A psychoacoustic model of harmonic cadences: a preliminary report

This report presents a psychoacoustically derived computational model of the perceived distance between any two major or minor triads, the degree of activity created by any given pair of triads, and the cadential effectiveness of three-triad progressions. It also provides statistical analyses of the ratings given by thirty-five participants for the "similarity" and "fit" of triads in a pair, and the "cadential effectiveness" of three-triad progressions. Multiple regressions show that the model provides highly significant predictions of the experimentally obtained ratings. Finally, it is argued that because the model is based upon psychoacoustic axioms, it is likely the regression equations represent true causal models. As such, the computational model and its associated theory question the plausibility of theoretical approaches to tonality that use only long-term memory and statistical features, as well as those approaches based upon symmetrical geometrical structures like the torus. It is hoped that the psychoacoustic approach proposed here may herald not only the return of psychoacoustic approaches to tonal music theory, but also the exploration of the tonal possibilities offered by non-standard tunings and non-harmonic timbres

Empirically testing <i>Tonnetz</i>, voice-leading, and spectral models of perceived triadic distance

We compare three contrasting models of the perceived distance between root-position major and minor chords and test them against new empirical data. The models include a recent psychoacoustic model called spectral pitch class distance, and two well-established music theoretical models – Tonnetz distance and voice-leading distance. To allow a principled challenge, in the context of these data, of the assumptions behind each of the models, we compare them with a simple “benchmark” model that simply counts the number of common tones between chords. Spectral pitch class and Tonnetz have the highest correlations with the experimental data and each other, and perform significantly better than the benchmark. The voice-leading model performs worse than the benchmark. We suggest that spectral pitch class distance provides a psychoacoustic explanation for perceived harmonic distance and its music theory representation, the Tonnetz. Scores and MIDI files of the stimuli, the experimental data, and the computational models are available in the online supplement

A Functional Taxonomy of Music Generation Systems

Digital advances have transformed the face of automatic music generation since its beginnings at the dawn of computing. Despite the many breakthroughs, issues such as the musical tasks targeted by different machines and the degree to which they succeed remain open questions. We present a functional taxonomy for music generation systems with reference to existing systems. The taxonomy organizes systems according to the purposes for which they were designed. It also reveals the inter-relatedness amongst the systems. This design-centered approach contrasts with predominant methods-based surveys and facilitates the identification of grand challenges to set the stage for new breakthroughs.Comment: survey, music generation, taxonomy, functional survey, survey, automatic composition, algorithmic compositio

Training of Tonal Similarity Ratings in Non-Musicians: A “Rapid Learning” Approach

Although cognitive music psychology has a long tradition of expert–novice comparisons, experimental training studies are rare. Studies on the learning progress of trained novices in hearing harmonic relationships are still largely lacking. This paper presents a simple training concept using the example of tone/triad similarity ratings, demonstrating the gradual progress of non-musicians compared to musical experts: In a feedback-based “rapid learning” paradigm, participants had to decide for single tones and chords whether paired sounds matched each other well. Before and after the training sessions, they provided similarity judgments for a complete set of sound pairs. From these similarity matrices, individual relational sound maps, intended to display mental representations, were calculated by means of non-metric multidimensional scaling (NMDS), and were compared to an expert model through procrustean transformation. Approximately half of the novices showed substantial learning success, with some participants even reaching the level of professional musicians. Results speak for a fundamental ability to quickly train an understanding of harmony, show inter-individual differences in learning success, and demonstrate the suitability of the scaling method used for learning research in music and other domains. Results are discussed in the context of the “giftedness” debate

Computational Tonality Estimation: Signal Processing and Hidden Markov Models

PhDThis thesis investigates computational musical tonality estimation from an audio signal. We present a hidden Markov model (HMM) in which relationships between chords and keys are expressed as probabilities of emitting observable chords from a hidden key sequence. The model is tested first using symbolic chord annotations as observations, and gives excellent global key recognition rates on a set of Beatles songs. The initial model is extended for audio input by using an existing chord recognition algorithm, which allows it to be tested on a much larger database. We show that a simple model of the upper partials in the signal improves percentage scores. We also present a variant of the HMM which has a continuous observation probability density, but show that the discrete version gives better performance. Then follows a detailed analysis of the effects on key estimation and computation time of changing the low level signal processing parameters. We find that much of the high frequency information can be omitted without loss of accuracy, and significant computational savings can be made by applying a threshold to the transform kernels. Results show that there is no single ideal set of parameters for all music, but that tuning the parameters can make a difference to accuracy. We discuss methods of evaluating more complex tonal changes than a single global key, and compare a metric that measures similarity to a ground truth to metrics that are rooted in music retrieval. We show that the two measures give different results, and so recommend that the choice of evaluation metric is determined by the intended application. Finally we draw together our conclusions and use them to suggest areas for continuation of this research, in the areas of tonality model development, feature extraction, evaluation methodology, and applications of computational tonality estimation.Engineering and Physical Sciences Research Council (EPSRC)

Proportion as it relates to interior design

The purpose of this study was to gain a better understanding of proportion as it pertains to interior design. The historical significance and conceptions of proportion in design today were reviewed. Broadly defined, proportion is the quantitative relationship between objects and areas. It is the relationship of sizes and shapes. Since scale refers to size it is related to proportion. Sizes of things are compared to sizes with which one is familiar. Basic to all sizes is the size of man. The human scale is the basis for design since all designs are executed for human beings to enjoy. The relationship of sizes between parts of an object, between objects, and between objects and the whole is of concern to a designer. In interior design proportion is concerned with the quantitative relationship of two or more of the design elements—line, form, color, texture, light, pattern, and space—to themselves, to each other, and to the whole

Schubert's harmonic language and Fourier phase space

This article introduces a type of harmonic geometry, Fourier phase space, and uses it to advance the understanding of Schubert’s tonal language and comment upon current topics in Schubert analysis. The space derives from the discrete Fourier transform on pitch-class sets developed by David Lewin and Ian Quinn but uses primarily the phases of Fourier components, unlike Lewin and Quinn, who focus more on the magnitudes. The space defined by phases of the third and fifth components closely resembles the Tonnetz and has a similar common-tone basis to its topology but is continuous and takes a wider domain of harmonic objects. A number of musical examples show how expanding the domain enables us to extend and refine some the conclusions of neo-Riemannian theory about Schubert’s harmony. Through analysis of the Trio and Adagio from Schubert’s String Quintet and other works using the geometry, the article develops a number of concepts for the analysis of chromatic harmony, including a geometric concept of interval as direction (intervallic axis), a novel approach to triadic voice leading (triadic orbits), and theories of tonal regions.Accepted manuscrip

Visualizing Harmony Using Chordal Glyphs and Color Mapping

Musical scores are frequently annotated with harmonic information, but widely used text-based methods rely on a limited number of visual channels. Though glyph-based methods exploit more channels, existing systems often violate perceptual design principles when employing color and rarely capture the frequency of chordal changes or their harmonic function. In this work, we introduce a new design idiom for augmenting sheet music through chordal glyphs embedded directly within musical staves. Harmonic concepts, weighted by saliency and categorized by data type, are mapped to visual channels ranked by discriminability. Preattentive processing is leveraged to support various user tasks, alongside redundant encodings of foundational harmonic elements to improve overall perceptual effectiveness. Key names and chord roots are displayed using parallel hue-based 12-step categorical colormaps. We then distill several design implications inherent in assigning colors to musical pitches regarding perceptual and linguistic effectiveness. Following this discussion, we outline open research directions