The Pythagorean comma and preference for a stretched octave

Pythagorean tuning, which derives all musical intervals of a Western chromatic scale from repeated application of a fifth and subsequent reduction (collapse) into a single octave, results in an octave interval slightly larger than 2:1, and the difference between a Pythagorean octave and a 2:1 octave is referred to as the Pythagorean comma. Empirical studies suggest that listeners prefer a stretched octave, that is, an octave interval slightly larger than 2:1. The magnitudes of the Pythagorean comma and of the stretch from a 2:1 ratio in preferred octave tuning are similar, and it is suggested these phenomena might be connected. Implications of such a similarity that are addressed include (a) how the Pythagorean comma predicts preference for a stretched octave, (b) rejection of uncertainties in tuning as a cause of preference for a stretched octave, (c) how the Pythagorean comma and preference for a stretched octave might be related to tension and musical aesthetics, (d) the nature of “scales” and “intervals” in musical and psychophysical senses, (e) the role of motion through auditory pitch space in the Pythagorean comma and preference for a stretched octave, and (f) incorporation of elements of Pythagorean tuning into the representation of the octave.

Space

This chapter investigates a variety of ways in which music might be thought to be essentially spatial in relatively literal ways. It begins by considering whether certain spaces or spatial features are essential to musical works or performances. These include the space of a work’s composition, performance spaces for which a work is composed or within which it is performed, and the spatial disposition of performers (e.g., off-stage instruments). It then considers spaces “within” music, paying special attention to the notion of “pitch space”—the space in which we experience musical tones as higher or lower than one another and melodic lines as moving.

Generic Sequences and the Generic Tonnetz

This article employs concepts from diatonic set theory, transformation theory, and neo-Riemannian theory in an investigation of sequences in generic (mod-7) pitch space. Sequences may be described using generic transposition operators, mod-7 analogs of the more familiar mod-12 chromatic transposition operators, and may be mapped in a generic Tonnetz analogous to the Tonnetz diagrams of neo-Riemannian theory. A complete classification of generic sequences with two-chord transposition blocks is derived from the paths described by the sequences in the Tonnetz. While it is tempting to regard generic space as “diatonic,” the examples demonstrate that generic structure governs many sequences that are not actually diatonic at all. The examples also show that the sequences found in the literature include many more types than are widely recognized and that generic sequence structure may apply in numerous other contexts besides patterns of chord roots.

Instrumental expertise and musical timbre modulate the spatial representation of pitch

Humans show a tendency to represent pitch in a spatial format. A classical finding supporting this spatial representation is the Spatial–Musical Association of Response Codes (SMARC) effect, reflecting faster responses to low tones when pressing a left/bottom-side key and to high tones when pressing a right/top-side key. Despite available evidence suggesting that the horizontal and vertical SMARC effect may be differently modulated by instrumental expertise and musical timbre, no study has so far directly explored this hypothesis in a unified framework. Here, we investigated this possibility by comparing the performance of professional pianists, professional clarinettists and non-musicians in an implicit timbre judgement task, in both horizontal and vertical response settings. Results showed that instrumental expertise significantly modulates the SMARC effect: whereas in the vertical plane a comparable SMARC effect was observed in all groups, in the horizontal plane the SMARC effect was significantly modulated by the specific instrumental expertise, with pianists showing a stronger pitch–space association compared to clarinettists and non-musicians. Moreover, the influence of pitch along the horizontal dimension was stronger in those pianists who started the instrumental training at a younger age. Results also showed an influence of musical timbre in driving the horizontal, but not the vertical, SMARC effect, with only piano notes inducing a pitch–space association. Taken together, these findings suggest that sensorimotor experience due to instrumental training and musical timbre affect the mental representation of pitch on the horizontal space, whereas the one on the vertical space would be mainly independent from musical practice.

The embodied and the cultural in the conceptualization of pitch space in Croatian

Most conceptual metaphors that conceptualize musical pitch rely heavily on human perception, images and experience structured through spatial and orientation image schemas such as the schema of verticality. The paper analyses conceptual metaphors that structure pitch relations in terms of vertical space, thickness and size as they appear in the Croatian musical terminology. The image schemas of verticality and size are analysed within the conceptual metaphors pitch relations are relations in vertical space and pitch relations are relations in size in order to define to what extent their motivation is embodied and universal, and what can be attributed to cross-cultural and cross-linguistic influences present in the creation and understanding of music terminology in Croatia.

Hemiolic Metric Space in Afro-Diasporic Popular Musics

Pitch spaces such as the circle of fifths model change through time in a composition, recording, or improvisation. Metric spaces theorized over the past twenty years do the same for changes (notated or not) in meter. Trajectories in either space and their potentially reinforcing or conflicting relationships contribute to deeper interpretations of musical form. None of the metric spaces proposed to date is well suited to Afrodiasporic popular musics, which characteristically tend to pose regularly uneven metric foregrounds against rigid and recursively even metric backgrounds. This article introduces a new metric space specifically applicable to such repertoires. The article opens with a brief review of existing metric spaces. Part 1 is an exploratory metric analysis of Afro-Brazilian guitarist Baden Powell’s 1967 recording of “Canto de Xangô,” which motivates the theoretical developments of part 2. Part 3 is a short analysis of his 1963 recording of “Sorongaio” that demonstrates both how hemiolic metric space can newly be analyzed in pure-duple environments and how this metric space can be isomorphic with pitch space.

Prokofiev’s Symphony no. 2, Yuri Kholopov, and the Theory of Twelve-Tone Chords

A small collection of works, including Prokofiev’s Symphony no. 2 (1924), include chords with all twelve pitch classes. Yuri Kholopov, the foremost late-Soviet theorist, considered twelve-tone chords a branch of twelve-tone technique. Taking Prokofiev and Kholopov as a starting point, and building on prior scholarship by Martina Homma, I assemble a history and theory of twelve-tone chords. The central theoretical problem is that of differentiation: as all twelve-tone chords contain the same twelve pitch classes, there is essentially only one twelve-tone chord. Yet twelve-tone chords can be categorized on the basis of their deployment in pitch space. Twelve-tone chords tend to exhibit three common features: they avoid doublings, they have a range of about 3 to 5.5 octaves, and their vertical interval structure follows some sort of pattern. This article contextualizes twelve-tone chords within the broader early-twentieth-century experimentation with aggregate-based composition.

Outline of Music Semantics

We provide the outline of a semantics for music. We take music cognition to be continuous with normal auditory cognition, and thus to deliver inferences about “virtual sources” of the music. As a result, sound parameters that trigger inferences about sound sources in normal auditory cognition produce related ones in music. But music also triggers inferences on the basis of the movement of virtual sources in tonal pitch space, which has points of stability, points of instability, and relations of attraction among them. We sketch a framework that aggregates inferences from normal auditory cognition and tonal inferences, by way of a theory of musical truth: a source undergoing a musical movement m is true of an object undergoing a series of events e just in case there is a certain structure-preserving map between m and e. This framework can help revisit aspects of musical syntax: Lerdahl and Jackendoff’s (1983) grouping structure can be seen to reflect the mereology (“partology”) of events that are abstractly represented in the music. Finally, we argue that this “refentialist” approach to music semantics still has the potential to provide an account of diverse emotional effects in music.