Download e-book for iPad: Computational Counterpoint Worlds: Mathematical Theory, by Octavio Alberto Agustín-Aquino, Julien Junod, Guerino

By Octavio Alberto Agustín-Aquino, Julien Junod, Guerino Mazzola

ISBN-10: 331911235X

ISBN-13: 9783319112350

ISBN-10: 3319112368

ISBN-13: 9783319112367

The mathematical thought of counterpoint used to be initially geared toward simulating the composition ideas defined in Johann Joseph Fux’s Gradus advert Parnassum. It quickly grew to become obvious that the algebraic gear utilized in this version serve to outline completely new structures of principles for composition, generated by way of new offerings of consonances and dissonances, which in flip bring about new regulations governing the succession of durations.

This is the 1st ebook bringing jointly contemporary advancements and views on mathematical counterpoint concept intimately. The authors contain fresh theoretical effects on counterpoint worlds, the extension of counterpoint to microtonal pitch structures, the singular homology of counterpoint versions, and the software program implementation of contrapuntal models.

The publication is acceptable for graduates and researchers. an excellent command of algebra is a prerequisite for knowing the development of the version.

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Additional resources for Computational Counterpoint Worlds: Mathematical Theory, Software, and Experiments

Example text

This turns the torus of thirds into a metric space. 1. , d g(x), g(y) = d(x, y). In other words, the distances in the torus of thirds are not altered when we transform it with a symmetry. Using this distance, we may define the diameter of a dichotomy (X/Y) as δ(X/Y) = 1 d(u, v). 2 u,v∈X and its span as σ(X/Y) = d u, p(u) . 2. Prove that the diameter and the span of all the elements of a strong dichotomy class (X/Y) are the same. In other words, if X = gX and Y = gY, then δ(X/Y) = δ(X /Y ) and σ(X/Y) = σ(X /Y ).

Who also focus on the γ-band 30 − 50 Hz) [72]. , the two tones of an interval are played one after the other without interruption. All the intervals were played in three orders: (1) all consonances, ordered according to their size; all dissonances, also ordered according to their size. (2) All consonances, ordered according to complementarity size (if possible); then all dissonances, ordered according to complementarity (if possible). (3) A mixed succession of all intervals according to a particular dodecaphonic all-interval series.

78 0,2,6 - 0,2,6 - 0,2,6 - 0,2,6 1,2 1,2 1,2 1,2 1,2 1,2 1,2 1,2 - 1,4 1,4 - 1,4 0,2,6 - 0,2,6 - 0,2,6 - 0,2,6 1 1,4 1 1,10 1 1,4 0,2,6 1,2 1,2 0,2,6 1 - 0 3 4 7 8 9 0 3,9 0,4,8 7 8 3,9 7 - Forbidden Successors for Dichotomy Class No. 1 Forbidden successors of counterpoint intervals for selected representatives of the six strong dichotomies, according to Hichert’s algorithm. For each dichotomy, in the column k appears their respective consonances, and in the columns with heading j = 0, . . 11 the intervals j + ε.

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Computational Counterpoint Worlds: Mathematical Theory, Software, and Experiments by Octavio Alberto Agustín-Aquino, Julien Junod, Guerino Mazzola

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