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
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

4. 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.

5. 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

6. 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

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

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

9. Procedure
33 composers from the Yale Classical Archives (ycac.yale.edu)
Byrd
Lully
Pachelbel 1653
Couperin 1653
Purcell
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
Hummel
Schubert
Mendelssohn
Chopin
Schumann
Liszt
Verdi
Wagner
Brahms
Saint-Saens
Tchaikovsky
Dvorák
Faure
Scriabin
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: f1 f2 f3 f4 f5 f6
0.399
0.817
0.785
0.767
0.821
0.629

11. 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

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