Tropical theory

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The tropical theory is an auxiliary system in twelve-tone composition developed by the composer Josef Matthias Hauer (1883–1959) . Its targeted compositional application is called tropical technique .

Hauer understands the so-called 44 tropes ("turns", "constellation groups") developed by him towards the end of 1921 as a system of order in the musical twelve-tone space, with the help of which it is possible to record all 479,001,600 permutations of the 12 tones (possibilities of forming Twelve-tone rows ) and to structure and order by summarizing common properties. Hauer always proceeds in such a way that he divides the twelve tones into any two six-tone groups and considers their interval relationships. For a “trope”, which is nothing more than a combination of two hexachords that complement each other to form a chromatic total, neither absolute pitches nor a specific arrangement of tones within the “tropical halves” are relevant. H. of the six-note groups. The order of the two halves of the tropics is also not mandatory; they can be interchanged as desired. The Hauer system proves to be complete and self-contained on the basis of calculations and systematic procedures carried out to derive the tropics.

Symmetry as the basis of the theory of the tropics

The content and meaning of the tropical theory consists in the consideration of the tropics' own interval relationships with the purpose of gaining knowledge relevant to typesetting. Different types of symmetries and other significant interval relationships (e.g. special sound structures) appear here on different levels, namely within the hexachords, between the two halves of a tropic, but also between entire tropics. In fact, there is not a single trope that cannot be described by its symmetries. This makes dividing the system into symmetry groups useful.

Based on the knowledge of a trope and its interval properties, it is possible to make statements about all twelve-tone rows that can be formed from it. This knowledge can also be used in a wide variety of compositional ways (“tropical technology”). For example, rows with special formal, harmonic or melodic properties can be formed, which in turn can be transferred to an entire composition through the use of special tropical technology. Overall, knowledge of the tropics can enable a composition to be precisely and according to plan, which creates the framework for the implementation of a compositional concept. Such a “concept” can be different and on different levels, be it for example: B. on a formal level (e.g .: creation of a "mirror cancer canon"), on a harmonic level (e.g .: the use of only certain sounds) or on the melodic level (e.g .: use of a " cantus firmus ") etc. ... It can also contain combinations of these options.

The 44 tropics are written down in so-called "tropical tables". A tropical table is an overview of the 44 tropics and optimally represents certain (or as many as possible) properties of the tropics. Since the arrangement of the tones within the tropical halves is arbitrary, there are just as many possible "tropical images", i. H. Notated visualization of a trope, such as tone arrangements - and therefore theoretically an almost infinite number of different tropic tables. As a rule, however, a tropical representation that is as clear and musically relevant as possible is recommended. Basically, the arrangement (numbering) of the tropics on a tropical table is also different, but the numbering of Hauer's table of August 11, 1948 has prevailed in practice due to its usefulness and its spread over the other existing possibilities.

Categories of the tropics

Because of their symmetries, the tropics can be divided into different categories. A combination of an exclusive and an inclusive view is suitable for optimally recording these symmetry properties. Overall, two basic types of symmetries can be distinguished:

  • Transposition of interval relationships: Two structures being compared show the same interval relationships on different pitches.
  • Mirroring (inversion) of interval relationships: Two structures, on different pitches, have the same interval relationships, but in the opposite direction.

These symmetries are considered on the three levels already mentioned:

  • Relationships between the tropics (reflection only)
  • Relationships between hexachords (transposition and / or mirror)
  • Relationships within hexachords (transposition and / or mirror)

Considering the possibility of cancer formation in the sense of a category for the morphological classification of the tropics would be inexpedient, since the two halves of a tropic as well as the tones within the tropical halves can be exchanged at will. So it is possible to develop your own cancer within each trope. It would be just as inexpedient to consider a transposition as a relationship between two whole tropes, because since the tropics are characterized by their interval relationships, but not by absolute pitches, every arbitrarily transposed tropic naturally results in itself. From the two possibilities mentioned above Symmetries on these three different levels result in the following relevant categories for the classification of the tropics:

Considerations at the tropical level

  • Reflection between two tropes: two tropes are mirrored to each other. All tropes that are not reversible by another tropic can be reversed by themselves. The Hauer student Sokolowski speaks of "exosymmetry" in this case.

18 tropics = 9 tropic pairs: Nos. 5–6, 15–16, 18–22, 19–21, 20–23, 24–25, 28–29, 31–33, 37–38;

Considerations at the hexachord level

  • Reflection, but no transposition between two hexachords of a tropics: Both halves of the tropics are mirrored to each other, but not in the transposition. Sokolowski calls these tropes "monosymmetrical". - 13 tropics: No. 2, 3, 9, 11, 12, 13, 26, 27, 30, 34, 39, 42, 43;
  • Transposition, but not an inversion between two hexachords of a trope: Both hexachords show the same interval relationships to one another, but at the same time are not in mirror to one another. - 2 tropics: No. 28, 29; These two tropes are mirrored to one another.
  • Mirroring and transposition between the hexachords at the same time: Both hexachords are mirrored to each other and at the same time also in the transposition. - 6 tropics: No. 1, 4, 10, 17, 41, 44;
  • The other 23 tropics, in which both halves are neither in transposition nor in mirror to each other, are either symmetrical to each other (8 tropic pairs: Nos. 5–6, 15–16, 18–22, 19–21, 20–23, 24-25, 31-33, 37-38) or they have mirror-symmetrical hexachords (7 tropes: No. 7, 8, 14, 32, 35, 36, 40), which Sokolowski calls "endosymmetrical".

Considerations within a hexachord

  • Reflection, but no transposition within both halves of the tropics: Both hexachords of a tropics are symmetrical in themselves and can be reversed. In some tropics, however, a tone would have to be doubled in an octave. It turns out that there is no such thing as a tropic where only one half of the tropics is symmetrical and the other is not. - 6 tropics: nos. 7, 14, 32, 35, 36, 40;
  • Transposition, but no reflection within both halves of the tropics: In both hexachords of a tropics there are (three-tone) sound structures that can be transposed from the rest of the tones within the tropical half in which they are located, with accurate intervals. - 11 tropics: No. 2, 3, 9, 15, 16, 28, 29, 30, 34, 39, 42;
  • Transposition and mirroring within both hexachords: In both hexachords, both transposable and reversible interval structures can be identified. There is no trope in which this possibility exists in just a single hexachord. - 7 tropics: No. 1, 4, 10, 17, 41, 44, but also No. 8;
  • Transposition within a hexachord: In a hexachord of a tropics there is a transposable (three-tone) sound structure in this half of the tropics. - 18 tropics: nos. 5, 6, 7, 14, 18, 19, 21, 22, 24, 25, 31, 32, 33, 35, 36, 37, 38, 40;
  • Tropics whose hexachords are not their own inversion and which also have no recognizable transposable (three-tone) structures. - 8 tropics: No. 11, 12, 13, 20, 23, 26, 27, 43;

In the mirror-image tropes, which have three-tone groups transposed in both halves (No. 2, 3, 9, 30, 39, 42), the three-tone group of the first hexachord can be shown in the second in mirrored form, but not in its original form. In the tropics 28 and 29, the three-tone structure in both hexachords is identical and cannot be represented in the mirror. In the tropics 1, 4, 10, 17, 41 and 44, a three-tone structure that is identical in both halves of the tropics can be mirrored or transposed as desired. In the other tropes (15, 16 and partly 8) the possible three-tone groups in the hexachords are different. It follows from the logic of the twelve-tone chromatic tone system that in a trope in which a hexachord is symmetrical in itself, the second tropical half can also be reversed by itself. For this reason, the category “reflection within a hexachord” does not exist. In the case of the categories that indicate a transposition within one or both hexachords, the possibility cannot be completely ruled out that there are other tropes beyond those mentioned above in which this property has not yet been discovered. Tropics No. 1, 4, 10, 17, 41 and 44. appear repeatedly on two levels. Both forms of symmetry (transposition and reflection) can be found in them at the same time, both on the hexachord level and when looking at the halves of the tropics in isolation. Referring to the terminology used by Hauer's student Victor Sokolowski (1911–1982), these six would be “polysymmetrical”, in the sense of symmetry on several levels. Overall, the above compilation shows how complex the tropical system is and how difficult it is to create a clear morphological classification, since neither an exclusive nor an inclusive one alone turns out to be complete. So only a combination of exclusive and inclusive considerations, as made here, seems to be optimal.

For tropes with transposing hexachords, Hauer used the adjective "contradicting". For tropes with mirrored hexachords, the term “mirror-like” can be used in reference to the tropical morphology of Othmar Steinbauer (1895–1962) (although he used the word with a different meaning). The pair of terms “mirror-like” - “mirror-like” is used, which is useful for tropical technology, only with regard to the ratio of the two hexachords of a tropic. So these two terms can be used to describe the tropical properties on the hexachordal level. Hence there are:

  • 8 contrasting tropes (1, 4, 10, 17, 28, 29, 41, 44),
  • 19 mirror-image tropes (1, 2, 3, 4, 9, 10, 11, 12, 13, 17, 26, 27, 30, 34, 39, 41, 42, 43, 44) and
  • 6 tropes that are identical and at the same time mirror-like (1, 4, 10, 17, 41, 44).

To the number of 44 tropics

The number of exactly 44 tropics results from the structural features of the hexachords. 44 tropics correspond to a total of 88 hexachord pairs. While the eight contrasting tropes consist of twice the same hexachord structure (each transposed by a certain interval), 36 tropics consist of two different hexachords. This confirms the number of 80 existing hexachord structures in the twelve-tone system, minus all possible transpositions and assignments of tones. If you want to include these, you should consider that of these 80 hexachords, 76 each have twelve transposition possibilities. Five hexachords, however - in the sense of Messiaen - can only be transposed to a limited extent (Hauer speaks of "tone sexes " in this context): The hexachords of the contradicting tropes 4, 17 and 44 each have six, four or two transposition possibilities. The two hexachords of the non-contradicting trope 39 each have six transpositions. So the calculation results in 75 · 12 + 3 · 6 + 4 + 2 = 924. Hauer gives this number as the total sum of all possible tone genders, which expresses the sum of all possible transposition of all possible hexachords. From this, the total number of possible twelve-tone rows is calculated if one includes the tropics as a combination of two hexachords: 924 · 6! · 6! = 924 x 720 x 720 = 479001600 = 12!


  1. "The study of compositional technique in twelve-tone music is very similar to that of harmony and counterpoint, the only difference lies in the greater possibilities, in the finer ramifications and ramifications. The entire compositional technique of twelve-tone music clings to the theory of the tropics, the requires intensive study. " Hauer, Josef Matthias: From Melos to Timpani. An introduction to twelve-tone music , Vienna 1925, p. 11.
  2. For a compilation of different tropical tables see Diederichs, Joachim, Fheodoroff, Nikolaus and Schwieger, Johannes (ed.): Josef Matthias Hauer. Fonts. Manifestos. Documents , Edition Österreichische Musikzeit, Vienna 2007, pp. 418, 440 and 442–447.
  3. See Sedivy, Dominik: Tropentechnik. Their application and their possibilities , Königshausen & Neumann, Würzburg 2012, pp. 47–50.
  4. ^ Hauer, Josef Matthias: From Melos to Kettledrum. An introduction to twelve-tone music , Vienna 1925, p. 14.
  5. See Neumann, Helmut (Ed.): The Sound Series Composition Theory according to Othmar Steinbauer (1895-1962) , Vol. 1, Peter Lang, Frankfurt / Vienna 2001, p. 195.
  6. ^ Hauer, Josef Matthias: From Melos to Kettledrum. An introduction to twelve-tone music , Vienna 1925, p. 14.
  7. Herbert Eimert also mentions the number 924 in the textbook of twelve-tone technology , Wiesbaden 1952, p. 20.


  • Barbara Boisits: Tropical theory. In: Oesterreichisches Musiklexikon . Online edition, Vienna 2002 ff., ISBN 3-7001-3077-5 ; Print edition: Volume 5, Verlag der Österreichischen Akademie der Wissenschaften, Vienna 2006, ISBN 3-7001-3067-8 .
  • Diederichs, Joachim, Fheodoroff, Nikolaus and Schwieger, Johannes (eds.): Josef Matthias Hauer. Fonts. Manifestos. Documents , Edition Österreichische Musikzeit, Vienna 2007, pp. 416–452.
  • Hauer, Josef Matthias: "Die Tropen", in: Musikblätter des Anbruch , Jhrg. 6/1, Universal Edition, Vienna 1924, pp. 18-21.
  • Hauer, Josef Matthias: From Melos to Timpani. An introduction to twelve-tone music , Universal Edition, Vienna 1925.
  • Hauer, Josef Matthias: twelve-tone technique. The Doctrine of the Tropics , Universal Edition, Vienna 1926.
  • Neumann, Helmut (ed.): The sound series composition theory according to Othmar Steinbauer (1895-1962) , vol. 1, Peter Lang, Frankfurt / Vienna 2001, pp. 167–197.
  • Sedivy, Dominik: Serial Composition and Tonality. An Introduction to the Music of Hauer and Steinbauer , edition mono, Vienna 2011, pp. 81–90.
  • Sedivy, Dominik: Tropical technology. Their application and their possibilities , Königshausen & Neumann, Würzburg 2012, pp. 44–53.
  • Sengstschmid, Johann: Between trope and twelve-tone play. JM Hauer's twelve-tone technique in selected examples , Gustav Bosse Verlag Regensburg 1980.
  • Weiss, Robert Michael: The twelve-tone play by Josef Matthias Hauer , housework at the University of Music and Performing Arts Vienna, Vienna 1980, p. 34ff.

See also

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