Toward a Music of Durational Indeterminacy

This chapter argues that the intense reduction of projective potential observed in Webern's op. 22 and carried even further by the brokenness of the projective field in Babbitt's Du invites comparison with efforts in the years following the Second World War to eliminate meter's hold on the attention and its involvement in the formation of those more or less determinate sonic durations called “phrases.” Boulez's le marteau sans maître's nine movements offers a great variety of approaches to musical continuity. Excerpts from this work thus lead to the end of the present study—a consideration of rhythm in music that has attempted to renounce meter's efficacy in the formation of phrase. An examination of these excerpts allows for a consideration of some more general questions of rhythm and some of the novel experiences offered by “the New Music.” The chapter also discusses the distinction of “constituent” and “phrase” in detail.

Problems of Meter in Early-Seventeenth-Century and Twentieth-Century Music

This chapter assesses meter in early-seventeenth-century and twentieth-century music. Specifically, it analyzes compositions by Monteverdi, Schütz, Webern, and Babbitt. Monteverdi's “Ohimè, se tanto amate” from the fourth book of madrigals presents a metrical subtlety rarely encountered in eighteenth- and nineteenth-century music. Here the projective field is very mobile, and mensural determinacy is restricted to relatively small measures. Meanwhile, Schütz's concertato motet “Adjoro vos, filiae Jerusalem” from the Symphoniae sacrae, Book I (1629), demonstrates extremely subtle rhythmic detail and great projective contrast used in the service of a compelling larger gesture. Here the repetition of small melodic figures is used for the creation of complex projective fields that serve the continuity of phrases and sections. The chapter then looks at the much smaller measures and much greater ambiguity in some music of the twentieth century.

Aperture referral in dioptric systems with stigmatic elements

A previous paper develops the general theory of aperture referral in linear optics and shows how several ostensibly distinct concepts, including the blur patch on the retina, the effective cornealpatch, the projective field and the field of view, are now unified as particular applications of the general theory. The theory allows for astigmatism and heterocentricity. Symplecticity and the generality of the approach, however, make it difficult to gain insight and mean that the material is not accessible to readers unfamiliar with matrices and linear algebra. The purpose of this paper is to examine whatis, perhaps, the most important special case, that in which astigmatism is ignored. Symplecticity and, hence, the mathematics become greatly simplified. The mathematics reduces largely to elementary vector algebra and, in some places, simple scalar algebra and yet retains the mathematical form of the general approach. As a result the paper allows insight into and provides a stepping stone to the general theory. Under referral an aperture under-goes simple scalar magnification and transverse translation. The paper pays particular attention to referral to transverse planes in the neighbourhood of a focal point where the magnification may be positive, zero or negative. Circular apertures are treated as special cases of elliptical apertures and the meaning of referred apertures of negative radius is explained briefly. (S Afr Optom 2012 71(1) 3-11)