Post-tonal music theory

Last updated October 16, 2022

Post-tonal music theory is the set of theories put forward to describe music written outside of, or 'after', the tonal system of the common practice period. It revolves around the idea of 'emancipating dissonance', that is, freeing the structure of music from the familiar harmonic patterns that are derived from natural overtones. As music becomes more complex, dissonance becomes indistinguishable from consonance.

Overview

In the latter part of the 19th century, composers began to move away from the tonal system. This is typified in Richard Wagner's music, especially Tristan und Isolde (the Tristan chord, for example). Arnold Schoenberg and his pupil Anton Webern proposed a theory on the emancipation of the dissonance to help analyse the general trend and, in particular, their own atonal music. Composers such as Charles Ives,^[1] Dane Rudhyar,^[2] and even Duke Ellington ^[3] and Lou Harrison,^[4] connected the emancipation of the dissonance with the emancipation of society and humanity.

The basic idea is that as time progresses, the ear becomes acclimatised to more and more complex sounds. This happens not just for individuals but also for societies as they start to write more complex music. Consonance and dissonance become indistinct from each other: dissonances slowly become heard as consonances. Jim Samson ^[5] explained it this way: "As the ear becomes acclimatized to a sonority within a particular context, the sonority will gradually become 'emancipated' from that context and seek a new one. The emancipation of the dominant-quality dissonances has followed this pattern, with the dominant seventh developing in status from a contrapuntal note in the sixteenth century to a quasi-consonant harmonic note in the early nineteenth. By the later nineteenth century the higher numbered dominant-quality dissonances had also achieved harmonic status, with resolution delayed or omitted completely. The greater autonomy of the dominant-quality dissonance contributed significantly to the weakening of traditional tonal function within a purely diatonic context."

Theory

Music written within the tonal system is generally analysed by defining a certain note as the primary or tonic note and the derived triad is the tonic chord. Other notes and chords are subservient to the tonic and in a strict hierarchy: the dominant note/chord is second in importance, others are lower down still. One example of this style of analysis is called Schenkerian analysis. However, this form of analysis cannot be applied to atonal music since the very point is to make all the notes and chords equal: there is no hierarchy. Instead, notes/chords can be described in terms of their properties and relationships at any particular moment: whether one note is higher than another, whether one chord has more notes than another, whether one chord is more widely spaced than another, and so on. One can also compare and contrast different strings of notes as transpositions (change in pitch) or inversions (change in note order) of each other. These terms are also used to compare chords. These methods of analysis have been used for centuries but became more important as music began to lose its tonal basis. One also needs to consider other aspects, such as how two or more simultaneous melodies relate to each other (counterpoint) and the same tools are used for this.

In the later 20th century, analysts started to adapt these tools to the yet more complex music being written. Musical set theory was first elaborated for tonal music^[6] but was quickly applied to atonal music^[7] since it simply provides concepts for categorizing musical objects (notes, chords, melodies and so on) and describing their relationship, without defining any particular note or chord as "primary". The later Transformational theory ^[8] uses a similar approach but concentrates on the relationships themselves. There are also theories which attempt to relate pitch and rhythm.

Application

Compositional applications of these theories are numerous, but in the present context of post-tonal music the most important is serialism . In this system, certain notes are chosen then written in an order e.g. E–F♯–C–B♭–G–F. (Usually there is no repetition, but this is not always observed.) These notes are then used as the basis for a composition by playing them in the original order, in reverse order (retrograde), in "upside down" order (Inversion i.e. upward intervals now go down, and vice versa), or both (retrograde inversion or "reversion" [Stravinsky's term]), and then transposed up or down. Chords can also be formed out of the series and these can be treated to similar techniques. Schoenberg used these methods in what has become known as twelve-tone technique. In this, all unique twelve notes of the musical scale are played once and once only in a specified order. The serial techniques described above are then applied.^[9] Later composers, such as Jean Barraqué and Pierre Boulez, sought to unify pitch and rhythm by organising the elements into sets of twelve, which resulted in what became known as total serialism.^[10] See also Formula composition which describes techniques used by Karlheinz Stockhausen.

Aside from serialism, other forms of compositional technique arose such as those based on chords utilizing fourths rather than the more traditional thirds (see quartal and quintal harmony and Synthetic chord), those based on other mathematical processes (see Schillinger System) and those based on specific scales (or "modes": see hexatonic scale, Heptatonic scale, Octatonic scale and Synthetic scale). Olivier Messiaen in his work The Technique of my Musical Language developed what he called modes of limited transposition which displayed a special type of symmetry and which he used in numerous compositions.

Further developments

Microtones and especially quarter tones have been used in music of the 20th and 21st centuries. These are the intervals between semitones. A full theory governing these has yet to be developed but the articles relating to these contain some of the most recent thoughts. (See 15 equal temperament, 19 equal temperament, 24 equal temperament, 34 equal temperament and 72 equal temperament.)

Examples

Transposition:

The notes A–B–C–D can be transposed downwards to A♭–B♭–C♭–D♭ (the ♭, called a flat, lowers the pitch by one semitone) or upwards to D–E–F–G (the note D is a perfect fourth higher than the note A, E the same amount higher than B and so on).
The chord C–E–G can be transposed upwards to C♯–E♯–G♯ (the ♯, called a sharp, raises the pitch by one semitone).

Inversion:

The notes D–F–E–B can be reversed to B–E–F–D; this form of inversion is called retrograde.
The upward intervals can become downward intervals and vice versa; this is the form that is properly called inversion.
These two can be combined and the result can be transposed: these are inclusive rather than exclusive processes.

When viewing the following musical examples, it may help to imagine a mirror being placed between the various versions:

How Retrograde and Inversion work Inversions.png — How Retrograde and Inversion work

The chord C–E–G can be inverted to E–G–C or G–C–E; this is called the inversion of a chord.
Two lines ("parts") of music can be "inverted" so that the original lower one is made to sound above the original higher one; this is inversion of counterpoint and can be applied to any number of parts.

Related Research Articles

In music, harmony is the process by which individual sounds are joined together or composed into whole units or compositions. Often, the term harmony refers to simultaneously occurring frequencies, pitches, or chords. However, harmony is generally understood to involve both vertical harmony (chords) and horizontal harmony (melody).

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.

In music theory, the tritone is defined as a musical interval composed of three adjacent whole tones. For instance, the interval from F up to the B above it is a tritone as it can be decomposed into the three adjacent whole tones F–G, G–A, and A–B. According to this definition, within a diatonic scale there is only one tritone for each octave. For instance, the above-mentioned interval F–B is the only tritone formed from the notes of the C major scale. A tritone is also commonly defined as an interval spanning six semitones. According to this definition, a diatonic scale contains two tritones for each octave. For instance, the above-mentioned C major scale contains the tritones F–B and B–F. In twelve-equal temperament, the tritone divides the octave exactly in half as 6 of 12 semitones or 600 of 1,200 cents.

Articles related to music include:

A fourth is a musical interval encompassing four staff positions in the music notation of Western culture, and a perfect fourth is the fourth spanning five semitones. For example, the ascending interval from C to the next F is a perfect fourth, because the note F is the fifth semitone above C, and there are four staff positions between C and F. Diminished and augmented fourths span the same number of staff positions, but consist of a different number of semitones.

In music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, or very nearly so.

A seventh chord is a chord consisting of a triad plus a note forming an interval of a seventh above the chord's root. When not otherwise specified, a "seventh chord" usually means a dominant seventh chord: a major triad together with a minor seventh. However, a variety of sevenths may be added to a variety of triads, resulting in many different types of seventh chords.

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.

A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent notes in a 12-tone scale. For example, C is adjacent to C♯; the interval between them is a semitone.

In classical music, a third is a musical interval encompassing three staff positions, and the major third is a third spanning four semitones. Along with the minor third, the major third is one of two commonly occurring thirds. It is qualified as major because it is the larger of the two: the major third spans four semitones, the minor third three. For example, the interval from C to E is a major third, as the note E lies four semitones above C, and there are three staff positions from C to E. Diminished and augmented thirds span the same number of staff positions, but consist of a different number of semitones.

The intervals from the tonic (keynote) in an upward direction to the second, to the third, to the sixth, and to the seventh scale degrees of a major scale are called major.

In music from Western culture, a sixth is a musical interval encompassing six note letter names or staff positions, and the major sixth is one of two commonly occurring sixths. It is qualified as major because it is the larger of the two. The major sixth spans nine semitones. Its smaller counterpart, the minor sixth, spans eight semitones. For example, the interval from C up to the nearest A is a major sixth. It is a sixth because it encompasses six note letter names and six staff positions. It is a major sixth, not a minor sixth, because the note A lies nine semitones above C. Diminished and augmented sixths span the same number of note letter names and staff positions, but consist of a different number of semitones.

The intervals from the tonic (keynote) in an upward direction to the second, to the third, to the sixth, and to the seventh scale degrees (of a major scale are called major.

In Western classical music, a minor sixth is a musical interval encompassing six staff positions, and is one of two commonly occurring sixths. It is qualified as minor because it is the smaller of the two: the minor sixth spans eight semitones, the major sixth nine. For example, the interval from A to F is a minor sixth, as the note F lies eight semitones above A, and there are six staff positions from A to F. Diminished and augmented sixths span the same number of staff positions, but consist of a different number of semitones.

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

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.

Modes of limited transposition are musical modes or scales that fulfill specific criteria relating to their symmetry and the repetition of their interval groups. These scales may be transposed to all twelve notes of the chromatic scale, but at least two of these transpositions must result in the same pitch classes, thus their transpositions are "limited". They were compiled by the French composer Olivier Messiaen, and published in his book La technique de mon langage musical.

In music, consonance and dissonance are categorizations of simultaneous or successive sounds. Within the Western tradition, some listeners associate consonance with sweetness, pleasantness, and acceptability, and dissonance with harshness, unpleasantness, or unacceptability, although there is broad acknowledgement that this depends also on familiarity and musical expertise. The terms form a structural dichotomy in which they define each other by mutual exclusion: a consonance is what is not dissonant, and a dissonance is what is not consonant. However, a finer consideration shows that the distinction forms a gradation, from the most consonant to the most dissonant. In casual discourse, as German composer and music theorist Paul Hindemith stressed, "The two concepts have never been completely explained, and for a thousand years the definitions have varied". The term sonance has been proposed to encompass or refer indistinctly to the terms consonance and dissonance.

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

Music theory analyzes the pitch, timing, and structure of music. It uses mathematics to study elements of music such as tempo, chord progression, form, and meter. The attempt to structure and communicate new ways of composing and hearing music has led to musical applications of set theory, abstract algebra and number theory.

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, an inversion is a type of change 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.

References

↑ Broyles, Michael (1996). "Charles Ives and the American Democratic Tradition". In J. Peter Burkholder (ed.). Charles Ives and His World. Princeton, New Jersey: Princeton University Press.
↑ Rudhyar, Dane (1928). Dissonant Harmony: A New Principle of Musical and Social Organization. Carmel, California: Hamsa Publications.
↑ Ellington, Duke (1993). "Interview in Los Angeles: On Jump for Joy, Opera, and Dissonance as a 'Way of Life"". In Mark Tucker (ed.). The Duke Ellington Reader.
↑ Harrison, Lou (1946). Oscar Baradinsky (ed.). About Carl Ruggles . Yonkers, New York: Alicat Book Shop Press.
↑ Samson, Jim (1977). Music in Transition: A Study of Tonal Expansion and Atonality, 1900–1920. New York City: W. W. Norton. ISBN 0-393-02193-9.
↑ Hanson, Howard. Harmonic Materials of Modern Music: Resources of the Tempered Scale. New York: Appleton-Century-Crofts.
↑ Forte, Allen. The Structure of Atonal Music. New Haven and London: Yale University Press. ISBN 0-300-01610-7.
↑ Lewin, David (1987). Generalized Musical Intervals and Transformations. New Haven, Connecticut: Yale University Press.
↑ Ross, Alex (2008). The Rest Is Noise . London: Fourth Estate. pp. 194–196. ISBN 978-1-84115-475-6.
↑ Ross 2008, pp. 363–364.

v t e Atonality and post-tonality
Scales and tuning	Equal temperament Mode of limited transposition Mystic chord Octatonic scale Whole tone scale
Genres and schools	Twelve-tone technique Row List Second Viennese School Serialism Spectral music
Techniques and concepts	Chromaticism Cyclic set Dissonant counterpoint Dynamic tonality Emancipation of the dissonance Klangfarbenmelodie Polymodal chromaticism Polytonality Distance model Quartal and quintal harmony Tone Clock Tone cluster Unified field
Compositions	List of atonal compositions