Lerdahl's Cognitive Constraints

Groups of constraints have much in common, and influence each other. Since groups of the constraints are related, I've put them here topically. The constraints tend to deal with:

Each constraint has a link to more detailed discussion.

In essence, though, the constraints themselves are hierarchical; most call on their immediate predecessors. For instance, constraint two (hierarchically structuring a 'musical surface') requires discrete events, which is the precise requirement of constraint one.

These dependancies are outlined in the picture below. I've broken the constraints into "families" of related constraints. As I began tracing the dependancies i noticed that there seemed to be three general types of constraints: constraints which deal with the cognition of time elements, constraints which deal with pitch elements, and some meta-level constraints which have to do with the types of structures the model itself expects to deal with (these are what Lerdahl calls "stability conditions"). Click in the picture for more detail:

It is also interesting to note that Lerdahl says: "constraints 13 - 17 may be variously jettisoned".

Grouping Constraints

  • 1. The musical surface must be capable of being parsed into a sequence of discrete events.

  • 2. The musical surface must be available for hierarchical structuring.

  • 3. The establishment of local grouping boundaries requires the presence of salient distinctive transitions at the musical surface.

  • 4. Projection of groups, especially at larger levels, depends on symmetry and on the establishment of musical parallelisms.

  • 5. The establishment of a metrical structure requires a degree of regularity in the placement of phenomenal accents.

  • 6. A complex time-span segmentation depends on the projection of a complex grouping and metrical structures.

  • 7. The projection of a time-span tree depends on a complex time-span segmentation in conjunction with a set of stability conditions.

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

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    Stability Conditions
  • 9. Stability conditions must operate on a fixed collection of elements.

  • 10. Intervals between elements of a collection arranged along a scale should fall within a certain [perceptible] range of magnitude.

  • 11. A pitch collection should recur at the octave to produce pitch classes.

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    Tuning and temperament
  • 12. There must be a strong psychoacoustical basis for stability conditions. For pitch collections, this entails intervals that proceed gradually from very small to comparatively large frequency ratios.

  • 13. Division of the octave into equal parts facilitates transposition and reduces memory load.

  • 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.

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    Pitch Space Representation
  • 15. Any but the most primitive stability conditions must be susceptible to multidimensional representation, where spatial distance correlates with cognitive distance.

  • 16. Levels of pitch space must be sufficiently available from musical surfaces to be internalized.

  • 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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