The title page of Jost Buergi’s Progress-Tabulen (1620) is not only an eye catching masterpiece of the art of printing and a substrate of the tables inside, it also reveals fundamental insights into the nature of numbers and number systems.
The circular arrangement of the two number sequences links algebra with geometry by mapping the exponential function on an angular scale of red numbers to black numbers in the logarithmic place value system. In retrospect, the step from Buergi's diagram to the circular slide rule for multiplication seems a small one.
The lecture is intended as a contribution to the history of circular diagrams and volvelles – paper constructs with rotating parts – that emerged during the Renaissance. In particular, we will take a look at visualisations in contemporary music theory, which deal with closely related questions.
Opportunities to historically underpin and unconventionally motivate topics in mathematics education – geometric sequences, inverse functions, modulo arithmetic and number systems – emerge casually. Making calculating discs for music theory will sensitise learners for the fundaments of calculation techniques and number systems, and it promotes historically anchored, discovery-based learning. The transfer achievements to be made will sharpen the view for interdisciplinary topics.
The short video documents the activities of the Study Group "Musical Diagrams" at the 21st Quinquennial Congress of the International Musicological Society (IMS), which took place in Athens from 22 to 26 August 2022. The work "Typology & Topology of Musical Diagrams (2020)" by Gerhard Dirmosers (Linz) consisting of 34 posters in a matrix arrangement has been displayed for the first time. These posters feature the "universe of diagrams" from the history of music theory (and color theory) as interrelated diagrams following various networks.
Several posters by Daniel Muzzulini and his collaborators - Susan Forscher Weiss, Christoph Reuter, Christoph Stähli) complement the exhibition. Corresponding interactive new exhibits from the Virtual Museum Sound Colour Space and a mechatronic instrument (E-Bow) were also shown at the exhibition.
Nach der geläufigen Auffassung wurden Logarithmen im ausgehenden 16. Jahrhundert entwickelt. In der Musiklehre pythagoreischer Prägung hingegen ist logarithmisches Denken Standard. Ausdruck und Grundlage dieses Denkens ist die terminologische Unterscheidung von Zahlenverhältnissen und Intervallen. So bezeichnet "Sesquialtera" das Zahlenverhältnis 3 : 2 und "Diapente" gibt das zugehörige Intervall der Quinte an.
Der Abstandsbegriff für Tonhöhen, der sich in den musikalischen Intervallen und der zugehörigen Sprache manifestiert, stellt ein logarithmisches Mass für Zahlenverhältnisse dar. Die Tatsache, dass wir gleiche Frequenzverhältnisse als identische Intervalle erkennen und transponierte Melodien wiedererkennen können, ist ein Beispiel für das Weber-Fechner-Gesetz der Psychophysik, das für verschiedene quantitative Wahrnehmungsvorgänge eine logarithmische Bewertung postuliert. In der Musiktheorie hat diese Erkenntnis schon früh zu einem intuitiven Umgang mit Logarithmen geführt. Der Beitrag geht den damit verbundenen Visualisierungen von Boethius bis Newton und Rameau nach.
Medieval methods to visualize sound developed in dialogue with geometrical, arithmetical, cosmological, grammatical, philosophical, logical, and theological knowledge. Located between the sensory and the imaginative, the quadrivium of musica, geometry, arithmetic, and cosmology was nonetheless grounded in the realm of the visual. Visualizations of the harmony of the spheres, the harmony of the human body, and bodily perceived sounds appear in manuscripts as diagrams, graphs, and line drawings using parchment, ink, and pigments. The talk focuses on two circular medieval diagrams from the 9th and 10th centuries. Close analysis of the diagrams and their codices will demonstrate how medieval scribes and scholars created new shapes and forms on parchment to move between dimensions.