Profile

Research Interest

Mathematical Music Theory. This research involves several mathematical areas used in scientific Musicology, such as Category Theory, Abstract Algebra, Algebraic Combinatorics on Words. It is cross disciplinary not only with Music, but with areas such as Computer Science and Cognitive Science.

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Teaching Interest

Bridge to higher Mathematics, Linear Algebra, Abstract Algebra, Multivariate Calculus, Topology.

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Ph.D. students

Rodrigo Castro López Vaal

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Selected Publications

(2013) Noll, T., Montiel, M. Glarean’s Dodecachordon Revisited. In Mathematics and Computation in Music J. Wild, J.Hust & J. A. Burgoyne (Eds.), pp. 151-166. Lecture Notes in Artificial Intelligence, Subseries of Lecture Notes in Computer Science, Springer Publishing Company. ISBN 978-3-642-39356-3

(2013) Norgaard, M., Spencer, J. & Montiel, M. Testing Cognitive Theories: a Probabilistic Based Algorithm for Melody and Rhythm in Jazz Improvisation. Psychomusicology: Music, Mind and Brain special edition on The Improvising Brain. Accepted for publication.

(2012) Aceff-Sánchez, F., Agustín, O., du Plessis, J., Lluis-Puebla, E. & Montiel, M. An Introduction to Group Theory with Applications to Mathematical Music Theory. Bookboon Ventus publishing Aps. ISBN: 978-87-403-0324-7. Online at:http://bookboon.com/en/textbooks/mathematics/an-introduction-to-group-theory

(2009) Agustín, O., du Plessis, J., Lluis-Puebla, E. & Montiel, M. (2009). Una Introducción a la Teoría de Grupos con Aplicaciones en la Teoría Matemática de la Música. Online publications of the Mexican Mathematical Society. ISBN: 968-9161-37-7 (online version); 968-9161-38-5 (CD version); 968-9161-36-9 (hard copy version).(textbook) http://www.smm.org.mx/smm/PETextos

(2004) Montiel, M. The Denotator: Its Structure, Construction, and Role in Mathematical Music Theory In Perspectives in Theoretical and Computational Music Theory, 89-105, Guerino Mazzola, Thomas Noll and Emilio Lluis-Puebla (Eds.), Epos, Universitat Osnabruck: Germany, ISBN:3-923486-57-x www.epos.uni-osnabrueck.de/music/books.

(2002) Montiel, M. Contributor to The Topos of Music: Geometric Logic of Concepts, Theory and Performance by Guerino Mazzola, Birkhauser Publishing Company, Berlin: Germany. ISBN-13: 978-3764357313 (book)

Teaching

Course Development

  • Development of a pilot course pairing: Mathematical Music Theory-Postonal Analysis (2014). These two courses are offered together to the same group of students, both mathematics/statistics and music majors, advanced undergraduate and graduate. This pilot is result of an award granted by the College of Arts and Sciences.
  • Development of a course on ‘Mathematical Music Theory’ (2014). The objectives include the use of specific aspects of Group Theory, Algebraic Combinatorics on Words, Topology, and Topos Theory in the analysis of general objects of Music (scales, chords, rhythmic patterns) as well as specific applications (development of software, analysis of pieces from different time periods and genres). The course was accepted by the Mathematic and Statistics Department at Georgia State University and will be part of the course offerings in the catalogue.
  • Development of a multidisciplinary course (mathematics and music students) on ‘Mathematical Music Theory’ as part of the Study Abroad with Peers program (2014). The course took place during Maymester 2014 at the East Chinese University of Science and Technology with Students from GSU and from ECUST.
  • Development and implementation of a mini- course (20 hours) on ‘Mathematical Music Theory’ (November 24- 29, 2013) for seniors and graduate students at the East Chinese University of Science and Technology.
  • Development and implementation of a workshop on modeling through Mathematical Music Theory for high school mathematics teachers. (June 2013).
  • Co-design of the doctoral course Learning Theories in Collegiate Mathematics Education (2010).
  • Online Linear Algebra course for "Georgia OnMyLine" (University System of Georgia)

Reading courses :

Linear Algebra videos, Math 4435/6435

Research & Publications

Books and Book Chapters :

  1. Aceff-Sánchez, F., Agustín, O., du Plessis, J., Lluis-Puebla, E. & Montiel, M. (2012). An Introduction to Group Theory with Applications to Mathematical Music Theory. Bookboon Ventus publishing Aps. ISBN: 978-87-403-0324-7. Online at: http://bookboon.com/en/textbooks/mathematics/an-introduction-to-group-theory
  2. Agustín, O., du Plessis, J., Lluis-Puebla, E. & Montiel, M. (2009). Una Introducción a la Teoría de Grupos con Aplicaciones en la Teoría Matemática de la Música. Online publications of the Mexican Mathematical Society. ISBN: 968-9161-37-7 (online version); 968-9161-38-5 (CD version); 968-9161-36-9 (hard copy version). (textbook) http://www.smm.org.mx/smm/PETextos
  3. Font, V., Montiel, M., Vidakovic, D. & Wilhelmi, M.R. (2011) Dimensional Analogy and Different Coordinate Systems through the Lens of Three Different Theoretical Perspectives. R. Nata (Ed). Progress in Education, Vol. 19, pp. 39-76.Nova Science Publishers, N.Y. (Book Chapter).
  4. Montiel, M. (2004) The Denotator: Its Structure, Construction, and Role in Mathematical Music Theory In Perspectives in Theoretical and Computational Music Theory, 89-105, Guerino Mazzola, Thomas Noll and Emilio Lluis-Puebla (Eds.), Epos, Universitat Osnabruck: Germany, ISBN:3-923486-57-x www.epos.uni-osnabrueck.de/music/books. (book invitation).
  5. Montiel, M. (2002) Contributor to The Topos of Music: Geometric Logic of Concepts, Theory and Performance by Guerino Mazzola, Birkhauser Publishing Company, Berlin: Germany. ISBN-13: 978-3764357313 (book)

Peer Reviewed Journal Articles :

  1. Montiel, M. & Gómez, F. (2014). Music in the Pedagogy of Mathematics. Journal of Mathematics and Music, Special Issue: Pedagogies of Mathematical Music Theory. Volume 8, Number 2, July 2014.
  2. (2013) Noll, T., Montiel, M. Glarean’s Dodecachordon Revisited. In Mathematics and Computation in Music J. Wild, J.Hust & J. A. Burgoyne (Eds.), pp. 151-166. Lecture Notes in Artificial Intelligence, Subseries of Lecture Notes in Computer Science, Springer Publishing Company. ISBN 978-3-642-39356-3
  3. Norgaard, M., Spencer, J. Montiel, M. (2013). Testing Cognitive Theories: a Probabilistic Based Algorithm for Melody and Rhythm in Jazz Improvisation. Psychomusicology: Music, Mind and Brain special edition on The Improvising Brain. December 2013, Volume 23, Number 4, pages 243-254.
  4. Montiel, M., Wilhelmi, M., Vidakovic, D. & Elstak, I. Vectors, Change of Basis and Matrix Representation: Onto-Semiotic Approach in the Analysis of Creating Meaning. International Journal of Mathematical Education in Science and Technology. Volume 43, Issue 1, January 2012, pages 11-32. Electronic reference: ISSN: 0020-739X , 1464-5211.
  5. Montiel, M., Wilhelmi, M., Vidakovic, D. & Elstak, I. (2011) Dimensional Analogy and Different Coordinate Systems, An Onto-Semiotic Approach. Mediterranean Journal for Research in Mathematics Education Vol 10, No. 1-2, pp. 131-168.
  6. Lacasta, E., Wilhelmi, M. & Montiel, M. (2010). Ostension et Rapport des Professeurs du Secondaire a la Limite de Fonctions. Quaderni di Ricerca in Didattica (Mathematics), n.1, N. 20, 305-328.
  7. Montiel, M., Bhatti, U. (June 15, 2010) Advanced Mathematics Online: Assessing Particularities in the Online Delivery of a Second Linear Algebra Course. Online Journal of Distance Learning Administration. http://www.westga.edu/~distance/ojdla/
  8. Montiel, M., Wilhelmi, M., Vidakovic, D. & Elstak, I. (Nov., 2009) Using the Onto-Semiotic Approach to Identify and Analyze Mathematical Meaning when Transiting between Different Coordinate Systems in a Multivariate Context, Educational Studies in Mathematics. Volume 72, Issue2, pp. 139-160. Electronic reference: Educ Stud Math. DOI 10.1007/s10649-009-9184-2.
  9. Montiel, M., Vidakovic, D., M.,Kabael, T. (2008). The Relationship Between Students' Understanding of Functions in Cartesian and Polar Coordinate Systems. Investigations in Mathematics Learning 1,2, 52-70.

Peer Reviewed Conference Proceedings :

  1. Norgaard, M., Montiel, M., Spencer, J. (2013) Chords not required: Incorporating horizontal and vertical aspects independently in a computer improvisation algorithm. International Symposium on Performance Science, to be held in Vienna, August 29-31 2013. Paper accepted.
  2. Spencer, Jonathan, Montiel, M., Norgaard, M. (2013) Testing Cognitive Theories by Creating a Probabilistic Based Algorithm for Melody and Rhythm in Jazz Improvisation. Poster presentation. Fourth International Conference on Mathematics and Computation in Music held at McGill University, Montreal, Canada, June 12-12 2103. Abstract published in J. Wild, J.Hust & J. A. Burgoyne (Eds.), pp. 151-166. Lecture Notes in Artificial Intelligence, Subseries of Lecture Notes in Computer Science, Springer Publishing Company. ISBN 978-3-642-39356-3
  3. Montiel, M. (2011) The RUBATO COMPOSER Software Concept for Learning Advanced Mathematics. Memoirs of the Fourth International Seminar on Mathematical Music Theory. Emilio Lluis-Puebla & Octavio Agustín-Aquino (Eds). Electronic Publications of the Mexican Mathematical Society. 77-96. Available at: http://smm.org.mx/publicaciones/pe/memorias/2011/v4/pdf/smm-pe-memorias-2011-v4.pdf
  4. Montiel, M. (2011) Presentación del Libro: Una Introducción a la Teoría de Grupos con Aplicaciones en la Teoría Matemática de la Música (Presentation of the Book: An Introduction to Group Theory with Applications to Mathematical Music Theory). Memoirs of the Fourth International Seminar on Mathematical Music Theory. Emilio Lluis-Puebla & Octavio Agustín-Aquino (Eds). Electronic Publications of the Mexican Mathematical Society. 163-165. Available at: http://smm.org.mx/publicaciones/pe/memorias/2011/v4/pdf/smm-pe-memorias-2011-v4.pdf
  5. Elstak. I., Vidakovic, D. & Montiel, M. (2010). Navigating Geometries with Different Metrics: College Students' Understanding of Taxicab-Geometry. In: Brosnan, P., Erchick, D. B., & Flevares, L. (Eds.). (2010). Proceedings of the 32nd annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Vol. VI, pp. 582-589 (peer reviewed). Columbus, OH: The Ohio State University.
  6. Montiel, M.; Wilhelmi, M. R.; Font, V. y Vidakovic, D. (2010). Dimensional analogy and coordinate systems. In Pinto,M.M.F.& Kawasaki, T.F.(Eds.). Proceedings of the 34th Conference of the International Group for the Psychology of Mathematics Education, Vol. 3, pp. 305-312. Belo Horizonte, Brazil: PME.
  7. Montiel, M., Vidakovic, D., Wilhelmi, M., & Elstak, I. (2009). Using the onto-semiotic approach: Different coordinate systems and dimensional analogy in multivariate calculus. In Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.), Proceedings of the 31st annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education; Vol. 5, pp. 95-103. Atlanta, GA: Georgia State University.
  8. Montiel, M., Wilhelmi, M., Vidakovic, D. & Elstak, I. (2009) Using the Onto-Semiotic Approach to Identify and Analyze Mathematical Meaning in a Multivariate Context. In Durand-Guerrier,V., Soury-Lavergne, S. & Azarello, F. (Eds). Proceedings from the Sixth Conference of European Research in Mathematics Education (CERME); pp. 2286-2295 Leon, France. http://www.inrp.fr/publications/edition-electronique/cerme6/cerme6.pdf
  9. Montiel, M., (2006) Mathematical Fluency Measured with the Four Parameters of Foreign Language Learning: Applications of the Integral in Alatorre, S., Cortina, J.L., Saiz, M, Mendez, A. (Eds), Proceedings from the Twenty Eighth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Merida, Mexico: Universidad Pedagogica Nacional.
  10. Montiel, M., Álgebra y Música, Aportaciones Matematicas (Mathematical Contributions), A publication of the Mexican Mathematical Society, 1997, pp. 117-128.

Peer Reviewed Encyclopedia Entries :

  1. Montiel, M. (2012). Mathematics: Discovery or Invention. Encyclopedia of Mathematics and Society, Salem Press.
  2. Montiel, M. (2012). Theoretical Mathematics. Encyclopedia of Mathematics and Society, Salem Press.

Translations :

  1. English to Spanish translation of the book: Nonnegative Matrices by Alexander Graham for the Economics Department of the National Autonomous University of Mexico (UNAM) (1995).
  2. Spanish to English translation of the article: Wilhelmi, M. R., Godino, J. D.; Lacasta, E. (2007). Configuraciones epistémicas asociadas a la noción de igualdad de números reales. Reserches en Didactique des Mathématiques, 27(1), 77–120. [Revised English version, by M. Montiel (2011): 'Epistemic configurations associated to the notion of equality in real numbers', Quaderni di Ricerca in Didattica (Mathematics) 21, 53–82. Retrieved from: http://math.unipa.it/~grim/Wilhelmi_Q21.pdf]
  3. Spanish to English translation of the book Introducción a la Teoría de Grupos con Aplicaciones en la Teoría Matemática de la Música. (An Introduction to Group Theory with Applications to Mathematical Music Theory). Agustín, O., du Plessis, J., Lluis-Puebla, E. & Montiel, M. (2012)

Conferences And Presentations