A trope or tropus may refer to a variety of different concepts in

medieval In the history of Europe, the Middle Ages or medieval period lasted approximately from the late 5th to the late 15th centuries, similar to the post-classical period of global history. It began with the fall of the Western Roman Empire a ...

, 20th-, and 21st-century music. The term ''trope'' derives from the Greek (''tropos''), "a turn, a change", related to the root of the verb (''trepein''), "to turn, to direct, to alter, to change". The Latinised form of the word is ''tropus''. In music, a trope is adding another section, or trope to a

plainchant Plainsong or plainchant (calque from the French ''plain-chant''; la, cantus planus) is a body of chants used in the liturgies of the Western Church. When referring to the term plainsong, it is those sacred pieces that are composed in Latin text ...

or section of plainchant, thus making it appropriate to a particular occasion or

festival A festival is an event ordinarily celebrated by a community and centering on some characteristic aspect or aspects of that community and its religion or cultures. It is often marked as a local or national holiday, mela, or eid. A festival co ...

Medieval music

From the 9th century onward, trope refers to additions of new music to pre-existing chants in use in the

Western Christian Church Western Christianity is one of two sub-divisions of Christianity (Eastern Christianity being the other). Western Christianity is composed of the Latin Church and Western Protestantism, together with their offshoots such as the Old Catholic C ...

. Three types of addition are found in music manuscripts: # new

melisma Melisma ( grc-gre, μέλισμα, , ; from grc, , melos, song, melody, label=none, plural: ''melismata'') is the singing of a single syllable of text while moving between several different notes in succession. Music sung in this style is ref ...

s without text (mostly unlabelled or called "trope" in manuscripts) # addition of a new text to a pre-existing melisma (more often called ''prosula'', ''prosa'', ''verba'' or ''versus'') # new verse or verses, consisting of both text and music (mostly called trope, but also ''laudes'' or ''versus'' in manuscripts).. The new verses can appear preceding or following the original material, or in between phrases. O God creator of all things, thou our merciful God eleyson, we pray to thee, O great king of kings, singing praises together to thee eleyson, to whom be praise, power, peace and dominion for ever without end eleyson, O Christ, sole king, O Son coeternal with the kind Father eleyson who saved mankind, being lost, giving life for death eleyson lest your pastured sheep should perish, O Jesus, good shepherd eleyson. Consoler of suppliant spirits below, we beseech thee eleyson, O Lord, our strength and our salvation for eternity eleyson, O highest God, grant to us the gifts of eternal life and have mercy upon us eleyson. The standard Latin-rite ninefold Kyrie is the backbone of this trope. Although the supplicatory format ('eleyson'/'have mercy') has been retained, the Kyrie in this troped format adopts a distinctly Trinitarian cast with a tercet address to the Holy Spirit which is not present in the standard Kyrie. ''Deus creator omnium'' is thus a fine example of the literary and doctrinal sophistication of some of the tropes used in the Latin rite and its derived uses in the mediæval period.

20th-century music

In certain types of

atonal Atonality in its broadest sense is music that lacks a tonal center, or key. ''Atonality'', in this sense, usually describes compositions written from about the early 20th-century to the present day, where a hierarchy of harmonies focusing on a ...

and serial music, a trope is an unordered collection of different pitches, most often of

cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...

six (now usually called an unordered

hexachord In music, a hexachord (also hexachordon) is a six-note series, as exhibited in a scale ( hexatonic or hexad) or tone row. The term was adopted in this sense during the Middle Ages and adapted in the 20th century in Milton Babbitt's serial theor ...

, of which there are two complementary ones in twelve-tone

equal temperament An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, ...

). Tropes in this sense were devised and named by

Josef Matthias Hauer Josef Matthias Hauer (March 19, 1883 – September 22, 1959) was an Austrian composer and music theorist. He is best known for developing, independent of and a year or two before Arnold Schoenberg, a method for composing with all 12 notes of the ...

in connection with his own

twelve-tone technique The twelve-tone technique—also known as dodecaphony, twelve-tone serialism, and (in British usage) twelve-note composition—is a method of musical composition first devised by Austrian composer Josef Matthias Hauer, who published his "law ...

, developed simultaneously with but overshadowed by

Arnold Schoenberg Arnold Schoenberg or Schönberg (, ; ; 13 September 187413 July 1951) was an Austrian-American composer, music theorist, teacher, writer, and painter. He is widely considered one of the most influential composers of the 20th century. He was as ...

's. Hauer discovered the 44 tropes, pairs of complementary hexachords, in 1921, allowing him to classify any of the 479,001,600 twelve-tone melodies into one of 44 types. The primary purpose of the tropes is not

analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...

(although it can be used for it) but composition. A trope is neither a

hexatonic scale In music and music theory, a hexatonic scale is a scale with six pitches or notes per octave. Famous examples include the whole-tone scale, C D E F G A C; the augmented scale, C D E G A B C; the Prometheus scale, C D E F A B C; and the blues sca ...

nor a

chord Chord may refer to: * Chord (music), an aggregate of musical pitches sounded simultaneously ** Guitar chord a chord played on a guitar, which has a particular tuning * Chord (geometry), a line segment joining two points on a curve * Chord ( ...

. Likewise, it is neither a pitch-class set nor an interval-class set. A trope is a framework of contextual interval relations. Therefore, the key information a trope contains is not the set of intervals it consists of (and by no means any set of pitch-classes), it is the relational structure of its intervals. Each trope contains different types of symmetries and significant structural intervallic relations on varying levels, namely within its hexachords, between the two halves of an hexachord and with relation to whole other tropes. Based on the knowledge one has about the intervallic properties of a trope, one can make precise statements about any twelve-tone row that can be created from it. A composer can utilize this knowledge in many ways in order to gain full control over the musical material in terms of form, harmony and melody. Invers canon á 3

The hexachords of trope no. 3 are related by inversion. Trope 3 is therefore suitable for the creation of inversional and retrograde inversional structures. Moreover, its primary formative intervals are the minor second and the major third/minor sixth. This trope contains ,2,6twice inside its first hexachord (e.g. F–G–B and G–A–C and ,4,6in the second one (e.g. A–C–D and B–D–E). Its

multiplications Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...

M₅ and M₇ will result in trope 30 (and vice versa). Trope 3 also allows the creation of an intertwined retrograde transposition by a major second and therefore of trope 17 (e.g., G–A–C–B–F–F–, –E–E–C–D–B–A → Bold pitches represent a hexachord of trope 17). In general, familiarity with the tropes enables a composer to precisely predetermine a whole composition according to almost any structural plan. For instance, an inversional twelve-tone row from this trope 3 (such as G–A–C–B–F–F–D–C–A–B–E–D) that is harmonized by the –3–3–3method as suggested by Hauer, will result in an equally inversional sequence of sonorities. This will enable the composer, for example, to write an inversional canon or a

mirror fugue A mirror fugue is a fugue, or rather two fugues, one of which is the mirror image of the other. It is as though a mirror were placed above or below an existing fugue, producing inversions of each interval in each part, as well as inverting the posi ...

easily (see example 1). The symmetry of a twelve-tone row can thus be transferred to a whole composition likewise. Consequently, trope technique allows the integration of a formal concept into both a twelve-tone row and a harmonic matrix—and therefore into a whole musical piece.

References

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Medieval music

20th-century music

See also

References

Further reading