Trope (music)#20th-century music

The tropics is a doctrine of the composer Josef Matthias Hauer (1883-1959) developed auxiliary system in the twelve-tone composition. Your targeted compositional application is called Tropical technology.

Hauer understands he developed towards the end of 1921 and 44 so-called tropical ( " twists ", " constellation groups " ) as a system of order in the musical Zwölftonraum, by means of which it is possible permutations of all 479 001 600 12 notes ( possibilities of forming to survey twelve-tone rows ) and to structure by grouping common properties and organize. This Hauer moves fundamentally so that it divides the twelve notes in any two Sechstongruppen and considers its interval ratios. For a " trope ", which therefore is nothing more than an action taken in this sense is the combination of two complementary chromatic Total hexachords, are neither absolute pitch still relevant a particular arrangement of tones within the " tropics halves ", ie the Sechstongruppen. Also, the order of the two halves tropics is not mandatory, they can arbitrarily be interchanged. The Hauer'sche system proves to be the basis of calculations and conducted systematic method for the derivation of the tropics as a complete and self-contained.

Symmetry as the basis of the tropics teaching

The content and meaning of the tropics doctrine consists in the contemplation of the tropical own interval relationships with the purpose of obtaining set of technically relevant findings. Here, show different types of symmetries and other significant interval relations (eg: special sound structures ) at different levels, namely within the hexachords, between the two halves of a trope, but also between entire tropics. Indeed, there is not a single trope, can not be described by their symmetries. Thus, the division of the system into symmetry groups is appropriate.

Based on the knowledge of a trope and its intervallic characteristics, it is possible to make statements about all of her malleable twelve-tone rows. Likewise, this knowledge can be compositionally very many ways to use ( " tropical art "). So, for example, rows with specific formal, harmonic or melodic properties can be formed, which can be through the use of special technical processes tropics in turn transmitted to an entire composition. Overall, the knowledge of the tropics can provide accurate and according to plan predetermination of a composition which provides a framework for the implementation of a compositional concept. Such a " concept " may turn out differently and at different levels, whether, for example, on a formal level (for example: creating a " mirror cancer canon " ), on harmonic level (eg: the use of only certain sounds ) or melodic level (eg: use of a " cantus firmus " ), etc. ... but it can also involve combinations of these options.

The 44 tropics are written together in so-called " tropical panels ". A tropical table is an overview of the 44 tropical and provides optimally, certain (or as many ) properties of the tropics dar. Since the arrangement of tones within the tropics halves is arbitrary, there are as many possible " tropical images ", that is listed on the visualization of a trope as Tonanordnungen - and therefore theoretically almost infinite number of different tropical panels. In general, however, offers a possible clear and musically relevant aspects displayed tropics representation. In principle, the arrangement ( numbering) of the tropics on a tropical panel is different, but the numbering of Hauer's board has prevailed from August 11, 1948 because of their usefulness and their distribution over the other existing options in practice.

Categories of tropical

Because of their symmetries, the tropics in different categories can be divided. To fully understand these symmetry properties, a combination suitable from an exclusionary and an inclusive view. A total of two basic types of symmetries are distinguishable:

  • Transposition of interval ratios: two compared structures have, at different pitch, the same interval ratios.
  • Reflection ( inversion) interval of ratios: Two structures have, at different pitch, Interval same conditions, but in the reverse direction on.

These symmetries are considered on the three above-mentioned levels:

  • Relationships between tropical (only mirroring)
  • Relationships between Hexachorden ( transposition and / or mirror )
  • Relationships within Hexachorden ( transposition and / or mirror )

The consideration of possibilities of cancer formation in terms of a category for morphological classification of the tropics would be impractical because the two halves of a trope can be freely exchanged as well as the tones within the tropics halves yes. Thus, it is possible to form their own cancer within each trope. Would be inappropriate to consider a transposition as a ratio between two whole tropics would be, for in the tropics are characterized by their interval relationships, but not by absolute pitch, any arbitrarily transposed trope now gives course again yourself from these above mentioned possibilities of two symmetries in these three different levels, the following relevant categories of the classification of the tropics:

Considerations on tropical level

  • Mirroring between two tropics: two tropical relate to each other in the mirror. All tropics, which are not reversible by another trope, can be reversed by yourself. The Hauer students Sokolowski speaks in this case of " Exosymmetrie ".

18 = 9 tropical tropics couples: No. 5-6, 15-16, 18-22, 19-21, 20-23, 24-25, 28-29, 31-33, 37-38;

Reflections on Hexachordebene

  • Mirroring, but no transposition between two Hexachorden a trope: both tropical halves to each other in the mirror, but not in the trans position. Sokolowski calls this tropical "mono symmetrical". - 13 tropical: 2, 3, 9, 11, 12, 13, 26, 27, 30, 34, 39, 42, 43;
  • Transposition, but no reversal between two Hexachorden a trope: both hexachords have the same interval relationships to each other, are not at the same time to each other in the mirror. - Tropical 2: No. 28, 29; These two tropics to each other in the mirror.
  • Reflection and transposition between the Hexachorden same time: Both hexachords each other in the mirror and at the same time in the transposition. - 6 tropical: Nos. 1, 4, 10, 17, 41, 44;
  • The remaining 23 tropics, where both halves are neither in the transposition, even in the mirror one another, are either symmetrical to each other (8 tropics couples: No. 5-6, 15-16, 18-22, 19-21, 20-23, 24-25, 31-33, 37-38 ) or they have to be mirror-symmetrical hexachords on (7 tropics: # 7, 8, 14, 32, 35, 36, 40 ), which Sokolowski calls " endosymmetrisch ".

Considerations within a Hexachordes

  • Mirroring, but no transposition within both halves tropics: Both hexachords a trope are symmetrical and can be reversed. In some tropical arising here, however, a sound would have to be oktavverdoppelt. It is found that there is no trope, wherein only one half is symmetrical tropics, but not the other. - 6 tropical: # 7, 14, 32, 35, 36, 40;
  • Transposition, but no mirroring within both halves tropics: In both Hexachorden a trope can be found ( three -note ) sound structures that can be faithfully transposed interval from the rest of the tones within the tropics half in which they stand. - 11 tropical: 2, 3, 9, 15, 16, 28, 29, 30, 34, 39, 42;
  • Transposition and reflection within both hexachords: In both Hexachorden can be both transposable and reversible interval structures make up. There is no trope in which this possibility exists in only a single hexachord. - 7 Tropical: Nos. 1, 4, 10, 17, 41, 44, but also No.8;
  • Transposition within a Hexachordes: In a Hexachord a trope there is a transposable in this tropical half ( three-note ) sound structure. - Tropical 18: No. 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 reversal and which also have no discernible transposable ( three-note ) structures. - Tropical 8: # 11, 12, 13, 20, 23, 26, 27, 43;

In the mirror-like tropics, the transposed into two halves Dreitongruppen have ( Nos. 2, 3, 9, 30, 39, 42) is the three-note of the first Hexachordes the second in a mirrored form, but not in their original form represented. In the tropics, 28 and 29, the Dreitonstruktur is identical in both Hexachorden and can not be represented in the mirror. In the tropics 1, 4, 10, 17, 41 and 44 may be identical in both halves of tropical Dreitonstruktur are shown transposed mirrored or desired. In the other tropical (15, 16 and partially 8) in the possible Dreitongruppen Hexachorden are different. It arises from the logic of the twelve-tone chromatic tonal system that, for a trope in which a hexachord is symmetrical in itself, the second half tropics can also be reversed by yourself. For this reason, not the category exists " mirroring within a Hexachordes ". For categories, indicating a transposition within one or both hexachords, can not rule out the possibility that there are other tropical about the above addition in which this property has not yet been discovered. Tropical Nos. 1, 4, 10, 17, 41 and 44 appear at two levels repeatedly. In them both symmetry forms ( transposition and mirroring), see also, both on the Hexachordebene as well when considering the tropics halves for themselves. Referring to the terminology when Hauer student Victor Sokolowski (1911-1982) were the six " polysymmetrisch ", in the sense of symmetry on several levels. Overall, it is apparent from the above cumulative and the complexity of the tropical system and how difficult is a unique morphological classification designed, since neither an exclusive nor an inclusive, taken alone, as fully proved. So it seems only a combination of ex - inclusive and observation, made ​​as here, to be optimal.

Hauer used for tropics with transposing Hexachorden the adjective " resist equal." For tropical with mirrored Hexachorden can in reference to the tropics morphology at Othmar Steinbauer (1895-1962) the term be " mirror images " used (although this word is used under a different meaning). The two terms " resist equal " - " mirror images " is what is appropriate for the tropics technique applied only with respect to the ratio of the two hexachords a trope. Thus, these two terms can be used on the Hexachordebene to describe the tropical characteristics. Consequently, there are:

  • 8 against tropical same (1, 4, 10, 17, 28, 29, 41, 44 ),
  • 19 mirror-like tropics ( 1, 2, 3, 4, 9, 10, 11, 12, 13, 17, 26, 27, 30, 34, 39, 41, 42, 43, 44) and
  • 6 tropics, which are (1, 4, 10, 17, 41, 44 ) reflected the same and at the same time reflecting the same.

For the number of 44 tropical

The number of exactly 44 tropical follows necessarily due to structural characteristics of hexachords. 44 correspond to a total of 88 tropical Hexachordpaaren. During the same eight resist tropics (but to each transposed a certain interval ) formed from the same twice Hexachordstruktur, there are 36 tropical from two different Hexachorden. Thus, the number of 80 existing Hexachordstrukturen in the twelve-tone system, minus any possible transpositions and rerouting of tones is confirmed. If you want these Anticipated, it would be considered that of these 80 Hexachorden turn have 76 twelve transposition possibilities. Five hexachords are, however - in the sense Messiaen - limited transposable ( Hauer speaks in this context of " Tongeschlechtern " ): The hexachords the same resist tropics 4, 17 and 44 each have six, four, or two transposition possibilities. The two hexachords the non-resistant same trope 39 each have six transpositions. So give the bill 75 × 12 3 × 6 4 2 = 924 This number indicates Hauer as the sum total of all possible Tongeschlechter on, so that the sum of all transposition possibilities of all possible hexachords is expressed. Solving the equations to calculate the total number of possible twelve-tone rows, if the tropics as a combination of two hexachords involving: 924 · 6! · 6! = 924 · 720 · 720 = 479 001 600 = 12!

Swell

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