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Getting Stylistic Information from Pitch-Class Distributions Using the DFT

This presentation explores the use of Discrete Fourier Transform (DFT) to analyze pitch-class distributions and extract stylistic information in music. It discusses characteristic functions, change of basis with DFT, phase spaces, and regression analysis. Results and future directions are also presented.

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Getting Stylistic Information from Pitch-Class Distributions Using the DFT

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  1. Getting Stylistic Information from Pitch-Class Distributions Using the DFT Jason Yust, Boston University Presentation to the Northeast Music Cognition Group, 2/2/2019

  2. Discrete Fourier Transform on Pcsets Lewin, David (1959). “Re: Intervallic Relations between Two Collections of Notes,” JMT 3/2. ——— (2001). “Special Cases of the Interval Function between Pitch Class Sets X and Y.” JMT 45/1. Quinn, Ian (2006–2007). “General Equal-Tempered Harmony,” Perspectives of New Music 44/2–45/1. Amiot, Emmanuel (2013). “The Torii of Phases.” Proceedings of the International Conference for Mathematics and Computation in Music, Montreal, 2013 (Springer). Yust, Jason (2015). “Schubert’s Harmonic Language and Fourier Phase Spaces.” JMT 59/1.

  3. Characteristic Functions And using non-integer values, the pc-vector can describe pc-distributions The characteristic function of a pcset is a 12-place vector with 1s for each pc and 0s elsewhere: By allowing other integer values, the characteristic function can also describe pc-multisets ( 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0 ) ( 2, 0, 0.5, 0.25, 0, 1, 0, 1, 0, 0.25, 0.5, 0 ) ( 2, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0 ) C C# D E∫ E F F# G G# A B∫ B

  4. DFT as a Change of Basis The DFT is a change of basis from a sum of pc spikes to a sum of discretized periodic (perfectly even) curves. The magnitudes of DFT components contain precisely the intervallic information of the set. They are equivalent under transposition, inversion, and Z-relations (homometry).

  5. Phase Spaces One-dimensional phase spaces are Quinn’s Fourier balances, superimposed n-cycles created by multiplying the pc-circle by n. Ph3 Ph2 Ph1 Ph6 Ph5 Ph4 N.B. counter-clockwise orientation

  6. Phase Spaces One-dimensional phase spaces are Quinn’s Fourier balances, superimposed n-cycles created by multiplying the pc-circle by n. Ph3 Ph2 Ph1 “Chromaticity” “Dyadicity” “Triadicity” N.B. counter-clockwise orientation

  7. Phase Spaces One-dimensional phase spaces are Quinn’s Fourier balances, superimposed n-cycles created by multiplying the pc-circle by n. “Octatonicty” Or: “Axis” Function “Whole-tone quality” “Diatonicity” Ph6 Ph5 Ph4

  8. Krumhansl’s Tonal Space is a DFT Phase Space  Ph3  Ph5 Toroidal MDS solution of key profiles from Krumhansl and Kessler 1982

  9. Procedure 33 composers from the Yale Classical Archives (ycac.yale.edu) Byrd 1547 Lully 1632 Pachelbel 1653 Couperin 1653 Purcell 1659 Couperin 1668 Vivaldi 1678 Telemann 1681 Rameau 1683 J.S. Bach 1685 Handel 1685 Scarlatti 1685 Zipoli 1688 Sammartini 1700 Haydn 1732 Cimarosa 1749 Mozart 1756 Beethoven 1770 Hummel 1778 Schubert 1797 Mendelssohn 1809 Chopin 1810 Schumann 1810 Liszt 1811 Verdi 1813 Wagner 1813 Brahms 1833 Saint-Saens 1835 Tchaikovsky 1840 Dvorák 1841 Faure 1845 Scriabin 1872 Rachmaninoff 1873

  10. Procedure • Pitch-class distributions transposed to C and averaged for each composer from —First 20 quarter-notes —Last 20 quarter-notes —Whole pieces • Distributions converted with DFT • Multiple regression on mode, date, date2, position, and interactions for each component • Each regression simplified by backwards elimination until all factors are significant Resulting R2:

  11. Results: Regression Resulting R2: Factors in the in the final model and their effect sizes

  12. Results: Component Magnitudes Magnitudes of all components (whole pieces): Major mode

  13. Results: Component Magnitudes Magnitudes of all components (whole pieces): Minor mode

  14. Results: Component Magnitudes Cubic regression of |f5| (diatonicity)

  15. Results: Component Magnitudes Major keys Minor keys Regression predictions separated by position for |f5|, |f3|, and |f4|

  16. Results: Phases Predicted Ph5/Ph3 separated by mode and position

  17. Results: Phases Predicted Ph4/Ph3 separated by mode and position

  18. Acknowledgments Thanks to Matthew Chiu who assembled the data for this study

  19. Future Directions: Times Series DFTs Example: Corelli, Op. 4/8 Sarabande Ph5 Ph3 Ph2

  20. Future Directions: Times Series DFTs Example: Corelli, Op. 4/8 Sarabande Ph5 Ph3 Ph2

  21. Getting Stylistic Information from Pitch-Class Distributions Using the DFT Jason Yust, Boston University Presentation to the Northeast Music Cognition Group, 2/2/2019

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