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

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.

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

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

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

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

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 “Diatonicity” “Whole-tone quality” Ph4 Ph5 Ph6

Krumhansl’s Tonal Space is a DFT Phase Space Toroidal MDS solution of key profiles from Krumhansl and Kessler 1982

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

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: f1 f2 f3 f4 f5 f6 0.399 0.817 0.785 0.767 0.821 0.629

Results: Regression Resulting R2: f1 f2 f3 f4 f5 f6 0.399 0.817 0.785 0.767 0.821 0.629 Factors in the in the final model and their effect sizes

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

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

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

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

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

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

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

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

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

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