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Then one imagines the notes up (or down) by the corresponding interval. A performer using this method may calculate each note individually, or group notes together (e.g. "a descending chromatic passage starting on F" might become a "descending chromatic passage starting on A" in the target key).
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 .
Transposition and inversion can be represented as elementary arithmetic operations. If x is a number representing a pitch class, its transposition by n semitones is written T n = x + n mod 12. Inversion corresponds to reflection around some fixed point in pitch class space.
The twelve-tone technique—also known as dodecaphony, twelve-tone serialism, and (in British usage) twelve-note composition—is a method of musical composition.The technique is a means of ensuring that all 12 notes of the chromatic scale are sounded equally often in a piece of music while preventing the emphasis of any one note [3] through the use of tone rows, orderings of the 12 pitch classes.
An interval is inverted by raising or lowering either of the notes by one or more octaves so that the higher note becomes the lower note and vice versa. For example, the inversion of an interval consisting of a C with an E above it (the third measure below) is an E with a C above it – to work this out, the C may be moved up, the E may be lowered, or both may be moved.
The next note is sharper than C by the assigned value in cents. Finally, the two notes are played simultaneously. Note that the JND for pitch difference is 5–6 cents. Played separately, the notes may not show an audible difference, but when they are played together, beating may be heard (for example if middle C and a note 10 cents higher are ...
In mm. 16, both hands transpose down three semitones to B ♭, C, E ♭ and E, G ♭, A respectively. Later on, in mm. 20, the right hand moves on to A− and the left back to E ♭ −. After repeating the structure of mm. 12–19 in mm. 29–34 the piece ends with the treble part returning to A− and the bass part returning to E ♭ .
Here is an example of non-permutation of trichords, using retrogradation, inversion, and retrograde-inversion, combined in each case with transposition, as found within the tone row (or twelve-tone series) from Anton Webern's Concerto: [4]