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Retrograde inversion is a musical term that literally means "backwards and upside down": "The 
Conventionally, inversion is carried out first, and the inverted form is then taken backward to form the retrograde inversion, so that the untransposed retrograde inversion ends with the pitch that began the prime form of the series. In his late twelve-tone works, however,
inverse
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of the series is sounded in reverse order." Retrograde reverses the order of the motif's pitches: what was the first pitch becomes the last, and vice versa. This is a technique used in music, specifically in twelve-tone technique, where the inversion
Inversion or inversions may refer to:
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* Inversion (music), a term with various meanings in music theory and musical set theory
* ...
and retrograde techniques are performed on the same tone row successively, " e inversion of the prime series in reverse order from last pitch to first."
Conventionally, inversion is carried out first, and the inverted form is then taken backward to form the retrograde inversion, so that the untransposed retrograde inversion ends with the pitch that began the prime form of the series. In his late twelve-tone works, however, Igor Stravinsky
Igor Fyodorovich Stravinsky (6 April 1971) was a Russian composer, pianist and conductor, later of French (from 1934) and American (from 1945) citizenship. He is widely considered one of the most important and influential composers of the ...
preferred the opposite order, so that his row charts use inverse retrograde (IR) forms for his source sets, instead of retrograde inversions (RI), although he sometimes labeled them RI in his sketches.
For example, the forms of the row from ''Requiem Canticles'' are as follows:
P0:
R0:
I0:
RI0:
IR0:
Note that IR is a transposition of RI, the pitch class between the last pitches of P and I above RI.
Other compositions that include retrograde inversions in its rows include works by Tadeusz Baird and Karel Goeyvaerts. One work in particular by the latter composer, '' Nummer 2'', employs retrograde of the recurring twelve-tone row B–F–F–E–G–A–E–D–A–B–D–C in the piano part.Herman Sabbe, ''Het muzikale serialisme als techniek en als denkmethode: Een onderzoek naar de logische en historische samenhang van de onderscheiden toepassingen van het seriërend beginsel in de muziek van de periode 1950–1975'' (Ghent: Rijksuniversiteit te Gent, 1977): 55. It is performed in both styles, particularly in the outer sections of the piece. The final movement of Hindemith's Ludus Tonalis, the Postludium, is an exact retrograde inversion of the work's opening Praeludium.
Sources
Musical symmetry Serialism {{Music-theory-stub