Ganymed's heavenly descent

Schubert's song “Ganymed” has attracted a great deal of interest from analysts due to its progressive tonal plan, often seen as a challenge to Schenkerian theories of tonal structure, and evocative text. This article draws upon a spatial theory of tonal meaning which helps both to resolve the epistemological impasse faced by reductive theories of tonal structure, and to better access Schubert’s interpretation of Goethe’s text through spatial metaphors that derive from the harmony of the song. It also highlights an allusion to Beethoven's Op. 53 “Waldstein” Piano Sonata in the song that has previously gone unremarked, and identifies this as part of a network of references to Beethoven’s sonata that act both as homage to and critique of Beethoven's middle-period style. These serve both as a window into the song, and into Schubert’s aesthetic stance vis-à-vis his most pre-eminent musical forebear. The theory of tonal space draws upon previous publications, but is re-explained in music-theoretical terms relating to diatonicity and triadicity here. It realizes latent directional metaphors in the diatonic sharp-flat and triadic dominant-subdominant dimensions, which are of hermeneutic value for tonal music. Such a theory helps us interpret Schubert’s tonal plan, explain his choices of keys, and better understand his reading of Goethe's text and aesthetic priorities in setting it to music.Accepted manuscrip

Harmonic qualities in Debussy's "Les sons et les parfums tournent dans l'air du soir"

This analysis of the fourth piece from Debussy's Préludes Book I illustrates typical harmonic techniques of Debussy as manipulations of harmonic qualities. We quantify harmonic qualities via the magnitudes and squared-magnitudes of the coefficients of the discrete Fourier transform (DFT) of pitch class sets, following Ian Quinn. The principal activity of the piece occurs in the fourth and fifth coefficients, the octatonic and diatonic qualities, respectively. The development of harmonic ideas can therefore be mapped out in a two-dimensional octatonic/diatonic phase space. Whole-tone material, representative of the sixth coefficient of the DFT, also plays an important role. I discuss Debussy's motivic work, how features of tonality – diatonicity and harmonic function – relate to his musical language, and the significance of perfectly balanced set classes, which are a special case of nil DFT coefficients.Accepted manuscrip

Harmony and Technology Enhanced Learning

New technologies offer rich opportunities to support education in harmony. In this chapter we consider theoretical perspectives and underlying principles behind technologies for learning and teaching harmony. Such perspectives help in matching existing and future technologies to educational purposes, and to inspire the creative re-appropriation of technologies

Restoring the structural status of keys through DFT phase space

One of the reasons for the widely felt influence of Schenker’s theory is his idea of long-range voice-leading structure. However, an implicit premise, that voice leading is necessarily a relationship between chords, leads Schenker to a reductive method that undermines the structural status of keys. This leads to analytical mistakes as demonstrated by Schenker’s analysis of Brahms’s Second Cello Sonata. Using a spatial concept of harmony based on DFT phase space, this paper shows that Schenker’s implicit premise is in fact incorrect: it is possible to model long-range voice-leading relationships between objects other than chords. The concept of voice leading derived from DFT phases is explained by means of triadic orbits. Triadic orbits are then applied in an analysis of Beethoven’s Heiliger Dankgesang, giving a way to understand the ostensibly “Lydian” tonality and the tonal relationship between the chorale sections and “Neue Kraft” sections

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

Automatic estimation of harmonic tension by distributed representation of chords

The buildup and release of a sense of tension is one of the most essential aspects of the process of listening to music. A veridical computational model of perceived musical tension would be an important ingredient for many music informatics applications. The present paper presents a new approach to modelling harmonic tension based on a distributed representation of chords. The starting hypothesis is that harmonic tension as perceived by human listeners is related, among other things, to the expectedness of harmonic units (chords) in their local harmonic context. We train a word2vec-type neural network to learn a vector space that captures contextual similarity and expectedness, and define a quantitative measure of harmonic tension on top of this. To assess the veridicality of the model, we compare its outputs on a number of well-defined chord classes and cadential contexts to results from pertinent empirical studies in music psychology. Statistical analysis shows that the model's predictions conform very well with empirical evidence obtained from human listeners.Comment: 12 pages, 4 figures. To appear in Proceedings of the 13th International Symposium on Computer Music Multidisciplinary Research (CMMR), Porto, Portuga

A constraint-based framework to model harmony for algorithmic composition

Music constraint systems provide a rule-based approach to composition. Existing systems allow users to constrain the harmony, but the constrainable harmonic information is restricted to pitches and intervals between pitches. More abstract analytical information such as chord or scale types, their root, scale degrees, enharmonic note representations, whether a note is the third or fifth of a chord and so forth are not supported. However, such information is important for modelling various music theories. This research proposes a framework for modelling harmony at a high level of abstraction. It explicitly represents various analytical information to allow for complex theories of harmony. It is designed for efficient propagation-based constraint solvers. The framework supports the common 12-tone equal temperament, and arbitrary other equal temperaments. Users develop harmony models by applying user-defined constraints to its music representation. Three examples demonstrate the expressive power of the framework: (1) an automatic melody harmonisation with a simple harmony model; (2) a more complex model implementing large parts of Schoenberg’s tonal theory of harmony; and (3) a composition in extended tonality. Schoenberg’s comprehensive theory of harmony has not been computationally modelled before, neither with constraints programming nor in any other way.

An end-to-end machine learning system for harmonic analysis of music

We present a new system for simultaneous estimation of keys, chords, and bass notes from music audio. It makes use of a novel chromagram representation of audio that takes perception of loudness into account. Furthermore, it is fully based on machine learning (instead of expert knowledge), such that it is potentially applicable to a wider range of genres as long as training data is available. As compared to other models, the proposed system is fast and memory efficient, while achieving state-of-the-art performance.Comment: MIREX report and preparation of Journal submissio