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{{Short description|Essay by Fred Lerdahl}}

{{Format footnotes|date=September 2020|reason=Parenthetical referencing has been [[WP:PARREF|deprecated]]; convert to [[Help:Shortened footnotes|shortened footnotes]].}}

'''"Cognitive Constraints on Compositional Systems"''' is an essay by [[Fred Lerdahl]] that cites [[Pierre Boulez]]'s ''[[Le Marteau sans Maîtremaître]]'' (1955) as an example of "a huge gap between compositional system and cognized result," though he "could have illustrated just as well with works by [[Milton Babbitt]], [[Elliott Carter]], [[Luigi Nono (composer)|Luigi Nono]], [[Karlheinz Stockhausen]], or [[Iannis Xenakis]]". (In [[semiologysemiotics|semiological]] terms, this is a gap between the [[esthesic and poietic]] processes.) To explain this gap, and in hopes of bridging it, Lerdahl proposes the concept of a [[musical grammar]], "a limited set of rules that can generate indefinitely large sets of musical events and/or their structural descriptions.". He divides this further into compositional grammar and listening grammar, the latter being one "more or less unconsciously employed by auditors, that generates mental representations of the music". He divides the former into natural and artificial compositional grammars. While the two have historically been fruitfully mixed, a natural grammar arises spontaneously in a culture while an artificial one is a conscious invention of an individual or group in a culture; the gap can arise only between listening grammar and artificial grammars. To begin to understand the listening grammar, Lerdahl and [[Ray Jackendoff]] created a theory of musical cognition, ''[[Generative theory of tonal music|A Generative Theory of Tonal Music]]'' (1983; {{ISBN|0-262-62107-X}}). That theory is outlined in the essay. Lerdahl's constraints on artificial compositional grammars are:

==Constraints on event sequences==

Lerdahl's constraints on artificial compositional grammars are:

===Event sequences===

*''Constraint 1'': The musical surface must be capable of being parsed into a sequence of discrete events.

**[counterexample: [[György Ligeti|Ligeti]], [[computer music]]]

*''Constraint 2'': The musical surface must be available for [[hierarchical]] structuring by the listening grammar.

**[through grouping structure, metrical structure, time-span reduction, and prolongational reduction. "Associational" factors such as [[Motif (music)|motivic]] development and [[timbre|timbral]] relations are ignored, but for "Timbral Hierarchies" see Lerdahl 1987]{{incomplete short citation|date=December 2021|reason=Or is 1988 meant?}}

*''Constraint 3'': The establishment of local grouping boundaries requires the presence of salient distinctive transitions at the musical surface.

**[counterexample: [[minimal music]]]

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*''Constraint 8'': The projection of a prolongational tree depends on a corresponding time-span tree in conjunction with a set of stability conditions.

==Constraints on underlying=Underlying materials===

*''Constraint 9'': Stability conditions must operate on a fixed collection of elements.

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*''Constraint 10'': [[Interval (music)|Intervals]] between elements of a collection arranged along a scale should fall within a certain range of magnitude.

*''Constraint 11'': A [[pitch (music)|pitch]] collection should recur at the [[octave]] to produce [[pitch class]]es.

**[ {{Bracket|[[octave equivalency]] ]}}

*''Constraint 12'': There must be a strong [[psychoacoustic]] basis for stability conditions. For pitch collections, that requires intervals that proceed gradually from very small to comparatively large frequency [[ratio]]s.

**[ {{Bracket|[[just intonation]] ]}}

*''Constraint 13'': Division of the octave into equal parts facilitates [[transposition (music)|transposition]] and reduces memory load.

**[ {{Bracket|[[equal temperament]] ]}}

*''Constraint 14'': Assume pitch sets of ''n''-fold equal divisions of the octave. Then subsets that satisfy uniqueness, coherence, and simplicity will facilitate ___location within the overall pitch space.

**[only certain divisions of the octave, 12 and 20 included, allow uniqueness, coherence, and transpositional simplicity, and only the diatonic and pentatonic subsets of the 12-tone chromatic set follow these constraints (Balzano, 1980, 1982){{incomplete short citation|date=December 2021}}]

===Pitch space===

*''Constraint 15'': Any but the most primitive stability conditions must be susceptible to multidimensional representation, where spatial distance correlates with cognitive distance.

*''Constraint 16'': Levels of pitch space must be sufficiently available from musical surfaces to be internalized.

*''Constraint 17'': A reductionally organized pitch space is needed to express the [[steps and skips]] by which cognitive distance is measured and to express degrees of melodic completeness.

**[completedness resembles implication-realization theory (Meyer, 1973{{incomplete short citation|date=December 2021}} and Narmour, 1977{{incomplete short citation|date=December 2021}}), the ''Zug'', ''[[Urlinie]]'', and ''[[Bassbrechung]]'' ([[Heinrich Schenker|Schenker]]){{incomplete short citation|date=December 2021}}.]

He concludes, "Some of these constraints seem to me binding, others optional. Constraints 9–12 are essential for the very existence of stability conditions. Constraints 13–17, on the other hand, can be variously jettisoned." Examples given are [[Carnatic music|South Indian music]], which doesn't [[modulation (music)|modulate]] and isn't equally tempered (13 & 14), and music such as that of [[Claude Debussy]], [[Béla Bartók]], and others who "have developed consonance-dissonance patterns directly from the [[total chromatic]]" (14–17).

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*''Aesthetic Claim 2'': The best music arises from an alliance of a compositional grammar with the listening grammar.

To these ends he proposes the use of the terms "[[complexity]]" and "complicatedness", complexity being hierarchical structural richness, and complicatedness being "numerous non-redundant events per unit time." On Lerdahl's view complexity has aesthetic value, while complicatedness is neutral. He writes, "All sorts of music satisfy these criteria —– for example, Indian [[raga]], Japanese [[Koto (musical instrument)|koto]], [[jazz]], and most Western [[art music]]. [[Rock music]] fails on grounds of insufficient complexity. Much [[contemporary music]] pursues complicatedness as compensation for a lack of complexity. In short, these criteria allow for infinite variety but only along certain lines."

"I find this conclusion both exciting and —– initially at least —– alarming...the constraints are tighter than I bargained for."

"My second aesthetic claim in effect rejects this ["progressivist"] attitude in favor of the older view that music-making should be based on "nature". For the ancients, nature may have resided in the music of the spheres, but for us it lies in the musical mind."

== Reception ==

Lerdahl's paper has elicited many responses.

Lerdahl's paper has elicited many responses. While writer [[Roger Scruton]], author of [[How to Be a Conservative]], praised it, calling it "hard-hitting" ([[#Scruton|Scruton 2016]]), there have been a number of negative reactions.

[[Roger Scruton]] praised it, calling it "hard-hitting".{{sfn|Scruton|2016}}

[[Nicholas Cook]] wrote, "The idea that music is a process of communication in which listeners decode structures that composers have encoded... is... based on several disputable assumptions: that people choose to listen grammatically; that there is, or ought to be, an equivalance between compositional and listening grammars; and, most fundamentally, that there is such a thing as musical grammar" ([[#Cook1994|Cook 1994, 88]]). He writes that Lerdahl

[[Nicholas Cook]] wrote, "The idea that music is a process of communication in which listeners decode structures that composers have encoded... is... based on several disputable assumptions: that people choose to listen grammatically; that there is, or ought to be, an equivalanceequivalence between compositional and listening grammars; and, most fundamentally, that there is such a thing as musical grammar" ([[#Cook1994.{{sfn|Cook |1994, |p=88]]).}} He writes that Lerdahl

<blockquote>...assume(s) that there should be a more or less linear relationship between the manner in which a composer conceives a composition and the manner in which a listener perceives it. ...Lerdahl's aim is to specify the conditions that must be fulfilled if there is to be conformity between 'compositional grammar' and 'listening grammar'. And... he ends up by measuring existing music against the stipulations of his theory, using this as a basis for aesthetic evaluation. The result is to write off not only the Darmstadt avant-garde and minimalism, but also huge swathes of non-Western and popular music. ([[#Cook1999|Cook 1999, 241]])</blockquote>

<blockquote>...assume([s)] that there should be a more or less linear relationship between the manner in which a composer conceives a composition and the manner in which a listener perceives it. ...Lerdahl's aim is to specify the conditions that must be fulfilled if there is to be conformity between '"compositional grammar'" and '"listening grammar'". And... he ends up by measuring existing music against the stipulations of his theory, using this as a basis for aesthetic evaluation. The result is to write off not only the Darmstadt avant-garde and minimalism, but also huge swathes of non-Western and popular music. ([[#Cook1999{{sfn|Cook |1999, |p=241]])}}</blockquote>

He asks:

<blockquote>What... does an article like Lerdahl's '"Cognitive Constraints on Compositional Systems'" actually do? By subordinating the production and reception of music to theoretically defined criteria of communicative success, it creates a charmed hermeneutic circle that excludes everything from critical musicology to social psychology. It slips imperceptibly from description to prescription, so reinforcing the hegemony of theory. In this way, while the literary genre of Lerdahl's article is the scientific paper—apaper – a genre predicated on the transparent representation of an external reality—itsreality – its substance lies at least equally in its illocutionary force. ([[#Cook1999{{sfn|Cook |1999, |p=252]])}}</blockquote>

(See further discussion in ([[#Cook2007|Cook 2007, 252]]).)

Vincent Meelberg wrote:

<blockquote>Lerdahl relates musical comprehension to the reconstruction of compositional methods. As soon as the listener finds out how a piece is composed, Lerdahl argues, s/he has comprehended the music. In other words: he seems to claim that there is a single, true grasp of music, namely the knowledge of the compositional method. Yet, while knowledge of these methods might be helpful, it is by far not the only means by which the listener can structure the music, and in so doing gains musical comprehension. Rather, musical comprehension can be established through... a process which allows for many different ways to comprehend the same musical piece. Musical comprehension depends on the relation between the (individual) listener and the musical work. Hence, the individual listener has a decisive influence on the way that work is grasped, which in turn results in the existence of many different musical structures by which the music can be grasped and comprehended. ([[#Meelberg2006|Meelberg 2006, 29]])</blockquote>

Morag Josephine Grant wrote, "The paradox of Lerdahl's argument... is that while it is perfectly acceptable to adopt the composer's own system when dealing with compositional-technical analysis, it seems equally acceptable to revert to musical thinking of a quite different type when the aural result comes to be analysed" ([[#Grant2001|Grant 2001, 218]]). She continued, "Lerdahl's argument that musical language, like spoken language, is generative in structure excludes the possibility of other, non-hierarchical methods of achieving musical coherence... Lerdahl's concentration on the audibility of the row... blinds or deafens him to the simple fact that the use of the row is itself a constraint, not just on the composer, but in the aid of comprehensibility as well" ([[#Grant2001|Grant 2001, 219]]).

John Bouz wrote that he "finds it concerning that some prominent perception-based theories tend to correlate 'good music' with that which can be used to best showcase the analytical system itself. All too often the application of these theories by theorists is done backwards: the theory is used to determine the value of music (and therefore constrains music), instead of being tested by the music. Lerdahl’s article 'Cognitive Constraints on Compositional Systems' is an example in which this type of dubious inversion occurs" ([[#Bouz2013|Bouz 2013, 94]]).

Robert Rowe wrote:

<blockquote>In my view, Fred Lerdahl should not be so surprised that Boulez’s serial technique, his "compositional grammar," is often treated as if it were irrelevant, particularly when he himself notes that much of the interest of the piece comes from the workings of a musical mind operating beyond the scope of the purely formal rules. He cannot have it both ways: he cannot maintain that what makes ''Le Marteau'' a great piece of music is Boulez's musicianship, his "intuitive constraints," and maintain at the same time that music cannot be great unless cognition is explicitly coded into the formal system. To say that "the best music arises from an alliance of a compositional grammar with the listening grammar" and at the same time to recognize ''Le Marteau'' as a "remarkable" work when no such alliance occurs must mean that the Aesthetic Claim 2 carries little force indeed ([[#Rowe1992|Rowe 1992, 105]]).</blockquote>

He continued:

<blockquote>What is important is the way listeners make sense of music, a sense employed by composers, performers, and listeners alike... Basing aesthetic claims, and establishing constraints on composition, on an incomplete account again amounts to overestimating the theory and shortchanging the mind’s capacity to deal with many different kinds of music ([[#Rowe1992|Rowe 1992, 105]]).</blockquote>

Morag Josephine Grant wrote, "The paradox of Lerdahl's argument... is that while it is perfectly acceptable to adopt the composer's own system when dealing with compositional-technical analysis, it seems equally acceptable to revert to musical thinking of a quite different type when the aural result comes to be analysed" ([[#Grant2001.{{sfn|Grant |2001, |p=218]]).}} She continued, "Lerdahl's argument that musical language, like spoken language, is generative in structure excludes the possibility of other, non-hierarchical methods of achieving musical coherence... Lerdahl's concentration on the audibility of the row... blinds or deafens him to the simple fact that the use of the row is itself a constraint, not just on the composer, but in the aid of comprehensibility as well" ([[#Grant2001.{{sfn|Grant |2001, |p=219]]).}}

John Croft's master's degree thesis examines Lerdahl's essay in depth. In his conclusion, he wrote:

<blockquote>we have plenty of music that does not conform to Lerdahl's grammar: what, then, are people who claim to find it as interesting as tonal-metrical music actually doing? Either they are deluding themselves, or they are lying, or they have non-human brains. None of these answers seems entirely satisfactory. But if we do not like any of these answers, then we must admit that it is a matter of exposure and acquired understanding after all, in which case we are certainly a far cry from innate psychological universals ([[#Croft1999|Croft 1999, 54]]) [...] Vague language and tacit assumptions can be brought into the service of conservativism and aesthetic authoritarianism. It points to the misguided nature of attempts to turn the question of the dissemination of post-tonal music from an aesthetic, political, and indeed economic issue into a cognitive-scientific one. In this age when words like 'accessibility' and 'communication' are used too frequently and with too little understanding, it is of some significance that at least one major attempt to give scientific respectability to the conservative side of the debate fails ([[#Croft1999|Croft 1999, 55]]).</blockquote>

John Croft's master's degree thesis examines Lerdahl's essay in depth. In his conclusion, he wrote:<blockquote>we have plenty of music that does not conform to Lerdahl's grammar: what, then, are people who claim to find it as interesting as tonal-metrical music actually doing? Either they are deluding themselves, or they are lying, or they have non-human brains. None of these answers seems entirely satisfactory. But if we do not like any of these answers, then we must admit that it is a matter of exposure and acquired understanding after all, in which case we are certainly a far cry from innate psychological universals ([[#Croft1999|Croft 1999, 54]]) [...] Vague language and tacit assumptions can be brought into the service of conservativismconservatism and aesthetic authoritarianism. It points to the misguided nature of attempts to turn the question of the dissemination of post-tonal music from an aesthetic, political, and indeed economic issue into a cognitive-scientific one. In this age when words like '"accessibility'" and '"communication'" are used too frequently and with too little understanding, it is of some significance that at least one major attempt to give scientific respectability to the conservative side of the debate fails ([[#Croft1999.{{sfn|Croft |1999, 55]]).|pp=54–55}}</blockquote>

Arved Ashby commented:

For additional opinions and discussion, see [[#Bauer2004{{harvtxt|Bauer 2004]], [[#Boros1995Boros|Boros 1995]]}}, [[#Boros1996{{harvtxt|Boros |1996]]}}, [[#Denham2009{{harvtxt|Denham |2009]]}}, [[#Dibben1996{{harvtxt|Dibben |1996]]}}, [[#Heinemann1993{{harvtxt|Heinemann |1993]]}}, [[#Heinemann1998{{harvtxt|Heinemann |1998]], [[#Horn2015|Horn 2015]]}}, and [[#Mosch2004{{harvtxt|Mosch |2004]]}}.

<blockquote>I don't believe music, even serial music, need be limited to a grammatical-linguistic conception of meaning. I would go further and say that modern art or art in any period, for that matter need not function grammatically. The work of Mallarme and the Symbolist poets tends toward the oblique: they take lexical items (words) and ask that we process them not lexically, denotatively, but at a musical (or at least metaphorical) level. If we were to take a Lerdahlian approach to these writers, we could reach some absurd parallel conclusions: by analogy to Lerdahl's tacitly intentionalist thinking, one could go so far as to try and discredit the Symbolists for being ungrammatical, or say that they produced works that are ''unsuccessful'' because of an anomaly between the language at their surface and the means by which they are to be taken in ([[#Ashby2004|Ashby 1992, 33]]).</blockquote>

== References ==

For additional opinions and discussion, see [[#Bauer2004|Bauer 2004]], [[#Boros1995|Boros 1995]], [[#Boros1996|Boros 1996]], [[#Denham2009|Denham 2009]], [[#Dibben1996|Dibben 1996]], [[#Heinemann1993|Heinemann 1993]], [[#Heinemann1998|Heinemann 1998]], [[#Horn2015|Horn 2015]], and [[#Mosch2004|Mosch 2004]].

{{reflist|15em}}

'''Sources'''

{{div col|colwidth=45em}}

* {{cite journal|last=Boros|first=James|year=1995|title=A 'New Totality'?|journal=[[Perspectives of New Music]]|volume=33|number=1/2|pages=538–553}}

* {{cite journal|last=Boros|first=James|year=1996|title=A Response to Lerdahl|journal=Perspectives of New Music|volume=34|number=1|pages=252–258|doi=10.2307/833505 |jstor=833505 }}

* {{wikicitecite book|reflast=Cook1994 Cook|referencefirst=Nicholas|author-link=Cook, Nicholas. Cook|year=1994. "|chapter=Perception: A Perspective from Music Theory." In ''|title=Musical Perceptions'', ed. |editor1=Rita Aiello|editor2=[[John with Sloboda|John A. Sloboda, ]]|pages=64–95. |___location=Oxford: |publisher=Oxford University Press.}}

* {{wikicitecite book|reflast=Cook1999 Cook|referencefirst=Cook, Nicholas. |year=1999. "|chapter=Analysing Performance and Performing Analysis." In ''|title=Rethinking Music'', ed. |editor1=Nicholas Cook and |editor2=[[Mark Everist, ]]|pages=239–261. |___location=Oxford: |publisher=Oxford University Press.}}

* {{wikicitecite thesis|reflast=Croft1999 Croft|referencefirst=Croft, John. |year=1999. ''|title=Musical memory, complexity, and Lerdahl’sLerdahl's cognitive constraints.'' A Thesis submitted for the degree of |type=Master of Music, thesis|publisher=Department of Music, The University of Sheffield.}}

* {{cite journal|last=Denham|first=A. E.|date=September 2009|title=The Future of Tonality|journal=[[British Journal of Aesthetics]]|volume=49 |issue=4 |pages=427–450|doi=10.1093/aesthj/ayp031 }}

* {{wikicitecite thesis|reflast=Dibben1996 Dibben|referencefirst=Dibben, Nicola. |year=1996. ''|title=The Role of Reductional Representations in the Perception of Atonal Music.'' |type=Unpublished doctoral dissertation, |publisher=Department of Music, The University of Sheffield.}}

* {{wikicitecite book|reflast=Grant2001 Grant|referencefirst=Grant, Morag Josephine. |year=2001. ''|title=Serial Music Serial Aesthetics: Compositional Theory in Post-War Europe.'' |series=Music in the Twentieth Century, |others=[[Arnold Whittall]], general editor. |___location=Cambridge: |publisher=Cambridge University Press.}}

* {{wikicitecite thesis|reflast=Heinemann1993 Heinemann|referencefirst=Heinemann, Stephen. |year=1993. "|title=Pitch-class Set Multiplication in Boulez’sBoulez's "'Le Marteau sans maitre"' with [Original Composition]"". A Thesis submitted for the degree of |type=Doctor of Musical Arts, thesis|publisher=Department of Music, University of WA.Washington}}

* {{wikicitecite journal|reflast=Heinemann1998 Heinemann|referencefirst=Heinemann, Stephen. |year=1998. "|title=Pitch-Class Set Multiplication in Theory and Practice". ''|journal=[[Music Theory Spectrum'' ]]|volume=20, no. |number=1: |pages=72–96|doi=10.2307/746157 |jstor=746157 }}

* {{wikicitecite book|reflast=Mosch2004 Mosch|referencefirst=Ulrich|author-link=Mosch, Ulrich. Mosch|year=2004. ''|title=Musikalisches Hören serieller Musik: Untersuchungen am Beispiel von Pierre Boulez’Boulez' «Le Marteau sans maître».'' |___location=Saarbrücken: |publisher=Pfau-Verlag.}}

* {{wikicitecite web|reflast=Scruton |referencefirst=Roger|author-link=Scruton, Roger. Scruton|year=2016. "[|url=http://www.futuresymphony.org/the-clothes-have-no-emperor |title=The Clothes Have No Emperor]". |website=Future Symphony Institute (accessed |access-date=12 December 2021).}}

{{div col end}}

==Further Sources reading==

{{div col|colwidth=45em}}

* {{wikicite |ref=Ashby2004 |reference=Ashby, Arved. 2004. "Intention and Meaning in Modernist Music." In ''The Pleasure of Modernist Music: Listening, Meaning, Intention, Ideology'', ed. Arved Ashby, 23-45. Rochester, NY: University of Rochester Press.}}

* {{wikicite |ref=Bauer2004 |reference=BauerAshby, AmyArved. 2004. "'Tone-Color, Movement, Changing Harmonic Planes': Cognition, Constraints,Intention and Conceptual BlendsMeaning in Modernist Music." In ''The Pleasure of Modernist Music: Listening, Meaning, Intention, Ideology'', ed. Arved Ashby, 121-15223–45. Rochester, NYNew York: University of Rochester Press.}}

* {{wikicite |ref=Ashby2004 |reference=AshbyBauer, ArvedAmy. 2004. "Intention'Tone-Color, Movement, Changing Harmonic Planes': Cognition, Constraints, and MeaningConceptual Blends in Modernist Music." In ''The Pleasure of Modernist Music: Listening, Meaning, Intention, Ideology'', ed. Arved Ashby, 23-45121–152. Rochester, NYNew York: University of Rochester Press.}}

* {{wikicite |ref=Boros1995 |reference=Boros, James. 1995. "A 'New Totality'?". ''Perspectives of New Music'' 33, no. 1/2: 538-553.}}

* {{wikicite |ref=Bouz2013 |reference=Bouz, John. 2013. ''Examining the Application of Principles of Auditory Perception to Music Analysis: Building a Perception-based Music Theory to Capture Musical Features of Palestrina’sPalestrina's Missa Aeterna Christi Munera.'' A Thesis submitted for the degree of Master of Arts thesis, Department of Music, University of Calgary.}}

* {{wikicite |ref=Boros1996 |reference=Boros, James. 1996. "A Response to Lerdahl". ''Perspectives of New Music'' 34, no. 1: 252-258.}}

* {{wikicite |ref=Cook2007[[Nicholas Cook|reference=Cook, Nicholas]]. 2007. ''Music, Performance, Meaning: Selected Essays.'' Ashgate Contemporary Thinkers on Critical Musicology Series. Aldershot: Ashgate.}}

* {{wikicite |ref=Bouz2013 |reference=Bouz, John. 2013. ''Examining the Application of Principles of Auditory Perception to Music Analysis: Building a Perception-based Music Theory to Capture Musical Features of Palestrina’s Missa Aeterna Christi Munera.'' A Thesis submitted for the degree of Master of Arts, Department of Music, University of Calgary.}}

* {{wikicite |ref=Horn2015 |reference=Horn, Walter. 2015. "Tonality, Musical Form, and Aesthetic Value". ''[[Perspectives of New Music]]'' 53, no. 2: 201-235201–235.}}

* {{wikicite |ref=Cook1994 |reference=Cook, Nicholas. 1994. "Perception: A Perspective from Music Theory." In ''Musical Perceptions'', ed. Rita Aiello with John A. Sloboda, 64–95. Oxford: Oxford University Press.}}

* [[Fred Lerdahl|Lerdahl, Fred]] (1988). "Cognitive Constraints on Compositional Systems." In ''Generative Processes in Music: The Psychology of Performance, Improvisation, and Composition'', ed. [[John Sloboda]], 231-59231–259. Oxford: Oxford University Press. Reprinted in ''Contemporary Music Review'' 6/2 (1992), pp.&nbsp;97–121. ([http://www.bussigel.com/lerdahl/pdf/Cognitive%20Constraints%20on%20Compositional%20Systems.pdf text])

* {{wikicite |ref=Cook1999 |reference=Cook, Nicholas. 1999. "Analysing Performance and Performing Analysis." In ''Rethinking Music'', ed. Nicholas Cook and Mark Everist, 239–261. Oxford: Oxford University Press.}}

* {{wikicite |ref=Meelberg2006 |reference=Meelberg, Vincent. 2006. ''New Sounds, New Stories: Narrativity in Contemporary Music.'' Leiden: Leiden University Press.}}

* {{wikicite |ref=Cook2007 |reference=Cook, Nicholas. 2007. ''Music, Performance, Meaning: Selected Essays.'' Ashgate Contemporary Thinkers on Critical Musicology Series. Aldershot: Ashgate.}}

* {{wikicite |ref=Rowe1992 |reference=Rowe, Robert. 1992. ''Interactive Music Systems.'' Cambridge, MAMassachusetts: MIT Press.}}

* {{wikicite |ref=Croft1999 |reference=Croft, John. 1999. ''Musical memory, complexity, and Lerdahl’s cognitive constraints.'' A Thesis submitted for the degree of Master of Music, Department of Music, The University of Sheffield.}}

{{div col end}}

* {{wikicite |ref=Denham2009 |reference=Denham, A.E. 2009. "The Future of Tonality". ''The British Journal of Aesthetics'' September 2009: 427-450.}}

* {{wikicite |ref=Dibben1996 |reference=Dibben, Nicola. 1996. ''The Role of Reductional Representations in the Perception of Atonal Music.'' Unpublished doctoral dissertation, Department of Music, The University of Sheffield.}}

* {{wikicite |ref=Grant2001 |reference=Grant, Morag Josephine. 2001. ''Serial Music Serial Aesthetics: Compositional Theory in Post-War Europe.'' Music in the Twentieth Century, Arnold Whittall, general editor. Cambridge: Cambridge University Press.}}

* {{wikicite |ref=Heinemann1993 |reference=Heinemann, Stephen. 1993. "Pitch-class Set Multiplication in Boulez’s "Le Marteau sans maitre" with [Original Composition]"". A Thesis submitted for the degree of Doctor of Musical Arts, Department of Music, University of WA.}}

* {{wikicite |ref=Heinemann1998 |reference=Heinemann, Stephen. 1998. "Pitch-Class Set Multiplication in Theory and Practice". ''Music Theory Spectrum'' 20, no. 1: 72–96.}}

* {{wikicite |ref=Horn2015 |reference=Horn, Walter. 2015. "Tonality, Musical Form, and Aesthetic Value". ''Perspectives of New Music'' 53, no. 2: 201-235.}}

*Lerdahl, Fred (1988). "Cognitive Constraints on Compositional Systems." In ''Generative Processes in Music: The Psychology of Performance, Improvisation, and Composition'', ed. John Sloboda, 231-59. Oxford: Oxford University Press. Reprinted in ''Contemporary Music Review'' 6/2 (1992), pp.&nbsp;97–121. [http://www.bussigel.com/lerdahl/pdf/Cognitive%20Constraints%20on%20Compositional%20Systems.pdf]

* {{wikicite |ref=Meelberg2006 |reference=Meelberg, Vincent. 2006. ''New Sounds, New Stories: Narrativity in Contemporary Music.'' Leiden: Leiden University Press.}}

* {{wikicite |ref=Mosch2004 |reference=Mosch, Ulrich. 2004. ''Musikalisches Hören serieller Musik: Untersuchungen am Beispiel von Pierre Boulez’ «Le Marteau sans maître».'' Saarbrücken: Pfau-Verlag.}}

* {{wikicite |ref=Rowe1992 |reference=Rowe, Robert. 1992. ''Interactive Music Systems.'' Cambridge, MA: MIT Press.}}

* {{wikicite|ref=Scruton |reference=Scruton, Roger. 2016. "[http://www.futuresymphony.org/the-clothes-have-no-emperor The Clothes Have No Emperor]". Future Symphony Institute (accessed 12 December 2021).}}

[[Category:Music cognition]]