Retrograde inversion is a musical term that literally means "backwards and upside down": "The inverse of the series is sounded in reverse order." Retrograde reverses the order of the motive'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 and retrograde techniques are performed on the same tone row successively, "[t]he 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 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: 0 2 t e 1 8 6 7 9 4 3 5 R0: 5 3 4 9 7 6 8 1 e t 2 0 I0: 0 t 2 1 e 4 6 5 3 8 9 7 RI0: 7 9 8 3 5 6 4 e 1 2 t 0 IR0: 5 7 6 1 3 4 2 9 e 0 8 t
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 employ retrograde and retrograde inversion with the recurring twelve-tone row (B F♯ F E G A♭ E♭ D A B♭ D♭ C). It is performed in both styles,[clarification needed] particularly in the outer sections of the piece.
- Bruce Benward and Marilyn Nadine Saker, Music: In Theory and Practice, seventh edition ([full citation needed], 2003): Vol. I, p.310. ISBN 978-0-07-294262-0.
- Steven G. Laitz, The Complete Musician: An Integrated Approach to Tonal Theory, Analysis, and Listening, third edition (Oxford and New York: Oxford University Press, 2012): p. 780.
- Benward & Saker (2003): p. 359.
- Arnold Whittall, The Cambridge Introduction to Serialism. Cambridge Introductions to Music (Cambridge and New York: Cambridge University Press, 2008): p. 139. ISBN 978-0-521-68200-8 (pbk).
- Paul Schuyler Phillips, "The Enigma of 'Variations': A Study of Stravinsky's Final Work for Orchestra", Music Analysis 3, no. 1 (March 1984): 69–89. Citation on p. 70, and p. 88, n. 6.
- 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):[page needed].
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