Music

We propose here a science of human body/embodiment. We discuss owings, and then offer an overall picture of this nascent science, as ies. We imagine a science whose core subject is the connection of the body -as it appears in an action-perception paradigm derived

We propose here a science of human body/embodiment. We discuss our motivations, provide scientific and artistic precedents/foreshadowings, and then offer an overall picture of this nascent science, as well as a sample, unsolved problem from the field of disability studies. We imagine a science whose core subject is the connection of

HE SHIFT from a set-theoretical to a transformational approach is one of the most intriguing and promising achievements of modern music theory. A study of the emergence of the group concept in music theory, analysis, and composition shows that the transformational per-spective ...

Based on modern information technology, mathematics and semiotics, systematic musicology has fundamentally changed. As a theoretical basis of this change, mathematical music theory offers a precise language and models to describe musicological facts. It resides on the three paradigms of symmetry, local-global gestalts and the Yoneda classification by "variation of perspectives". Operational- ization of the theory is an inte gral

We give an overview of mathematical music theory as it has been de-veloped in the past twenty years. The present theory includes a formal language for musical and musicological objects and relations. This lan-guage is built upon topos theory and its logic. Various models of ...

Die von Finsler in [1] betrachteten totalendlichen Mengen bilden einen interessanten Bereich ~ von endlichen Mengen, der die ,,natfirlichen Zahlen" 0, (0}, ({0}}, ... enthhlt: Die gew6hnliche, ffir natfirliche Zahlen definierte Addition und Multiplikation kann auf ganz ~• ausgedehnt werden; ~-ist nicht-kommutative reguliire Halbgruppe mit Neutralelement bei beiden Operationen, welche u.a. durch ein Linksdistributivgesetz verkniipft sind. Zu den Elementen einer 0,Zahlentheorie" in o~ geh6ren Shtze fiber die Zerfhllbarkeit von Zahlen aus ~-in Produkte und Summen von multiplikativ resp. additiv unzerlegbaren Zahlen. Unsere wichtigsten Ergebnisse in dieser Richtung finden sich in den Siitzen 11, 18, 20, 23 und 25 weiter unten. Der Bequemlichkeit halber sind die meisten der in [1] aufgestellten S~ttze hier nochmals kurz bewiesen worden. I. Definitionen Die Wendung ,,dann und nur dann" kfirzen wir im folgenden mit ,,falls" ab. Unter I,.)a verstehen wir wie fiblich die Menge {x I ex. y mit xeyea}. Oft schreiben wir statt Ua auch t,.Jx~,. Die Menge der natfirlichen Zahlen kiirzen wir mit ~o ab. Sei a eine Menge. DEFINITION I: Sei neoJ. Wir definieren die n-te Vereinigung U"a rekursiv durch (i) U~ =a. (ii) U J "+ a: = U (U"a). DEFINITION II: a heisst totalendlich (oder Finslerzahl), falls ein no existiert, sodass U"Oa=0. DEFINITION III: Das kleinste n omit U"~ heisst die Stufe yon a. Bezeichnung: no= :la].

We give statistical evidence that specific rhythmic, melodic, and harmonic analyses are dominating shaping structures for the agogical streams from 28 performances of Schumann's "Tr/iumerei", as measured by Repp [1]. These analyses are provided by the Rubato analysis and performance workstation. The statistical model is based on regression analysis and realizes shaping of agogics by a second-order linear differential operator. More precisely, the logarithm of the tempo function T(E) at onset time E can be expressed as a linear function of a sequence of particular smooth "weight" functions WI(E) and their first and second derivatives. These numerical functions are generated by Rubato's harmonic, melodic, and rhythmical analyses and include an averaging process over a natural grouping hierarchy defined by Schumann's score. (~) 2000 Elsevier Science Ltd. All rights reserved. ieywords-Music performance, Tempo curves, Time series analysis, Regression, Hierarchical smoothing.

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Our study sought to determine whether, where and how the consonance or dissonance of musical intervals might be reflected in the electrical activity (depth EEG) of the human brain. Presenting an adequate musical test program to a patient with depth electrodes implanted within the planum temporale (near Heschl's gyrus) and the hippocampal formations of both sides, we found that the reactions shown by the depth EEG of the limbic system correspond in precise and distinctly differing patterns to the consonances and dissonances presented. The main findings are: (1) the EEG of the left hippocampus reflects the consonancedissonance dichotomy; (2) the activity of the left hippocampus but not the right is affected by a previous sequence of dissonances in responding to a single consonance; (3) the EEG activity of the right Heschl's gyrus (the left Heschl's gyrus was not analyzed), but neither the left nor the right hippocampus, responded differently to a given succession of intervals and its pitch inversion. These results may help to understand the different emotional responses which usually arise in response to consonant or dissonant musical intervals.

Musical performance theory and the theory of musical structure in general is a rapidly developing field of musicology that has wide practical implications. Due to the complex nature of music, statistics is likely to play an important role. In spite of this, up to the present, applications of statistical methods to music have been rare and mostly limited to a formal confirmation of results obtained by other methods. The present paper introduces a statistical approach to the analysis of metric, melodic and harmonic structures of a score and their influence on musical performance. Examples by Schumann, Webern and Bach illustrate the proposed method of numerical encoding and hierarchical decomposition of score information. Application to performance data is exemplified by the analysis of tempo data for Schumann's "Träumerei" op. 15/7. The paper demonstrates why statistics should play a major active part in performance research. The results obtained here are only a starting point and should, hopefully, stimulate a fruitful discussion between statisticians, musicologists, computer scientists and other researchers interested in the area.

Riassunto  Il punto di partenza di questo lavoro è l'interpretazione delle topologie di Grothendieck nel linguaggio dei sistemi relazionali (cfr. [5]); ciò permette uno studio aritmetico di tali topologie. Attraverso nuove tecniche, qui introdotte, si ottengono risultati non privi di interesse, tra i quali una caratterizzazione degli spazi topologici noetheriani, ed ulteriori strutture topologiche di tipo locale (siti tangenti), per le

In abstract mathematical music theory, the data format of denotators is associated with set-valued presheaves over module categories. In this paper, we present an implementation of this concept framework in the Rubato Composer system, a Java application comprising a GUI for manipulation and combination of rubettes. These are plug-ins that can be added and connected for communication of denotator data. Rubato Composer is a GPL software and is accessible to the computer music community for download and collaboration. In this paper, the functionality, architecture, concept framework, and the implemented mathematical operators are presented and illustrated with a "functorial composition".

We give an overview of mathematical music theory as it has been developed in the past twenty years. The present theory includes a formal language for musical and musicological objects and relations. This language is built upon topos theory and its logic. Various models of musical phenomena have been developed. They include harmony (function theory, cadences, and modulations), classical counterpoint (Fux rules), rhythm, motif theory, and the theory of musical performance. Most of these models have also been implemented and evaluated in computer applications. Some models have been tested empirically in neuro-musicology and the cognitive science of music. The mathematical nature of this modeling process canonically embedds the given historical music theories in a variety of fictitious theories and thereby enables a qualification of historical reality against potential variants. As a result, the historical realizations often turn out to be some kind of "best possible world" and thus reveals a type of "anthropic principle" in music. These models use different types of mathematical approaches, such as-for instance-enumeration combinatorics, group and module theory, algebraic geometry and topology, vector fields and numerical solutions of differential equations, Grothendieck topologies, topos theory, and statistics. The results lead to good simulations of classical results of music and performance theory. There is a number of classifiaction theorems of determined categories of musical structures. The overview concludes by a discussion of mathematical and musicological challenges which issue from the investigation of music by mathematics, including the project of "Grand Unification" of harmony and counterpoint and the classification of musical performance.

In this paper, first musical compositions are presented, which are created using the mathematical counterpoint theory of Guerino Mazzola and his collaborators. These compositions also use the RUBATOr software’s components for counterpoint constructions. The present work aims at opening new “exotic” directions of contrapuntal composition in non-Fuxian worlds. The authors would like to receive first impressions about these compositions, which are available as scores and audio files. Introduction A mathematical model of classical counterpoint for first species as codified by [5] was first published in [7] and later described in detail in [8, 9]. The original model was extended to five “exotic” counterpoint worlds, where the Fux theory model can be restated mutatis mutandis for five new dichotomies of intervals in the 12-tempered chromatic tuning. The context of 12-tempered tuning was extended to “microtonal” environments, together with a sophisticated embedding of contrapuntal theories...

The mathematical modeling of classical counterpoint in [Maz02] complements the classical Fux system [Fux42] by five other “exotic” systems of rules for composition. They all derive from the same algebraic model, but only the traditional Fuxian system has been used by musicians, leaving five unexplored counterpoint worlds for the composition of new music. In this paper, we present and discuss concepts, algorithms, and the implementation in software modules of the Rubato composer system [MM06] for the possible morphing transitions between these systems. This enables the creation of exotic contrapuntal compositions by structural transport. To do so, we represent authorisations and interdictions within different counterpoint systems by directed graphs. Morphing contrapuntal compositions from one world into contrapuntal compositions compliant with the rules of another world then reduces to the construction of a set of distinguished digraph morphisms. This shows that the worlds do not onl...