LEADER 04264nam 22005655 450 001 9911007483103321 005 20250601130241.0 010 $a3-031-82236-6 024 7 $a10.1007/978-3-031-82236-0 035 $a(CKB)39160567800041 035 $a(MiAaPQ)EBC32142873 035 $a(Au-PeEL)EBL32142873 035 $a(DE-He213)978-3-031-82236-0 035 $a(EXLCZ)9939160567800041 100 $a20250601d2025 u| 0 101 0 $aeng 135 $aur||||||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aPersistent Homology and Discrete Fourier Transform $eAn Application to Topological Musical Data Analysis /$fby Victoria Callet-Feltz 205 $a1st ed. 2025. 210 1$aCham :$cSpringer Nature Switzerland :$cImprint: Springer,$d2025. 215 $a1 online resource (312 pages) 225 1 $aComputational Music Science,$x1868-0313 311 08$a3-031-82235-8 327 $a1 Introduction -- Part I The two-dimensional Discrete Fourier Transform -- 2 The DFT for modeling basic musical structures -- 3 Generalization of theoretical results -- Part II Persistent homology on musical bars -- 4 Mathematical background -- 5 Musical scores and filtration -- Part III Musical applications -- 6 The DFT as a metric on the set of notes and chord -- 7 Harmonization of Pop songs -- 8 Classification of musical style -- 9 A different approach: the Hausdorff distance -- 10 Conclusion and perspectives for future research. 330 $aThis book proposes contributions to various problems in the field of topological analysis of musical data: the objects studied are scores represented symbolically by MIDI files, and the tools used are the discrete Fourier transform and persistent homology. The manuscript is divided into three parts: the first two are devoted to the study of the aforementioned mathematical objects and the implementation of the model. More precisely, the notion of DFT introduced by Lewin is generalized to the case of dimension two, by making explicit the passage of a musical bar from a piece to a subset of Z/tZ×Z/pZ, which leads naturally to a notion of metric on the set of musical bars by their Fourier coefficients. This construction gives rise to a point cloud, to which the filtered Vietoris-Rips complex is associated, and consequently a family of barcodes given by persistent homology. This approach also makes it possible to generalize classical results such as Lewin's lemma and Babitt's Hexachord theorem. The last part of this book is devoted to musical applications of the model: the first experiment consists in extracting barcodes from artificially constructed scores, such as scales or chords. This study leads naturally to song harmonization process, which reduces a song to its melody and chord grid, thus defining the notions of graph and complexity of a piece. Persistent homology also lends itself to the problem of automatic classification of musical style, which will be treated here under the prism of symbolic descriptors given by statistics calculated directly on barcodes. Finally, the last application proposes a encoding of musical bars based on the Hausdorff distance, which leads to the study of musical textures. The book is addressed to graduate students and researchers in mathematical music theory and music information research, but also at researchers in other fields, such as applied mathematicians and topologists, who want to learn more about mathematical music theory or music information research. 410 0$aComputational Music Science,$x1868-0313 606 $aMusic$xMathematics 606 $aMathematics 606 $aMusic 606 $aMathematics in Music 606 $aApplications of Mathematics 606 $aMusic 615 0$aMusic$xMathematics. 615 0$aMathematics. 615 0$aMusic. 615 14$aMathematics in Music. 615 24$aApplications of Mathematics. 615 24$aMusic. 676 $a780.0519 700 $aCallet-Feltz$b Victoria$01826786 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9911007483103321 996 $aPersistent Homology and Discrete Fourier Transform$94394783 997 $aUNINA