Transposition (music)

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. 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. Transposition example from Koch.png
Transposition example from Koch Loudspeaker.svg Play top   Loudspeaker.svg 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.

In music, transposition refers to the process or operation of moving a collection of notes (pitches or pitch classes) up or down in pitch by a constant interval.

Contents

The shifting of a melody, a harmonic progression or an entire musical piece to another key, while maintaining the same tone structure, i.e. the same succession of whole tones and semitones and remaining melodic intervals.

Musikalisches Lexicon, 879 (1865), Heinrich Christoph Koch (trans. Schuijer) [1]

For example, one 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 Tn(A), representing the addition (mod 12) of an integer n to each of the pitch class integers of the set A. [1] 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.

Scalar transpositions

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.

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 semitones. For instance, transposing the pitches C4–E4–G4 upward by four semitones, one obtains the pitches E4–G4–B4.

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 signatures). For example, transposing the pitches C4–E4–G4 up two steps in the familiar C major scale gives the pitches E4–G4–B4. Transposing the same pitches up by two steps in the F major scale instead gives E4–G4–B4.

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. [1] For example, the pitch A4, or 9, transposed by a major third, or the pitch interval 4:

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

.

Sight transposition

Excerpt of the trumpet part of Symphony No. 9 of Antonin Dvorak, where sight transposition is required. Dvorak 9, trumpet part exerpt.png
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.

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 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 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.

Transpositional equivalence

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:

or

For pitch class transposition by a pitch class interval:

[2]

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:

where to is any integer 0–11 inclusive, where, of course, the to remains fixed for a given transposition. The + sign indicates ordinary transposition. Here To is the transposition corresponding to to (or o, according to Schuijer); pi,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]

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

See also

Sources

  1. 1 2 3 4 Schuijer, Michiel (2008). Analyzing Atonal Music, pp. 52–54. ISBN   978-1-58046-270-9.
  2. Rahn, John (1987). Basic atonal theory. New York: Schirmer Books. pp. &#91, page needed &#93, . ISBN   0-02-873160-3. OCLC   54481390.
  3. 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.
  4. 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.
  5. 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.

Related Research Articles

In music theory, a scale is any set of musical notes ordered by fundamental frequency or pitch. A scale ordered by increasing pitch is an ascending scale, and a scale ordered by decreasing pitch is a descending scale.

In music theory, an interval is a difference in pitch between two sounds. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or harmonic if it pertains to simultaneously sounding tones, such as in a chord.

Chromatic scale Musical scale with twelve pitches separated by semitone intervals

The chromatic scale is a set of twelve pitches used in tonal music, with notes separated by the interval of a semitone. Almost all western musical instruments, such as the piano, are made to produce the chromatic scale, while other instruments capable of continuously variable pitch, such as the trombone and violin, can also produce microtones, or notes between those available on a piano.

Enharmonic

In modern musical notation and tuning, an enharmonic equivalent is a note, interval, or key signature that is equivalent to some other note, interval, or key signature but "spelled", or named differently. Thus, the enharmonic spelling of a written note, interval, or chord is an alternative way to write that note, interval, or chord. The term is derived from Latin enharmonicus, from Late Latin enarmonius, from Ancient Greek ἐναρμόνιος (enarmónios), from ἐν (en)+ἁρμονία (harmonía).

Twelve-tone technique Musical composition method using all 12 chromatic scale notes equally often & not in a key

The twelve-tone technique—also known as dodecaphony, twelve-tone serialism, and twelve-note composition—is a method of musical composition first devised by Austrian composer Josef Matthias Hauer, who published his "law of the twelve tones" in 1919. In 1923, Arnold Schoenberg (1874–1951) developed his own, better-known version of 12-tone technique, which became associated with the "Second Viennese School" composers, who were the primary users of the technique in the first decades of its existence. The technique is a means of ensuring that all 12 notes of the chromatic scale are sounded as often as one another in a piece of music while preventing the emphasis of any one note through the use of tone rows, orderings of the 12 pitch classes. All 12 notes are thus given more or less equal importance, and the music avoids being in a key. Over time, the technique increased greatly in popularity and eventually became widely influential on 20th-century composers. Many important composers who had originally not subscribed to or actively opposed the technique, such as Aaron Copland and Igor Stravinsky, eventually adopted it in their music.

An octatonic scale is any eight-note musical scale. However, the term most often refers to the symmetric scale composed of alternating whole and half steps, as shown at right. In classical theory, this scale is commonly called the octatonic scale, although there are a total of 42 enharmonically non-equivalent, transpositionally non-equivalent eight-note sets.

Pitch class

In music, a pitch class (p.c. or pc) 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. "The pitch class C stands for all possible Cs, in whatever octave position." Important to musical set theory, a pitch class is "all pitches related to each other by octave, enharmonic equivalence, or both." Thus, using scientific pitch notation, the pitch class "C" is the set

Set theory (music)

Musical set theory provides concepts for categorizing musical objects and describing their relationships. Howard Hanson first elaborated many of the concepts for analyzing tonal music. Other theorists, such as Allen Forte, further developed the theory for analyzing atonal music, drawing on the twelve-tone theory of Milton Babbitt. The concepts of musical set theory are very general and can be applied to tonal and atonal styles in any equal temperament tuning system, and to some extent more generally than that.

Complement (music)

In music theory, complement refers to either traditional interval complementation, or the aggregate complementation of twelve-tone and serialism.

Interval cycle

In music, an interval cycle is a collection of pitch classes created from a sequence of the same interval class. In other words, a collection of pitches by starting with a certain note and going up by a certain interval until the original note is reached. In other words, interval cycles "unfold a single recurrent interval in a series that closes with a return to the initial pitch class". See: wikt:cycle.

Multiplication (music)

The mathematical operations of multiplication have several applications to music. Other than its application to the frequency ratios of intervals, it has been used in other ways for twelve-tone technique, and musical set theory. Additionally ring modulation is an electrical audio process involving multiplication that has been used for musical effect.

Set (music)

A set in music theory, as in mathematics and general parlance, is a collection of objects. In musical contexts the term is traditionally applied most often to collections of pitches or pitch-classes, but theorists have extended its use to other types of musical entities, so that one may speak of sets of durations or timbres, for example.

Interval vector

In musical set theory, an interval vector is an array of natural numbers which summarize the intervals present in a set of pitch classes. Other names include: ic vector, PIC vector and APIC vector

Diatonic and chromatic Terms in music theory to characterize scales

Diatonic and chromatic are terms in music theory that are most often used to characterize scales, and are also applied to musical instruments, intervals, chords, notes, musical styles, and kinds of harmony. They are very often used as a pair, especially when applied to contrasting features of the common practice music of the period 1600–1900.

In music theory, the word inversion describes certain types of changes to intervals, chords, voices, and melodies. In each of these cases, "Inversion" has a distinct but related meaning. The concept of inversion also plays an important role in musical set theory.

Fritz Heinrich Klein was an Austrian composer.

Forte number

In musical set theory, a Forte number is the pair of numbers Allen Forte assigned to the prime form of each pitch class set of three or more members in The Structure of Atonal Music. The first number indicates the number of pitch classes in the pitch class set and the second number indicates the set's sequence in Forte's ordering of all pitch class sets containing that number of pitches.

In music theory, equivalence class is an equality (=) or equivalence between properties of sets (unordered) or twelve-tone rows. A relation rather than an operation, it may be contrasted with derivation. "It is not surprising that music theorists have different concepts of equivalence [from each other]..." "Indeed, an informal notion of equivalence has always been part of music theory and analysis. Pitch class set theory, however, has adhered to formal definitions of equivalence." Traditionally, octave equivalency is assumed, while inversional, permutational, and transpositional equivalency may or may not be considered.

All-interval twelve-tone row

In music, an all-interval twelve-tone row, series, or chord, is a twelve-tone tone row arranged so that it contains one instance of each interval within the octave, 1 through 11. A "twelve-note spatial set made up of the eleven intervals [between consecutive pitches]." There are 1,928 distinct all-interval twelve-tone rows. These sets may be ordered in time or in register. "Distinct" in this context means in transpositionally and rotationally normal form, and disregarding inversionally related forms.

Pitch interval

In musical set theory, a pitch interval is the number of semitones that separates one pitch from another, upward or downward.