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

ISBN-10: 331911235X

ISBN-13: 9783319112350

ISBN-10: 3319112368

ISBN-13: 9783319112367

The mathematical idea of counterpoint was once 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 can also serve to outline solely new platforms of ideas for composition, generated by way of new offerings of consonances and dissonances, which in flip result in new regulations governing the succession of periods.

This is the 1st e-book bringing jointly fresh advancements and views on mathematical counterpoint idea intimately. The authors comprise fresh theoretical effects on counterpoint worlds, the extension of counterpoint to microtonal pitch platforms, the singular homology of counterpoint types, and the software program implementation of contrapuntal models.

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

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8. Let D = (VD , AD ) be a strict digraph. The full equivalence relation ∼ f on the vertex set VD is deﬁned as the universal equivalence. 3. 4. 4 5 Two relations based on the same deﬁnition, but applied to two diﬀerent vertex sets. 1 on page 494 in [33]. 52 4 Graphs 84 94 ··· 31 010 42 ··· 37 (0, 3, 4, 7, 8, 9)0|1 Fig. 4 The full quotient digraph as a complete singleton K1 (below). Presence, absence, and orientation of arrows in the strict digraph (above) do not play a role. All Fuxian consonances are identiﬁed whatever their connections.

30) for any j ∈ {0, . . n}. 6. Any cycle of the digraph D is contained inside a strong component, turning the strong quotient digraph, also called condensation in the literature, into a directed acyclic graph. 54 4 Graphs 03 09 43 15 84 87 57 ··· ··· 21 08 46 510 85 10 82 02 (0)3|6 (1, 4)1|2 ··· (8)1|3 ··· (2, 5)1|3 (1, 4)0|2 (0)2|6 (8)2|3 Fig. 6 Partial representation of the twelve-tone counterpoint world in class 75 of [57]. A few consonances of the strict digraph (above) and their enclosing strong components (below).

This construction is repeated for all recording positions and the frequency bands θ, α, β; band δ is omitted since it may be aﬀected by noise. -L. S. , Q? 2. Observe the similarity of distribution of these quotients for our patients with respect to homologous recordings and frequency bands. This means that for consonances, participation lowers more when the second tone appears than for dissonances. 4 Polarity To test the response to the (K/D)-polarity, we confronted each consonant interval X with all the dissonant intervals Y, and we looked for particular responses in cases where Y was the interval that should correspond to X according to the polarity formula Y = 2X + 5.

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

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