58,492 research outputs found
The Distance Geometry of Music
We demonstrate relationships between the classic Euclidean algorithm and many
other fields of study, particularly in the context of music and distance
geometry. Specifically, we show how the structure of the Euclidean algorithm
defines a family of rhythms which encompass over forty timelines
(\emph{ostinatos}) from traditional world music. We prove that these
\emph{Euclidean rhythms} have the mathematical property that their onset
patterns are distributed as evenly as possible: they maximize the sum of the
Euclidean distances between all pairs of onsets, viewing onsets as points on a
circle. Indeed, Euclidean rhythms are the unique rhythms that maximize this
notion of \emph{evenness}. We also show that essentially all Euclidean rhythms
are \emph{deep}: each distinct distance between onsets occurs with a unique
multiplicity, and these multiplicies form an interval . Finally,
we characterize all deep rhythms, showing that they form a subclass of
generated rhythms, which in turn proves a useful property called shelling. All
of our results for musical rhythms apply equally well to musical scales. In
addition, many of the problems we explore are interesting in their own right as
distance geometry problems on the circle; some of the same problems were
explored by Erd\H{o}s in the plane.Comment: This is the full version of the paper: "The distance geometry of deep
rhythms and scales." 17th Canadian Conference on Computational Geometry (CCCG
'05), University of Windsor, Canada, 200
Designing the sound of a cut-off drum
The spectral action in noncommutative geometry naturally implements an ultraviolet cut-off, bycounting the eigenvalues of a (generalized) Dirac operator lower than an energy of unification.Inverting the well known question \u201chow to hear the shape of a drum\u201d, we ask what drum can bedesigned by hearing the truncated music of the spectral action ? This makes sense because thesame Dirac operator also determines the metric, via Connes distance. The latter thus offers anoriginal way to implement the high-momentum cut-off of the spectral action as a short distancecut-off on space. This is a non-technical presentation of the results of
A space for inflections: following up on JMM's special issue on mathematical theories of voice leading
Journal of Mathematics and Music's recent special issue 7(2) reveals substantial common ground between mathematical theories of harmony advanced by Tymoczko, Hook, Plotkin, and Douthett. This paper develops a theory of scalar inflection as a kind of voice-leading distance using quantization in voice-leading geometries, which combines the best features of different approaches represented in the special issue: it is grounded in the concrete sense of voice-leading distance promoted by Tymoczko, invokes scalar contexts in a similar way as filtered point-symmetry, and abstracts the circle of fifths like Hook's signature transformations. The paper expands upon Tymoczko's âgeneralized signature transformâ showing the deep significance of generalized circles of fifths to voice-leading properties of all collections. Analysis of Schubert's Notturno for Piano Trio and âNacht und TrĂ€umeâ demonstrate the musical significance of inflection as a kind of voice leading, and the value of a robust geometrical understanding of it.Accepted manuscrip
Persistent topology for natural data analysis - A survey
Natural data offer a hard challenge to data analysis. One set of tools is
being developed by several teams to face this difficult task: Persistent
topology. After a brief introduction to this theory, some applications to the
analysis and classification of cells, lesions, music pieces, gait, oil and gas
reservoirs, cyclones, galaxies, bones, brain connections, languages,
handwritten and gestured letters are shown
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The influences of environmental conditions on source localisation using a single vertical array and their exploitation through ground effect inversion
The performance of microphone arrays outdoors is influenced by the environmental conditions. Numerical simulations indicate that, while horizontal arrays are hardly affected, direction-of-arrival (DOA) estimation with vertical arrays becomes biased in presence of ground reflections and sound speed gradients. Turbulence leads to a huge variability in the estimates by reducing the ground effect. Ground effect can be exploited by combining classical source localization with an appropriate propagation model (ground effect inversion). Not only does this allow the source elevation and range to be determined with a single vertical array but also it allows separation of sources which can no longer be distinguished by far field localization methods. Furthermore, simulations provide detail of the achievable spatial resolution depending on frequency range, array size and localization algorithm and show a clear advantage of broadband processing. Outdoor measurements with one or two sources confirm the results of the numerical simulations
Generalized Tonnetze and Zeitnetze, and the topology of music concepts
The music-theoretic idea of a Tonnetz can be generalized at different levels: as a network of chords relating by maximal intersection, a simplicial complex in which vertices represent notes and simplices represent chords, and as a triangulation of a manifold or other geometrical space. The geometrical construct is of particular interest, in that allows us to represent inherently topological aspects to important musical concepts. Two kinds of music-theoretical geometry have been proposed that can house Tonnetze: geometrical duals of voice-leading spaces and Fourier phase spaces. Fourier phase spaces are particularly appropriate for Tonnetze in that their objects are pitch-class distributions (real-valued weightings of the 12 pitch classes) and proximity in these space relates to shared pitch-class content. They admit of a particularly general method of constructing a geometrical Tonnetz that allows for interval and chord duplications in a toroidal geometry. This article examines how these duplications can relate to important musical concepts such as key or pitch height, and details a method of removing such redundancies and the resulting changes to the homology of the space. The method also transfers to the rhythmic domain, defining Zeitnetze for cyclic rhythms. A number of possible Tonnetze are illustrated: on triads, seventh chords, ninth chords, scalar tetrachords, scales, etc., as well as Zeitnetze on common cyclic rhythms or timelines. Their different topologies â whether orientable, bounded, manifold, etc. â reveal some of the topological character of musical concepts.Accepted manuscrip
Topology of Networks in Generalized Musical Spaces
The abstraction of musical structures (notes, melodies, chords, harmonic or
rhythmic progressions, etc.) as mathematical objects in a geometrical space is
one of the great accomplishments of contemporary music theory. Building on this
foundation, I generalize the concept of musical spaces as networks and derive
functional principles of compositional design by the direct analysis of the
network topology. This approach provides a novel framework for the analysis and
quantification of similarity of musical objects and structures, and suggests a
way to relate such measures to the human perception of different musical
entities. Finally, the analysis of a single work or a corpus of compositions as
complex networks provides alternative ways of interpreting the compositional
process of a composer by quantifying emergent behaviors with well-established
statistical mechanics techniques. Interpreting the latter as probabilistic
randomness in the network, I develop novel compositional design frameworks that
are central to my own artistic research
Inner-sphere complexation of cations at the rutile-water interface: A concise surface structural interpretation with the CD and MUSIC model
Acidâbase reactivity and ion-interaction between mineral surfaces and aqueous solutions is most frequently investigated at the macroscopic scale as a function of pH. Experimental data are then rationalized by a variety of surface complexation models. These models are thermodynamically based which in principle does not require a molecular picture. The models are typically calibrated to relatively simple solid-electrolyte solution pairs and may provide poor descriptions of complex multi-component mineralâaqueous solutions, including those found in natural environments. Surface complexation models may be improved by incorporating molecular-scale surface structural information to constrain the modeling efforts. Here, we apply a concise, molecularly-constrained surface complexation model to a diverse suite of surface titration data for rutile and thereby begin to address the complexity of multi-component systems. Primary surface charging curves in NaCl, KCl, and RbCl electrolyte media were fit simultaneously using a charge distribution (CD) and multisite complexation (MUSIC) model [Hiemstra T. and Van Riemsdijk W. H. (1996) A surface structural approach to ion adsorption: the charge distribution (CD) model. J. Colloid Interf. Sci. 179, 488â508], coupled with a Basic Stern layer description of the electric double layer. In addition, data for the specific interaction of Ca2+ and Sr2+ with rutile, in NaCl and RbCl media, were modeled. In recent developments, spectroscopy, quantum calculations, and molecular simulations have shown that electrolyte and divalent cations are principally adsorbed in various inner-sphere configurations on the rutile 1 1 0 surface [Zhang Z., Fenter P., Cheng L., Sturchio N. C., Bedzyk M. J., Predota M., Bandura A., Kubicki J., Lvov S. N., Cummings P. T., Chialvo A. A., Ridley M. K., BĂ©nĂ©zeth P., Anovitz L., Palmer D. A., Machesky M. L. and Wesolowski D. J. (2004) Ion adsorption at the rutileâwater interface: linking molecular and macroscopic properties. Langmuir 20, 4954â4969]. Our CD modeling results are consistent with these adsorbed configurations provided adsorbed cation charge is allowed to be distributed between the surface (0-plane) and Stern plane (1-plane). Additionally, a complete description of our titration data required inclusion of outer-sphere binding, principally for Cl- which was common to all solutions, but also for Rb+ and K+. These outer-sphere species were treated as point charges positioned at the Stern layer, and hence determined the Stern layer capacitance value. The modeling results demonstrate that a multi-component suite of experimental data can be successfully rationalized within a CD and MUSIC model using a Stern-based description of the EDL. Furthermore, the fitted CD values of the various inner-sphere complexes of the mono- and divalent ions can be linked to the microscopic structure of the surface complexes and other data found by spectroscopy as well as molecular dynamics (MD). For the Na+ ion, the fitted CD value points to the presence of bidenate inner-sphere complexation as suggested by a recent MD study. Moreover, its MD dominance quantitatively agrees with the CD model prediction. For Rb+, the presence of a tetradentate complex, as found by spectroscopy, agreed well with the fitted CD and its predicted presence was quantitatively in very good agreement with the amount found by spectroscopy
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