Profiles of Pitch Classes --- Circularity of Relative Pitch and Key: Experiments, Models, Music Analysis, and Perspectives

Hendrik Purwins
Technische Universität Berlin, Germany (October, 2005)


The doubly-circular inter-relation of the major and minor keys on all twelve pitch classes can be depicted in toroidal models of inter-key relations (TOMIR). We demonstrate convergence of derivations on the explanatory levels of a) an experiment in music psychology, b) geometrical considerations in music theory, and c) computer implementation of musical listening scenarios. Generalizing Shepard (1964) to full overtone spectra, circular perception of relative pitch is experimentally verified and mathematically modeled as the spectrum of pitch differences, derived from virtual pitch (Terhardt 1998). Musical examples of circular pitch, tempo, and loudness are analyzed. For each pitch class calculating the intensity in
a musical recording, our constant quotient (CQ-)profile method is
a) consistent with psychological probe tone ratings,
b) highly efficient,
c) computable in real-time,
d) stable with respect to sound quality,
e) applicable to transposition,
f) free of musical presupposition, except approximately equal temperament, and
g) sensitive to substantial musical features (style, composer, tendency towards chromaticism, and major/minor alteration) in a highly compact reduction. In Bach, Chopin, Alkan, Scriabin, Hindemith, and Shostakovich, the latter features are extracted from overall CQ-profiles by classification (support vector machine [SVM], regularized discriminant analysis) and clustering. Their inter-relations are visualized by a technique called Isomap. Leman (1995) models acquisition of inter-key relations. Artificial cadences are preprocessed through modeling the auditory periphery. Then, in the toroidal self organizing feature map (SOM) a TOMIR is arrived at. We extend this approach by
a) using a great many actual performances of music as input, and/or
b) not presupposing toroidal topology a priori. Visualizing CQ-profiles from Bach's WTC by correspondence analysis (CA) and Isomap reveals the circle of fifths. TOMIRs evolve from
a) average CQ-profiles of Chopin's Preludes and toroidal SOMs,
b) overall annotated duration profiles of Bach's WTC I from score and CA, and
c) formalization of music theoretical derivation from Weber's (1817) key chart by the topographic ordering map introduced here. These results are consistent with Krumhansl's (1982) visualization of music psychology ratings. As auxiliary results, we discuss fuzzy distance, spatial and systematized synesthetic visualization in conjunction with beat weight detection on multiple time scales and suggested applications to automatic tone center tracking. Furthermore, statistics on the key preference of various composers are collected. Based on the latter, CA visualizes composer inter-relations. This thesis substantially contributes to content retrieval (MPEG-7), automated analysis, interactive audio-based computer music, and musicology.

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