Complementary harmonics

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The Complementary harmony is one of the German-Bohemian composer Heinrich Simbriger developed (1903-1976) twelve-tone composition theory. It was created on the basis of Josef Matthias Hauer's theory of the tropics as a systematization and expansion of his building block technology.

Simbriger has presented his complementary harmonics primarily in his main theoretical works, the writing Complementary Harmonics and The Sound Management in Twelve-Tone Music .

Basics

The basis of the complementary harmony is not a twelve-tone row , but the division of the twelve-tone material into tone groups, which always complement each other to a total of twelve tones ("supplementary groups"): " What is 'complementary harmony'? The division of the total complex of 12 tones into 2 , 3, 4, 5 ... etc. up to 12 individual groups. Each division accordingly complements the overall complex. Within each type of division, all individual groups are mutually 'complementary' with regard to the twelve-tone complex. "

Simbriger found out that there are general basic types of sounds (sound formation possibilities) in the twelve-tone room 351, whereby in the systematics he established the first type corresponds to a "monophonic chord" (i.e. a single tone) and the 351st basic type corresponds to a twelve-part chord, which is all chromatic Includes tones. In between there are all other polyphonic sound formation possibilities of two (6 types), three (19 types), four (43), five (66 types), six-part (80 types) etc., minus all transpositions and inversions: For example the chords CEG, GEC, C-Es-As or D-F-sharp-B-F sharp-B-D flat are all assigned to the same type (No. 25), which corresponds to the major triad.

It is obvious that certain basic types can always be combined in groups in such a way that they " complement " each other to form the chromatic total. Simbriger has examined and systematized the most important two-, three- and four-part group formations in detail. Due to the nature of the twelve-tone system, it has been shown that there are numerous correspondences and symmetries both within and between the supplementary groups. Consequently, these can serve as the basis for the systematic classification within the complementary harmony (cf. the different mirror relationships, e.g. in the section "Typology 6 + 6"). In complementary harmony, such groups, which always complement each other to form a twelve-tone wholeness, and their properties are consciously used in a compositional way.

Simbriger states that the important properties of the complementary harmony are that they enable "orderly sound management" to an even higher degree than the old (= tonal) music, since logically sounding chord connections can be achieved even in very dissonant areas. In the melodic, this theory enables a motivic and thematic development of music as well as the almost arbitrary formation of imitations , reflections and crab walks . Another characteristic is also a big advantage in working with complementary groups of sounds, consists in the possibility of deducting pitch spaces against each other and thus musical Formblidung allow to diverse ways and especially large formal. He sees it as a synthesis of old and new musical design principles that a composer can fall back on at will and as needed.

Two-part grouping

Two supplement groups:
1 + 11 2 + 10 3 + 9 4 + 8 5 + 7 6 + 6

Of the six two-part groups, only the last two, 5 + 7 (or 7 + 5) and especially 6 + 6, are seen as significant for compositional practice.

Typology 6 + 6

With this division of the twelve-tone material into two hexachords, there are a total of 44 complementary sound groups ("complex ions"). These 44 cases correspond exactly to Josef Matthias Hauer's tropes . Historically, Simbriger has actually created the most precise system of the 44 tropics in the entire 20th century as part of his investigation of the grouping 6 + 6. In doing so, he developed the following system:

  • Type A : Both supplementary groups are structurally identical, only transposed against each other. (8 complex ions)
  • Type B : Both supplementary groups are mutually mirrored. (13 complex ions)
  • Type C : Both supplementary groups are inherently different, but each is symmetrical in itself. (7 complex ions)
  • Type D : Both supplementary groups are both unequal and inherently asymmetrical. Two complex ions each combine to form a double mirror pair. (8 pairs of complexes)

Each of the 44 complex ions of the grouping 6 + 6 is assigned to one of these four types. Each type implies certain possibilities of compositional treatment, which Simbriger describes in his theoretical texts. He pays particular attention to the possibilities of creating melodies and harmonies, as well as the use of symmetries in the context of canons . Of all the existing possibilities of complexation, Simbriger attaches the greatest importance to the grouping 6 + 6, as this allows the best overview of the sound material. Accordingly, he goes into this type in the greatest detail.

Typology 5 + 7

  • Type A : Both supplementary groups are different from one another, but each is symmetrical in itself. At the same time, the five-tone complex is contained in the seven-tone complex, transposed two or three times, and also in the mirror. (10 complex ions)
  • Type B : Both groups are inherently asymmetrical. Two pairs of complexes of this type are mirrored to one another. At the same time, the five-tone complex is always contained in mirror form in the seven-tone complex. (28 pairs of complexes)

This division serves as an important addition to the grouping 6 + 6. The peculiarity is that within the scope of this grouping there are other possibilities of symmetry formation and thus compositional usability.

Three-part grouping

Three supplement groups:
1 + 1 + 10 1 + 2 + 9 1 + 3 + 8 1 + 4 + 7 1 + 5 + 6
2 + 2 + 8 2 + 3 + 7 2 + 4 + 6 2 + 5 + 5
3 + 3 + 6 3 + 4 + 5
4 + 4 + 4

Of the three-part schemes, Simbriger emphasizes the particular relevance of the balanced grouping 4 + 4 + 4, while he does not attach particular importance to the other complexions due to the great differences in the number of tones. At most, the scheme 3 + 4 + 5 may still offer a certain incentive as an independent type.

In addition, the other groups can be understood as subtypes of two-part groups (e.g. 1 + 2 + 9 as a derivative of 3 + 9; 1 + 3 + 8 and 2 + 2 + 8 of 4 + 8; 1 + 4 + 7, 2 + 3 +7 and 2 + 5 + 5 as subgroups of 5 + 7; 1 + 5 + 6, 2 + 4 + 6 and 3 + 3 + 6 as subtypes of 6 + 6). Here the two-part types offer more options and are preferred to the restrictive three-parting in most cases.

Typology 4 + 4 + 4

With the order 4 + 4 + 4 Simbriger created its own typology of 499 complexions of sounds and differentiated according to symmetry types. Since this large number is actually unmanageable and is therefore at the very edge of practicality, Simbriger has limited itself to the most significant types. Nevertheless, there are some symmetry relationships that do not exist in the other groups and therefore enable other compositional applications.

  • Type A : All three tetrachords are identical and at the same time symmetrical in themselves. (6 complex ions)
  • Type B : Other, as a whole symmetrical groups:
1) Two tetrachords are the same, the third is different. All three are symmetrical in themselves. (27 complex ions)
2) Two tetrachords are in the mirror, the third, different from it, is symmetrical in itself. (52 complex ions)
3) All three supplementary groups are different from one another, but symmetrical in themselves. (10 complex ions)
  • Type C : Asymmetrical groupings:
1) Two tetrachords are the same, the third is different.
a) There are three mirror relationships. (12 complex ions)
b) There are two mirror relationships. (16 complex ions)
2) All three tetrachords are different from one another.
a) Three are symmetrical in themselves. (4 complex ions)
b) Two are symmetrical in themselves. (46 complex ions)
c) Only a tetrachord is symmetrical in itself. (198 complex ions)
d) All three tetrachords are asymmetrical. (128 complex ions)

Four-part grouping

Among the four-part groups, the division of the chromatic tone material into four three-part tone groups (3 + 3 + 3 + 3) is seen as useful, while the others are primarily of theoretical value, but hardly practical. In general, Simbriger sees the division into four as the limit of the practically usable area.

Four supplement groups:
1 + 1 + 1 + 9 1 + 2 + 2 + 9 2 + 2 + 2 + 6
1 + 1 + 2 + 8 1 + 2 + 3 + 6 2 + 2 + 3 + 5
1 + 1 + 3 + 7 1 + 2 + 4 + 5 2 + 2 + 4 + 4
1 + 1 + 4 + 6 1 + 3 + 3 + 5 2 + 3 + 3 + 4
1 + 1 + 5 + 5 1 + 3 + 4 + 4 3 + 3 + 3 + 3

Typology 3 + 3 + 3 + 3

Since the number of complex ions in the division 3 + 3 + 3 + 3 is extremely large (almost a thousand) and because two trichords can always be formed from a hexachord, there is a typological relationship with the grouping 6 + 6 as a superordinate type. Simbriger has only summarized the essential types here:

  • Type A : All trichords have the same structure. (5 complex ions)
  • Type B : three trichords are the same, the fourth is different.
1) All four triads are symmetrical in themselves. (6 complex ions)
2) The fourth triad is the only one that is symmetrical in itself. (unknown number)
  • Type C : Two triads are the same.
1) The second triad pair is the inverse of the first pair. (13 complex ions)
2) Every two trichords are structurally the same. (4 complex ions)
3) Two triads are equal and symmetrical in themselves. The other two triads are unequal and unsymmetrical in themselves. But they each form the opposite of one another. (22 complex ions)
  • Type D : All trichords are unequal.
1) There are two pairs of mirror-like trichords (24 complex ions)
2) Two trichords form the mirror to each other, the other two are symmetrical in themselves. (23 complex ions)

application

Since the complementary harmony is a direct elaboration of the building block technique of Josef Matthias Hauer, it can also be proven in the works of several composers, although these mostly do not refer to Simbriger, but to Hauer. So a working method in the sense of building block technology / complementary harmonics u. a. can be proven in works by the following composers:

References

  1. ^ Heinrich Simbriger, Complementary Harmonics , Esslingen 1980, p. 31.
  2. ↑ Heinrich Simbriger, Complementary Harmonics , pp. 6–12.
  3. ↑ See Heinrich Simbriger, Complementary Harmonics , pp. 2 and 13.
  4. ^ Heinrich Simbriger, Complementary Harmonics , pp. 209f.
  5. ↑ Heinrich Simbriger, Complementary Harmonics , p. 39f.
  6. ↑ Heinrich Simbriger, Complementary Harmonics , pp. 38–157.
  7. ^ Heinrich Simbriger: Complementary Harmonics. Pp. 158-175.
  8. ↑ Heinrich Simbriger, Complementary Harmonics , p. 176.
  9. ↑ Heinrich Simbriger, Complementary Harmonics , pp. 177–200.
  10. ^ Heinrich Simbriger, Complementary Harmonics , p. 201.
  11. ↑ Heinrich Simbriger, Complementary Harmonics , pp. 201–207.
  12. ↑ cf. Hans Ulrich Götte, Josef Matthias Hauer's composition techniques with special consideration of deterministic methods , Kassel a. a. 1989, p. 69.

literature

  • Heinrich Simbriger: Complementary Harmonics , Die Künstlergilde, Esslingen 1979, 2nd edition 1980
  • Heinrich Simbriger: The sound guidance in twelve-tone music. Peritonale Harmonik , Die Künstlergilde, Esslingen o. J. (1991)
  • Hermann Heiss: Elements of musical composition , Hochstein & Co, Heidelberg 1949
  • Thomas Emmerig (ed.): Theory and Analysis. Studies on the work of Heinrich Simbriger with three first publications from the estate , ConBrio, Regensburg 2011.
  • Thomas Emmerig (ed.): "Above all, I am a composer ..." Biography and work of Heinrich Simbriger with a first publication from the estate and a sound documentation , ConBrio, Regensburg 2012.

See also