F. Rigaud, B. David, and L. Daudet, A parametric model and estimation techniques for the inharmonicity and tuning of the piano, The Journal of the Acoustical Society of America, vol.133, issue.5, pp.3107-3118, 2013.
DOI : 10.1121/1.4799806
URL : https://hal.archives-ouvertes.fr/hal-00856735

F. Rigaud, B. David, and L. Daudet, A parametric model of piano tuning, Proceedings of the 14th International Conference on Digital Audio Effects (DAFx), pp.393-399, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00727395

F. Rigaud, B. David, and L. Daudet, Piano sound analysis using non-negative matrix factorization with inharmonicity constraint, Proceedings of the 20th European Signal Processing Conference (EUSIPCO), pp.2462-2466, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00727398

F. Rigaud, A. Drémeau, B. David, and L. Daudet, A probabilistic line spectrum model for musical instrument sounds and its application to piano tuning estimation, 2013 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics, 2013.
DOI : 10.1109/WASPAA.2013.6701879

F. Rigaud, A. Falaize, B. David, and L. Daudet, Does inharmonicity improve an NMF-based piano transcription model?, 2013 IEEE International Conference on Acoustics, Speech and Signal Processing, pp.11-15, 2013.
DOI : 10.1109/ICASSP.2013.6637599
URL : https://hal.archives-ouvertes.fr/hal-00856734

J. B. Allen, Short term spectral analysis, synthesis, and modification by discrete Fourier transform, IEEE Transactions on Acoustics, Speech, and Signal Processing, vol.25, issue.3, pp.235-238, 1977.
DOI : 10.1109/TASSP.1977.1162950

M. Aramaki, J. Bensa, L. Daudet, P. Guillemain, and R. Martinet, Resynthesis of Coupled Piano String Vibrations Based on Physical Modeling, Journal of New Music Research, vol.30, issue.3, pp.30213-226, 2001.
DOI : 10.1076/jnmr.30.3.213.7472

R. Badeau, N. Bertin, and E. Vincent, Stability Analysis of Multiplicative Update Algorithms and Application to Nonnegative Matrix Factorization, IEEE Transactions on Neural Networks, vol.21, issue.12, pp.1869-1881, 2010.
DOI : 10.1109/TNN.2010.2076831
URL : https://hal.archives-ouvertes.fr/inria-00555984

R. Badeau, B. David, R. , and G. , High-resolution spectral analysis of mixtures of complex exponentials modulated by polynomials, IEEE Transactions on Signal Processing, vol.54, issue.4, pp.1341-1350, 2006.
DOI : 10.1109/TSP.2006.870556
URL : https://hal.archives-ouvertes.fr/hal-00479781

B. Bank and H. Lehtonen, Perception of longitudinal components in piano string vibrations, The Journal of the Acoustical Society of America, vol.128, issue.3, pp.117-123, 2010.
DOI : 10.1121/1.3453420

B. Bank and L. Sujbert, Generation of longitudinal vibrations in piano strings: From physics to sound synthesis, The Journal of the Acoustical Society of America, vol.117, issue.4, pp.2268-2278, 2005.
DOI : 10.1121/1.1868212

B. Bank, S. Zambon, and F. Fontana, A Modal-Based Real-Time Piano Synthesizer, IEEE Transactions on Audio, Speech, and Language Processing, vol.18, issue.4, pp.809-821, 2010.
DOI : 10.1109/TASL.2010.2040524

J. P. Bello, L. Daudet, and M. B. Sandler, Automatic Piano Transcription Using Frequency and Time-Domain Information, IEEE Transactions on Audio, Speech and Language Processing, vol.14, issue.6, pp.142241-2251, 2008.
DOI : 10.1109/TASL.2006.872609

E. Benetos, Automatic Transcription of Polyphonic Music Exploiting Temporal Evolution, 2012.

E. Benetos and S. Dixon, Joint Multi-Pitch Detection Using Harmonic Envelope Estimation for Polyphonic Music Transcription, IEEE Journal of Selected Topics in Signal Processing, vol.5, issue.6, pp.1111-1123, 2011.
DOI : 10.1109/JSTSP.2011.2162394

E. Benetos and C. Kotropoulos, A tensor-based approach for automatic music genre classification, Proceedings of the 16th European Signal Processing Conference (EUSIPCO), 2008.

E. Benetos, M. Kotti, K. , and C. , Musical instrument classification using non-negative matrix factorization algorithms, 2006 IEEE International Symposium on Circuits and Systems, pp.221-224, 2006.
DOI : 10.1109/ISCAS.2006.1692967

J. Bensa, Analyse et synthèse de sons de piano par modèles physiques et de signaux, 2003.

J. Bensa, S. Bilbao, R. Kroland-martinet, and J. O. Smith, The simulation of piano string vibration: From physical models to finite difference schemes and digital waveguides, The Journal of the Acoustical Society of America, vol.114, issue.2, pp.1095-1107, 2003.
DOI : 10.1121/1.1587146

N. Bertin, Les factorisations en matrices non-négatives. Approches contraintes et probabilistes, application à la transcription automatique de musique polyphonique, 2009.

N. Bertin, R. Badeau, and E. Vincent, Fast bayesian nmf algorithms enforcing harmonicity and temporal continuity in polyphonic music transcription, 2009 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics, pp.29-32, 2009.
DOI : 10.1109/ASPAA.2009.5346531
URL : https://hal.archives-ouvertes.fr/inria-00601355

N. Bertin, R. Badeau, and E. Vincent, Enforcing Harmonicity and Smoothness in Bayesian Non-Negative Matrix Factorization Applied to Polyphonic Music Transcription, IEEE Transactions on Audio, Speech, and Language Processing, vol.18, issue.3, pp.538-549, 2010.
DOI : 10.1109/TASL.2010.2041381
URL : https://hal.archives-ouvertes.fr/inria-00557088

E. Blackham, The Physics of the Piano, Scientific American, vol.213, issue.6, pp.88-99, 1965.
DOI : 10.1038/scientificamerican1265-88

E. Blanco-martín, F. J. Casajus-quiros, and L. I. Ortiz-berenguer, An improved pattern-matching method for piano multi-pitch detection, 124th Convention of the Audio Engineering Society (AES), 2008.

B. Bremmer, Aural tuning tests for 2:1, 4:2 and 6:3 type octaves, 2007.

B. Bremmer, Midrange piano tuning, 2007.

B. Capleton, Theory and Practice of Piano Tuning. A Manual on the Art, Techniques and Theory, pp.1-626, 2007.

J. Chabassier, Modélisation et simulation numérique d'un piano par modèles physiques, 2012.

A. Chaigne and A. Askenfelt, Numerical simulations of piano strings. II. Comparisons with measurements and systematic exploration of some hammer???string parameters, The Journal of the Acoustical Society of America, vol.95, issue.3, pp.1631-1640, 1993.
DOI : 10.1121/1.408549

A. Chaigne and J. Kergomard, Acoustique des instruments de musique, Belin, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00455011

S. S. Chen, D. L. Donoho, and M. A. Saunders, Atomic Decomposition by Basis Pursuit, SIAM Journal on Scientific Computing, vol.20, issue.1, pp.33-61, 1998.
DOI : 10.1137/S1064827596304010

A. Cichocki, R. Zdunek, and S. Amari, Csisz??r???s Divergences for Non-negative Matrix Factorization: Family of New Algorithms, 6 th International Conference on Independent Component Analysis and Blind Signal Separation (ICA'06), pp.32-39, 2006.
DOI : 10.1007/11679363_5

A. Cichocki, R. Zdunek, and S. Amari, Nonnegative matrix and tensor factorization, IEEE Signal Processing Magazine, vol.25, issue.1, pp.142-145, 2008.
DOI : 10.1002/9780470747278

A. Cichocki, R. Zdunek, A. H. Phan, and S. Amari, Nonnegative Matrix and Tensor Factorizations. Applications to Exploratory Multi-way Data Analysis and Blind Source Separation, 2009.

J. E. Cohen and U. G. Rothblum, Nonnegative ranks, decompositions, and factorizations of nonnegative matrices, Linear Algebra and its Applications, vol.190, pp.149-168, 1993.
DOI : 10.1016/0024-3795(93)90224-C

P. Comon, Independent component analysis, A new concept?, Signal Processing, vol.36, issue.3, pp.287-314, 1994.
DOI : 10.1016/0165-1684(94)90029-9
URL : https://hal.archives-ouvertes.fr/hal-00417283

J. Conklin and H. A. , Design and tone in the mechanoacoustic piano. Part II. Piano structure, The Journal of the Acoustical Society of America, vol.100, issue.2, pp.695-708, 1996.
DOI : 10.1121/1.416233

J. Conklin and H. A. , Design and tone in the mechanoacoustic piano. Part III. Piano strings and scale design, The Journal of the Acoustical Society of America, vol.100, issue.3, pp.1286-1298, 1996.
DOI : 10.1121/1.416017

J. Conklin and H. A. , Generation of partials due to nonlinear mixing in a stringed instrument, The Journal of the Acoustical Society of America, vol.105, issue.1, pp.536-545, 1999.
DOI : 10.1121/1.424589

A. Cont, Realtime Audio to Score Alignment for Polyphonic Music Instruments, using Sparse Non-Negative Constraints and Hierarchical HMMS, 2006 IEEE International Conference on Acoustics Speed and Signal Processing Proceedings, 2006.
DOI : 10.1109/ICASSP.2006.1661258
URL : https://hal.archives-ouvertes.fr/hal-00723224

M. Davy, S. J. Godsill, and J. Idier, Bayesian analysis of polyphonic western tonal music, The Journal of the Acoustical Society of America, vol.119, issue.4, pp.2498-2517, 2006.
DOI : 10.1121/1.2168548
URL : https://hal.archives-ouvertes.fr/inria-00120240

A. P. Dempster, N. M. Laird, R. , and D. B. , Maximum likelihood from incomplete data via the EM algorithm, Journal of the Royal Statistical Society, Serie B, vol.39, issue.1, pp.1-38, 1977.

A. Dessein, A. Cont, and G. Lemaitre, Real-time polyphonic music transcription with non-negative matrix factorization and beta-divergence, Proceedings of the 11th International Society for Music Information Retrieval conference (ISMIR), pp.489-494, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00708682

I. S. Dhillon and S. Sra, Generalized nonnegative matrix approximations with Bregman divergences, Proceedings of the Neural Information Processing Systems (NIPS) Conference, pp.283-290, 2006.

S. Dixon, M. Mauch, and D. Tidhar, Estimation of harpsichord inharmonicity and temperament from musical recordings, The Journal of the Acoustical Society of America, vol.131, issue.1, pp.878-887, 2012.
DOI : 10.1121/1.3651238

D. Donoho and V. Stodden, When does non-negative matric factorization give a correct decomposition into parts, Proceedings of the conference Advances in neural information processing systems, 2004.

B. Doval and X. Rodet, Estimation of fundamental frequency of musical sound signals, [Proceedings] ICASSP 91: 1991 International Conference on Acoustics, Speech, and Signal Processing, pp.3657-3660, 1991.
DOI : 10.1109/ICASSP.1991.151067

B. Doval and X. Rodet, Fundamental frequency estimation and tracking using maximum likelihood harmonic matching and HMMs, IEEE International Conference on Acoustics Speech and Signal Processing, pp.221-224, 1993.
DOI : 10.1109/ICASSP.1993.319095

J. S. Downie, The Music Information Retrieval Evaluation eXchange (MIREX), D-Lib Magazine, vol.12, issue.12, 2006.
DOI : 10.1045/december2006-downie
URL : https://hal.archives-ouvertes.fr/inria-00544675

Z. Duan, B. Pardo, and C. Zhang, Multiple fundamental frequency estimation by modeling spectral peaks and non-peak regions, IEEE Transactions on Audio, Speech, and Language Processing, issue.8, pp.182121-2133, 2010.

J. Durrieu, B. David, R. , and G. , A Musically Motivated Mid-Level Representation for Pitch Estimation and Musical Audio Source Separation, IEEE Journal of Selected Topics in Signal Processing, vol.5, issue.6, pp.1180-1191, 2011.
DOI : 10.1109/JSTSP.2011.2158801

J. Durrieu, G. Richard, B. David, and C. Févotte, Source/Filter Model for Unsupervised Main Melody Extraction From Polyphonic Audio Signals, IEEE Transactions on Audio, Speech, and Language Processing, vol.18, issue.3, pp.564-575, 2010.
DOI : 10.1109/TASL.2010.2041114

K. Ege, La table d'harmonie du piano. Etudes modales en basses et moyennes fréquences, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00994586

K. Ege, X. Boutillon, D. , and B. , High-resolution modal analysis, Journal of Sound and Vibration, vol.325, pp.4-5852, 2009.
DOI : 10.1016/j.jsv.2009.04.019
URL : https://hal.archives-ouvertes.fr/hal-00413250

J. Eggert and E. Körner, Sparse coding and NMF, 2004 IEEE International Joint Conference on Neural Networks (IEEE Cat. No.04CH37541), pp.2529-2533, 2004.
DOI : 10.1109/IJCNN.2004.1381036

B. Elie, F. Gautier, D. , and B. , Estimation of mechanical properties of panels based on modal density and mean mobility measurements, Mechanical Systems and Signal Processing, vol.40, issue.2, pp.40628-644, 2013.
DOI : 10.1016/j.ymssp.2013.06.009

V. Emiya, Transcription automatique de la musique de piano, 2008.
URL : https://hal.archives-ouvertes.fr/pastel-00004867

V. Emiya, R. Badeau, D. , and B. , Multipitch estimation of quasi-harmonic sounds in colored noise, Proceedings of the 10th International Conference on Digital Audio Effects (DAFx), pp.93-98, 2007.
URL : https://hal.archives-ouvertes.fr/inria-00545615

V. Emiya, R. Badeau, D. , and B. , Multipitch Estimation of Piano Sounds Using a New Probabilistic Spectral Smoothness Principle, IEEE Transactions on Audio, Speech, and Language Processing, vol.18, issue.6, pp.1643-1654, 2010.
DOI : 10.1109/TASL.2009.2038819
URL : https://hal.archives-ouvertes.fr/inria-00510392

V. Emiya, N. Bertin, B. David, and R. Badeau, MAPS -A piano database for multipitch estimation and automatic transcription of music, 2010.
URL : https://hal.archives-ouvertes.fr/inria-00544155

V. Emiya, B. David, and R. Badeau, A Parametric Method for Pitch Estimation of Piano Tones, 2007 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP '07, pp.249-252, 2007.
DOI : 10.1109/ICASSP.2007.366663
URL : https://hal.archives-ouvertes.fr/inria-00544147

J. Engelbrecht, A. Mägi, S. , and A. , Grand piano manufacturing in Estonia: the problem of piano scaling, Proceedings of the Estonian Academy of Sciences, vol.5, issue.2, pp.155-167, 1999.

S. Essid, A single-class SVM based algorithm for computing an identifiable NMF, 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp.2053-2056, 2012.
DOI : 10.1109/ICASSP.2012.6288313

S. Ewert and M. Müller, Using score-informed constraints for NMF-based source separation, 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp.129-132, 2012.
DOI : 10.1109/ICASSP.2012.6287834

S. Ewert, M. Müller, and M. Sandler, Efficient data adaption for musical source separation methods based on parametric models, 2013 IEEE International Conference on Acoustics, Speech and Signal Processing, pp.46-50, 2013.
DOI : 10.1109/ICASSP.2013.6637606

G. Fant, Acoustic Theory of Speech Production, 1960.
DOI : 10.1515/9783110873429

J. A. Fessler and A. O. Hero, Space-alternating generalized expectation-maximization algorithm, IEEE Transactions on Signal Processing, vol.42, issue.10, pp.422664-2677, 2010.
DOI : 10.1109/78.324732
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.50.701

C. Févotte, N. Bertin, and J. Durrieu, Nonnegative Matrix Factorization with the Itakura-Saito Divergence: With Application to Music Analysis, Neural Computation, vol.14, issue.3, pp.793-830, 2009.
DOI : 10.1016/j.sigpro.2007.01.024

C. Févotte and A. T. Cemgil, Nonnegative matrix factorization as probabilistic inference in composite models, Proceedings of the 17th European Signal Processing Conference (EUSIPCO), pp.1913-1917, 2009.

C. Févotte and J. Idier, Algorithms for Nonnegative Matrix Factorization with the ??-Divergence, Neural Computation, vol.11, issue.9, pp.2421-2456, 2011.
DOI : 10.1109/TASL.2009.2034186

D. Fitzgerald, M. Cranitch, and E. Coyle, On the use of the beta divergence for musical source separation, IET Irish Signals and Systems Conference (ISSC 2009), pp.1-6, 2009.
DOI : 10.1049/cp.2009.1711

J. L. Flanagan and R. M. Golden, Phase Vocoder, Bell System Technical Journal, vol.45, issue.9, pp.1493-1509, 1966.
DOI : 10.1002/j.1538-7305.1966.tb01706.x

H. Fletcher, Normal Vibration Frequencies of a Stiff Piano String, The Journal of the Acoustical Society of America, vol.36, issue.1, pp.203-209, 1964.
DOI : 10.1121/1.1918933

H. Fletcher, E. D. Blackham, and R. Stratton, Quality of Piano Tones, The Journal of the Acoustical Society of America, vol.34, issue.6, pp.749-761, 1962.
DOI : 10.1121/1.1918192

N. H. Fletcher and T. D. Rossing, The physics of musical instruments, 1998.

J. Fritsch and M. D. Plumbey, Score informed audio source separation using constrained nonnegative matrix factorization and score synthesis, 2013 IEEE International Conference on Acoustics, Speech and Signal Processing, pp.888-891, 2013.
DOI : 10.1109/ICASSP.2013.6637776

A. Galembo and A. Askenfelt, Signal representation and estimation of spectral parameters by inharmonic comb filters with application to the piano, IEEE Transactions on Speech and Audio Processing, vol.7, issue.2, pp.197-203, 1999.
DOI : 10.1109/89.748124

A. Gelman, J. B. Carlin, H. S. Stern, R. , and D. B. , Bayesian Data Analysis, Chapman & Hall/CRC Texts in Statistical Science, 2003.

N. Giordano, Mechanical impedance of a piano soundboard, The Journal of the Acoustical Society of America, vol.103, issue.4, pp.2128-2133, 1998.
DOI : 10.1121/1.421358

S. Godsill and M. Davy, Bayesian computational models for inharmonicity in musical instruments, IEEE Workshop on Applications of Signal Processing to Audio and Acoustics, 2005., pp.283-286, 2005.
DOI : 10.1109/ASPAA.2005.1540225

E. Gómez, Tonal Description of Polyphonic Audio for Music Content Processing, INFORMS Journal on Computing, vol.18, issue.3, pp.294-304, 2006.
DOI : 10.1287/ijoc.1040.0126

M. Goto, T. Nishimura, H. Hashiguchi, and R. Oka, RWC music database: Music genre database and musical instrument sound database, Proceedings of the 4th International Society for Music Information Retrieval conference (ISMIR), pp.229-230, 2003.

C. E. Gough, The theory of string resonances on musical instruments, Acta Acustica with Acustica, vol.49, issue.2, pp.124-141, 1981.

R. Gribonval and E. Bacry, Harmonic decomposition of audio signals with matching pursuit, IEEE Transactions on Signal Processing, vol.51, issue.1, pp.101-111, 2003.
DOI : 10.1109/TSP.2002.806592
URL : https://hal.archives-ouvertes.fr/inria-00576203

E. Hayashi, M. Yamame, and H. Mori, Behavior of piano-action in a grand piano. I. Analysis of the motion of the hammer prior to string contact, The Journal of the Acoustical Society of America, vol.105, issue.6, pp.3534-3544, 1999.
DOI : 10.1121/1.424678

H. V. Helmholtz, On the Sensation of Tone As a Physiological Basis for the Theory of Music, 1863.

R. Hennequin, Décomposition de spectrogrammes musicaux informée par des modèles de synthèse spectrale. Modélisation des variations temporelles dans les éléments sonores, 2011.

R. Hennequin, R. Badeau, D. , and B. , Time-dependent parametric and harmonic templates in non-negative matrix factorization, Proceedings of the 13th International Conference on Digital Audio Effects (DAFx-10), pp.246-253, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00945292

R. Hennequin, R. Badeau, D. , and B. , NMF with time-frequency activations to model non stationary audio events, 2010 IEEE International Conference on Acoustics, Speech and Signal Processing, pp.744-753, 2011.
DOI : 10.1109/ICASSP.2010.5495733
URL : https://hal.archives-ouvertes.fr/hal-00945293

R. Hennequin, R. Badeau, D. , and B. , Score informed audio source separation using a parametric model of non-negative spectrogram, 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2011.
DOI : 10.1109/ICASSP.2011.5946324
URL : https://hal.archives-ouvertes.fr/hal-00945294

H. Hinrichsen, Entropy-based tuning of musical instruments, Revista Brasileira de Ensino de F??sica, vol.34, issue.2, pp.2301-2302, 2012.
DOI : 10.1590/S1806-11172012000200004

M. Hirschkorn, J. Mcphee, and S. Birkett, Dynamic Modeling and Experimental Testing of a Piano Action Mechanism, Journal of Computational and Nonlinear Dynamics, vol.1, issue.1, pp.47-55, 2006.
DOI : 10.1115/1.1951782

M. Hodgkinson, J. Wang, J. Timoney, and V. Lazzarini, Handling inharmonic series with median-adjustive trajectories, Proceedings of the 12th International Conference on Digital Audio Effects (DAFx-09), pp.471-477, 2009.

H. Hotelling, Analysis of a complex of statistical variables into principal components, Journal of Educational Psychology, vol.24, issue.6, pp.471-441, 1933.

P. O. Hoyer, Non-negative matrix factorization with sparseness constraints, Journal of Machine Learning Research, vol.5, pp.1457-1469, 2004.

F. Itakura and S. Saito, Analysis-synthesis telephony based upon the maximum likelihood method, Proceedings of the 6th International Congress on Acoustics, pp.17-20, 1968.

H. Järveläinen, V. Välimäki, and M. Karjalainen, Audibility of the timbral effects of inharmonicity in stringed instrument tones, Acoustics Research Letters Online, vol.2, issue.3, pp.79-84, 2001.
DOI : 10.1121/1.1374756

H. Järveläinen, T. Verma, and V. Välimäki, Perception and Adjustment of Pitch in Inharmonic String Instrument Tones, Journal of New Music Research, vol.31, issue.4, pp.311-319, 2002.
DOI : 10.1076/jnmr.31.4.311.14167

C. Joder, F. Weninger, D. Virette, and B. Schuller, A comparative study on sparsity penalties for NMF-based speech separation: Beyond LP-norms, 2013 IEEE International Conference on Acoustics, Speech and Signal Processing, pp.858-862, 2013.
DOI : 10.1109/ICASSP.2013.6637770

F. Kaiser and T. Sikora, Music structure discovery in popular music using nonnegative matrix factorization, Proceedings of the 11th International Society for Music Information Retrieval conference (ISMIR), pp.429-434, 2010.

H. Kameoka, T. Nishimoto, and S. Sagayama, Multi-pitch detection algorithm using constrained gaussian mixture model and information criterion for simultaneous speech, Proceedings of Speech Prosody (SP2004), pp.533-536, 2004.

A. Klapuri and M. Davy, Signal Processing Methods for Music Transcription, 2006.
DOI : 10.1007/0-387-32845-9
URL : https://hal.archives-ouvertes.fr/inria-00120226

A. P. Klapuri, Multipitch estimation and sound separation by the spectral smoothness principle, 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.01CH37221), pp.3381-3384, 2001.
DOI : 10.1109/ICASSP.2001.940384

B. Klingenberg, J. Curry, and A. Dougherty, Non-negative matrix factorization: Ill-posedness and a geometric algorithm, Pattern Recognition, vol.42, issue.5, pp.918-928, 2009.
DOI : 10.1016/j.patcog.2008.08.026

A. Kobzantsev, D. Chazan, and Y. Zeevi, Automatic transcription of piano polyphonic music, ISPA 2005. Proceedings of the 4th International Symposium on Image and Signal Processing and Analysis, 2005., pp.414-418, 2005.
DOI : 10.1109/ISPA.2005.195447

R. Kompass, A Generalized Divergence Measure for Nonnegative Matrix Factorization, Neural Computation, vol.39, issue.3, pp.780-791, 2007.
DOI : 10.1162/089976602320264033

S. Kullback and R. A. Leibler, On Information and Sufficiency, The Annals of Mathematical Statistics, vol.22, issue.1, pp.79-86, 1951.
DOI : 10.1214/aoms/1177729694

H. Laurberg, M. G. Christensen, M. D. Plumbley, L. K. Hansen, and S. H. Jensen, Theorems on Positive Data: On the Uniqueness of NMF, Computational Intelligence and Neuroscience, vol.5, pp.1-9, 2008.
DOI : 10.1109/LSP.2002.800502

L. Carrou, J. L. Gautier, F. Dauchez, N. , G. et al., Modelling of sympathetic string vibrations, Acta Acustica united with Acustica, vol.91, issue.2, pp.277-288, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00474982

L. Roux, J. Kameoka, H. Ono, N. De-cheveigné, A. Sagayama et al., Computational auditory induction by missing-data non-negative matrix factorization, Speech Communication. Special issue on Perceptual and Statistical Audition, issue.5, pp.53658-676, 2011.

D. D. Lee and H. S. Seung, Learning the parts of objects by non-negative matrix factorization, Nature, vol.401, pp.788-791, 1999.

D. D. Lee and H. S. Seung, Algorithms for non-negative matrix factorization, Advances in Neural Information Processing Systems (NIPS), pp.556-562, 2000.

H. Lehtonen, H. Penttinen, J. Rauhala, and V. Välimäki, Analysis and modeling of piano sustain-pedal effects, The Journal of the Acoustical Society of America, vol.122, issue.3, pp.1787-1797, 2007.
DOI : 10.1121/1.2756172

P. Leveau, E. Vincent, G. Richard, and L. Daudet, Instrument-Specific Harmonic Atoms for Mid-Level Music Representation, IEEE Transactions on Audio, Speech, and Language Processing, vol.16, issue.1, pp.116-128, 2008.
DOI : 10.1109/TASL.2007.910786
URL : https://hal.archives-ouvertes.fr/inria-00544175

C. Lin, Projected Gradient Methods for Nonnegative Matrix Factorization, Neural Computation, vol.5, issue.10, pp.2756-2779, 2007.
DOI : 10.1007/BF01584660

W. Liu and K. Yuan, Sparse p-norm Nonnegative Matrix Factorization for clustering gene expression data, International Journal of Data Mining and Bioinformatics, vol.2, issue.3, pp.236-249, 2008.
DOI : 10.1504/IJDMB.2008.020524

A. Liutkus, R. Badeau, R. , and G. , Gaussian Processes for Underdetermined Source Separation, IEEE Transactions on Signal Processing, vol.59, issue.7, pp.3155-3167, 2011.
DOI : 10.1109/TSP.2011.2119315
URL : https://hal.archives-ouvertes.fr/hal-00643951

S. Mallat, A Wavelet Tour of Signal Processing, Third Edition: The Sparce Way, 2008.

S. Mallat and Z. Zhang, Matching pursuits with time-frequency dictionaries, IEEE Transactions on Signal Processing, vol.41, issue.12, pp.413397-3415, 1993.
DOI : 10.1109/78.258082
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.335.5769

A. Mamou-mani, J. Frelat, and C. Besnainou, Numerical simulation of a piano soundboard under downbearing, The Journal of the Acoustical Society of America, vol.123, issue.4, pp.2401-2406, 2008.
DOI : 10.1121/1.2836787

M. Marolt, A Connectionist Approach to Automatic Transcription of Polyphonic Piano Music, IEEE Transactions on Multimedia, vol.6, issue.3, pp.439-449, 2004.
DOI : 10.1109/TMM.2004.827507

D. W. Martin and W. D. Ward, Subjective Evaluation of Musical Scale Temperament in Pianos, The Journal of the Acoustical Society of America, vol.33, issue.5, pp.582-585, 1961.
DOI : 10.1121/1.1908730

R. J. Mcaulay and T. F. Quatieri, Speech analysis/Synthesis based on a sinusoidal representation, IEEE Transactions on Acoustics, Speech, and Signal Processing, vol.34, issue.4, pp.744-754, 1986.
DOI : 10.1109/TASSP.1986.1164910

G. Monti and M. Sandler, Automatic polyphonic piano note extraction using fuzzy logic in a blackboard system, Proceedings of the 5th International Conference on Digital Audio Effects (DAFx), pp.39-44, 2002.

B. C. Moore, B. R. Glasberg, and R. W. Peters, Relative dominance of individual partials in determining the pitch of complex tones, The Journal of the Acoustical Society of America, vol.77, issue.5, pp.1853-1860, 1984.
DOI : 10.1121/1.391936

P. M. Morse, Vibration and Sound, 1948.

G. J. Mysore, P. Smaragdis, R. , and B. , Non-negative Hidden Markov Modeling of Audio with Application to Source Separation, Proceedings of the 9th International Conference on Latent Variable Analysis and Signal Separation, pp.140-148, 2010.
DOI : 10.1007/978-3-642-15995-4_18

B. Niedermayer, Non-negative matrix division for the automatic transcription of polyphonic music, Proceedings of the 9th International Society for Music Information Retrieval conference (ISMIR), pp.544-549, 2008.

K. Ochiai, H. Kameoka, and S. Sagayama, Explicit beat structure modeling for non-negative matrix factorization-based multipitch analysis, 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp.133-136, 2012.
DOI : 10.1109/ICASSP.2012.6287835

L. I. Ortiz-berenguer, F. J. Casajús-quirós, M. Torres-guijarro, and J. A. Beracoechea, Piano transcription using pattern recognition: aspects on parameter extraction, Proceedings of the 7th International Conference on Digital Audio Effects, pp.212-216, 2004.

L. I. Ortiz-berenguer and F. J. Casajús-quirós, Polyphonic transcription using piano modeling for spectral pattern recognition, Proceedings of the 5th International Conference on Digital Audio Effects (DAFx-02), pp.45-50, 2002.

A. Ozerov and C. Févotte, Multichannel Nonnegative Matrix Factorization in Convolutive Mixtures for Audio Source Separation, IEEE Transactions on Audio, Speech, and Language Processing, vol.18, issue.3, pp.550-563, 2010.
DOI : 10.1109/TASL.2009.2031510

P. Paatero and U. Tapper, Positive matrix factorization: A non-negative factor model with optimal utilization of error estimates of data values, Environmetrics, vol.18, issue.2, pp.111-126, 1994.
DOI : 10.1002/env.3170050203

R. M. Parry and I. Essa, Incorporating Phase Information for Source Separation via Spectrogram Factorization, 2007 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP '07, pp.661-664, 2007.
DOI : 10.1109/ICASSP.2007.366322

R. M. Parry and I. Essa, Phase-Aware Non-negative Spectrogram Factorization, Proceedings of the 7th International Conference on Independent Component Analysis and Signal Separation, pp.536-543, 2007.
DOI : 10.1007/978-3-540-74494-8_67

M. Parvaix and L. Girin, Informed Source Separation of Linear Instantaneous Under-Determined Audio Mixtures by Source Index Embedding, IEEE Transactions on Audio, Speech, and Language Processing, vol.19, issue.6, pp.1721-1733, 2011.
DOI : 10.1109/TASL.2010.2097250
URL : https://hal.archives-ouvertes.fr/hal-00695763

V. P. Pauca, F. Shahnaz, M. W. Berry, and R. J. Plemmons, Text Mining using Non-Negative Matrix Factorizations, Proceedings of the 4th SIAM International Conference on Data Mining, pp.452-456, 2004.
DOI : 10.1137/1.9781611972740.45

J. Paulus and T. Virtanen, Drum transcription with non-negative spectrogram factorization, Proceedings of the 13th European Signal Processing Conference, 2005.

R. Plomp, Pitch of Complex Tones, The Journal of the Acoustical Society of America, vol.41, issue.6, pp.1526-1533, 1967.
DOI : 10.1121/1.1910515

G. E. Poliner and D. Ellis, A Discriminative Model for Polyphonic Piano Transcription, EURASIP Journal on Advances in Signal Processing, vol.2007, issue.1, p.2007, 2007.
DOI : 10.1162/089976601750264965

B. Raj, T. Virtanen, S. Chaudhuri, and R. Singh, Non-negative matrix factorization based compensation of music for automatic speech recognition, Proceeding of the 11th Annual Conference of the International Speech Communication Association (INTERSPEECH), pp.717-720, 2010.

R. A. Rasch and V. Heetvelt, String Inharmonicity and Piano Tuning, Music Perception: An Interdisciplinary Journal, vol.3, issue.2, pp.171-190, 1985.
DOI : 10.2307/40285331

J. Rauhala, H. M. Lehtonen, and V. Välimäki, Fast automatic inharmonicity estimation algorithm, The Journal of the Acoustical Society of America, vol.121, issue.5, pp.184-189, 2007.
DOI : 10.1121/1.2719043.1

J. Rauhala, H. M. Lehtonen, and V. Välimäki, Toward Next-Generation Digital Keyboard Instruments, IEEE Signal Processing Magazine, vol.24, issue.2, pp.12-20, 2007.
DOI : 10.1109/MSP.2007.323260

J. Rauhala and V. Välimäki, F0 estimation of inharmonic piano tones using partial frequencies deviation method, Proceedings of the International Computer Music Conference (ICMC'07), pp.453-456, 2007.

C. Rijsbergen, Information Retrieval, Butterworths, 1979.

R. J. Ritsma, Frequencies Dominant in the Perception of the Pitch of Complex Sounds, The Journal of the Acoustical Society of America, vol.42, issue.1, pp.191-198, 1967.
DOI : 10.1121/1.1910550

C. Roads, L'audionumérique. Musique et informatique (The Computer Music Tutorial). Dunod, 2007.

T. D. Rossing and N. H. Fletcher, Principles of vibration and sound, 1995.

X. Serra and J. Smith, Spectral Modeling Synthesis: A Sound Analysis/Synthesis System Based on a Deterministic Plus Stochastic Decomposition, Computer Music Journal, vol.14, issue.4, pp.12-24, 1990.
DOI : 10.2307/3680788

P. Smaragdis and J. C. Brown, Non-negative matrix factorization for polyphonic music transcription, 2003 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics (IEEE Cat. No.03TH8684), pp.177-180, 2003.
DOI : 10.1109/ASPAA.2003.1285860
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.475.7518

P. Smaragdis, B. Raj, and M. Shashanka, Sparse and shift-invariant feature extraction from non-negative data, 2008 IEEE International Conference on Acoustics, Speech and Signal Processing, pp.2069-2072, 2008.
DOI : 10.1109/ICASSP.2008.4518048

A. Stulov, Hysteretic model of the grand piano hammer felt, The Journal of the Acoustical Society of America, vol.97, issue.4, pp.2577-2585, 1995.
DOI : 10.1121/1.411912

A. Stulov, Physical modelling of the piano string scale, Applied Acoustics, vol.69, issue.11, pp.69977-984, 2008.
DOI : 10.1016/j.apacoust.2007.07.008

H. Suzuki, Model analysis of a hammer-string interaction, Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, pp.1285-1288, 1986.

H. Suzuki, Vibration and sound radiation of a piano soundboard, The Journal of the Acoustical Society of America, vol.80, issue.6, pp.1573-1582, 1986.
DOI : 10.1121/1.394321

C. Valette and C. Cuesta, Mécanique de la corde vibrante, 1993.

E. Vincent, N. Bertin, and R. Badeau, Harmonic and inharmonic Nonnegative Matrix Factorization for Polyphonic Pitch transcription, 2008 IEEE International Conference on Acoustics, Speech and Signal Processing, pp.109-112, 2008.
DOI : 10.1109/ICASSP.2008.4517558
URL : https://hal.archives-ouvertes.fr/inria-00544183

E. Vincent, N. Bertin, and R. Badeau, Adaptive Harmonic Spectral Decomposition for Multiple Pitch Estimation, IEEE Transactions on Audio, Speech, and Language Processing, vol.18, issue.3, pp.528-537, 2010.
DOI : 10.1109/TASL.2009.2034186
URL : https://hal.archives-ouvertes.fr/inria-00544094

E. Vincent and M. D. Plumbey, Low Bit-Rate Object Coding of Musical Audio Using Bayesian Harmonic Models, IEEE Transactions on Audio, Speech and Language Processing, vol.15, issue.4, pp.1273-1282, 2007.
DOI : 10.1109/TASL.2006.889792

T. Virtanen, Monaural Sound Source Separation by Nonnegative Matrix Factorization With Temporal Continuity and Sparseness Criteria, IEEE Transactions on Audio, Speech and Language Processing, vol.15, issue.3, pp.1066-1074, 2007.
DOI : 10.1109/TASL.2006.885253

T. Virtanen, T. Cemgil, and S. Godsill, Bayesian extensions to non-negative matrix factorisation for audio signal modelling, 2008 IEEE International Conference on Acoustics, Speech and Signal Processing, pp.1825-1828, 2008.
DOI : 10.1109/ICASSP.2008.4517987

T. Virtanen and A. Klapuri, Separation of harmonic sounds using linear models for the overtone series, Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), pp.1757-1760, 2002.

W. Wang and X. Zou, Non-negative matrix factorization based on projected nonlinear conjugate gradient algorithm, Proceedings of ICA Research Network International Workshop (ICARN), pp.5-8, 2008.

G. Weinreich, Coupled piano strings, The Journal of the Acoustical Society of America, vol.63, issue.S1, pp.1474-1484, 1977.
DOI : 10.1121/1.2016722

K. W. Wilson, B. Rah, P. Smaragdis, and A. Divakaran, Speech denoising using nonnegative matrix factorization with priors, 2008 IEEE International Conference on Acoustics, Speech and Signal Processing, pp.4029-4032, 2008.
DOI : 10.1109/ICASSP.2008.4518538

C. Yeh and A. Röbel, Adaptive noise level estimation, Proceedings of the 9th International Conference on Digital Audio Effects (DAFx-06), pp.145-148, 2006.
URL : https://hal.archives-ouvertes.fr/hal-01161362

T. Yoshioka and D. Sakaue, Log-normal matrix factorization with application to speech-music separation, Proceedings of the 2012 Workshop on Statistical and Perceptual Audition Organized Jointly with the Speech Communication with Adaptive Learning Consortium (SAPA-SCALE 2012), pp.80-85, 2012.

R. W. Young, Inharmonicity of Plain Wire Piano Strings, The Journal of the Acoustical Society of America, vol.24, issue.3, pp.267-273, 1952.
DOI : 10.1121/1.1906888

R. Zdunek and A. Cichocki, Nonnegative matrix factorization with constrained second-order optimization, Signal Processing, vol.87, issue.8, pp.1904-1916, 2007.
DOI : 10.1016/j.sigpro.2007.01.024

Y. Zhang and Y. Fang, A NMF algorithm for blind separation of uncorrelated signals, 2007 International Conference on Wavelet Analysis and Pattern Recognition, pp.999-1003, 2007.
DOI : 10.1109/ICWAPR.2007.4421577