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A Matrix Technique for Analysing Musical Tuning Systems

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Any musical interval may be treated as a vector quantity based upon arbitrarily selected unit vector intervals. A matrix technique is developed whereby entire tuning systems may be derived and described in vector terms. This approach is compared to and contrasted with conventional numerical techniques which involve the fractional and logarithmic manipulation of musical intervals.


Jedes musikalische Intervall kann als Vektorgröße auf der Grundlage willkürlich ausgewählter Einheitsvektor-Intervalle behandelt werden. Es wird eine Matrix-Technik entwickelt, wobei ganze Tonsysteme abgeleitet und durch Vektoren beschrieben werden können. Dieser Weg wird konventionellen numerischen Techniken verglichen und gegenübergestellt, welche von der Verknüpfung musikalischer Intervalle mittels Frequenzverhältnissen und deren Logarithmen Gebrauch machen.


Tout intervalle musical peut être traité comme une grandeur vectorielle définie dans une base dont les vecteurs unitaires sont des intervalles arbitrairement choisis. A partir de cette remarque on a développé une technique de matrices dont peuvent se déduire des systèmes complets d'accordage décrits en termes de vecteurs. On compare et oppose cette nouvelle méthode aux techniques numériques classiques où les intervalles musicaux sont traités par fractions et logarithmes.
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Document Type: Research Article

Publication date: February 1, 1984

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  • Acta Acustica united with Acustica, published together with the European Acoustics Association (EAA), is an international, peer-reviewed journal on acoustics. It publishes original articles on all subjects in the field of acoustics, such as general linear acoustics, nonlinear acoustics, macrosonics, flow acoustics, atmospheric sound, underwater sound, ultrasonics, physical acoustics, structural acoustics, noise control, active control, environmental noise, building acoustics, room acoustics, acoustic materials, acoustic signal processing, computational and numerical acoustics, hearing, audiology and psychoacoustics, speech, musical acoustics, electroacoustics, auditory quality of systems. It reports on original scientific research in acoustics and on engineering applications. The journal considers scientific papers, technical and applied papers, book reviews, short communications, doctoral thesis abstracts, etc. In irregular intervals also special issues and review articles are published.
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