Developing quadrature themes

Readers will most likely be aware of the works of Johann Sebastian Bach in the field of music, particularly of his Goldberg variations and the changes that can be rung, wherein aesthetically appealing alterations to structure produce a raft of colourful sounding themes. Similarly in the field of quadrature it is possible to revisit and re-establish well-known formulae by developing variations on the theme of a three-point interpolating quadratic formed to represent a function that is to be integrated.

Madeleine Thien's Chinese Encyclopedia: Facts, Musics, Sympathies

This article explores the “encyclopedic” properties of Madeleine Thien's Do Not Say We Have Nothing (2016), seeking to define the novel as inherently comparative—that is, providing, in Edward Said's words, “a comparative or, better, a contrapuntal perspective” on the world with no need for a second counterpart text to draw cross-literary parallels. Written from a transpacific narratorial stance of a millennial Vancouver-based daughter of Chinese immigrants, the narrative communicates her secondhand knowledge about the traumatic twentieth-century history of the People's Republic of China, accumulated in multiple alternating substories of ordinary individuals’ “practical past” as opposed to official historiography. The article likens Thien's patchwork storytelling to Jorge Luis Borges's apocryphal “Chinese” encyclopedia and novel, to the premodern equation between language and reality discussed in Michel Foucault's “archaeology of knowledge,” to classical Chinese novels as described by Goethe and Franco Moretti, and to J. S. Bach's polyphonic layout of the Goldberg Variations. Constructing sympathetic networks of music and literature, Do Not Say We Have Nothing facilitates readerly immersion, yet its fictional storyworld may not feel universally plausible. Sharing its writer's experience of teaching Thien in Hong Kong, the article suggests that a critique of the novel's Western, nearly Orientalist standpoint with respect to sensitive issues of recent Chinese history does not dismiss the contrapuntal outlook Thien's readers are invited to adopt beyond their experiential backgrounds. Reading Thien, one learns to hear the world's polyphony. That, and not a comprehensive multitude of facts summarizing a national mentality and coherent knowledge about the world, makes Do Not Say We Have Nothing encyclopedic.

The Vertical Construction in Canons from Bach’s Goldberg Variations (BWV 988; BWV 1087): Some Aspects

В статье сопоставлены две группы канонов И. С. Баха из Гольдберг-вариаций (BWV 988 и BWV 1087): собственно каноны-вариации и 14 зашифрованных бесконечных канонов на восемь начальных басовых нот Арии — темы вариационного цикла. Объектом сравнения выступила гармония как один из основных общескрепляющих элементов вариационного цикла. Опора вариаций на сохраняемую гармонию требует специфических решений при построении канонов: об интервалах, направлении и времени вступления, об их сорасположении, о прямом и инверсионном движении, наконец, о точности воспроизведения басовой мелодии, в том числе о фигурировании и о взаимодействии баса с прочими голосами. Предписанные темой гармонические вертикали влияют на способ развертывания горизонталей и на состав опорных тонов их мелодии; изменение гармонии происходит как следствие и имитационного движения верхних голосов, и изменений баса, и в результате использования минорной тональности. В 14 канонах начальные басовые звуки Арии повторяются без фигурационного варьирования, притом чаще всего в одном или нескольких нижних голосах. В этом видится основание к более точному воспроизведению тематической гармонии практически во всех канонах с их разнообразными видами и соответственно имитационно-контрапунктическими приемами. Взаимозависимость горизонтали (баса / cantus firmus и голосов имитации) и вертикали, формирующейся при построении канона, — основа корреляции канонообразующих и контрапунктирующих голосов с гармонией темы. The article gives a comparison of the two groups of canon-variations from the J. S. Bach’s cycle Aria with 30 Variations (Goldberg Variations, BWV 988) and Diverse Canons on the first eight Foundation Notes of the preceding Aria (BWV 1087). The latter is 14 canons in the abbreviated notation using the ground bass line from the subject of Goldberg Variations in so called Handexemplar (Christoph Wolff), where they (the canons) were included by Bach himself. Perpetual canons comprising the group of 14 vary in thematic material, number of parts, devices of imitational and contrapuntal treatment. The bass notes of the Aria are used in the canonic parts as well as cantus firmus in free counterpointing part. Moreover, parts of the subject become constitutional elements for almost every part in the canons, and similar, contrary, and retrograde motion creates simple, double and one triple canon. Analysis of the first bars of the canon-variations (on the first eight notes) and the decryptions of the encrypted canons published by Christoph Wolff show the interrelatedness between the horizontal (primarily eight-sound melody / сantus firmus) and various verticals emerging as a result of canonical progression. Comparison of these canons let us clarify the means of intervallic and chordal correlation of the canonic parts: with varying degrees of accuracy thematic harmonic course of the beginning of the Aria is reproduced in both examined groups. Changes in the thematically relevant vertical construction among other reasons are seen as a result of imitative-contrapuntal progression within the variation cycle on the one hand and of interaction between different techniques of contrapuntal writing in the encrypted canons — on the other one.

Bach the Teacher

This chapter examines the works of Bach’s later years, including several published collections, in the context of his teaching. The latter, considered in its broadest sense, included not only instruction in the St. Thomas School but private lessons and mentorship for university students and younger professional musicians. To these activities Bach added the revision and publication of compositions that could serve as examples for study and emulation. Among the latter are the four volumes of Clavierübung, including the harpsichord partitas, Italian Concerto, and Goldberg Variations; the Schemelli Chorales and Canonic Variations for organ; and the Musical Offering and Art of Fugue. Also instructive, in a profound sense, are the great vocal works of these years: the passions, oratorios, and Latin church music, including the B-Minor Mass.

An artistic portrait of Glenn Gould based on tHe analysis of his recordings of the Goldberg Variations by Johann Sebastian bach from 1955 and 1981, part 2

The article is the continuation of the text on the artistic profile of Glenn Gould based on the analysis of recordings of The Goldberg Variations from 1955 and 1981. The first part of the article, published in the previous issue of “Notes Muzyczny”, describing the life and style of the Cana- dian pianist and also the origin, construction and general characteristics of The Goldberg Variations by J.S. Bach, was sort of an introduction to the detailed analysis of both recordings, which is the main subject of the author’s deliberations. Additionally, apart from the mentioned analysis, the article presents general characteristics of each variation plus the circumstances when both versions of The Goldberg Variations were written. An important part of the article is touching on the problem of performance and perception of this great composition – as it had been believed for centuries that The Variations were challenging not only for a performer but also for a listener. For that reason, both versions of The Variations performed by Glenn Gould, which are commonly considered as masterpieces, are worth knowing.