{"title":"为什么拓扑?","authors":"Dmitri Tymoczko","doi":"10.1080/17459737.2020.1799563","DOIUrl":null,"url":null,"abstract":"Music theorists have modeled voice leadings as paths through higher-dimensional configuration spaces. This paper uses topological techniques to construct two-dimensional diagrams capturing these spaces’ most important features. The goal is to enrich set theory’s contrapuntal power by simplifying the description of its geometry. Along the way, I connect homotopy theory to “transformational theory,” show how set-class space generalizes the neo-Riemannian transformations, extend the Tonnetz to arbitrary chords, and develop a simple contrapuntal “alphabet” for describing voice leadings. I mention several compositional applications and analyze short excerpts from Gesualdo, Mozart, Wagner, Stravinsky, Schoenberg, Schnittke, and Mahanthappa.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":"13 1","pages":"114 - 169"},"PeriodicalIF":0.5000,"publicationDate":"2020-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"6","resultStr":"{\"title\":\"Why topology?\",\"authors\":\"Dmitri Tymoczko\",\"doi\":\"10.1080/17459737.2020.1799563\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Music theorists have modeled voice leadings as paths through higher-dimensional configuration spaces. This paper uses topological techniques to construct two-dimensional diagrams capturing these spaces’ most important features. The goal is to enrich set theory’s contrapuntal power by simplifying the description of its geometry. Along the way, I connect homotopy theory to “transformational theory,” show how set-class space generalizes the neo-Riemannian transformations, extend the Tonnetz to arbitrary chords, and develop a simple contrapuntal “alphabet” for describing voice leadings. I mention several compositional applications and analyze short excerpts from Gesualdo, Mozart, Wagner, Stravinsky, Schoenberg, Schnittke, and Mahanthappa.\",\"PeriodicalId\":50138,\"journal\":{\"name\":\"Journal of Mathematics and Music\",\"volume\":\"13 1\",\"pages\":\"114 - 169\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2020-05-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"6\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Mathematics and Music\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1080/17459737.2020.1799563\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematics and Music","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1080/17459737.2020.1799563","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
Music theorists have modeled voice leadings as paths through higher-dimensional configuration spaces. This paper uses topological techniques to construct two-dimensional diagrams capturing these spaces’ most important features. The goal is to enrich set theory’s contrapuntal power by simplifying the description of its geometry. Along the way, I connect homotopy theory to “transformational theory,” show how set-class space generalizes the neo-Riemannian transformations, extend the Tonnetz to arbitrary chords, and develop a simple contrapuntal “alphabet” for describing voice leadings. I mention several compositional applications and analyze short excerpts from Gesualdo, Mozart, Wagner, Stravinsky, Schoenberg, Schnittke, and Mahanthappa.
期刊介绍:
Journal of Mathematics and Music aims to advance the use of mathematical modelling and computation in music theory. The Journal focuses on mathematical approaches to musical structures and processes, including mathematical investigations into music-theoretic or compositional issues as well as mathematically motivated analyses of musical works or performances. In consideration of the deep unsolved ontological and epistemological questions concerning knowledge about music, the Journal is open to a broad array of methodologies and topics, particularly those outside of established research fields such as acoustics, sound engineering, auditory perception, linguistics etc.