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A brief review is given of psychological results demonstrating that these music-theoretic definitions describe implicit knowledge that listeners have of tonal-harmonic music. This is followed by a brief historical summary of maps that were proposed previously of keys, tones, and chords, and a hybrid tone-chord map. The article then summarizes a number of issues that arise in connection with geometric representations. Finally, the innovations in the three proposed models are highlighted, particularly their contributions to visualizing the dynamics and temporal relationships that exist in extended musical pieces.<\/jats:p>","DOI":"10.1145\/1095534.1095542","type":"journal-article","created":{"date-parts":[[2005,11,7]],"date-time":"2005-11-07T16:00:45Z","timestamp":1131379245000},"page":"1-14","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":6,"title":["The geometry of musical structure"],"prefix":"10.1145","volume":"3","author":[{"given":"Carol L.","family":"Krumhansl","sequence":"first","affiliation":[{"name":"Cornell University, Ithaca, NY"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2005,10]]},"reference":[{"key":"e_1_2_1_1_1","unstructured":"Chew E. 2000. Towards a Mathematical Model of Tonality. Ph.D. dissertation Operations Research Center MIT. Cambridge MA.  Chew E. 2000. 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(Revised edition originally published in 1885)."},{"key":"e_1_2_1_6_1","volume-title":"The New Grove Dictionary of Music and Musicians","author":"Hyer B."},{"key":"e_1_2_1_7_1","unstructured":"Journal of Music Theory. Special Issue on Neo-Riemannian Theory 42 2 (1998).  Journal of Music Theory. Special Issue on Neo-Riemannian Theory 42 2 (1998)."},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1016\/0010-0285(79)90016-1"},{"key":"e_1_2_1_9_1","unstructured":"Krumhansl C. L. 1990. Cognitive Foundations of Musical Pitch. Oxford University Press New York.  Krumhansl C. L. 1990. Cognitive Foundations of Musical Pitch. 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MIT Press Cambridge MA."},{"key":"e_1_2_1_17_1","first-page":"221","article-title":"On interpreting Bach","volume":"6","author":"Longuet","year":"1971","journal-title":"Machine Intelligence"},{"key":"e_1_2_1_18_1","unstructured":"Mattheson J. 1980. Kleine General-Bass-Schule. Laaber-Verlag (Originally published in 1735).  Mattheson J. 1980. Kleine General-Bass-Schule. Laaber-Verlag (Originally published in 1735)."},{"key":"e_1_2_1_19_1","unstructured":"Rasch R. 1983. Three Musical Circles for Keyboard. The Diapason Press Utrecht The Netherlands.  Rasch R. 1983. Three Musical Circles for Keyboard. The Diapason Press Utrecht The Netherlands."},{"key":"e_1_2_1_20_1","unstructured":"Schoenberg A. 1969. Structural Functions of Harmony. (Rev. ed.). Norton New York (Originally published in 1954).  Schoenberg A. 1969. Structural Functions of Harmony. (Rev. ed.). 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