Chord Function Research Articles

Parametric competition in non-autonomous Hamiltonian systems

In this work we use the formalism of chord functions (i.e. characteristic functions) to analytically solve quadratic non-autonomous Hamiltonians coupled to a reservoir composed by an infinity set of oscillators, with Gaussian initial state. We analytically obtain a solution for the characteristic function under dissipation, and therefore for the determinant of the covariance matrix and the von Neumann entropy, where the latter is the physical quantity of interest. We study in details two examples that are known to show dynamical squeezing and instability effects: the inverted harmonic oscillator and an oscillator with time dependent frequency. We show that it will appear in both cases a clear competition between instability and dissipation. If the dissipation is small when compared to the instability, the squeezing generation is dominant and one can see an increasing in the von Neumann entropy. When the dissipation is large enough, the dynamical squeezing generation in one of the quadratures is retained, thence the growth in the von Neumann entropy is contained.

Schenkerian Theory, Neo-Riemannian Theory and Late Schubert: A Lesson from Tovey

Current analytical studies on Schubert’s tonality have tended to favour either Schenkerian theory or, more recently, neo-Riemannian theory to explain the composer’s signature harmonic progressions. What remains unclear with respect to these two prevailing analytical purviews is the extent to which one may relate to the other.This article offers a new way of understanding how Schenkerian and neo-Riemannian views of Schubert’s late tonal practices may be complementary, using Tovey’s concept of key-relations from his article ‘Tonality in Schubert’ of 1928. It suggests that Tovey’s key-relations can function as a bridge between these two theories because they approximate parsimonious voice-leading operations while preserving chord function within a tonal hierarchy. In forming an intermediate pathway between Schenkerian diatony and neo-Riemannian theory’s parsimonious voice-leading operations, Tovey’s key-relations highlight the important contributions that Schenkerian theory and neo-Riemannian theory offer to our understanding of Schubert’s tonality.

Analysis of a Multiflap Control System for a Swashplateless Rotor

An analytical study was conducted examining the feasibility of a swashplateless rotor controlled through two trailing-edge flaps (TEFs), where the cyclic and collective controls were provided by separate TEFs. This analysis included a parametric study examining the impact of various design parameters on TEF deflections. Blade pitch bearing stiffness; blade pitch index; and flap chord, span, location, and control function of the inboard and outboard flaps were systematically varied on a utility-class rotorcraft trimmed in steady level flight. Gradient-based optimizations minimizing flap deflections were performed to identify single- and two-TEF swashplateless rotor designs. Steady, forward flight analysis suggest that a two-TEF swashplateless rotor where the outboard flap provides cyclic control and inboard flap provides collective control can reduce TEF deflection requirements without a significant impact on power, compared to a single-TEF swashplateless rotor design.

Priming paradigm reveals harmonic structure processing in congenital amusia

Deficits for pitch structure processing in congenital amusia has been mostly reported for melodic stimuli and explicit judgments. The present study investigated congenital amusia with harmonic stimuli and a priming task. Amusic and control participants performed a speeded phoneme discrimination task on sung chord sequences. The target phoneme was sung either on a functionally important chord (tonic chord, referred to as “related target”) or a less important one (subdominant chord, referred to as “less-related target”). Correct response times were faster when the target phoneme was sung on tonic chords rather than on subdominant chords, and this effect was less pronounced, albeit significant, in amusic participants. These data report for the first time a deficit in congenital amusia for chord processing, but also provide evidence that, despite this deficit, amusic individuals have internalized sophisticated syntactic-like functions of chords in the Western tonal musical system. This finding suggests that thanks to this musical knowledge, amusic individuals could develop expectancies for musical events, and, presumably, follow the tension-relaxation schemas in Western tonal music, which also influence emotional responses to music.

From Hipparchus to a helix ruler

This paper gives a brief history of the chord function — a precursor of the sine function — from the mathematics/astronomy of Hipparchus and Ptolemy to the early twentieth century.

Shadows of music–language interaction on low frequency brain oscillatory patterns

Electrophysiological studies investigating similarities between music and language perception have relied exclusively on the signal averaging technique, which does not adequately represent oscillatory aspects of electrical brain activity that are relevant for higher cognition. The current study investigated the patterns of brain oscillations during simultaneous processing of music and language using visually presented sentences and auditorily presented chord sequences. Music-syntactically regular or irregular chord functions were presented in sync with syntactically or semantically correct or incorrect words. Irregular chord functions (presented simultaneously with a syntactically correct word) produced an early (150–250 ms) spectral power decrease over anterior frontal regions in the theta band (5–7 Hz) and a late (350–700 ms) power increase in both the delta and the theta band (2–7 Hz) over parietal regions. Syntactically incorrect words (presented simultaneously with a regular chord) elicited a similar late power increase in delta–theta band over parietal sites, but no early effect. Interestingly, the late effect was significantly diminished when the language-syntactic and music-syntactic irregularities occurred at the same time. Further, the presence of a semantic violation occurring simultaneously with regular chords produced a significant increase in later delta–theta power at posterior regions; this effect was marginally decreased when the identical semantic violation occurred simultaneously with a music syntactical violation. Altogether, these results show that low frequency oscillatory networks get activated during the syntactic processing of both music and language, and further, these networks may possibly be shared.

Nonlinear semiclassical dynamics of open systems

A semiclassical approximation for an evolving density operator, driven by a 'closed' Hamiltonian and 'open' Markovian Lindblad operators, is reviewed. The theory is based on the chord function, i.e. the Fourier transform of the Wigner function. It reduces to an exact solution of the Lindblad master equation if the Hamiltonian is a quadratic function and the Lindblad operators are linear functions of positions and momenta. The semiclassical formulae are interpreted within a (real) double phase space, generated by an appropriate classical double Hamiltonian. An extra 'open' term in the double Hamiltonian is generated by the non-Hermitian part of the Lindblad operators in the general case of dissipative Markovian evolution. Decoherence narrows the relevant region of double phase space to the neighbourhood of a caustic for both the Wigner function and the chord function. This difficulty is avoided by the definition of a propagator, here developed in both representations. Generalized asymptotic equilibrium solutions are thus presented for the first time.

On generalized buffon needle problem for lattices

Two new concepts, the generalized support function and restricted chord function, both referring to a convex set, were introduced in [1]. General formulae to yield the kinematic measure of a segment of fixed length in a convex set were established based on these concepts. In this article, using the partial intersection method, we consider the generalized Buffon problem for three kinds of lattices. We determine the probability of intersection of a body test needle of length l, l < α.

Differences in Electric Brain Responses to Melodies and Chords

The music we usually listen to in everyday life consists of either single melodies or harmonized melodies (i.e., of melodies "accompanied" by chords). However, differences in the neural mechanisms underlying melodic and harmonic processing have remained largely unknown. Using EEG, this study compared effects of music-syntactic processing between chords and melodies. In melody blocks, sequences consisted of five tones, the final tone being either regular or irregular (p = .5). Analogously, in chord blocks, sequences consisted of five chords, the final chord function being either regular or irregular. Melodies were derived from the top voice of chord sequences, allowing a proper comparison between melodic and harmonic processing. Music-syntactic incongruities elicited an early anterior negativity with a latency of approximately 125 msec in both the melody and the chord conditions. This effect was followed in the chord condition, but not in the melody condition, by an additional negative effect that was maximal at approximately 180 msec. Both effects were maximal at frontal electrodes, but the later effect was more broadly distributed over the scalp than the earlier effect. These findings indicate that melodic information (which is also contained in the top voice of chords) is processed earlier and with partly different neural mechanisms than harmonic information of chords.

Semiclassical theory for small displacements

Characteristic functions contain complete information about all the moments of a classical distribution and the same holds for the Fourier transform of the Wigner function: a quantum characteristic function, or the chord function. However, knowledge of a finite number of moments does not allow for accurate determination of the chord function. This provides the overlap of the state with all its possible rigid translations (or displacements). We here present a semiclassical approximation of the chord function for large Bohr-quantized states, which is accurate right up to a caustic, beyond which the chord function becomes evanescent. It is verified to pick out blind spots, which are displacements for zero overlaps. These occur even for translations within a Planck area of the origin. We derive a simple approximation for the closest blind spots, depending on the Schrödinger covariance matrix, which is verified for Bohr-quantized states.

‘A Science of Tonal Love’? Drive and Desire in Twentieth‐Century Harmony: the Erotics of Skryabin

ABSTRACTLeonid Sabaneyev attested that Skryabin's compositions contained within them ‘a science of tonal love’, and Skryabin himself described his two Op. 57 pieces –Désir and Caresse dansée– as ‘new ways of making love’. But what makes this music so erotic in nature? The composer theorised about the nature of desire and sexuality in his writings, but this discussion rarely spills over into analysis of his compositional system. Given that Skryabin was so steeped in psychology throughout his life, I appeal to the work of Freud and Jacques Lacan, and particularly to their distinction between drive and desire (essentially, the fundamental instinct of the id versus its imaginary representation), a distinction found in Skryabin's own philosophical writings. But the progression between these two states bears comparison with both his philosophy and his harmonic processes, and I thus focus on the function of the dominant chord, exploring ways in which it can replicate the structures of drive and desire. In so doing, I scrutinise several piano miniatures to show that part of Skryabin's method of embodying drive in music lays out ambiguous chord structures which bear simultaneous tendencies to move in a number of different directions, as multivalent as the drive in the human subject. Further, I attempt to show that, out of mystical sonorities, Skryabin temporally unfolds a dialogue of different dominant ‘drives’, and eventually selects and nurtures a single one at the expense of others, a motion equivalent to desire.

Harmonic Stasis and Oscillation in Björk’s Medúlla

Björk Guðmundsdóttir’s 2004 all-vocal album Medúlla provides an opportunity to explore processes of harmonic oscillation, the repetition of two or more alternating harmonies, and stasis, the continuous repetition of a single harmony. Qualities of timelessness are evoked through harmonic stasis, which allows for intricate vocal textures to come to the fore in songs such as “Komið” and “Öll Birtan.” “Who Is It” illustrates how local-level and global processes of harmonic oscillation interact and how traditional chord function may be negated through oscillation. To account for harmonic progressions that suggest tonal centricity, concepts of repose and tension are applied as interpretive models to passages from “Triumph of a Heart.” This paper offers a model to describe oscillation and combines models developed by Tim Hughes to explore other uses of harmonic oscillation and stasis. Overarching oscillations support extra-musical narratives suggested by song lyrics and music videos. These models of analysis can be applied to other songs by Björk, as well as other popular music, as an alternative to traditional methods of understanding tonal harmony, which break down for such songs.

Determination of convex bodies by−∞-chord functions

AbstractIn 1998 A. Soranzo introduced the notions of +∞- and −∞-chord functions (see [

Semiclassical evolution of dissipative Markovian systems

A semiclassical approximation for an evolving density operator, driven by a ‘closed’ Hamiltonian operator and ‘open’ Markovian Lindblad operators, is obtained. The theory is based on the chord function, i.e. the Fourier transform of the Wigner function. It reduces to an exact solution of the Lindblad master equation if the Hamiltonian operator is a quadratic function and the Lindblad operators are linear functions of positions and momenta. Initially, the semiclassical formulae for the case of Hermitian Lindblad operators are reinterpreted in terms of a (real) double phase space, generated by an appropriate classical double Hamiltonian. An extra ‘open’ term is added to the double Hamiltonian by the non-Hermitian part of the Lindblad operators in the general case of dissipative Markovian evolution. The particular case of generic Hamiltonian operators, but linear dissipative Lindblad operators, is studied in more detail. A Liouville-type equivariance still holds for the corresponding classical evolution in double phase space, but the centre subspace, which supports the Wigner function, is compressed, along with expansion of its conjugate subspace, which supports the chord function. Decoherence narrows the relevant region of double phase space to the neighbourhood of a caustic for both the Wigner function and the chord function. This difficulty is avoided by a propagator in a mixed representation, so that a further ‘small-chord’ approximation leads to a simple generalization of the quadratic theory for evolving Wigner functions.

Amygdala activity can be modulated by unexpected chord functions during music listening

Numerous earlier studies have investigated the cognitive processing of musical syntax with regular and irregular chord sequences. However, irregular sequences may also be perceived as unexpected, and therefore have a different emotional valence than regular sequences. We provide behavioral data showing that irregular chord functions presented in chord sequence paradigms are perceived as less pleasant than regular sequences. A reanalysis of functional MRI data showed increased blood oxygen level-dependent signal changes bilaterally in the amygdala in response to music-syntactically irregular (compared with regular) chord functions. The combined data indicate that music-syntactically irregular events elicit brain activity related to emotional processes, and that, in addition to intensely pleasurable music or highly unpleasant music, single chord functions can also modulate amygdala activity.

Still More on Displacement

My thanks to Don Traut for his very intriguing comments on the larger role of displacement in Stravinsky's music. A strong parallel does indeed seem to exist between the metrical displacement of repeated motives and chords and the displacement of contrapuntal lines. Especially revealing is Traut's account of a passage from Stravinsky's Piano Concerto (Ex. 3). Schenker's earlier commentary on this passage (Ex. 4) is shown to have been a good deal more savvy about the nature of Stravinsky's art than his outright rejection of it would have led one to believe. The overlapping of tonic and dominant harmony, pervasive in the composer's neoclassical works from the time of Pulcinella (1919), has frequently been remarked upon, of course, as have the static, non developmental implications of the practice. By way of the Piano Sonata (Ex. 8), Traut includes it as a form of displacement. The sense of motion associated with the actual progression of these chord functions is more or less cancelled out by their superimposition. Harmony is flattened out in the process. My view of the Piano Concerto is that, at least on a local scale in the first movement, many of the techniques cited by Traut derive from Bach's use of the non-harmonic, metrically accented passing tone (another from of displacement, of course). Bach's keyboard music served as a constant point of reference for Stravinsky during the 1920s, even as a kind of daily staple for a while. The anti-modernist slogan back to Bach dates from this era. Underscored by Traut as well is the governing role that displace ment assumes in Stravinsky's music, whatever the stylistic orientation, presumably Russian, neoclassical, or serial. Overlooked by him, however, are the contrapuntal shifts that result as a matter of course from strati fication. In Stravinsky's layered structures, superimposed fragments, registrally and often instrumental!/ fixed, repeat according to spans or cycles that vary independently of one another. The result is an

Effects of Unexpected Chords and of Performer's Expression on Brain Responses and Electrodermal Activity

BackgroundThere is lack of neuroscientific studies investigating music processing with naturalistic stimuli, and brain responses to real music are, thus, largely unknown.Methodology/Principal FindingsThis study investigates event-related brain potentials (ERPs), skin conductance responses (SCRs) and heart rate (HR) elicited by unexpected chords of piano sonatas as they were originally arranged by composers, and as they were played by professional pianists. From the musical excerpts played by the pianists (with emotional expression), we also created versions without variations in tempo and loudness (without musical expression) to investigate effects of musical expression on ERPs and SCRs. Compared to expected chords, unexpected chords elicited an early right anterior negativity (ERAN, reflecting music-syntactic processing) and an N5 (reflecting processing of meaning information) in the ERPs, as well as clear changes in the SCRs (reflecting that unexpected chords also elicited emotional responses). The ERAN was not influenced by emotional expression, whereas N5 potentials elicited by chords in general (regardless of their chord function) differed between the expressive and the non-expressive condition.Conclusions/SignificanceThese results show that the neural mechanisms of music-syntactic processing operate independently of the emotional qualities of a stimulus, justifying the use of stimuli without emotional expression to investigate the cognitive processing of musical structure. Moreover, the data indicate that musical expression affects the neural mechanisms underlying the processing of musical meaning. Our data are the first to reveal influences of musical performance on ERPs and SCRs, and to show physiological responses to unexpected chords in naturalistic music.

Uniform approximation for the overlap caustic of a quantum state with its translations

The semiclassical Wigner function for a Bohr-quantized energy eigenstate is known to have a caustic along the corresponding classical closed phase space curve in the case of a single degree of freedom. Its Fourier transform, the semiclassical chord function, also has a caustic along the conjugate curve defined as the locus of diameters, i.e. the maximal chords of the original curve. If the latter is convex, so is its conjugate, resulting in a simple fold caustic. Indentations in the quantized curve generate self-intersections of the closed fold line and isolated cusp points. The uniform approximation through the fold caustic, that is derived here, describes the transition undergone by the overlap of the state with its translation, from an oscillatory regime for small chords, to evanescent overlaps, rising to a maximum near the caustic. The diameter caustic for the Wigner function is also treated.

The tonal function of a task-irrelevant chord modulates speed of visual processing

Harmonic priming studies have provided evidence that musical expectations influence sung phoneme monitoring, with facilitated processing for phonemes sung on tonally related (expected) chords in comparison to less-related (less-expected) chords [Bigand, Tillmann, Poulin, D’Adamo, and Madurell (2001). The effect of harmonic context on phoneme monitoring in vocal music. Cognition, 81, B11–B20]. This tonal relatedness effect has suggested two interpretations: (a) processing of music and language interact at some level of processing; and (b) tonal functions of chords influence task performance via listeners’ attention. Our study investigated these hypotheses by exploring whether the effect of tonal relatedness extends to the processing of visually presented syllables (Experiments 1 and 2) and geometric forms (Experiments 3 and 4). For Experiments 1–4, visual target identification was faster when the musical background fulfilled listeners’ expectations (i.e., a related chord was played simultaneously). In Experiment 4, the addition of a baseline condition (i.e., without an established tonal center) further showed that the observed difference was due to a facilitation linked to the related chord and not to an inhibition or disruption caused by the less-related chord. This outcome suggests the influence of musical structures on attentional mechanisms and that these mechanisms are shared between auditory and visual modalities. The implications for research investigating neural correlates shared by music and language processing are discussed.

Short-term effects of processing musical syntax: An ERP study

We investigated influences of short-term experience on music-syntactic processing, using a chord-sequence paradigm in which sequences ended on a harmony that was syntactically either regular or irregular. In contrast to previous studies (in which block durations were rather short), chord sequences were presented to participants for around 2 h while they were watching a silent movie with subtitles. Results showed that the music-syntactically irregular chord functions elicited an early right anterior negativity (ERAN), and that the ERAN amplitude significantly declined over the course of the experiment. The ERAN has previously been suggested to reflect the processing of music-syntactic irregularities, and the present data show that the cognitive representations of musical regularities are influenced by the repeated presentation of unexpected, irregular harmonies. Because harmonies were task-irrelevant, the data suggest that cognitive representations of musical regularities can change implicitly, i.e., even when listeners do not attend to the harmonies, and when they are presumably oblivious of the changes of such representations. Although the ERAN amplitude was significantly reduced, it was still present towards the end of the experiment at the right anterior electrodes, indicating that cognitive representations of basic music-syntactic regularities cannot easily be erased.