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The aim of this work is to study the interaction between two classical objects of mathematics: the modular group, and the absolute Galois group. The latter acts on the category of finite index subgroups of the modular group. However, it is a task out of reach do understand this action in this generality. We propose a lattice which parametrizes a certain system of &rdquo / geometric&rdquo / elements in this category. This system is setwise invariant under the Galois action, and there is a hope that one can explicitly understand the pointwise action on the elements of this system. These elements admit moreover a combinatorial description as quadrangulations of the sphere, satisfying a natural nonnegative curvature condition. Furthermore, their connections with hypergeometric functions allow us to realize these quadrangulations as points in the moduli space of rational curves with 8 punctures. These points are conjecturally defined over a number field and our ultimate wish is to compare the Galois action on the lattice elements in the category and the corresponding points in the moduli space.

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The compositional peculiarities of the Grave of Johann Sebastian Bach’s Toccata in C, BWV 564 make authoritative performance practice of the same quite difficult. Often, disparate solutions concerning the performance practice thereof, primarily those that concern registration and interpretation of the work, result. Upon discussing the resemblance of BWV 564 in its entirety with that of the genre of the Italian concerto, I assess the precise nature of Italian concerto influence upon the work, assigning primary compositional influence to Tomaso Albinoni, specifically via Johann Gottfried Walther’s transcription of Concerto IV of Albinoni’s Concerto a cinque, Op. 2 of 1700, effectively placing such influence prior to Bach’s c. 1713-1714 encounter with the concerted works of Antonio Vivaldi and thus assigning Bach’s composition of the work to his initial years as court organist and chamber musician at the Weimar ducal chapel. I also assess the resemblance of the sectional character of BWV 564 with that of the early Baroque multisectional Praeludium, exemplified in Dieterich Buxtehude’s Praeludium in A, BuxWV 151, ultimately to assert the rhetorical orientation of BWV 564 and to raise the registration implications thereof. Given the numerous renovations to the Compenius organ in the Weimar ducal chapel that occurred before, during, and after Bach’s tenure as court organist and chamber musician, I set forth a likely specification set for the Compenius organ of the Weimar ducal chapel during the initial years of Bach’s first appointment at Weimar. With the compositional timeframe and physical conditions reasonably established, and on account of both the accompanimental textures and musical idiomatic allusions present in the movement, I advance not only a one-manual performance practice of the Adagio of BWV 564 in its entirety but also a single general registration scheme resonant with continuo performance practices that were contemporaneous with the composition of the work. / text

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