[[Image:Transposition example from Koch.png|thumb|350px|right|Transposition example from Koch

For example, a music transposer might transpose an entire piece of music into another key. Similarly, one might transpose a tone row or an unordered collection of pitches such as a chord so that it begins on another pitch.

The transposition of a set A by n semitones is designated by T<sub>n</sub>(A), representing the addition (mod 12) of an integer n to each of the pitch class integers of the set A.

Twelve-tone transposition

Milton Babbitt defined the "transformation" of transposition within the twelve-tone technique as follows:

By applying the transposition operator (T) to a [twelve-tone] set we will mean that every p of the set P is mapped homomorphically (with regard to order) into a T(p) of the set T(P) according to the following operation:

:<math>\boldsymbol{T}_o(p_{i,j})=p_{i,j}+t_o</math>

where t<sub>o</sub> is any integer 0–11 inclusive, where, of course, the t<sub>o</sub> remains fixed for a given transposition. The + sign indicates ordinary transposition. Here T<sub>o</sub> is the transposition corresponding to t<sub>o</sub> (or o, according to Schuijer); p<sub>i,j</sub> is the pitch of the ith tone in P belong to the pitch class (set number) j.

:

Allen Forte defines transposition so as to apply to unordered sets of other than twelve pitches:

:the addition mod 12 of any integer k in S to every integer p of P.

thus giving, "12 transposed forms of P".

Fuzzy transposition

Joseph Straus created the concept of fuzzy transposition, and fuzzy inversion, to express transposition as a voice-leading event, "the 'sending' of each element of a given PC [pitch-class] set to its T<sub>n</sub>-correspondent...[enabling] him to relate PC sets of two adjacent chords in terms of a transposition, even when not all of the 'voices' participated fully in the transpositional move.". A transformation within voice-leading space rather than pitch-class space as in pitch class transposition.

See also

  • List of music software
  • Modulation (music)
  • Pitch shift
  • Transposing instrument
  • Capo

References