View source for Transposition (music)

{{Short description|Operation in music}}
[[Image:Transposition example from Koch.png|thumb|350px|right|Transposition example from Koch<ref name="Schuijer"/> {{audio|Transposition example from Koch top.mid|Play top}} {{audio|Transposition example from Koch bottom.mid|Play bottom}}. In this chromatic transposition, the melody on the first line is in the key of D, while the melody on the second line is identical except that it is a [[major third]] lower, in the key of B{{music|b}}.]]

In [[music]], '''transposition''' refers to the process or [[operation (music)|operation]] of moving a [[set (music)|collection]] of [[Musical note|note]]s ([[pitch (music)|pitches]] or [[pitch class]]es) up or down in [[pitch (music)|pitch]] by a constant [[interval (music)|interval]].
{{blockquote|The shifting of a [[melody]], a [[chord progression|harmonic progression]] or an entire musical piece to another key, while maintaining the same tone structure, i.e. the same succession of [[major second|whole tones]] and [[semitone]]s and remaining melodic intervals.|''Musikalisches Lexicon'', 879 (1865), [[Heinrich Christoph Koch]] (trans. Schuijer)<ref name="Schuijer">Schuijer, Michiel (2008). ''Analyzing Atonal Music'', pp. 52–54. {{ISBN|978-1-58046-270-9}}.</ref>}}
For example, a '''music transposer''' might transpose an entire [[piece (music)|piece]] of music into another [[key (music)|key]].  Similarly, one might transpose a [[tone row]] or an unordered collection of pitches such as a [[chord (music)|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 ([[modular math|mod 12]]) of an integer ''n'' to each of the pitch class integers of the set ''A''.<ref name="Schuijer"/> Thus the set (''A'') consisting of 0–1–2 transposed by 5 semitones is 5–6–7 ('''''T'''''<sub>5</sub>(''A'')) since {{nowrap|0 + 5 {{=}} 5}}, {{nowrap|1 + 5 {{=}} 6}}, and {{nowrap|2 + 5 {{=}} 7}}.

==Scalar transpositions==
In scalar transposition, every pitch in a collection is shifted up or down a fixed number of [[musical scale|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.

===Chromatic transposition===
Chromatic transposition is scalar transposition within the [[chromatic scale]], implying that every pitch in a collection of notes is shifted by the same number of [[semitone]]s.  For instance, transposing the pitches C<sub>4</sub>–E<sub>4</sub>–G<sub>4</sub> upward by four semitones, one obtains the pitches E<sub>4</sub>–G{{music|#}}<sub>4</sub>–B<sub>4</sub>.

===Diatonic transposition===
Diatonic transposition is scalar transposition within a [[diatonic scale]] (the most common kind of scale, indicated by one of a few standard [[key signature]]s).  For example, transposing the pitches C<sub>4</sub>–E<sub>4</sub>–G<sub>4</sub> up two steps in the familiar C major scale gives the pitches E<sub>4</sub>–G<sub>4</sub>–B<sub>4</sub>.  Transposing the same pitches up by two steps in the F major scale instead gives E<sub>4</sub>–G<sub>4</sub>–B{{music|b}}<sub>4</sub>.

==Pitch and pitch class transpositions==
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.<ref name="Schuijer"/> For example, the pitch A<sub>4</sub>, or 9, transposed by a major third, or the pitch interval 4:
:<math>9 + 4 = 13</math>
while that pitch class, 9, transposed by a major third, or the pitch class interval 4:
:<math>9 + 4  =13 \equiv 1\pmod{12}</math>.

==Sight transposition==
[[File:Dvorak 9, trumpet part exerpt.png|thumb|Excerpt of the [[trumpet]] part of [[Symphony No. 9 (Dvořák)|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 instrument]]s 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.

===Interval===
First one determines the interval between the written key and the target key. 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).

===Clef===
[[Clef]] transposition is routinely taught (among other places) in Belgium and France. One imagines a different clef and a different key signature than the ones printed. The change of clef is used so that the lines and spaces correspond to different notes than the lines and spaces of the original score. Seven clefs are used for this: treble (2nd line G-clef), bass (4th line F-clef), baritone (3rd line F-clef or 5th line C-clef, although in France and Belgium sight-reading exercises for this clef, as a preparation for clef transposition practice, are always printed with the 3rd line F-clef), and C-clefs on the four lowest lines; these allow any given [[staff position]] to correspond to each of the seven [[Musical note|note]] names A through G. The signature is then adjusted for the actual accidental (natural, sharp or flat) one wants on that note. The octave may also have to be adjusted (this sort of practice ignores the conventional octave implication of the clefs), but this is a trivial matter for most musicians.

===Numbers===
Transposing by numbers means, one determines the [[degree (music)|scale degree]] of the written note (e.g. first, fourth, fifth, etc.) in the given key. The performer then plays the corresponding scale degree of the target chord.

=={{anchor|Transpositional equivalency|Transpositional invariance|Transpositionally equivalent}}Transpositional equivalence==<!--[[Transpositional invariance]], [[Transpositional equivalency]], and [[Transpositionally equivalent]] redirect directly here.-->

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 [[set theory (music)|musical set theory]]. The terms ''transposition'' and ''transposition equivalence'' allow the concept to be discussed as both an [[operation (music)|operation]] and [[relation (music)|relation]], an activity and a state of being. Compare with [[modulation (music)|modulation]] and [[closely related key|related key]].

Using [[integer notation]] and [[modular arithmetic|modulo]] 12, to transpose a pitch ''x'' by ''n'' semitones:
:<math>\boldsymbol{T}^p_n (x) = x+n</math>
or
:<math>\boldsymbol{T}^p_n (x) \rightarrow x+n</math>
For pitch class transposition by a pitch class interval:
:<math>\boldsymbol{T}_n (x) = x+n \pmod{12}</math>
<ref>{{Cite book |last=Rahn |first=John |title=Basic atonal theory |publisher=Schirmer Books |location=New York |year=1987 |pages={{Page needed|date=May 2010}} |isbn=0-02-873160-3 |oclc=54481390 |url=https://archive.org/details/basicatonaltheor0000rahn }}</ref>

==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 ''i''th tone in ''P'' belong to the pitch class (set number) ''j''.
:<ref>Babbitt (1992). ''The Function of Set Structure in the Twelve-Tone System'', p. 10. PhD dissertation, Princeton University [1946]. cited in Schuijer (2008), p. 55. ''p'' = element, ''P'' = twelve-tone series, ''i'' = order number, ''j'' = pitch-class number.</ref>

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''".<ref>Forte (1964). "A Theory of Set-Complexes for Music", p. 149, ''Journal of Music Theory'' 8/2:136–83. cited in Schuijer (2008), p. 57. ''p'' = element, ''P'' = pitch class set, ''S'' = universal set.</ref>

==Fuzzy transposition==
Joseph Straus created the concept of '''fuzzy transposition''', and [[fuzzy inversion]], to express transposition as a [[Voice leading|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.".<ref>Straus, Joseph N. (April 11, 2003). "Voice Leading in Atonal Music", unpublished lecture for the Dutch Society of Music Theory. Royal Flemish Conservatory of Music, Ghent, Belgium. or Straus, Joseph N. (1997). "Voice Leading in Atonal Music" in ''Music Theory in Concept and Practice'', ed. James M. Baker, David W. Beach, and Jonathan W. Bernard, 237–74. Rochester, NY: University of Rochester Press. Cited in Schuijer (2008), pp. 61–62.</ref> A transformation within voice-leading space rather than [[pitch-class space]] as in pitch class transposition.

==See also==
* [[List of music software]]
*[[Modulation (music)]]
*[[Music transposer]]
*[[Pitch shift]]
*[[Transposing instrument]]
*[[Capo (musical device)|Capo]]

==References==
<references/>

==External links==
* [http://tabtransposer.com/ Chords transposition in song sheets plus showing these chords for different instruments]
* [http://chords.kytara.cz/transpose-music/ Chords transposition]
* [http://www.statistics101.net/chordsmith/ ChordSmith: Java program to transpose chords in song sheets]
* [http://www.transposer.org/ Online Tool to transpose songs]
* [http://www.chordchanger.com/ Chordchanger.com: online tool to transpose guitar chords]
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[[Category:Musical techniques]]
[[Category:Pitch (music)]]