The Butterfly Schema as a Product of the Tendency for Congruence and Hierarchical Selection in the Instrumental Musical Grammar of the Classical Period

Diverging explanations of local multiparametric schemata are found in music of the common practice period (c. 1600–c. 1900). Associative statistical theories describe schemata as situated structures in particular times and places, whereas generative theories present these constructions as features formed through stability in universal and general rule systems. Associative-statistical theories of schemata elucidate the culturally conditioned relationships between features (distinctive attributes commonly used in grammars and schemata), but do not show the influence of universal psychological constraints; generative theories reveal the implicit structure of music, but do not formalise particular grammatical features and contexts. A synthesis of generative and associative-statistical approaches is necessary to model the interaction between universal and particular constraints of grammars and schemata. This dissertation focuses on a novel localised schema formed in the Classical instrumental grammar, termed the butterfly schema. It is posited that the butterfly schema is generated by a tendency for congruence that is manifest in and between the particular features of this grammar.

Computational musicology and psychology provide interdisciplinary insight on the formal possibilities and limitations of grammatical structure. Computational models of schemata and grammars show how the congruent features of musical structure can be represented and formalised. However, they also highlight the difficulties found in the automatic analyses of multiparametric relationships, and may be limited on account of their inductive frameworks. Psychological approaches are important for establishing universal laws of cognition, but are limited in their potential to account for the diversity of musical structuring in grammars. The synthesis of associative-statistical and generative approaches in the present dissertation permits modelling the combination of the universal and particular attributes of butterfly schemata. Butterfly schemata are dependent on the particular grammars of periods of history, but are constrained by the tendency for congruence, which is proposed to be a cognitive universal. The features of the butterfly schema and the Classical instrumental grammar are examined and compared against the features of the Baroque and Romantic grammars, showing how they are formed from diverse types of congruent structuring. The butterfly schema is a congruent grammatical category of the Classical instrumental grammar that comprises: chords that are close to the tonic in pitch space (with a chiastic tension curve starting and ending on the tonic); a textural and metrical structure that is regular and forms a regular duple hierarchy at the level of regular functional harmonic change and at two immediately higher levels; and simple harmonic-rhythm ratios (1:1 and 3:1).

A survey conducted using arbitrary corpora in European instrumental music, c. 1750–c.1850, shows the distribution of butterfly schemata. Butterfly schemata are more common in the Classical-period sample (c. 1750–c. 1800) than in the Romantic-period sample (c. 1800–c.1850), suggesting that the tendency for congruence manifest in and between the features common in the Classical grammar generates butterfly schemata. A second component to the statistical analysis concerns the type of schemata observed, since the tendency for congruence is presumed to also apply to the type of features that form in butterfly schemata. Maximally congruent features are generated more commonly than minimally congruent features, indicating the influence of the tendency for congruence. This dissertation presents a formulation of the Classical instrumental grammar as a multiparametrically congruent system, and a novel explanation and integration of the concepts of grammars and schemata. A final component to the dissertation poses that the features of the Classical instrumental grammar and butterfly schema follow a distinct order of dependency, governed by the mechanism of selection in culture. Although the tendency for congruence governs all features of a grammar, features are also formed by the top-down action of culture which selects those features. Thus, a top-down hierarchical selection model is presented which describes how the butterfly schema is formed through the order of selection of features in the Classical instrumental grammar.