Discrete phase retrieval in musical structures

Journal of Mathematics and Music

Volumn 5, Issue 2, 2011, Pages 99-116

  Mandereau, John ^a,b   Ghisi, Daniele ^c   Amiot, Emmanuel ^c   Andreatta, Moreno ^b   Agon, Carlos ^b  

^a UNIVERSITY OF PISA   (Italy)

^b IRCAM   (France)

^c CPGE   (France)

Author keywords

GIS (generalized interval systems); homometry; interval vector; k deck; Patterson function; phase retrieval; spectral units; Z relation

Indexed keywords

EID: 80053592401     PISSN: 17459737     EISSN: 17459745     Source Type: Journal    
DOI: 10.1080/17459737.2011.608820     Document Type: Article

References (25)

1. J.A. Bondy, A graph reconstructor's manual, in Surveys in Combinatorics 1991, D. Keedwell, ed., London Mathematical Society Lecture Notes Series, Cambridge University Press, Cambridge, 1991, pp. 221-252. Available at http://www.ecp6.jussieu.fr/pageperso/bondy/research/papers/recon.pdf. MR 1161466 (93e:05071).
2. J.A. Bondy and R.L. Hemminger, Graph reconstruction-a survey, J. Graph Theory 1 (1977), pp. 227-268. MR 0480189 (58 #372).
3. J. Mandereau, D. Ghisi, E. Amiot, M. Andreatta, and C. Agon, Z-relation and homometry in musical distributions, J. Math. Music 5 (2011), pp. 83-98.
4. D. Lewin, Intervallic relations between two collections of notes, J. Music Theory 3(2) (1959), pp. 298-301.
5. I. Quinn, A unified theory of chord quality in equal temperaments, Ph.D. thesis, Eastman School of Music, 2004. Available at http://music101.pbworks.com/ f/Quinn,Ian-Diss.pdf
6. E. Amiot, Eine Kleine Fourier Musik, in Mathematics and Computation in Music, no. 37-3 in Communications in Computer and Information Science, SMCM, Springer, Berlin, 2007, pp. 469-476. Available at http://www. springerlink.com/content/rv46154m2n668630/
7. E. Amiot, Discrete Fourier transform and Bach's good temperament, Music Theor. Online 15 (2009). Available at http://mto.societymusictheory.org/issues/ mto.09.15.2/mto.09.15.2.amiot.html
8. E. Amiot and W.A. Sethares, An algebra for periodic rhythms and scales, preprint (2009). Available at http://canonsrythmiques.free.fr/pdf/AlgScales.pdf
9. H. Reiter, Classical Harmonic Analysis and Locally Compact Groups, Oxford University Press, Oxford, 1968. MR 0306811 (46 #5933).
10. J. Rosenblatt and P.D. Seymour, The structure of homometric sets, SIAM J. Algebr. Discret. Methods 3 (1982), pp. 343-350. MR 0666859 (83m:12029).
11. E. Amiot, On the group of rational spectral units with finite order, (2009). Available at arXiv, http://arxiv.org/abs/0907.0857
12. J. Mandereau, Étude des ensembles homométriques et leur application en théorie mathématique de la musique et en composition assistée par ordinateur, Master's thesis, Université Pierre-et-Marie-Curie, 2009. Available at http://recherche.ircam.fr/equipes/ repmus/moreno/MasterMandere au.pdf
13. D. Lewin, Generalized Musical Intervals and Transformations, Yale University Press, New Haven, 1987, 2nd ed., Oxford University Press, Oxford, 2007.
14. N. Collins, Uniqueness of pitch class spaces, minimal bases and Z partners, Proceedings of the Diderot Forum on Mathematics and Music, Vienna, 1999.
15. D. Ghisi,Vettori intervallari: Non degenerazione e Z-relation, Bachelor's thesis, Università di Milano Bicocca, Italy, 2006.
16. A.J. Radcliffe andA.D. Scott, Reconstructing subsets of Zn, J. Comb. TheoryA 83 (1998), pp. 169-187, MR 1636968 (99k:05158).
17. P. Jaming, The phase retrieval problem for cyclotomic crystals, in Topics on the Interface between Harmonic Analysis and Number Theory, T. Erdelyi, B. Saffari, and G. Tenenbaum, eds., CIRM, Marseille, 2005. Available at http://www.univ-orleans.fr/mapmo/membres/jaming/recherche/habilitation/art17. pdf, cannot find volume publication reference nor editor on the WWW.
18. P. Jaming and M.N. Kolountzakis, Reconstruction of functions from their triple correlations, New York J. Math. 9 (2003), pp. 149-164. Available at http://hal.archives-ouvertes.fr/hal-00005817/
19. L. Pebody, The reconstructibility of finite Abelian groups, Comb. Probab. Comput. 13 (2004), pp. 867-892. MR 2102413 (2005m:05153).
20. L. Pebody, Reconstructing odd necklaces, Comb. Probab. Comput. 16 (2007), pp. 503-514. MR 2334582 (2008f:05123).
21. G. Lachaussée, Théorie des ensembles homométriques, Undergraduate thesis, École Polytechnique, France, 2010. Available at http://recherche.ircam.fr/equipes/repmus/moreno/MasterLachaussee.pdf
22. F.A. Grünbaum and C.C. Moore, The use of higher-order invariants in the determination of generalized Patterson Cyclotomic Sets, Acta Crystallogr. A 51 (1995), pp. 310-323. MR 1331196 (96e:82087).
23. T.Althuis and F. Göbel, Z-related Pairs in Microtonal Systems, Memorandum 1524, University of Twente, Enschede, 2000.
24. A.J. Radcliffe and A.D. Scott, Reconstructing under group actions, Graphs Comb. 22(3) (2006), pp. 399-419. Available at http://people.maths.ox.ac.uk/ ∼scott/Papers/actions.pdf. MR 2264862 (2007h:20042).
25. V.B. Mnukhin, The k-orbit reconstruction for Abelian and Hamiltonian groups, Acta Appl. Math. 52 (1998), pp. 149-162. MR 1649695 (99i:20006).