Editing Music theory (section)

==As academic discipline==
The scholarly study of music theory in the twentieth century has a number of different subfields, each of which takes a different perspective on what are the primary phenomenon of interest and the most useful methods for investigation.

===Analysis===
{{Main|Musical analysis|Schenkerian analysis|Transformational theory}}
[[File:Debussy Pelleas et Melisande prelude opening.PNG|thumb|upright=1.3|Typically a given work is analyzed by more than one person and different or divergent analyses are created. For instance, the first two bars of the prelude to [[Claude Debussy]]'s ''[[Pelléas et Mélisande (opera)|Pelléas et Melisande]]''
are analyzed differently by Leibowitz, Laloy, van Appledorn, and Christ. Leibowitz analyses this succession harmonically as D minor:I–VII–V, ignoring melodic motion, Laloy analyses the succession as D:I–V, seeing the G in the second measure as an [[musical ornamentation|ornament]], and both van Appledorn and Christ analyse the succession as D:I–VII.
{{audio|Debussy Pelleas et Melisande-prelude opening.mid|Play}}]]
Musical analysis is the attempt to answer the question ''how does this music work?'' The method employed to answer this question, and indeed exactly what is meant by the question, differs from analyst to analyst, and according to the purpose of the analysis. According to [[Ian Bent]], "analysis, as a pursuit in its own right, came to be established only in the late 19th century; its emergence as an approach and method can be traced back to the 1750s. However, it existed as a scholarly tool, albeit an auxiliary one, from the [[Middle Ages]] onwards."{{sfn|Bent|1987|loc=6}}{{incomplete short citation|date=December 2021}} [[Adolf Bernhard Marx]] was influential in formalising concepts about composition and music understanding towards the second half of the 19th century. The principle of analysis has been variously criticized, especially by composers, such as [[Edgard Varèse]]'s claim that, "to explain by means of [analysis] is to decompose, to mutilate the spirit of a work".<ref>Quoted in {{harvnb|Bernard|1981|loc=1}}</ref>

[[Schenkerian analysis]] is a method of musical analysis of tonal music based on the theories of [[Heinrich Schenker]] (1868–1935). The goal of a Schenkerian analysis is to interpret the underlying structure of a tonal work and to help reading the score according to that structure. The theory's basic tenets can be viewed as a way of defining [[tonality]] in music. A Schenkerian analysis of a passage of music shows hierarchical relationships among its pitches, and draws conclusions about the structure of the passage from this hierarchy. The analysis makes use of a specialized symbolic form of musical notation that Schenker devised to demonstrate various [[Schenkerian analysis#Techniques of prolongation|techniques of elaboration]]. The most fundamental concept of Schenker's theory of tonality may be that of ''tonal space''.<ref>Schenker described the concept in a paper titled ''Erläuterungen'' ("Elucidations"), which he published four times between 1924 and 1926: ''Der Tonwille'' (Vienna, Tonwille Verlag, 1924) vol. 8–9, pp. 49–51, vol. 10, pp. 40–42; ''Das Meisterwerk in der Musik'' (München, Drei Masken Verlag), vol. 1 (1925), pp. 201–05; 2 (1926), pp. 193–97. English translation, ''Der Tonwille'', Oxford University Press, vol. 2, pp. 117–18 (the translation, although made from vols. 8–9 of the German original, gives as original pagination that of ''Das Meisterwerk'' 1; the text is the same). The concept of tonal space is still present in {{harvtxt|Schenker|1979|loc=especially p. 14, § 13}}, but less clearly than in the earlier presentation.</ref> The intervals between the notes of the tonic triad form a ''tonal space'' that is filled with passing and neighbour notes, producing new triads and new tonal spaces, open for further elaborations until the surface of the work (the score) is reached.

Although Schenker himself usually presents his analyses in the generative direction, starting from the [[fundamental structure]] (''Ursatz'') to reach the score, the practice of Schenkerian analysis more often is reductive, starting from the score and showing how it can be reduced to its fundamental structure. The graph of the ''Ursatz'' is arrhythmic, as is a strict-counterpoint cantus firmus exercise.{{sfn|Schenker|1979|loc=p. 15, § 21}} Even at intermediate levels of the reduction, rhythmic notation (open and closed noteheads, beams and flags) shows not rhythm but the hierarchical relationships between the pitch-events. Schenkerian analysis is ''subjective''. There is no mechanical procedure involved and the analysis reflects the musical intuitions of the analyst.{{sfn| Snarrenberg|1997|loc={{Page needed|date=July 2015}}}} The analysis represents a way of hearing (and reading) a piece of music.

Transformational theory is a branch of music theory developed by [[David Lewin]] in the 1980s, and formally introduced in his 1987 work, ''Generalized Musical Intervals and Transformations''. The theory, which models [[Transformation (music)|musical transformations]] as elements of a [[Group theory|mathematical group]], can be used to analyze both [[tonality|tonal]] and [[atonal music]]. The goal of transformational theory is to change the focus from musical objects—such as the "C [[major chord]]" or "G major chord"—to relations between objects. Thus, instead of saying that a C major chord is followed by G major, a transformational theorist might say that the first chord has been "transformed" into the second by the "[[dominant (music)|Dominant]] operation." (Symbolically, one might write "Dominant(C major) = G major.") While traditional [[set theory (music)|musical set theory]] focuses on the makeup of musical objects, transformational theory focuses on the [[interval (music)|intervals]] or types of musical motion that can occur. According to Lewin's description of this change in emphasis, "[The transformational] attitude does not ask for some observed measure of extension between reified 'points'; rather it asks: 'If I am ''at'' s and wish to get to t, what characteristic ''gesture'' should I perform in order to arrive there?'"{{sfn|Lewin|1987|loc=159}}

===Music perception and cognition===
{{Further|Music psychology|Fred Lerdahl|Ray Jackendoff}}

Music psychology or the psychology of music may be regarded as a branch of both [[psychology]] and [[musicology]]. It aims to explain and understand musical [[behavior]] and [[experience]], including the processes through which music is perceived, created, responded to, and incorporated into everyday life.{{sfn|Tan, Peter, and Rom|2010|loc=2}}{{sfn|Thompson|n.d.|loc=320}} Modern music psychology is primarily [[Empirical research|empirical]]; its knowledge tends to advance on the basis of interpretations of data collected by systematic [[observation]] of and interaction with [[Human subject research|human participants]]. Music psychology is a field of research with practical relevance for many areas, including music [[musical technique|performance]], [[music composition|composition]], [[music education|education]], [[music criticism|criticism]], and [[music therapy|therapy]], as well as investigations of human [[aptitude]], skill, [[intelligence]], creativity, and [[social behavior]].

Music psychology can shed light on non-psychological aspects of [[musicology]] and musical practice. For example, it contributes to music theory through investigations of the [[perception]] and [[Cognitive musicology|computational modelling]] of musical structures such as [[melody]], [[harmony]], [[tonality]], [[rhythm]], [[Meter (music)|meter]], and [[Musical form|form]]. Research in [[music history]] can benefit from systematic study of the history of [[musical syntax]], or from psychological analyses of composers and compositions in relation to perceptual, affective, and social responses to their music.

===Genre and technique===
{{Main|Music genre|Musical technique}}
[[File:Bundesarchiv B 145 Bild-F008930-0019, Beethovenhalle Bonn, Kammerkonzert Kölner Trio.jpg|thumb|A Classical [[piano trio]] is a group that plays [[chamber music]], including [[sonata]]s. The term "piano trio" also refers to works composed for such a group.]]
A music genre is a conventional category that identifies some pieces of music as belonging to a shared tradition or set of conventions.{{sfn|Samson|n.d.}} It is to be distinguished from ''[[musical form]]'' and ''musical style'', although in practice these terms are sometimes used interchangeably.{{sfn|Wong|2011}}{{Failed verification|date=December 2014}}<!--An electronic search of this web article finds only one occurrence of the word "form", as part of the compound expression; the word "style" does not occur at all. As a result, this citation confirms neither the claim that genre should be distinguished from form and style, nor the claim that they are sometimes used interchangeably in practice.-->

Music can be divided into different [[genres]] in many different ways. The artistic nature of music means that these classifications are often subjective and controversial, and some genres may overlap. There are even varying academic definitions of the term ''genre ''itself. In his book ''Form in Tonal Music'', Douglass M. Green distinguishes between genre and [[Musical form|form]]. He lists [[madrigal (music)|madrigal]], [[motet]], [[canzona]], [[ricercar]], and dance as examples of genres from the [[Renaissance music|Renaissance]] period. To further clarify the meaning of ''genre'', Green writes, "Beethoven's Op. 61 and Mendelssohn's Op. 64 are identical in genre—both are violin concertos—but different in form. However, Mozart's Rondo for Piano, K. 511, and the ''Agnus Dei'' from his Mass, K. 317 are quite different in genre but happen to be similar in form."{{sfn|Green|1979|loc=1}} Some, like [[Peter van der Merwe (musicologist)|Peter van der Merwe]], treat the terms ''genre'' and ''style'' as the same, saying that ''genre'' should be defined as pieces of music that came from the same style or "basic musical language."{{sfn|van der Merwe|1989|loc=3}}

Others, such as Allan F. Moore, state that ''genre'' and ''style'' are two separate terms, and that secondary characteristics such as subject matter can also differentiate between genres.{{sfn|Moore|2001|loc=432–33}} A music genre or subgenre may also be defined by the [[musical technique]]s, the style, the cultural context, and the content and spirit of the themes. Geographical origin is sometimes used to identify a music genre, though a single geographical category will often include a wide variety of subgenres. Timothy Laurie argues that "since the early 1980s, genre has graduated from being a subset of popular music studies to being an almost ubiquitous framework for constituting and evaluating musical research objects".{{sfn|Laurie|2014|loc=284}}

Musical technique is the ability of [[musical instrument|instrumental]] and vocal musicians to exert optimal control of their instruments or [[vocal cords]] to produce precise musical effects. Improving technique generally entails practicing exercises that improve muscular sensitivity and agility. To improve technique, musicians often practice fundamental patterns of notes such as the [[Natural minor|natural]], [[Minor scale|minor]], [[Major scale|major]], and [[chromatic scale]]s, [[Minor triad|minor]] and [[major triad]]s, [[Dominant seventh chord|dominant]] and [[diminished seventh]]s, formula patterns and [[arpeggio]]s. For example, [[Triad (music)|triads]] and [[Seventh chord|sevenths]] teach how to play chords with accuracy and speed. [[Scale (music)|Scales]] teach how to move quickly and gracefully from one note to another (usually by step). Arpeggios teach how to play [[broken chord]]s over larger intervals. Many of these components of music are found in compositions, for example, a scale is a very common element of classical and romantic era compositions.{{citation needed|date=July 2015}}

[[Heinrich Schenker]] argued that musical technique's "most striking and distinctive characteristic" is [[repetition (music)|repetition]].{{sfn|Kivy|1993|loc=327}} Works known as [[étude]]s (meaning "study") are also frequently used for the improvement of technique.

===Mathematics===
{{Main|Music and mathematics}}
Music theorists sometimes use mathematics to understand music, and although music has no [[axiomatic]] foundation in modern mathematics, mathematics is "the basis of sound" and sound itself "in its musical aspects... exhibits a remarkable array of number properties", simply because nature itself "is amazingly mathematical".{{sfn|Smith Brindle|1987|loc=42–43}} 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]]. Some composers have incorporated the [[golden ratio]] and [[Fibonacci numbers]] into their work.{{sfn|Smith Brindle|1987|loc=chapter 6, ''passim''}}{{sfn|Garland and Kahn|1995|loc={{Page needed|date=July 2015}}}} There is a long history of examining the relationships between music and mathematics. Though ancient Chinese, Egyptians and Mesopotamians are known to have studied the mathematical principles of sound,{{sfn|Smith Brindle|1987|loc=42}} the [[Pythagoreanism|Pythagoreans]] (in particular [[Philolaus]] and [[Archytas]]){{sfn|Purwins|2005|loc=22–24}} of ancient Greece were the first researchers known to have investigated the expression of [[musical scale]]s in terms of numerical [[ratio]]s.
[[File:HarmonicIdentities.Names.Frequencies.svg|thumb|right|400px|The first 16 harmonics, their names and frequencies, showing the exponential nature of the octave and the simple fractional nature of non-octave harmonics]]
In the modern era, musical [[set theory]] uses the language of mathematical set theory in an elementary way to organize musical objects and describe their relationships. To analyze the structure of a piece of (typically atonal) music using musical set theory, one usually starts with a set of tones, which could form motives or chords. By applying simple operations such as [[transposition (music)|transposition]] and [[Melodic inversion|inversion]], one can discover deep structures in the music. Operations such as transposition and inversion are called [[isometries]] because they preserve the intervals between tones in a set. Expanding on the methods of musical set theory, some theorists have used abstract algebra to analyze music. For example, the pitch classes in an equally tempered octave form an [[abelian group]] with 12 elements. It is possible to describe [[just intonation]] in terms of a [[free abelian group]].{{sfn|Wohl|2005}}

===Serial composition and set theory===
[[File:Berg's Lyric Suite Mov. I tone row B-P.PNG|thumb|Tone row from [[Alban Berg]]'s ''[[Lyric Suite (Berg)|Lyric Suite]]'', movement I[[File:Berg's Lyric Suite Mov. I tone row B-P.mid]]]]
{{Further|Serialism|Set theory (music)|Arnold Schoenberg|Milton Babbitt|David Lewin|Allen Forte}}
In music theory, serialism is a method or technique of [[Musical composition|composition]] that uses a series of values to manipulate different [[aspect of music|musical elements]]. Serialism began primarily with [[Arnold Schoenberg]]'s [[twelve-tone technique]], though his contemporaries were also working to establish serialism as one example of [[atonality|post-tonal]] thinking. Twelve-tone technique orders the twelve notes of the [[chromatic scale]], forming a [[tone row|row]] or series and providing a unifying basis for a composition's [[melody]], [[harmony]], structural progressions, and [[variation (music)|variations]]. Other types of serialism also work with [[set (music)|sets]], collections of objects, but not necessarily with fixed-order series, and extend the technique to other musical dimensions (often called "[[parameter (music)|parameters]]"), such as [[duration (music)|duration]], [[Dynamics (music)|dynamics]], and [[timbre]]. The idea of serialism is also applied in various ways in the visual arts, design, and architecture<ref>{{harvnb|Bandur|2001|loc=5, 12, 74}}; {{harvnb|Gerstner|1964|loc=passim}}</ref>

"Integral serialism" or "total serialism" is the use of series for aspects such as duration, dynamics, and register as well as pitch.{{Sfn|Whittall|2008|loc=273}} Other terms, used especially in Europe to distinguish post-World War II serial music from twelve-tone music and its American extensions, are "general serialism" and "multiple serialism".{{sfn|Grant|2001|loc=5–6}}

Musical set theory provides concepts for categorizing musical objects and describing their relationships. Many of the notions were first elaborated by [[Howard Hanson]] (1960) in connection with tonal music, and then mostly developed in connection with atonal music by theorists such as [[Allen Forte]] (1973), drawing on the work in twelve-tone theory of Milton Babbitt. The concepts of set theory are very general and can be applied to tonal and atonal styles in any equally tempered tuning system, and to some extent more generally than that.{{Citation needed|date=July 2015}}

One branch of musical set theory deals with collections (sets and permutations) of pitches and pitch classes (pitch-class set theory), which may be ordered or unordered, and can be related by musical operations such as [[Transposition (music)|transposition]], [[Melodic inversion|inversion]], and [[Complement (music)|complementation]]. The methods of musical set theory are sometimes applied to the analysis of rhythm as well.{{Citation needed|date=July 2015}}

===Musical semiotics===
{{Further|Music semiology|Jean-Jacques Nattiez}}
[[File:Roman Jakobson.jpg|thumb|upright|Semiotician [[Roman Jakobson]]]]
Music semiology ([[semiotics]]) is the study of signs as they pertain to music on a variety of levels. Following [[Roman Jakobson]], [[Kofi Agawu]] adopts the idea of musical semiosis being introversive or extroversive—that is, musical signs within a text and without.{{Citation needed|date=July 2015}} "Topics", or various musical conventions (such as horn calls, dance forms, and styles), have been treated suggestively by Agawu, among others.{{Citation needed|date=July 2015}} The notion of [[Musical Gestures|gesture]] is beginning to play a large role in musico-semiotic enquiry.{{Citation needed|date=July 2015}}

:"There are strong arguments that music inhabits a semiological realm which, on both [[ontogenetic]] and [[phylogenetic]] levels, has developmental priority over verbal language."{{sfn|Middleton|1990|loc=172}}{{sfn|Nattiez|1976}}{{sfn|Nattiez|1990}}{{sfn|Nattiez1989}}{{sfn|Stefani|1973}}{{sfn|Stefani|1976}}{{sfn|Baroni|1983}}{{sfn|''Semiotica''|1987|loc=66:1–3}}{{incomplete short citation|date=December 2021}}{{clarify |date=November 2018|reason=The quotation cannot stem from all these references at once!}}

Writers on music semiology include Kofi Agawu (on topical theory,{{Citation needed|date=July 2015}} [[Heinrich Schenker]],{{sfn|Dunsby|Stopford|1981|loc=49–53}}{{sfn|Meeùs|2017|loc=81–96}} Robert Hatten (on topic, gesture){{Citation needed|date=July 2015}}, [[Raymond Monelle]] (on topic, musical meaning){{Citation needed|date=July 2015}}, [[Jean-Jacques Nattiez]] (on introversive taxonomic analysis and ethnomusicological applications){{Citation needed|date=July 2015}}, [[Anthony Newcomb]] (on narrativity){{Citation needed|date=July 2015}}, and [[Eero Tarasti]]{{Citation needed|date=July 2015}}.

[[Roland Barthes]], himself a semiotician and skilled amateur pianist, wrote about music in ''Image-Music-Text,''{{Full citation needed|date=July 2015}} ''The Responsibilities of Form,''{{Full citation needed|date=July 2015}} and ''Eiffel Tower,''{{Full citation needed|date=July 2015}} though he did not consider music to be a semiotic system{{Citation needed|date=August 2015}}.

Signs, meanings in music, happen essentially through the connotations of sounds, and through the social construction, appropriation and amplification of certain meanings associated with these connotations. The work of [[Philip Tagg]] (''Ten Little Tunes'',{{Full citation needed|date=July 2015}} ''Fernando the Flute'',{{Full citation needed|date=July 2015}} ''Music's Meanings''{{Full citation needed|date=July 2015}}) provides one of the most complete and systematic analysis of the relation between musical structures and connotations in western and especially popular, television and film music. The work of [[Leonard B. Meyer]] in ''Style and Music''{{Full citation needed|date=July 2015}} theorizes the relationship between ideologies and musical structures and the phenomena of style change, and focuses on romanticism as a case study.

===Education and careers===
[[File:Pat Carpenter picture.JPG|thumb|[[Columbia University]] music theorist [[Patricia Carpenter (music theorist)|Pat Carpenter]] in an undated photo]]
Music theory in the practical sense has been a part of education at conservatories and music schools for centuries, but the status music theory currently has within academic institutions is relatively recent. In the 1970s, few universities had dedicated music theory programs, many music theorists had been trained as composers or historians, and there was a belief among theorists that the teaching of music theory was inadequate and that the subject was not properly recognised as a scholarly discipline in its own right.{{sfn|McCreless|n.d.}} A growing number of scholars began promoting the idea that music theory should be taught by theorists, rather than composers, performers or music historians.{{sfn|McCreless|n.d.}} This led to the founding of the [[Society for Music Theory]] in the United States in 1977. In Europe, the French ''Société d'Analyse musicale'' was founded in 1985. It called the First European Conference of Music Analysis for 1989, which resulted in the foundation of the ''Société belge d'Analyse musicale'' in Belgium and the ''Gruppo analisi e teoria musicale'' in Italy the same year, the ''Society for Music Analysis'' in the UK in 1991, the ''Vereniging voor Muziektheorie'' in the Netherlands in 1999 and the ''Gesellschaft für Musiktheorie'' in Germany in 2000.{{sfn|Meeùs|2015|loc=111}} They were later followed by the Russian Society for Music Theory in 2013, the Polish Society for Music Analysis in 2015 and the ''Sociedad de Análisis y Teoría Musical'' in Spain in 2020, and others are in construction. These societies coordinate the publication of music theory scholarship and support the professional development of music theory researchers. They formed in 2018 a network of European societies for Theory and/or Analysis of Music, the [https://europeanmusictheory.eu/ EuroT&AM]

As part of their initial training, music theorists will typically complete a [[B.Mus]] or a [[Bachelor of Arts|B.A.]] in music (or a related field) and in many cases an M.A. in music theory. Some individuals apply directly from a bachelor's degree to a PhD, and in these cases, they may not receive an M.A. In the 2010s, given the increasingly interdisciplinary nature of university graduate programs, some applicants for music theory PhD programs may have academic training both in music and outside of music (e.g., a student may apply with a B.Mus. and a Masters in Music Composition or Philosophy of Music).

Most music theorists work as instructors, lecturers or professors in colleges, universities or [[Music school|conservatories]]. The job market for tenure-track professor positions is very competitive: with an average of around 25 tenure-track positions advertised per year in the past decade, 80–100 PhD graduates are produced each year (according to the Survey of Earned Doctorates) who compete not only with each other for those positions but with job seekers that received PhD's in previous years who are still searching for a tenure-track job.<!--How competitive is "very"? As competitive as for, say, a position as an airline pilot?--> Applicants must hold a completed PhD or the equivalent degree (or expect to receive one within a year of being hired—called an "ABD", for "[[All But Dissertation]]" stage) and (for more senior positions) have a strong record of publishing in peer-reviewed journals. Some PhD-holding music theorists are only able to find insecure positions as [[sessional lecturer]]s. The job tasks of a music theorist are the same as those of a professor in any other humanities discipline: teaching undergraduate and/or graduate classes in this area of specialization and, in many cases some general courses (such as [[Music appreciation]] or Introduction to Music Theory), conducting research in this area of expertise, publishing research articles in peer-reviewed journals, authoring book chapters, books or textbooks, traveling to conferences to present papers and learn about research in the field, and, if the program includes a graduate school, supervising M.A. and PhD students and giving them guidance on the preparation of their theses and dissertations. Some music theory professors may take on senior administrative positions in their institution, such as [[Dean (education)|Dean]] or Chair of the School of Music.