Key profile optimisation for the computational modelling of tonal centre

Vermeulen, Hendrik Johannes (2012-12)

Thesis (MPhil)--Stellenbosch University, 2012.


ENGLISH ABSTRACT: Tonality cognition incorporates a number of diverse and multidisciplinary aspects, including music cognition, acoustics, culture, computer-aided modelling, music theory and brain science. Current research shows growing emphasis on the use of computational models implemented on digital computers for music analysis, particularly with reference to the analysis of statistical properties, form and tonal properties. The applications of these analytical techniques are numerous, including the classification of genre and style, Music Information Retrieval (MIR), data mining and algorithmic composition. The research described in this document focuses on three aspects of tonality analysis, namely music cognition, computational modelling and music theory, particularly from the perspectives of statistical analysis and key-finding. Mathematical formulations are presented for a number of computational algorithms for analysing the statistical and tonal properties of music encoded in symbolic format. These include algorithms for determining the distributions of note durations, pitch intervals and pitch classes for statistical analysis and for template-based key-finding for tonal analysis. The implementation and validation of these computational algorithms on the Matlab software platform are subsequently discussed. The software application is used to determine whether a more optimal combination of pitch class weighing model and key profile template for the template-based key-finding algorithm can be derived, using the 24 preludes from Bach’s Well-tempered Clavier Book I, the Courante from Bach's Cello Suite in C major and the Gavotte from Bach's French Suite No. 5 in G major (BWV 816) as test material. Four pitch class weighing models, namely histogram weighing, flat weighing, linear durational weighing and durational accent weighing, are investigated. Two prominent key profile templates proposed in literature are considered, namely a key profile derived from tonality cognition experiments and a key profile based on classical music theory principles. The results show that the key-finding performances of all the combinations of the pitch class weighing models and existing key profile templates depend on the nature of the test material and that none of the combinations perform optimally for all test material. The software application is subsequently used to determine whether a more optimal key profile template can be derived using a pattern search parameter estimation algorithm. This investigation was conducted for diverse sets of search conditions, including unconstrained and constrained key profile coefficients, different pitch class weighing models, various key resolutions and different search algorithm parameters. Using the same sample material as for the key-finding evaluations, the investigation showed that a more optimal key profile, compared to existing profiles, can be derived. In comparing the average key-finding scores for all of the test material, the optimised profiles outperform the existing profiles substantially. The optimised key profiles introduce new pitch class hierarchies where the supertonic and the subdominant rate higher at the expense of the mediant in the major profile to improve the tracking of key modulations.

AFRIKAANSE OPSOMMING: Kognitiewe tonaliteit behels 'n aantal uiteenlopende en multidissiplinêre aspekte, insluitende musiek, akoestiek, kultuur, rekenaargesteunde modelering, musiekteorie en breinwetenskap. Huidige navorsing toon toenemende klem op die gebruik van berekenende modelering wat op digitale rekenaars geimplimenteer is vir musiekanalise, veral met verwysing na die analise van statistiese eienskappe, vorm en tonale eienskappe. Die aanwending van hierdie analitiese tegnieke is veelvoudig, insluitende die klassifikasie van genre of styl, onttrekking van musiekinformasie, dataversameling en algoritmiese komposisie. Die navorsing wat in hierdie dokument beskryf word fokus op drie aspekte van tonaliteit analise, naamlik musiekkognisie, berekenende modelering en musiekteorie, veral vanuit die perspektiewe van statistiese analise and toonsoortsoek. Wiskundige formulerings word aangebied vir 'n aantal berekeningalgoritmes vir die analise van die statistiese en tonale eienskappe van musiek wat in simboliese formaat ge-enkodeer is. Hierdie sluit algoritmes in vir die bepaling van die verspreidings van nootlengtes, toonintervalle en toonklasse vir statistiese analise en vir templaatgebaseerde toonsoortsoek vir tonale analise. Die implementering en validering van hierdie berekeningalgoritmes op die Matlab programmatuur platvorm word vervolgens bespreek. Die programmatuur toepassing word vervolgens gebruik om te bepaal of 'n meer optimale kombinasie van toonklas weegmodel en toonsoortprofiel templaat vir die templaat-gebaseerde toonsoortsoek algoritme afgelei kan word, deur gebruik te maak van Bach se Well-tempered Clavier Book I, die Courante van Bach se Cello Suite in C major en die Gavotte van Bach se French Suite No. 5 in G major (BWV 816) as toetsmateriaal. Vier toonklas weegmodelle, naamlik histogram weging, plat weging, lineêre duurtyd weging en duurtyd aksent weging, word ondersoek. Twee prominente toonsoortprofiel template uit die literatuur word oorweeg, naamlik 'n toonsoortprofiel wat van tonaliteit kognisie eksperimente afgelei is en 'n toonsoortprofiel gebaseer op klassieke musiekteoretiese beginsels. Die resultate wys dat die toonsoortsoek prestasies van al die kombinasies van die toonklas weegmodelle en bestaande toonsoortprofiel template afhang van die aard van die toetsmateriaal en dat geen van die kombinasies optimaal presteer vir alle toetsmateriaal nie. Die programmatuur toepassing word vervolgens aangewend om vas te stel of 'n meer optimale toonsoortprofiel afgelei kan word deur gebruik te maak van 'n patroonsoek parameterestimasie algoritme. Hierdie ondersoek is uitgevoer vir uiteenlopende stelle soektoestande, insluitende onbeperkte en beperkte toonsoortprofiel koëffisiënte, verskillende toonklas weegmodelle, 'n verskeidenheid toonsoort resolusies en verskillende soekalgoritme parameters. Deur gebruik te maak van dieselfde toetsmateriaal as vir die toonsoortsoek evaluerings, toon die ondersoek dat 'n meer optimale toonsoortprofiel, in vergelyking met bestaande profiele, afgegelei kan word. In 'n vergelyking van die gemiddelde toonsoortsoek prestasie vir al die toetsmateriaal, presteer die geoptimeerde profiele aansienlik beter as die bestaande profiele. The ge-optimeerde toonsoortprofiele lei tot nuwe toonklas hiërargiee waar die supertonikum en die subdominant hoër rangposissies beklee ten koste van die mediant in die majeur profiel, ten einde die navolg van toonsoort modulasies te verbeter.

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