An algebra for periodic rhythms and scales

Journal of Mathematics and Music

Volumn 5, Issue 3, 2011, Pages 149-169

  Amiot, Emmanuel ^a   Sethares, William A ^a  

^a University of Wisconsin Madison   (United States)

Author keywords

circulating matrices; DFT; FLID; Fourier; homometry; Lewin; linear combination of scales; periodic rhythm; scale matrix; scale vector; scales; sum of scales; tilings; Z relation

Indexed keywords

EID: 84857538789     PISSN: 17459737     EISSN: 17459745     Source Type: Journal    
DOI: 10.1080/17459737.2011.640469     Document Type: Article

References (25)

1. W.A. Sethares, Tuning, Timbre, Spectrum, Scale, 2nd edn., Springer-Verlag, London, 2004.
2. W.A. Sethares, Rhythm and Transforms, Springer-Verlag, London, 2007.
3. A. King, Employments of the 'standard pattern' in Yoruba music, African Music 2(3) (1961), pp. 51-54.
4. D. Tymoczko, Scale theory, serial theory, and voice-leading, Music Anal. 27/i (2008).
5. J. Hook, Response to Dmitri Tymoczko, Conference on Modeling Music Systems, in Memory of John Clough. Chicago, IL, July 9, 2005.
6. D. Lewin, Intervallic relations between two collections of notes, J. Music Theory 3 (1959), pp. 298-301.
7. E.J. Candès and T. Tao, Near-optimal signal recovery from random projections: Universal encoding strategies, IEEE Trans. Inform. Theory 52 (2004), pp. 5406-5425.
8. Available at http://homepages.cae.wisc.edu/~sethares/software/ AlgebraScalesExamples.nb
9. J. Clough and G. Myerson, Variety and multiplicity in diatonic systems, J. Music Theory 29(2), pp. 249-270.
10. G. Strang and T. Nguyen, Wavelets and Filter Banks,Wellesley College Press,Wellesley, MA, 1996.
11. E. Amiot, À propos des canons rythmiques, Gazette des Math. 106 (2005), pp. 43-67.
12. R.M. Gray, "Toeplitz and Circulant Matrices: A review" Foundations and Trends in Communications and Information Theory, 2(3) (2006), pp. 155-239.
13. Ian Quinn, "Listening to Similarity Relations," Perspectives of New Music 39/2 (2001), 108-158.
14. J. Rosenblatt, Phase retrieval, Commun. Math. Phys. 95 (1984), pp. 317-343.
15. J. Mandereau, D. Ghisi, E. Amiot, M. Andreatta, C. Agon, Z-relation and homometry in musical distributions, J. Math. Music 3 (2011), pp. 83-98.
16. E. Amiot, On the group of rational spectral units with finite order, preprint.Available at http://arxiv4.library.cornell. edu/abs/0907.0857v1 (2009).
17. J. Wild, Flat interval distribution and other generalisations of the all-interval property, Lecture at the MCM2007 in Berlin (unpublished).
18. C.S. Burrus, R.A. Gopinath, and H. Guo, Wavelets and Wavelet Transforms, Prentice-Hall, Englewood Cliffs, NJ, 1998.
19. P.J. Huber, Projection pursuit, Ann. Statist. 13(2) (1985), pp. 435-475.
20. S.G. Mallat and Z. Zhang, Matching pursuits with time-frequency dictionaries, IEEE Trans. Signal Processing 41(12) (1993).
21. I. Daubechies, Time-frequency localization operators: A geometric phase space approach, IEEE Trans. Information Theory 34(4) (1988), pp. 605-612.
22. E.J. Candès, M.Wakin, and S. Boyd, Enhancing sparsity by reweighted L1 minimization, J. Fourier Anal. Appl. 14 (2007), pp. 877-905.
23. K. Krumhansl, Cognitive Foundations of Musical Pitch, Oxford University Press, Oxford, 1990.
24. E. Amiot, Newperspectives on rhythmic canons and the spectral conjecture, J.Math. Music, special issue on rhythmic canons, C. Agon and M. Andreatta dir. (2009), pp. 71-84.
25. M. Kolountzakis, private communication.