Beats that Commute: Algebraic and Kinesthetic Models for Math-Rock Grooves

Math rock’s most salient compositional facet is its cyclical repetition of grooves featuring changing and odd-cardinality meter. These unconventional grooves deform the conventional rhythmic structures of rock, such as backbeat and steady pulse, in such a way that a listener’s sense of metric organization is initially thwarted. Using transcriptions from math-rock artists such as Radiohead, The Mars Volta, and The Chariot, the author demonstrates a new analytical apparatus aimed at making sense of the ways listeners and performers process these changing pulse levels: the pivot pulse. The pivot pulse is defined as the slowest temporal level preserved in a given meter change. The author suggests that the preservation or disruption of the primary pulse level (that is, the temporal level at which a listener’s or performer’s primary kinesthetic involvement happens, such as dancing or foot-tapping) is of paramount importance. For example, a change from 4/4 to 3/4, which preserves the quarter-note pulse, will be less disruptive to a listener’s metric entrainment than a change from 4/4 to 7/8 or 7/8 to 15/16, both of which split the primary pulse in half. In order to formalize pivot-pulse methodology, the author presents an algebraic model based on the commutative operations greatest common denominator and lowest common multiple. Pivot-pulse methodology is also applied metaphorically to the kinesthetic interpretations of performers and listeners to better understand the complex movements incited by math-rock grooves

Beats that Commute: Algebraic and Kinesthetic Models for Math-Rock Grooves

Math rock’s most salient compositional facet is its cyclical repetition of grooves featuring changing and odd-cardinality meter. These unconventional grooves deform the conventional rhythmic structures of rock, such as backbeat and steady pulse, in such a way that a listener’s sense of metric organization is initially thwarted. Using transcriptions from math-rock artists such as Radiohead, The Mars Volta, and The Chariot, the author demonstrates a new analytical apparatus aimed at making sense of the ways listeners and performers process these changing pulse levels: the pivot pulse. The pivot pulse is defined as the slowest temporal level preserved in a given meter change. The author suggests that the preservation or disruption of the primary pulse level (that is, the temporal level at which a listener’s or performer’s primary kinesthetic involvement happens, such as dancing or foot-tapping) is of paramount importance. For example, a change from 4/4 to 3/4, which preserves the quarter-note pulse, will be less disruptive to a listener’s metric entrainment than a change from 4/4 to 7/8 or 7/8 to 15/16, both of which split the primary pulse in half. In order to formalize pivot-pulse methodology, the author presents an algebraic model based on the commutative operations greatest common denominator and lowest common multiple. Pivot-pulse methodology is also applied metaphorically to the kinesthetic interpretations of performers and listeners to better understand the complex movements incited by math-rock grooves

BEATS THAT COMMUTE: ALGEBRAIC AND KINESTHETIC MODELS FOR MATH-ROCK GROOVES

Math rock’s most salient compositional facet is its cyclical repetition of ostinati featuring changing and odd-cardinality meters

Notes on explicit and inversion formulas for the Chebyshev polynomials of the first two kinds

Feng Qi, Da-Wei Niu, and Dongkyu Lim, Notes on explicit and inversion formulas for the Chebyshev polynomials of the first two kinds, Miskolc Mathematical Notes 20 (2019), no. 2, 1129--1137; available online at https://doi.org/10.18514/MMN.2019.2976.International audienceIn the paper, starting from the Rodrigues formulas for the Chebyshev polynomials of the first and second kinds, by virtue of the Fa\`a di Bruno formula, with the help of two identities for the Bell polynomials of the second kind, and making use of a new inversion theorem for combinatorial coefficients, the authors derive two nice explicit formulas and their corresponding inversion formulas for the Chebyshev polynomials of the first and second kinds

On the calculation of the gauge volume size for energy-dispersive X-ray diffraction

Equations for the calculation of the dimensions of a gauge volume, also known as the active volume or diffraction lozenge, in an energy-dispersive diffraction experiment where the detector is collimated by two ideal slits have been developed. Equations are given for equatorially divergent and parallel incident X-ray beams, assuming negligible axial divergenc

Vibrating quantum billiards on Riemannian manifolds

Quantum billiards provide an excellent forum for the analysis of quantum chaos. Toward this end, we consider quantum billiards with time-varying surfaces, which provide an important example of quantum chaos that does not require the semiclassical ($\hbar \longrightarrow 0$) or high quantum-number limits. We analyze vibrating quantum billiards using the framework of Riemannian geometry. First, we derive a theorem detailing necessary conditions for the existence of chaos in vibrating quantum billiards on Riemannian manifolds. Numerical observations suggest that these conditions are also sufficient. We prove the aforementioned theorem in full generality for one degree-of-freedom boundary vibrations and briefly discuss a generalization to billiards with two or more degrees-of-vibrations. The requisite conditions are direct consequences of the separability of the Helmholtz equation in a given orthogonal coordinate frame, and they arise from orthogonality relations satisfied by solutions of the Helmholtz equation. We then state and prove a second theorem that provides a general form for the coupled ordinary differential equations that describe quantum billiards with one degree-of-vibration boundaries. This set of equations may be used to illustrate KAM theory and also provides a simple example of semiquantum chaos. Moreover, vibrating quantum billiards may be used as models for quantum-well nanostructures, so this study has both theoretical and practical applications.Comment: 23 pages, 6 figures, a few typos corrected. To appear in International Journal of Bifurcation and Chaos (9/01

Temporal moments of a tracer pulse in a perfectly parallel flow system

Perfectly parallel groundwater transport models partition water flow into isolated one-dimensional stream tubes which maintain total spatial correlation of all properties in the direction of flow. The case is considered of the temporal moments of a conservative tracer pulse released simultaneously into N stream tubes with arbitrarily different advective–dispersive transport and steady flow speeds in each of the stream tubes. No assumptions are made about the form of the individual stream tube arrival-time distributions or about the nature of the between-stream tube variation of hydraulic conductivity and flow speeds. The tracer arrival-time distribution g(t,x) is an N-component finite-mixture distribution, with the mean and variance of each component distribution increasing in proportion to tracer travel distance x. By utilising moment relations of finite mixture distributions, it is shown (to r=4) that the rth central moment of g(t,x) is an rth order polynomial function of x or φ, where φ is mean arrival time. In particular, the variance of g(t,x) is a positive quadratic function of x or φ. This generalises the well-known quadratic variance increase for purely advective flow in parallel flow systems and allows a simple means of regression estimation of the large-distance coefficient of variation of g(t,x). The polynomial central moment relation extends to the purely advective transport case which arises as a large-distance limit of advective–dispersive transport in parallel flow models. The associated limit g(t,x) distributions are of N-modal form and maintain constant shapes independent of travel distance. The finite-mixture framework for moment evaluation is also a potentially useful device for forecasting g(t,x) distributions, which may include multimodal forms. A synthetic example illustrates g(t,x) forecasting using a mixture of normal distributions