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

The mathematical thought of counterpoint used to be 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 serve to outline totally new structures of principles for composition, generated by means of new offerings of consonances and dissonances, which in flip bring about new regulations governing the succession of durations.

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

The e-book is acceptable for graduates and researchers. a great command of algebra is a prerequisite for realizing the development of the version.

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2. It is important to stress that the previous lemma is false if g H. 2. It is clear that p0(K/D) [ε] is a polarity of (gK[ε]/gD[ε]) = (K[ε]/D[ε]). 2 · 5 ◦ T 1 · 1 = p0(K/D) [ε] ◦ g. This lemma yields a quick criterion to check if gX[ε] is polarized by p0 [ε]. 11) must hold, because α = γ and β = δ. 8). If ρ = gcd(v, 2k), then 2k −1 ρ T jρ+k−up(s) · u(X) ∩ X . X ∩ X = X = k2 . 6. Now observe that the cardinality of T y−up(s) · uX ∩ X cannot exceed k − 1, because (X/Y) is strong and y − p(s) 0, since p is the polarity.

The following result relates a subset of the endomorphisms of K7 [ε] (the consonant intervals from the ﬁfth perspective) with the Riemann dichotomy. 2 (T. Noll). The set W of the endomorphisms of K7 [ε] of the form T x · y coincide with R. Proof. b ∈ K7 [ε]. kb ∈ K7 [ε] because kb ∈ K7 , K7 being a multiplicative monoid, whence R ⊆ W. y ∈ K7 [ε], which implies that y ∈ K7 and henceforth T x (y) ∈ R. b. We see that ν−1 (K7 [ε]) = T Z12 K7 = R = W. 1. Every consonant interval is isomorphic to a product of endomorphisms of the major triad, and isomorphic to an endomorphism of the set of consonant contrapuntal intervals K7 [ε].

Hence δ = β and p0 [ε]g = gp0 [ε]. 2. It is important to stress that the previous lemma is false if g H. 2. It is clear that p0(K/D) [ε] is a polarity of (gK[ε]/gD[ε]) = (K[ε]/D[ε]). 2 · 5 ◦ T 1 · 1 = p0(K/D) [ε] ◦ g. This lemma yields a quick criterion to check if gX[ε] is polarized by p0 [ε]. 11) must hold, because α = γ and β = δ. 8). If ρ = gcd(v, 2k), then 2k −1 ρ T jρ+k−up(s) · u(X) ∩ X . X ∩ X = X = k2 . 6. Now observe that the cardinality of T y−up(s) · uX ∩ X cannot exceed k − 1, because (X/Y) is strong and y − p(s) 0, since p is the polarity.