Transposition (music)

In scalar transposition, every pitch in a collection is shifted up or down a fixed number of scale steps within some scale. The pitches remain in the same scale before and after the shift. This term covers both chromatic and diatonic transpositions as follows.

There are two further kinds of transposition, by pitch interval or by pitch interval class, applied to pitches or pitch classes, respectively. Transposition may be applied to pitches or to pitch classes.[1] For example, the pitch A₄, or 9, transposed by a major third, or the pitch interval 4:

${\displaystyle 9+4=13}$

while that pitch class, 9, transposed by a major third, or the pitch class interval 4:

${\displaystyle 9+4=13\equiv 1{\pmod {12}}}$.

Excerpt of the trumpet part of Symphony No. 9 of Antonín Dvořák, where sight transposition is required.

Although transpositions are usually written out, musicians are occasionally asked to transpose music "at sight", that is, to read the music in one key while playing in another. Musicians who play transposing instruments sometimes have to do this (for example when encountering an unusual transposition, such as clarinet in C), as well as singers' accompanists, since singers sometimes request a different key than the one printed in the music to better fit their vocal range (although many, but not all, songs are printed in editions for high, medium, and low voice).

There are three basic techniques for teaching sight transposition: interval, clef, and numbers.

Two musical objects are transpositionally equivalent if one can be transformed into another by transposition. It is similar to enharmonic equivalence, octave equivalence, and inversional equivalence. In many musical contexts, transpositionally equivalent chords are thought to be similar. Transpositional equivalence is a feature of musical set theory. The terms transposition and transposition equivalence allow the concept to be discussed as both an operation and relation, an activity and a state of being. Compare with modulation and related key.

Using integer notation and modulo 12, to transpose a pitch x by n semitones:

${\displaystyle {\boldsymbol {T}}_{n}^{p}(x)=x+n}$

or

${\displaystyle {\boldsymbol {T}}_{n}^{p}(x)\rightarrow x+n}$

For pitch class transposition by a pitch class interval:

${\displaystyle {\boldsymbol {T}}_{n}(x)=x+n{\pmod {12}}}$

[2]

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:

${\displaystyle {\boldsymbol {T}}_{o}(p_{i,j})=p_{i,j}+t_{o}}$

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

[3]

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".[4]

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_n-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.".[5] A transformation within voice-leading space rather than pitch-class space as in pitch class transposition.

• Modulation (music)
• Music transposer
• Pitch shift
• Transposing instrument
• Capo

• Chords transposition in song sheets plus showing these chords for different instruments
• Chords transposition
• ChordSmith: Java program to transpose chords in song sheets
• Online Tool to transpose songs
• Chordchanger.com: online tool to transpose guitar chords