Cognitive Constraints On Compositional Systems

"Cognitive Constraints on Compositional Systems" is an essay by Fred Lerdahl that 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 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, '' A Generative Theory of Tonal Music'' (1983; ). That theory is outlined in the essay.

Constraints

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.
** Ligeti, computer music">György_Ligeti.html" ;"title="ounterexample: 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 motivic development and timbral relations are ignored, but for "Timbral Hierarchies" see Lerdahl 1987]
*''Constraint 3'': The establishment of local grouping boundaries requires the presence of salient distinctive transitions at the musical surface.
** ounterexample: minimal music">minimal_music.html" ;"title="ounterexample: minimal music">ounterexample: minimal music*''Constraint 4'': Projections of groups, especially at larger levels, depends on symmetry and on the establishment of musical parallelism (music), parallelisms.
*''Constraint 5'': The establishment of a Metre (poetry), metrical structure requires a degree of regularity in the placement of phenomenal accents.
*''Constraint 6'': A complex time-span segmentation depends on the projection of complex grouping and metrical structures.
*''Constraint 7'': The projection of a time-span tree depends on a complex time-span segmentation in conjunction with a set of stability conditions.
*''Constraint 8'': The projection of a prolongational tree depends on a corresponding time-span tree in conjunction with a set of stability conditions.

Underlying materials

*''Constraint 9'': Stability conditions must operate on a fixed collection of elements.
** sually pitches or rather fundamentals with harmonic partials*''Constraint 10'': Intervals between elements of a collection arranged along a scale should fall within a certain range of magnitude.
*''Constraint 11'': A pitch collection should recur at the octave to produce pitch classes.
**
*''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 ratios.
**
*''Constraint 13'': Division of the octave into equal parts facilitates transposition and reduces memory load.
**
*''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)]

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.
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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 South Indian music