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Peter Lackner/Tropical Investigations

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Patterns of Intuition

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Abstract

Peter Lackner’s practice features an innovative approach to the systematisation of twelve-tone rows. In the second section of this chapter together with Harald Fripertinger a complete classification of tone rows in the twelve-tone scale is described, which is based on the notion of group actions. The main objects in this project are the orbits of tone rows under the action of the direct product of two dihedral groups. This means that tone rows are considered to be equivalent if and only if they can be constructed by transposing, inversion, retrograde, and/or time shift (rotation) from a single row. We determine the orbit, the normal form, the stabilizer class of a tone row, or its trope structure.

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Notes

  1. 1.

    Biographical introduction and texts from the composer translated from the German by Tamara Friebel.

  2. 2.

    Austrian painter, flautist and composer (1931–2013).

  3. 3.

    Hermann Markus Preßl (1939–1994). Austrian composer, professor at the University of Music and Performing Arts Graz.

  4. 4.

    “[\(\ldots \)] Est autem hoc proprium formae substantialis quod det materiae esse simpliciter. Ipsa enim est per quam res hoc ipsum quod est; [\(\ldots \)] Si qua igitur forma est qua non det materiae esse simpliciter, sed adveniat materiae jam existenti in actu per aliquam formam, non erit forma substantialis.” (“De Anima”; Thomas of Aquinas, Paris 1269). English translation from [25].

  5. 5.

    Kanon für Violine und Klavier translates as “canon for violin and piano”.

  6. 6.

    A certain array of elements, which are balanced in a square grid.

  7. 7.

    Hexentanz für Violine und Klavier translates as “witch-dance for violin and piano”.

  8. 8.

    See the definition from Fripertinger in section “Tropes” (see Sect. Tropes).

  9. 9.

    Kanon für Violine solo translates as “canon for solo violin”.

  10. 10.

    American composer and music theorist (1915–2009).

  11. 11.

    Two tropes are called connectable (see Sect. Tropes) if for each of the two hexachords of the first trope there exists a (uniquely determined) hexachord of the second trope so that the two hexachords have exactly five pitch classes in common.

  12. 12.

    Kanon für drei Bratschen translates as “canon for three violas”.

  13. 13.

    Kanon für zwei Gitarren translates as “canon for two guitars”.

  14. 14.

    From now on this implies twelve-tone rows.

  15. 15.

    Already in 1924 (cf. [15]) Hauer introduced the circular representation of tone rows. Instead of the chromatic order of the pitch classes he used the order according to the quint-circle.

  16. 16.

    Already in 1924 (cf. [15]) Hauer introduced this matrix representation for tone rows.

  17. 17.

    Accessed July 5, 2014.

References

  1. Colbourn CJ, Read RC (1979) Orderly algorithms for graph generation. Int J Comput Math 7:167–172

    Article  MATH  MathSciNet  Google Scholar 

  2. de Bruijn NG (1964) Pólya’s theory of counting. In: Beckenbach EF (ed) Applied combinatorial mathematics, Chap. 5. Wiley, New York, pp 144–184

    Google Scholar 

  3. de Bruijn NG (1971) A survey of generalizations of Pólya’s enumeration theorem. Nieuw Archief voor Wiskunde 2(XIX):89–112

    Google Scholar 

  4. de Bruijn NG (1972) Enumeration of mapping patterns. J Comb Theory (A) 12:14–20

    Article  MATH  Google Scholar 

  5. Florey H (1965/67) Tropentafel mit drehbaren Farbkreisen (Neuordnung der Tropen in Gestalt von \(35\) Grundbildern nach dem Verhältnis der Tropenhälften zu Gegenhälfte und Spiegelbild, woraus sich \(5\) Tropen-Typen ableiten). Picture, Acryl/Papierelemente, Karton; Plexiglas, Novopan, Schraubverbindungen \(98 \times 67\) cm

    Google Scholar 

  6. Florey H (1988) Analytische Bemerkungen zu Josef Matthias Hauers letztem Zwölftonspiel. Beilage zu einer Schallplatte herausgegeben von der Hochschule für Musik und darstellende Kunst in Graz

    Google Scholar 

  7. Fripertinger H (1992) Enumeration in musical theory. In: Beiträge zur Elektronischen Musik Heft 1. Hochschule für Musik und Darstellende Kunst, Graz

    Google Scholar 

  8. Fripertinger H (1992) Enumeration in musical theory. Séminaire Lotharingien de Combinatoire 476(S–26):29–42

    Google Scholar 

  9. Fripertinger H (1993) Endliche Gruppenaktionen auf Funktionenmengen. Das Lemma von Burnside–Repräsentantenkonstruktionen–Anwendungen in der Musiktheorie. Bayreuth Math Schr 45:19–135

    Google Scholar 

  10. Fripertinger H, Lackner P (2015) Database on tone rows and tropes, a short user’s guide. J Math Music (to appear)

    Google Scholar 

  11. Fripertinger H, Lackner P (2015) Tone rows and tropes. J Math Music (to appear)

    Google Scholar 

  12. GAP—Groups, Algorithms, and Programming, Version 4.5.7 (2012) The GAP Group. http://www.gap-system.org

  13. Golomb SW, Welch LR (1960) On the enumeration of polygons. Am Math Mon 87:349–353

    Article  MathSciNet  Google Scholar 

  14. Harary F, Palmer EM (1966) The power group enumeration theorem. J Comb Theory 1:157–173

    Article  MATH  MathSciNet  Google Scholar 

  15. Hauer JM (1924) Zur Lehre vom atonalen Melos. In: Westheim P (ed) Das Kunstblatt, vol 8(12), pp 353–360

    Google Scholar 

  16. Hunter DJ, von Hippel PT (2003) How rare is symmetry in musical 12-tone rows? Am Math Mon 110(2):124–132

    Article  MATH  Google Scholar 

  17. Ilomäki T (2008) On the similarity of twelve-tone rows. Studia Musica, vol 30. Sibelius Academy, Helsinki

    Google Scholar 

  18. Kerber A (1999) Applied finite group actions. Algorithms and combinatorics, vol 19. Springer, Berlin

    Google Scholar 

  19. Pólya G (1937) Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen. Acta Mathematica 68:145–254

    Article  MathSciNet  Google Scholar 

  20. Pólya G, Read RC (1987) Combinatorial enumeration of groups, graphs and chemical compounds. Springer, New York

    Book  Google Scholar 

  21. Read RC (1978) Every one a winner or how to avoid isomorphism search when cataloguing combinatorial configurations. Ann Discret Math 2:107–120

    Article  MATH  MathSciNet  Google Scholar 

  22. Read RC (1994) Four new mathematical problems in music theory. An essay written in fulfilment of the requirements for the A.R.C.T. In: Theory of the royal conservatory of music, Toronto

    Google Scholar 

  23. Read RC (1997) Combinatorial problems in the theory of music. Discret Math 167–168(1–3):543–551

    Article  MathSciNet  Google Scholar 

  24. Reiner DL (1985) Enumeration in music theory. Am Math Mon 92:51–54

    Article  MATH  MathSciNet  Google Scholar 

  25. Rowan JP (1949) The soul, a translation of St. Thomas Aquinas’ De anima. B. Herder Book Co, St. Louis

    Google Scholar 

  26. Sims CC (1970) Computational methods in the study of permutation groups. In: Leech J (ed) Computational problems in abstract algebra. Proc Conf, Oxford, 1967. Pergamon Press, Oxford, pp 169–183

    Google Scholar 

  27. Skreiner W (1981) Hans Florey: Farbtotalität in 35 Grundbildern: Neue Galerie am Landesmuseum Joanneum, 9. Juli-27. August 1981, Graz

    Google Scholar 

  28. SYMMETRICA—A program system devoted to representation theory, invariant theory and combinatorics of finite symmetric groups and related classes of groups (1987). Lehrstuhl II für Mathematik Universität Bayreuth, Bayreuth. http://www.algorithm.uni-bayreuth.de/en/research/SYMMETRICA/

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Authors and Affiliations

  1. Institute for Composition, Music Theory, Music History and Conduction, University of Music and Performing Arts Graz, Graz, Austria

    Peter Lackner

  2. Institute for Mathematics and Scientific Computing, University of Graz, Graz, Austria

    Harald Fripertinger

  3. Institute of Electronic Music and Acoustics, University of Music and Performing Arts Graz, Graz, Austria

    Gerhard Nierhaus

Authors
  1. Peter Lackner
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  2. Harald Fripertinger
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  3. Gerhard Nierhaus
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Correspondence to Gerhard Nierhaus .

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Editors and Affiliations

  1. University of Music and Performing, Institute of Electronic Music and Acoustics, Graz, Austria

    Gerhard Nierhaus

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Lackner, P., Fripertinger, H., Nierhaus, G. (2015). Peter Lackner/Tropical Investigations. In: Nierhaus, G. (eds) Patterns of Intuition. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9561-6_13

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  • DOI: https://doi.org/10.1007/978-94-017-9561-6_13

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