A Signal-Based Model of Teleology in Tonal Music

Brand, Mark Andre (2016-03)

Thesis (MEng)--Stellenbosch University, 2016.

Thesis

ENGLISH ABSTRACT: The operationalisation of music’s affective agency is an open problem. This thesis notes a long tradition of associating that agency with goal-directed dynamics in the music, but also that those dynamical forces are consistently described in terms of a theoretical framework which privileges Western tonal music by its reliance on the constructs of pitch and rhythm. Against the backdrop of a revival in interest regarding the question of musical universals, an alternate, geometric intuition is then proposed to account for the privileged status of the perfect cadence in tonal music in terms of a least action principle. An extensive review of harmonic priming experiments underscores the robustness of that status, regardless of a subject’s level of musical education or experience, and strongly suggests a degree of universality. A similarly extensive review of theoretical approaches to sub-Saharan African music reveals how such a geometric model might also account for teleology therein. The proposed model is operationalised by building a signal processing pipeline containing a recurrent, complex-valued, gradient-frequency artificial neural network, where each neuron implements non-linear dynamics according to a fully expanded canonical form describing the behaviour of excitatory / inhibitory neural populations near a Hopf bifurcation. In the absence of any suitable measure of network energy in the literature, an energy balance relation is constructed whereby to monitor energy flows into and out of the network. In each of four experiments, the network is first primed with a one second stimulus containing the pitches of a dominant quartad in closed position on G4. Seven targets, being the available diatonic triadic continuations to the quartad, are then each presented to the network in turn, and the individual components of the energy balance relation are recorded as the network seeks out its equilibrium. The first experiment employs sinusoidal stimuli, and wholly excludes inter-neural connectivity. This results in all targets taking the network to the same averaged rate of power supply in each case, being equal to the rate of dissipation at equilibrium, and provides no evidence of an advantage of any single target over the others. The second experiment implements typical “train-and-predict” semantics by allowing non-linear Hebbian learning to take place during the priming phase, holding learned connections fixed during the presentation of the target. This produces differentiation between the levels of supplied energy, favouring targets which contain the pitches of the prime, but does not thereby support our model. The third experiment allows learning to continue throughout both the priming and target phases, and thereby obtains a lower level of supplied energy in respect of the tonic target as compared to all others, a result which provides tenuous, but enticing support for our model. The fourth experiment employed piano tones instead of sinusoidal stimuli, and produced inconclusive results, thereby suggesting that a more complex network might be required to deal with real-world audio.

AFRIKAANSE OPSOMMING: Dit bly 'n ope vraag hoe dat musiek se invloed oor die menslike gemoed bestuur mag word. In hierdie tesis word daar gelet op 'n lang-durende assosiasie met 'n doelgerigte dinamika, maar verder ook dat sulke kragte deurgaans beskryf word in terme van Westerse musiekteorie, by wyse van hul afhanklikheid van begrippe soos toonhoogte en ritme. Gegewe hernude belangstelling in musikale universaliteit, word daar dan hierin 'n alternatiewe, geometriese intuïsie voorgelê wat poog om die bevoorregte status van die volmaakte kadens in tonale musiek te bereken in terme van 'n algemene tendens tot maksimale doeltreffendheid. 'n Uit - gebreide oorsig van sogenaamde “harmonic priming” eksperimente onderstreep die robuustheid van hierdie status, afgesien van 'n hoorder se musikale opvoeding of ondervinding, en dui sterk op universaliteit. 'n Ver -dere studie van teoretiese benaderings tot Afrikaanse musiek, suid van die Sahara, wys dat so 'n geometriese model ook hier soortgelyke doelgerigtheid mag ontbloot. Die voorgestelde model word in werking gestel deur die bou van 'n seinverwerkingspyplyn wat 'n herha - lende, kompleks-gewaardeerde, gradiënt-frekwensie neurale netwerk bevat, waar elke neuron nie-lineêre dinamika implementeer volgens 'n volledig uitgebreide kanoniese vorm wat die gedrag van stimulerende / inhiberende neurale populasie beskryf naby 'n Hopf bifurkasie. By gebrek aan 'n geskikte meetinstrument om die netwerk se energie te bepaal, word 'n energiebalansverhouding opgerig waarmee energievloei na en van die netwerk waargeneem kan word. In elk van vier eksperimente, word die netwerk eers voorberei deur middel van 'n een-sekonde stimulus wat bestaan uit die note van 'n dominantvierklank in geslote posisie op G4. Sewe teikens, wesenlik die beskikbare diatoniese drieklanke wat sou kon volg op die vierklank, word dan elk op sy beurt ingevoer, en die afsonderlike komponente van die energiebalansverhouding word aangeteken soos die netwerk sy ewewig uitsoek. Die eerste eksperiment maak gebruik van sinusvormige stimuli, en sluit inter-neurale verbindings heeltemaal uit. Gevolglik neem elke teikenstimulus die netwerk na dieselfde gemiddelde drywing, dié gelyk aan die dissipasiekoers by ewewig, en verskaf dus geen bewys van enige voordeel wat een teiken mag geniet bo die ander nie. Die tweede eksperiment implementeer die tipiese “rig-af-en-voorspel” siklus deurdat nielineêre Hebbian leerprosesse mag plaasvind tydens die eerste fase, terwyl hierdie verbindings staties gehou word tydens die invoer van die teiken. Hierdeur word 'n onderskeid gemerk tussen vlakke van energietoe - voer, met voorkeur aan teikens wat toonhoogtes bevat wat tydens die eerste fase gebruik is, en dus steun hierdie resultate nie ons model nie. Die derde eksperiment laat die leerproses toe om deurgaans voort te gaan, en verskaf sodoende 'n laer vlak van energietoevoer ten opsigte van die tonikum teiken, vergeleke met alle ander. Die resultaat verskaf skrale, dog aanloklike rugsteun vir ons model. Die vierde eksperiment maak gebruik van klaviernote, eerder as sinusgolwe, en verskaf onbeduidende resultate, 'n aanduiding dat 'n meer komplekse netwerk dalk benodig sal word om nie-kunsmatige klanke so te kan verwerk.

Please refer to this item in SUNScholar by using the following persistent URL: http://hdl.handle.net/10019.1/98644
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