Trois poèmes de Pierre Reverdy, one of the last works by René Leibowitz (1913–72), is a set of three short songs for four voices and piano. Composed in 1971, a year before the composer's death, the work raises interesting questions about form and style. We will first consider a few excerpts to acquaint the reader with the unique and unmistakably French flavor of Leibowitz's musical language. We will then discuss Leibowitz the theorist and proceed to analyze his Trois poèmes, calling upon compositional and analytical concepts discussed in Leibowitz's unpublished treatise on form in serial music, “Traité de la composition avec douze sons” (ca. 1950).

Example 12.1 shows the beginning of the first movement. The opening piano phrase in measure 1 sounds disjunct and Webernian. It serves to introduce the choral phrase that follows in measures 2–3, which is in contrast very smooth, with much oblique and stepwise motion. This pairing of a piano solo measure with a continuing choral phrase is replicated in the following measures (4–6), suggesting a simple principle of form-building and raising questions about what will follow.

Example 12.2 shows the beginning of the second movement. Its texture (double canon) in a simple declamatory rhythm recalls examples from the choral music of Mendelssohn, Brahms, Schoenberg, and Webern. In contrast to the opening of the first movement, whose phrases are clearly demarcated, the phrases here almost all overlap.

The texture of the third movement is completely different. The piano opening shown in example 12.3a makes a clear reference to the first Gymnopédie of Erik Satie. Finally, the end of the third movement, shown in example 12.3b, includes a succession of major and minor triads, concluding the work on an extended tonal harmony with a distinctly “French” sound (a major triad in second inversion with added augmented fourth in the last measure). These excerpts cause us to wonder: how can the relatively conservative use of a single twelve-tone row be made to assume such different musical characters; how do the final chords at the end of the third movement come about; and how does Leibowitz articulate form in such different environments?

How did Renaissance composers learn their craft? They could have learned much of their technique from treatises, especially from those portions devoted to counterpoint. Today, we often think of counterpoint as consisting primarily of rules of voice leading. Such rules, which are found in virtually every music treatise of the time, teach the student to regulate the melodic motions of lines in relation to the simultaneous intervals between them (e.g., conditions for approaching perfect consonances or for preparing and resolving dissonances). They were learned by young singers for the purpose of improvisation, and following them would have been as natural as speaking in grammatically correct sentences. Yet just as the art of oratory consists of more than correct grammar, so musical composition goes far beyond mere voice leading. Composers had to choose between many large-scale contrapuntal techniques involving texture, motivic and structural repetition, and variation. While there has been extensive study by scholars of the rules of voice leading in Renaissance music, less consideration has been given to these more advanced compositional techniques.

In this chapter, then, we will examine some of these compositional techniques by reviewing some two dozen treatises written between the mid-fifteenth and mid-seventeenth centuries. We will see that the real challenge for a Renaissance composer consisted not of employing “correct” contrapuntal voice leading but rather of elaborating primary musical material – sometimes called a soggetto – by varying it or combining it with some other melodic material. (As we will see, a soggetto need not be simply a melodic subject in equal or mixed rhythmic values; it could also be a duo, or, in the case of parody technique, even an entire polyphonic composition.)

Introduction

The linear stability analysis presented in the last chapter gives the critical Rayleigh number for the onset of thermal convection under a variety of conditions. However, because the governing equations have been linearized, the solutions cannot predict the magnitude of finite-amplitude convective flows. In order to do this it is necessary to retain nonlinear terms in the governing equations.

Even in the simplest thermal convection problems the governing equations are sufficiently complex that analytical solutions cannot be found. There are basically two methods for obtaining nonlinear solutions. The first is to make approximations and the second is to obtain fully numerical solutions. We will address the former method in this chapter and the latter method in the subsequent two chapters.

In this chapter we will consider four approximations used to obtain a better understanding of thermal convection. We first consider an eigenmode expansion of the basic equations. This approach provides one of the methods used to obtain fully numerical solutions. However, in this chapter we consider only severe truncations of the full set of eigenmode equations. Retention of only the lowest-order nonlinear terms leads to the Lorenz (1963) equations. This set of equations is of great interest because its solution was the first demonstration of deterministic chaos. In this approximate approach we address the question:

Question 8.1: Is mantle convection chaotic?

This questions leads directly to a second question:

Question 8.2: Is mantle convection turbulent?

The second approximate approach we consider is boundary layer theory. This approach reproduces the basic structure of thermal convection cells at high Rayleigh numbers. The third approach is the mean field approximation.

Introduction

In this chapter we review what is currently known about the structure and the bulk composition of the mantle, with emphasis on how these may influence the style of convection, and how they may be influenced by it. Brief descriptions of the crust and the core are also included, again with an emphasis on those aspects of each region most germane to mantle convection. There are several monographs dealing with Earth structure and composition in a general context, e.g., Ringwood (1975, 1979) and Anderson (1989).

The study of mantle structure is by tradition the province of seismology, while mantle composition has historically been a subject for high-pressure and high-temperature mineralogy, petrology, and geochemistry. A great many important advances have recently been made in these areas, with the result that the studies of mantle composition, structure, and dynamics are now closely related. New findings from seismology, mineral physics, and isotope geochemistry are quickly applied as constraints on models of mantle dynamics. There are also interactions in the other direction. Each new step in understanding the physics of convection is quickly incorporated into new interpretations of mantle structure. This interdisciplinary style of research is perhaps the single most important reason for the emerging view of the deep Earth as a unified physical and chemical system.

Spherically Averaged Earth Structure

The determination of elastic parameters and density throughout the Earth using observations of seismic waves and other constraints is the prototype inverse problem in geophysics. Like many inverse problems, it is formally nonunique and suffers from practical difficulties such as incomplete sampling and errors in the data.