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ACIGR ACIRG ACGIR ACGRI ACRGI ACRIG AICGR AICRG AIGCR AIGRC AIRGC AIRCG

ACIGR ACIRG ACGIR ACGRI ACRGI ACRIG AICGR AICRG AIGCR AIGRC AIRGC AIRCG AGICR AGIRC AGCIR AGCRI AGRCI AGRIC ARIGC ARICG ARGIC ARGCI ARCGI ARCIG CAIGR CAIRG CAGIR CAGRI CARGI CARIG CIAGR CIARG CIGAR CIGRA CIRGA CIRAG CGIAR CGIRA CGCIA CGARI CGRAI CGRIA CRIGA CRIAG CRGIA CRGAI CRAGI CRAIG

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ACIGR ACIRG ACGIR ACGRI ACRGI ACRIG AICGR AICRG AIGCR AIGRC AIRGC AIRCG

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  1. ACIGR ACIRG ACGIR ACGRI ACRGI ACRIG AICGR AICRG AIGCR AIGRC AIRGC AIRCG AGICR AGIRC AGCIR AGCRI AGRCI AGRIC ARIGC ARICG ARGIC ARGCI ARCGI ARCIG CAIGR CAIRG CAGIR CAGRI CARGI CARIG CIAGR CIARG CIGAR CIGRA CIRGA CIRAG CGIAR CGIRA CGCIA CGARI CGRAI CGRIA CRIGA CRIAG CRGIA CRGAI CRAGI CRAIG GCIAR GCIRA GCAIR GCARI GCRAI GCRIA GICAR GICRA GIACR GIARC GIRAC GIRCA GAICR GAIRC GACIR GACRI GARCI GARIC GRIAC GRICA GRGIA GRACI GRCAI GRCIA ICAGR ICARG ICGAR ICGRA ICRGA ICRAG IACGR IACRG IAGCR IAGRC IARGC IARCG IGACR IGARC IGCAR IGCRA IGRCA IGRAC IRAGC IRACG IRGAC IRGCA IRCGA IRCAG RCIGA RCIAG RCGIA RCGAI RCAGI RCAIG RICGA RICAG RIGCA RIGAC RIAGC RIACG RGICA RGIAC RGCIA RGCAI RGACI RGAIC RAIGC RAICG RAGIC RAGCI RACGI RACIG

  2. A Quantitative Measure of Melodic Structure: Computational Infrastructure and Cognitive Implications Craig Graci, Cognitive Science Program State University of New York at Oswego, USA

  3. What? I will talk about a metric which is intended to assess the degree to which structural interpretations of melody are plausible with respect to tonal theory.

  4. Why? • The problem of modeling melodic structure in tonal music has relevance to the study of listening (Berger, 2004), performance (Clarke, 2005) and composition (Marsella & Schmidt, 1999). • Consequently, the problem of measuring the degree to which a model actually captures melodic structure in tonal music should be of interest.

  5. Outline • Introduction • The Metric: Conception and Evaluation • Computational Framework for the Metric • Cognitive Relevance of the Metric • Conclusion

  6. Prelude to a Talk Generative Theory of Tonal Music ✗ Grouping Well-Formedness Rules ✗ Grouping Preference Rules ✗ Gestalt Principles of Organization ✗ Knowledge Representation ✗ Structural Generality ✗ Lisp ✗ Java ✗ Correlation ✗ Analysis of Covariance ✗ Microworld ✗ Cognitive Artifact

  7. Introduction

  8. The Grouping Problem • A grouping structure for a melody is what results from recursively partitioning the sequence of notes which constitute the melody into subsequences of notes. • The grouping problem for a tonal melody is to determine a psychologically plausible grouping structure for a given melody.

  9. Two Very DifferentGrouping Structures

  10. Two Rather SimilarGrouping Structures

  11. GTTM Chapter 3 • Perception - The process of finding meaningful patterns in sensory information. • Gestalt Principles - Ideas (e.g., proximetry, similarity, “good form”) pertaining to how things are perceptually grouped. • Grouping Preference Rules - With respect to grouping in tonal music, the GTTM GPRs are a manifestation of various Gestalt principles.

  12. Sample GTTM Grouping Preference Rule Applications Proximity Similarity “Good Form”

  13. Preference Rule Conflict Similarity Symmetry

  14. GTTM GPR Summary • GPR 1 Singleton “Avoidance” • GPR 2 Proximity • GPR 3 Similarity • GPR 4 IntensificationGPR 5 SymmetryGPR 6 ParallelismGPR 7 Time-Span and Prolongational Stability

  15. Gamma Gamma is a metric which computes the degree to which a structural interpretation of a melody is consistent with the Gestalt principles of perceptual organization, as manifested in the GTTM GPRs.

  16. The Metric:Conception and Evaluation

  17. Definition of Gamma γ = ω1 γ1 + ω2 γ2 + ω3 γ3 + ω5 γ5 + ω6 γ6 • γi is the GPRi factor - a function mapping a structural interpretation of the melody onto a real number between 0 and 1. • ωi are weights - real numbers which sum to 1.0 where

  18. Two Very DifferentGrouping Structures γ = 0.349 γ = 0.622

  19. Two Rather SimilarGrouping Structures γ = 0.592 γ = 0.560

  20. Claims NOT MadeAbout Gamma • Gamma is really good at doing what it is intended to do (which is to measure the quality of grouping structures for tonal melody). • There is such a thing as a fixed “one size fits all” metric for judging the quality of a grouping structure for all tonal melodies.

  21. Claims MadeAbout Gamma • Gamma is a useful tool for investigating phenomena surrounding the grouping problem in tonal melody. • Gamma is useful as an analytical tool for helping to determine sound grouping structures in tonal melody, and helping to learn about structural interpretation.

  22. Example Gamma Computation Little Tune (Kavalevsky) γ = ω1 γ1 + ω2 γ2 + ω3 γ3 + ω5 γ5 + ω6 γ6 = 1.0*0.1 + 0.22*0.42 + 0.22*0.5 + 0.2*1.0 + 0.34*0.75 = 0.667

  23. γ2 computation for a Little Tune Interpretation E D E C D C D2 E D E C D2 D2 E D E C D C D2 E D E D C2 C2 γ2 = 5.41/13 = 0.416

  24. γ3 computation for a Little Tune Interpretation E D E C D C D2 E D E C D2 D2 E D E C D C D2 E D E D C2 C2 γ3 = 6.93/14 = 0.495

  25. Computational Frameworkfor the Metric

  26. Clay? • Clay is a simple symbolic language which can be adapted to manipulate different sorts of virtual objects. • Clay has been adapted to manipulate rectangles, coinsanddice, number sequences, and notes to obtain Mondrian, Chance, Number Theory, and Music “Worlds”.

  27. Characteristics of Clay As a music knowledge representation Clay possesses a number of significant properties: • Clay is executable • Clay is procedural ... • Clay is “structurally general” with respect to grouping

  28. Structural Generality • According to Wiggins and Smaill (2000), structural generality “measures the amount of information about musical structure which can be encoded explicitly.” • Clay facilitates study of grouping structure in tonal melody by virtue of its ability to explicitly encode melodic structure.

  29. Clay and the Note • Clay, as a music knowledge representation language, features a note. • The note has lots of properties, including a scale, pitch (degree within the scale), duration (with respect to one beat), amplitude, and timbre. • Melodies are modeled by playing, resting, and manipulating the state of the note.

  30. Some “Lower Level” Clay Primitives • P - play the note • R - rest the note • X2 / X3 / X5 / X7 - expand the duration • S2 / S3 / S5 / S7 - shrink the duration • RP / LP - raise/lower the pitch a scale degree

  31. Lower Level Clay Interaction Examples ? P P P X3 P S3 C C C C3 ? P LP LP P RP P RP P C \ A / B / C

  32. Some “Higher Level” Clay Primitives • PL - play the note for twice its duration • PS - play the note for half its duration • PD - play the note for 1.5 times its duration • RP2 / RP3 / RP4 ... - raise the pitch of the note the number of scale degrees specified • LP2 / LP3 / LP4 ... - lower the pitch of the note the number of scale degrees specified

  33. Higher Level Clay Interaction Examples ? RP2 P LP P RP P LP2 P / E \ D \ E \ C ? RP P LP P RP PL LP / D \ C / D2

  34. Clay Programming Example ? G1 = RP2 P LP P RP P LP2 P ? G2 = RP P LP P RP PL LP ? PH1 = G1 G2 ? PH1 / E \ D / E \ C / D \ C / D2

  35. Clay Programming and Structural Interpretation ? G1 = RP2 P LP P RP P LP2 P ? G2 = RP P LP P RP PL LP ? PH1 = G1 G2 As a rule, a nonprimitive Clay command corresponds to a group.

  36. MxM: Music Exploration Machine • MxM is the host computational environment for Clay • Lurking within MxM, right along side Clay, are MetaClay commands for displaying, sketching, scoring, and analyzing Clay commands.

  37. “Little Tune” in Clay LT = P1 P2P1 = PH1 PH2P2 = PH1 PH3PH1 = G1 G2PH2 = G1 G3PH3 = G4 G5G1 = RP2 P LP P RP P LP2 PG2 = RP P LP P RP PL LPG3 = RP PL PL LPG4 = RP2 P LP P RP P LP P LPG5 = PL PL

  38. Text / Tree

  39. The Score

  40. Gamma as a MetaClay Command

  41. Gamma X: Gamma with Explanation

  42. Cognitive Relevanceof the Metric

  43. Two Small Studies • Study 1: Correlational study comparing Gamma and ratings of grouping structure. • Study 2: Quasi-experiment investigating the role that computational modeling, informed by Gamma, may play in developing structural grouping knowledge and ability.

  44. The Correlational Study This study was designed to empirically investigate the validity of Gamma as a measure of grouping structure.

  45. Method • Three melodies. Twenty-six structural interpretations of each melody. • Ratings from five musical people. • Gamma values. • The correlation between the average ratings and the values was calculated.

  46. The Three Melodies German Folk Song (GFS) Dona Nobis Pacem (DNP) Ecossaise (Beethoven) (ECOS)

  47. The Musical People • Recording engineer / horn player / Music Department faculty member • HCI Graduate Student with MIR experience • School teacher / linguist who did chorus and band throughout high school • Network administrator who did chorus and band throughout high school • Psycholinguistics professor / banjo picker

  48. The “Rating” Procedure • Introduction / Instruction (30 min) • Melody 1: Listen/Study (4 min) followed by Evaluations (26 min) • Melody 2: Listen/Study (4 min) followed by Evaluations (26 min) • Melody 3: Listen/Study (4 min) followed by Evaluations (26 min)

  49. “Marginally Bad” Example

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