Transposition (music)

# Transposition (music)

Discussion

Encyclopedia
In music
Music
Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...

transposition refers to the process, or operation, of moving a collection
Set (music)
A set in music theory, as in mathematics and general parlance, is a collection of objects...

of note
Note
In music, the term note has two primary meanings:#A sign used in musical notation to represent the relative duration and pitch of a sound;#A pitched sound itself....

s (pitches
Pitch (music)
Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...

or pitch class
Pitch class
In music, a pitch class is a set of all pitches that are a whole number of octaves apart, e.g., the pitch class C consists of the Cs in all octaves...

es) up or down in pitch
Pitch (music)
Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...

by a constant interval
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

.

For example, one might transpose an entire piece of music into another key
Key (music)
In music theory, the term key is used in many different and sometimes contradictory ways. A common use is to speak of music as being "in" a specific key, such as in the key of C major or in the key of F-sharp. Sometimes the terms "major" or "minor" are appended, as in the key of A minor or in the...

. Similarly, one might transpose a tone row
Tone row
In music, a tone row or note row , also series and set, refers to a non-repetitive ordering of a set of pitch-classes, typically of the twelve notes in musical set theory of the chromatic scale, though both larger and smaller sets are sometimes found.-History and usage:Tone rows are the basis of...

or an unordered collection of pitches such as a chord
Chord (music)
A chord in music is any harmonic set of two–three or more notes that is heard as if sounding simultaneously. These need not actually be played together: arpeggios and broken chords may for many practical and theoretical purposes be understood as chords...

so that it begins on another pitch.

The transposition of a set A by n semitones is designated by Tn(A), representing the addition (mod 12) of an integer n to each of the pitch class integers of the set A. Thus the set (A) consisting of 0-1-2 transposed by 5 semitones is 5-6-7 (T5(A)) since 0+5=5, 1+5=6, and 2+5=7.

### Chromatic and scalar (diatonic) transposition

There are two different kinds of transposition, depending on whether one is measuring intervals according to the chromatic scale or some other scale. In chromatic transposition one shifts every pitch in a collection of notes by a fixed number of semitones. For instance, if one transposes the pitches C4-E4-G4 upwards by four semitones, one obtains the pitches E4-G4-B4. In scalar transposition one shifts every pitch in a collection by a fixed number of scale steps
Musical scale
In music, a scale is a sequence of musical notes in ascending and descending order. Most commonly, especially in the context of the common practice period, the notes of a scale will belong to a single key, thus providing material for or being used to conveniently represent part or all of a musical...

relative to some scale. For example, if one transposes the pitches C4-E4-G4 up by two steps relative to the familiar C major scale, one obtains the pitches E4-G4-B4. If one transposes the same pitches up by two steps relative to the F major scale, one obtains instead E4-G4-B4. Scalar transposition is sometimes called diatonic transposition, but this term can be misleading, as it suggests transposition with respect to a diatonic scale. However, scalar transposition can occur with respect to any type of scale, not just the diatonic.

### Pitch and pitch class

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. For example the pitch A4, or 9, transposed by a major third, or the pitch interval 4:
while that pitch class, 9, tranposed by a major fourth, or the pitch class interval 4:
since
.

## Sight transposition

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
Transposing instrument
A transposing instrument is a musical instrument for which written notes are read at a pitch different from the corresponding concert pitch, which a non-transposing instrument, such as a piano, would play. Playing a written C on a transposing instrument will produce a note other than concert C...

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.

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
Clef
A clef is a musical symbol used to indicate the pitch of written notes. Placed on one of the lines at the beginning of the staff, it indicates the name and pitch of the notes on that line. This line serves as a reference point by which the names of the notes on any other line or space of the staff...

transposition is routinely taught in Belgium and France. One imagines a different clef than the one printed so that the lines and spaces correspond to different notes. Seven clefs are used for this: treble, bass, baritone, and C-clefs on the four lowest lines; these allow any given staff position to correspond to each of the seven note
Note
In music, the term note has two primary meanings:#A sign used in musical notation to represent the relative duration and pitch of a sound;#A pitched sound itself....

names A through G. The octave may also have to be adjusted, but this is a trivial matter for most musicians.

### Numbers

Transposing by numbers means, one determines the scale degree
Degree (music)
In music theory, a scale degree or scale step is the name of a particular note of a scale in relation to the tonic...

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 key.

## Transpositional equivalency

Two musical objects are transpositionally equivalent if one can be transformed into another by transposition. It is similar to enharmonic equivalence and octave 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
Modulation (music)
In music, modulation is most commonly the act or process of changing from one key to another. This may or may not be accompanied by a change in key signature. Modulations articulate or create the structure or form of many pieces, as well as add interest...

and related key.

Using integer notation and modulo
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain valueâ€”the modulus....

12, to transpose a pitch x by n semitones:
or
For pitch class transposition by a pitch class interval:

## Twelve-tone transposition

Milton Babbitt
Milton Babbitt
Milton Byron Babbitt was an American composer, music theorist, and teacher. He is particularly noted for his serial and electronic music.-Biography:...

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 homomophically (with regard to order) into a T(p) of the set T(P) according to the following operation:
where To is any integer 0-11 inclusive, where, of course, the To remains fixed for a given transposition. The + sign indicates ordinary transposition.

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

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 set to its Tn-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.