Papers by Daniel Muzzulini
Zenodo (CERN European Organization for Nuclear Research), 2015
Zurich/Lucerne * The author wishes to thank Martin Neukom (ICST Zurich) and Roman Oberholzer (KSA... more Zurich/Lucerne * The author wishes to thank Martin Neukom (ICST Zurich) and Roman Oberholzer (KSALP Lucerne) for their useful and critical comments, Lesley Paganetti (Basel) for proofreading and interesting debates, and Benjamin Wardhaugh (All Souls College, Oxford) for proofreading the nal text. This essay was written in the course of research for the project Sound-Colour-Space:
Empirical Musicology Review, Jun 28, 2021
Springer eBooks, 2022
Boethius and his followers used diagrammatic methods to estimate musical intervals with epimoric ... more Boethius and his followers used diagrammatic methods to estimate musical intervals with epimoric ratios, they determined geometric number sequences with triangular tables, and they treated the converse problem of dividing musical intervals equally. The collection of mathematical manuscripts Codex Basel F II 33 (ca. 1360) contains treatises by Nicolaus Oresme, Jordanus Nemorarius and others. Images in Nemorarius' treatise combine number triangles into complex spider webs and they display recursive algorithms. Oresme diagrams make use of irrational ratios. These little known images and their relationship to music theory are the focus of this paper.
Computers & Mathematics with Applications, 1989
This work presents (1) a mathematical model of classical counterpoint, based on distinguished sym... more This work presents (1) a mathematical model of classical counterpoint, based on distinguished symmetries between consonant and dissonant musical intervals and derived local symmetries, together with (2) an investigation of the electrical activity (depth EEG) of the human brain in relation with consonant and dissonant musical stimuli. Presenting a musical test program to 13 patients with electrodes implanted within different brain areas [hippocampal formations of both sides, planum temporale (near Heschl's gyms) and/or placed epicortically at mediobasal limbic structures], we found that the reaction of depth EEG corresponds in a precise and quantified way to the postulates of mathematical counterpoint theory. The main results are: (1) The EEG of the hippocampus reflects the consonance-dissonance dichotomy for simultaneous intervals in a predominant way. (2) Within the right Heschl's gyrus, the EEG response to the distinguished symmetry between consonances and dissonances is significant. (3) The EEG of right hemispheric locations dominate the processing of music related to Gestalt perception in space-time (pitch/onset-time), in particular of successive intervals. (4) The geometrically distinguished pair of dichotomies (the consonance-dissonance dichotomy and dichotomy of proper tonal intervals from the major tonic) is reflected within the spectral density data of the classical 0-, ct-, and p-frequency bands. These findings may help to understand the relation between music and emotion.
CERN European Organization for Nuclear Research - Zenodo, 2022
Boethius and his followers used diagrammatic methods to estimate musical intervals with epimoric ... more Boethius and his followers used diagrammatic methods to estimate musical intervals with epimoric ratios, they determined geometric number sequences with triangular tables, and they treated the converse problem of dividing musical intervals equally. The collection of mathematical manuscripts Codex Basel F II 33 (ca. 1360) contains treatises by Nicolaus Oresme, Jordanus Nemorarius and others. Images in Nemorarius' treatise combine number triangles into complex spider webs and they display recursive algorithms. Oresme diagrams make use of irrational ratios. These little known images and their relationship to music theory are the focus of this paper.
Lecture Notes in Computer Science, 2022
Boethius and his followers used diagrammatic methods to estimate musical intervals with epimoric ... more Boethius and his followers used diagrammatic methods to estimate musical intervals with epimoric ratios, they determined geometric number sequences with triangular tables, and they treated the converse problem of dividing musical intervals equally. The collection of mathematical manuscripts Codex Basel F II 33 (ca. 1360) contains treatises by Nicolaus Oresme, Jordanus Nemorarius and others. Images in Nemorarius' treatise combine number triangles into complex spider webs and they display recursive algorithms. Oresme diagrams make use of irrational ratios. These little known images and their relationship to music theory are the focus of this paper.
Ohio State University. Libraries, 2021
European Mathematical Society Magazine, 2022
In this paper, we solve an interesting Diophantine equation that is born from classical questions... more In this paper, we solve an interesting Diophantine equation that is born from classical questions of music theory.
Preda Mihăilescu, Daniel Muzzulini, A Diophantine equation concerning epimoric ratios. Eur. Math. Soc. Mag. 124 (2022), pp. 16–22
By investigating the conceptual field of sound, tone, pitch, and timbre in its relation to visual... more By investigating the conceptual field of sound, tone, pitch, and timbre in its relation to visual phenomena and geometrical concepts, the project Sound Colour Space – A Virtual Museumcontributes to an interdisciplinary field of research and explores its adequate modes of representation and communication. Many scientists and philosophers from antiquity to modern times have studied the relationships between sound, light and geometry. Many of their visualisations of acoustical, optical and perceptual topics speak to the eye and can be studied comparatively. These pictures are interesting because of their diagrammatic structure, in the way they combine text, images and spatial structures on a flat surface and in the way they address topological, philosophical and psychological questions. They often have an aesthetic value of their own. In addition to the development of an exemplary online publication, interactive audiovisual examples were created, which also were used for artistic projects
Die Befragung von Theoretikertexten aus vier Jahrhunderten zum Wesen der Klangfarbe, die Untersuc... more Die Befragung von Theoretikertexten aus vier Jahrhunderten zum Wesen der Klangfarbe, die Untersuchung der Dialektik von mathematisch-naturwissenschaftlichen und musikalisch- philosophischen Ideologien und Paradigmata wirft Licht auf eine komplexe, facherubergreifende Theoriedynamik. Als ein Hauptergebnis kann eine bemerkenswerte Nicht-Linearitat des wissenschaftlichen Fortschritts konstatiert werden. Die zentralen Konzepte und Theorien zur Klangfarbe, die als Errungenschaften des 19. Jahrhunderts gelten, sind alle schon fruher bedacht worden. Bereits im 17. Jahrhundert werden die Existenz der Obertone, Fragen der Spektraldynamik, die Knotenbildung bei Flageoletttonen, der Einfluss der spektralen Zusammensetzung auf die Klangqualitat, das Resonanzverhalten der Basilarmembran sowie die Mehrfachresonanz in der menschlichen Stimme zur Sprache gebracht. Die Mathematik entwickelt zeitgleich die Infinitesimalrechnung, den Funktions- und den abstrakten Dimensionsbegriff. Diese Errungenschaf...
Empirical Musicology Review (EMR), 2020
In 1665 Isaac Newton wrote a notebook in which he collected materials for a musical treatise whic... more In 1665 Isaac Newton wrote a notebook in which he collected materials for a musical treatise which was never completed. He investigated ways of approximately representing just intonation scales by dividing the octave into many equally sized intervals. Strictly speaking, equal divisions of the octave are incompatible with just intonation, and just intonation intervals are audibly different from the intervals played on a modern equally tempered modern piano. By increasing the number of parts of an equal division, just intonation can be approximated arbitrarily well. Scales with more than 60 microtonal steps per octave, however, never gained wide acceptance in music theory or practice. Newton divided the octave into 612 equal parts so that he could represent the syntonic chromatic scale very accurately and he studied several equal divisions of the octave with fewer parts. His approximation problem is looked at in three ways: (1) A reconstruction of how he determined the many EDO-repres...
Acta Musicologica, 2015
Abstract: The aim of this essay is to create a geometrical link between the music theory and the ... more Abstract: The aim of this essay is to create a geometrical link between the music theory and the mathematics of the early 17th century by studying and comparing diagrams which directly or indirectly refer to mathematical logarithms.The focus is on the relationships between ratios of numbers referring to sounds and related concepts of perception. The relationship between frequency and pitch is a paradigmatic case of the Weber-Fechner law of psychophysics, stating that equal frequency ratios are perceived as equally sized musical intervals. The Weber-Fechner law maintains that many perceptual phenomena are logarithmic by their very nature.The circular diagrams studied here are by Descartes (1618), Robert Fludd (1618) and Jost Bürgi (1620). Descartes’s diagrams have recently attracted the attention of authors from different fields. A second type of geometric diagrams related to musical arithmetic is looked at in the final section of this article.
Diagrams Conference (2021) : Poster, 2021
The 14 century diagrams in the focus of this paper deal with geometric sequences, fractional powe... more The 14 century diagrams in the focus of this paper deal with geometric sequences, fractional powers and recursion, which are mathematical topics relevant also for music theory: defining consonant intervals, comparing the sizes of musical intervals and dividing them into equal parts, constructing musical scales with several equal steps. These diagrams might have paved the way to mathematical concepts further developed only in the 16 and 17 centuries.
Journal of Music Theory, 1995
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Papers by Daniel Muzzulini
Preda Mihăilescu, Daniel Muzzulini, A Diophantine equation concerning epimoric ratios. Eur. Math. Soc. Mag. 124 (2022), pp. 16–22
Preda Mihăilescu, Daniel Muzzulini, A Diophantine equation concerning epimoric ratios. Eur. Math. Soc. Mag. 124 (2022), pp. 16–22