization of Musical Rhythm , 2007 .

[86]  Dmitri Tymoczko,et al.  The Geometry of Musical Chords , 2006, Science.

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

[89]  Peter C. Fishburn.  Convex Polygons with Few Intervertex Distances , 1995, Comput. Geom..

[90]  Godfried T. Toussaint.  Algorithmic, Geometric, and Combinatorial Problems in Computational Music Theory , 2003 .

[91]  A. Gibbons.  Algorithmic Graph Theory , 1985 .

[92]  Joel E. Cohen,et al.  Information Theory and Esthetic Perception. , 1967 .

[93]  Esko Ukkonen,et al.  Geometric algorithms for transposition invariant content based music retrieval , 2003, ISMIR.

[94]  John Rahn,et al.  Basic Atonal Theory , 1980 .

[95]  James A. Standifer.  The Tuareg: Their Music and Dances , 1988 .

[96]  Warren D. Smith,et al.  Reconstructing Sets From Interpoint Distances , 2003 .

[97]  Timothy M. Chan,et al.  Necklaces, Convolutions, and X+Y , 2006, Algorithmica.

[98]  P. ERDijS.  ON SOME METRIC AND COMBINATORIAL GEOMETRIC PROBLEMS , 1986 .

[99]  Godfried T. Toussaint,et al.  Geometric proximity graphs for improving nearest neighbor methods in instance-based learning and data mining , 2005, Int. J. Comput. Geom. Appl..

[100]  Timothy M. Chan,et al.  Fun-Sort--or the chaos of unordered binary search , 2004, Discret. Appl. Math..

[101]  Paul Erdös.  On some problems of elementary and combinatorial geometry , 1975 .

[102]  Robin Wilson,et al.  Music and Mathematics: From Pythagoras to Fractals , 2006 .

[103]  Esther M. Arkin,et al.  An efficiently computable metric for comparing polygonal shapes , 1991, SODA '90.

[104]  Tomomi Matsui,et al.  Computing a Geometric Measure of the Similarity Between two Melodies , 2003 .

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

[106]  S. Arom,et al.  L’aksak. Principes et typologie , 2004 .

[107]  Trudi Hammel Garland,et al.  Math and Music: Harmonious Connections , 1995 .

[108]  G. Toussaint,et al.  Perception of Rhythmic Similarity in Flamenco Music: Comparing Musicians and Non-Musicians. , 2009 .

[109]  Frank Ruskey,et al.  An efficient algorithm for generating necklaces with fixed density , 1999, SODA '99.

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

[111]  David Temperley,et al.  An Evaluation System for Metrical Models , 2004, Computer Music Journal.

[112]  David G. Stork,et al.  Pattern Classification , 1973 .

[113]  Jens Gregor,et al.  Efficient dynamic programming alignment of cyclic strings by shift elimination , 1996, Pattern Recognit..

[114]  M. Kendall.  Rank Correlation Methods , 1949 .

[115]  Marc Chemillier,et al.  Ethnomusicology, Ethnomathematics. The Logic Underlying Orally Transmitted Artistic Practices , 2002 .

[116]  Leo Moser,et al.  On The Different Distances Determined By n Points , 1952 .

[117]  A. L. Patterson.  Ambiguities in the X-Ray Analysis of Crystal Structures , 1944 .

[118]  Godfried T. Toussaint,et al.  Faster Algorithms for Computing Distances between One-Dimensional Point Sets , 2005 .

[119]  Timothy A. Johnson.  Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals , 2003 .

[120]  Lee Hye-ku.  Quintuple Meter in Korean Instrumental Music , 1981 .

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

[122]  David Lewin,et al.  Generalized Musical Intervals and Transformations , 1987 .

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

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

[125]  Timothy M. Chan,et al.  Necklaces, Convolutions, and X + Y , 2006, ESA.

[126]  John R. Pierce,et al.  The science of musical sound , 1983 .

[127]  Richard A. Chrisman.  Describing Structural Aspects of Pitch-Sets Using Successive-Interval Arrays , 1977 .

[128]  David Sankoff,et al.  Comparison of musical sequences , 1990, Comput. Humanit..

[129]  Kuo-Liang Chung.  An Improved Algorithm for Solving the Banded Cyclic String-to-String Correction Problem , 1998, Theor. Comput. Sci..

[130]  H. Witsenhausen.  On the Maximum of the Sum of Squared Distances Under a Diameter Constraint , 1974 .

[131]  Craig G. Nevill-Manning,et al.  Distance metrics and indexing strategies for a digital library of popular music , 2000, 2000 IEEE International Conference on Multimedia and Expo. ICME2000. Proceedings. Latest Advances in the Fast Changing World of Multimedia (Cat. No.00TH8532).

[132]  S. Mardix.  Homometrism in close-packed structures , 1990 .

[133]  János Pach,et al.  Research problems in discrete geometry , 2005 .

[134]  Marjorie Senechal.  A point set puzzle revisited , 2008, Eur. J. Comb..

[135]  E. Altman,et al.  Some Theorems on Convex Polygons , 1972, Canadian Mathematical Bulletin.

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

[137]  D. Wright,et al.  Mathematics and Music , 2009 .

[138]  Erhan Erkut,et al.  A comparison of p-dispersion heuristics , 1994, Comput. Oper. Res..

[139]  Joseph N. Straus.  Introduction to Post-Tonal Theory , 1989 .

[140]  J. Szabados,et al.  On a problem of P. Erdős , 1966 .

[141]  Douglas J. White.  The maximal dispersion problem and the "First point outside the neighbourhood" heuristic , 1991, Comput. Oper. Res..

[142]  Riccardo Poli,et al.  Topological crossover for the permutation representation , 2005, GECCO '05.

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

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

[145]  Vladimir A. Kulyukin,et al.  Generalized Hamming Distance , 2002, Information Retrieval.

[146]  Donncha O Maidín.  A geometrical algorithm for melodic difference , 1998 .

[147]  Bernard Monjardet,et al.  On the comparison of the Spearman and Kendall metrics between linear orders , 1998, Discret. Math..

[148]  Gerhard Widmer,et al.  Evaluating Rhythmic descriptors for Musical Genre Classification , 2004 .

[149]  Ning Hu,et al.  A comparison of melodic database retrieval techniques using sung queries , 2002, JCDL '02.

[150]  Richard Cohn.  Square Dances with Cubes , 1998 .

[151]  Godfried T. Toussaint,et al.  Efficient Many-To-Many Point Matching in One Dimension , 2007, Graphs Comb..

[152]  Heikki Mannila,et al.  Distance measures for point sets and their computation , 1997, Acta Informatica.

[153]  Jakob Teitelbaum and Godfried Toussaint.  RHYTHMOS: An Interactive System for Exploring Rhythm from the Mathematical and Musical Points of View , 2006 .

[154]  Proofs and generalizations of Patterson's theorems on homometric complementary sets , 1976 .

[155]  Norman Carey,et al.  Self-Similar Pitch Structures, Their Duals, and Rhythmic Analogues , 1996 .

[156]  Marc Chemillier.  Periodic musical sequences and Lyndon words , 2004, Soft Comput..

[157]  Milton Babbitt,et al.  Twelve-Tone Rhythmic Structure and the Electronic Medium , 1962 .

[158]  Christopher Francis Hasty.  Meter As Rhythm , 1997 .

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

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

[161]  C. Chuan.  Tone and Voice : A Derivation of the Rules of Voice-Leading from Perceptual Principles , 2001 .

[162]  Godfried T. Toussaint.  Computational Geometric Aspects of Musical Rhythm , 2004 .

[163]  Shigeo Abe DrEng.  Pattern Classification , 2001, Springer London.

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

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

[166]  M. J. Buerger.  Interpoint distances in cyclotomic sets , 1978 .

[167]  M. Fischer,et al.  STRING-MATCHING AND OTHER PRODUCTS , 1974 .

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

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

[170]  S. Soderberg.  Z-Related Sets as Dual Inversions , 1995 .

[171]  Richard M. Karp,et al.  Two special cases of the assignment problem , 1975, Discrete Mathematics.

[172]  Steven K. Blau.  The Hexachordal Theorem: A Mathematical Look at Interval Relations in Twelve-Tone Composition , 1999 .

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

[174]  Richard Teitelbaum.  Intervallic Relations in Atonal Music , 1965 .

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

[176]  Dmitri Tymoczko.  Scale Theory, Serial Theory and Voice Leading , 2008 .

[177]  Minghui Jiang.  A Linear-Time Algorithm for Hamming Distance with Shifts , 2007, Theory of Computing Systems.

[178]  Josep Lladós,et al.  Finding rotational symmetries by cyclic string matching , 1997, Pattern Recognit. Lett..

[179]  Eleanor Selfridge-Field,et al.  Melodic Similarity : concepts, procedures, and applications , 1998 .

[180]  Erik D. Demaine,et al.  The Distance Geometry of Deep Rhythms and Scales , 2005, CCCG.

[181]  D. Lewin.  Some Ideas about Voice-Leading between PCSets , 1998 .

[182]  P. Erdös.  On Sets of Distances of n Points , 1946 .

[183]  David Lewin,et al.  Re: Intervallic Relations between Two Collections of Notes , 1959 .

[184]  W. Stolz.  Information Theory and Esthetic Perception. , 1967 .