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[[Fred Lerdahl]]'s '''"Cognitive Constraints on Compositional Systems"''' cites [[Pierre Boulez]]'s ''[[Le Marteau sans Maître]]'' (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]], [[Karlheinz Stockhausen]], or [[Iannis Xenakis]]". In [[semiology|semiological]] terminology this may be called gap between the [[esthesic and poietic]] processes. To explain this gap, and eventuallyin tohopes helpof endbridging 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 further 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 and; the "gap" maycan only exist between listening grammar and artificial grammars. To begin to understand the listening grammar Lerdahl and [[Ray Jackendoff]] created a theory of musical cognition, ''A Generative Theory of Tonal Music'' (1983); ISBN 0-262-62107-X). ThisThat theory is outlined in the essay,. andLerdahl's constraints on artificial compositional grammars are quoted below:
==Constraints on event sequences==
**[ [[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 that only the diatonic and pentatonic subsets of the 12 -tone chromatic set follow these constraints (Balzano, 1980, 1982)]
==Pitch Space==
**[completedness resembles implication-realization theory (Meyer, 1973 and Narmour, 1977), the ''Zug'', ''[[Urlinie]]'', and ''Bassbrechung'' (Schenker).]
He concludes, "Some of these constraints seem to me binding, otherothers 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).
==Comprehensibility and value==
*''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 positively being hierarchical structural richness, and complicatedness neutrally being musical surfaces which contain "numerous non-redundant events per unit time...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 favourfavor 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."
==Source==
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