''Ausmultiplikation'' (literally, "multiplying-out") is a German term used by the composer,

Karlheinz Stockhausen Karlheinz Stockhausen (; 22 August 1928 – 5 December 2007) was a German composer, widely acknowledged by critics as one of the most important but also controversial composers of the 20th and early 21st centuries. He is known for his groun ...

to describe a

technique Technique or techniques may refer to: Music * The Techniques, a Jamaican rocksteady vocal group of the 1960s * Technique (band), a British female synth pop band in the 1990s * ''Technique'' (album), by New Order, 1989 * ''Techniques'' (album), by ...

in which a

long Long may refer to: Measurement * Long, characteristic of something of great duration * Long, characteristic of something of great length * Longitude (abbreviation: long.), a geographic coordinate * Longa (music), note value in early music mens ...

note Note, notes, or NOTE may refer to: Music and entertainment * Musical note, a pitched sound (or a symbol for a sound) in music * ''Notes'' (album), a 1987 album by Paul Bley and Paul Motian * ''Notes'', a common (yet unofficial) shortened versi ...

is replaced by shorter "

melodic A melody (), also tune, voice, or line, is a linear succession of musical tones that the listener perceives as a single entity. In its most literal sense, a melody is a combination of pitch and rhythm, while more figuratively, the term c ...

configurations, internally animated around central tones", resembling the

ornamental Ornamental may refer to: *Ornamental grass, a type of grass grown as a decoration *Ornamental iron, mild steel that has been formed into decorative shapes, similar to wrought iron work *Ornamental plant, a plant that is grown for its ornamental qua ...

technique of

divisions Division may refer to: Mathematics *Division (mathematics), the inverse of multiplication * Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting of 10,000 t ...

(also called "

diminution In Western culture, Western music and music theory, diminution (from Medieval Latin ''diminutio'', alteration of Latin ''deminutio'', decrease) has four distinct meanings. Diminution may be a form of embellishment (music), embellishment in whic ...

s") in

Renaissance music Renaissance music is traditionally understood to cover European music of the 15th and 16th centuries, later than the Renaissance era as it is understood in other disciplines. Rather than starting from the early 14th-century ''ars nova'', the mus ...

. Stockhausen first described this technique in connection with his "opus 1", ''

Kontra-Punkte ''Kontra-Punkte'' (Counter-Points, or Against-Points) is a composition for ten instruments by Karlheinz Stockhausen which resolves contrasts among six instrumental timbres, as well as extremes of note values and dynamic levels, into a homogeneou ...

'', composed in 1952–53, but in his later

formula composition Formula composition is a serialism, serially derived technique encountered principally in the music of Karlheinz Stockhausen, involving the projection, expansion, and ''Ausmultiplikation'' of either a single melody-formula, or a two- or three-voice ...

there is a related method of substituting a complete or partial formula for a single tone that is very long in a much slower, "more

background Background may refer to: Performing arts and stagecraft * Background actor * Background artist * Background light * Background music * Background story * Background vocals * ''Background'' (play), a 1950 play by Warren Chetham-Strode Record ...

" projection of the formula. When this is done at more than one level, the result is reminiscent of a

fractal In mathematics, a fractal is a Shape, geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scale ...

Notes and references

Notes References

Sources

* * * Post-tonal music theory Musical techniques Karlheinz Stockhausen {{music-theory-stub