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

The mathematical thought of counterpoint was once initially aimed toward simulating the composition ideas defined in Johann Joseph Fux’s Gradus advert Parnassum. It quickly turned obvious that the algebraic equipment utilized in this version can also serve to outline solely new platforms of ideas for composition, generated by means of new offerings of consonances and dissonances, which in flip result in new regulations governing the succession of durations.

This is the 1st publication bringing jointly fresh advancements and views on mathematical counterpoint conception intimately. The authors contain contemporary 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 e-book is acceptable for graduates and researchers. an outstanding command of algebra is a prerequisite for knowing the development of the version.

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**Example text**

3. Allowed steps are sent to allowed steps. An allowed step involves two consonances, and according to the deﬁnition of a strict digraph, ⎧ ⎪ ⎨ ζ0 , ζ1 ∈ VD , ¯ ∀ζ0 , ζ1 ∈ S , σ(ζ0 , ζ1 ) = 1 ⇒ ⎪ ⎩ (ζ , ζ ) A . 0 1 D Because the strict morphism φ always preserves non-adjacency, ∀ζ0 , ζ1 ∈ S¯ , σ(ζ0 , ζ1 ) = 1 ⇒ σ φ(ζ0 ), φ(ζ1 ) = 1. 4. Forbidden steps are sent to forbidden steps. Any forbidden step between two consonances ξ0 and ξ1 generates an arrow (ξ0 , ξ1 ) ∈ AD . By deﬁnition, φ preserves adjacency, ∀ζ0 , ζ1 ∈ VD , σ(ζ0 , ζ1 ) = 0 ⇒ σ φ(ζ0 ), φ(ζ1 ) = 0.

We made use of the spectral participation vector S (E) = (P(E), P(E)/Pδ (E), P(E)/Pθ (E), P(E)/Pα (E), P(E)/Pβ (E)), of an event E and its associated participation value v(E) = P(E)/Pθ (E) + P(E)/Pα (E) + P(E)/Pβ (E), which is a length measure. Here, P(E) is the total spectral power [76], Pδ (E) is the δ-power of event E, Pθ (E) is the θ-power, Pα (E) is the α-power,and Pβ (E) is the β-power, see also [63]. From the results obtained thus far, we conclude that this representation is wellsuited to give an adequate picture of a possible semantic charge of EEG signals.

Consonances belonging to a same strong component seem to show much tighter bonds than just cycles. , maximal complete subgraphs, in all worlds at least up to the twelve-note worlds. 1. The strong components of a strict digraph related to a counterpoint world form cliques. The claim has been veriﬁed only for the worlds with lower octave division (up to 2k = 14). In the general case, a proof or a counterexample is still needed for the higher microtonal cases. Despite the theoretical interest of strong digraphs, we will turn to a fourth equivalence relation, which will be used extensively in this chapter and the next.