At Timaeus 35b1–36b6 the divine craftsman creates a mixture out of which he fashions the world soul: forming an intermediate blend consisting of three ingredients, namely Sameness, Difference and Essence, he divides it according to a progression that gives rise to the sequence 1, 2, 3, 4, 9, 8, 27 (original sequence) and, after inserting the arithmetic and harmonic means into each double and triple interval, he generates intervals of 3/2, 4/3 and 9/8, which, in ancient music theory, correspond to the intervals of the fifth, the fourth and the whole tone. Aware that a fourth consists of two whole tones and a semitone, the demiurge fills all the fourths (4/3) with the interval of the whole tone (9/8) (36b1: τῷ τοῦ ἐπογδόου διαστήματι τὰ ἐπίτριτα πάντα συνεπληροῦτο) and ends up with leftovers that correspond to the fraction 256/243, the so-called λεῖμμα or δίεσις that approximates the semitone. The series 1, 4/3, 3/2, 2, 8/3, 3, 4, 9/2, 16/3, 6, 8, 9, 27/2, 18, 27 (diatonic sequence) has long been identified as a description of the intervals that make up the diatonic scale which, traditionally construed, encompasses four octaves and a major sixth. According to the dominant interpretation, the decision to end the procession at a musically arbitrary point indicates that Plato is not primarily concerned with issues relating to musical harmony (L. Brisson, Le Même et l'Autre [1998], pp. 314–32; F.M. Cornford, Plato's Cosmology, [1935], p. 69). Plato's Timaeus and the Missing Fourth Guest challenges the received view and highlights the indispensability of musical theory for an adequate interpretation of the relevant part of the Timaeus’ psychogonia.

The volume comprises an introduction, eight chapters and ten appendices. The introduction provides the background necessary for an appreciation of the musicological nuances of the dialogue and a summary of hermeneutical approaches that are favourable to the premise that the Timaean scale is worthy of serious consideration. A.A. discusses the ancient Greek conceptions of the octave and entertains the possibility that the fourth guest reported as missing in the opening frame of the dialogue (17a) is tied to the indeterminacy of the λεῖμμα in the intervals of the fourth and the fifth of the diatonic scale (pp. 2, 12). This rather ingenious suggestion, however, does not find textual support since none of the relevant terms is ever used in the Timaeus (A. Barker, Early Timaeus Commentaries [2003], p. 80 n. 27).

Chapter 1 provides an overview of several troublesome topics broached in the Timaeus, while pressing the point that the All Perfect Animal that serves as the ideal copy of the cosmos is to be identified with the Decad (p. 30), a concept germane to the Pythagorean τετρακτύς. A.A. reverts to a tradition that was represented in the Academy by Speusippus (fr. 28 Tarán) and claims to have unearthed a ‘decadic pattern of generation’ (p. 30) or a ‘decadic grammar’ (p. 35) that dictates the ordo of generation in the Timaeus. In addition, A.A. calls attention to what she terms ‘the Crantor matrix’, i.e. Crantor's method of arranging the numbers of the original sequence in the fashion of a lambda, which she progressively fills with additional numbers to make up the intervals necessary for the emergence of musical scales.

Chapter 2 initiates the argument on which the subsequent analysis builds. A.A. returns to the generation of the world soul in order to pinpoint her departure from the dominant interpretation, according to which the text at 36b1 suggests that the direction to fill the intervals of fourths with whole tones applies only to terms of the original sequence (p. 77). A.A. employs an ambiguity in the text and emphasises that the diatonic sequence includes intervals of doubles and triples that are not endemic to the original sequence, so that, for example, whole tones might also need to be inserted within new intervals of fourth (pp. 78–9). Proclaiming the text to be ‘a riddle with a correct solution’ (p. 81), A.A. notices the cumbersomeness that such a proliferation of intervals of fourths would produce and seeks ‘a justifiable principle’ (p. 83) that defines a particular set thereof to be supplemented.

This task is taken up in Chapters 3 and 4, in which A.A. comes up with a device for calculating the aforementioned set: she starts by multiplying each member of the diatonic sequence by 4/3 in order to identify fourteen intervals of fourth (p. 99) and proceeds to insert two tones and a diesis within them, according to the multiplication pattern (9/8 x 9/8 x 256/243) indicated by Plato (p. 103). A.A. arranges the new numbers that result from the above operations in the Crantor matrix and so extends the set of the original sequence to a total of 196 terms, which she lays out in 27 rows (pp. 108–9). Chapter 5, which constitutes the bulk of the argument, explores the musical relations that obtain within the members of the number set: in keeping with a tradition that is attested already in ancient commentaries, A.A. identifies the number 384 as the starting tone for the first complete octave of an ascending Dorian (or descending Lydian) diatonic scale and the number 1536 as the end tone for a second octave of the same sort, marking thus the emergence of a perfect disdiapason, i.e. two octaves that include no chromatic elements (pp. 133–8). She devotes the rest of Chapter 5 and all of Chapter 6 to explaining how the numbers give rise to eight additional complete octaves which, after being successively intruded by chromatic elements, degenerate into a series of ten incomplete octaves to a final triad, exploring all the while the possibility that the Timaeus scale allows for modulation in all three genera (diatonic, chromatic, enharmonic). In Chapter 7 A.A. returns to the myth of the creation of the world soul and proposes that a mathematical interpretation of the musical data provides the key for understanding how the demiurge forms the letter χ (‘chi’) out of the soul's fabric and reattaches its ends so as to constitute the cosmic orbits. Almost as an afterthought, Chapter 8 examines the concept of harmonia as an organising principle employed by Plato in his attempt to construct an ideal constitution in the Laws vis-à-vis his project of constructing an ideal cosmos in the Timaeus.

A.A. has put together an impressive volume of numerical evidence, and it can safely be argued that this volume will remain the most exhaustive musicological study of Timaeus 35a–36d. The book is written in straightforward prose, but it requires significant intellectual effort to follow the thread of A.A.'s quantitative analyses. A.A. works with T. Taylor's translation of the Timaeus, whereas the Greek text, when cited, is given in the footnotes. Leaving aside the infelicitous mistakes in the first occurrence of Greek cited in the volume as not relevant to the core of the argument (‘εἶς’ instead of ‘εἷς’, ‘έστια· τόρων’ instead of ‘ἑστιατόρων’ on p. 1 n. 2; cf. χαλεπτόν instead of χαλεπόν on p. 20), it should be noted that the adherence to Taylor's translation leads her to take Cornford to task erroneously for his interpretation of Timaeus 35a (p. 61; cf. Cornford [1935], pp. 60–1). There are, of course, other areas of dispute. For one, the idea that Timaeus 36a–b is a number generator is mostly assumed rather than argued for: A.A. does away with Plato's perplexing talk of magnitude regarding soul stuff by proposing that we ‘might think of the demiurge, instead, as simply generating numbers from the soul stuff’ (p. 64). The term διάστημα, favoured by Plato at Timaeus 35b1–36b6, is reminiscent of Archytas fr. 2 – where it probably refers to a musical interval – and has led to extended discussions with which A.A. could have fruitfully engaged (C.A. Huffman, Archytas of Tarentum [2005], p. 169). Finally, there is the occasional impression that A.A. works from within a neo-Pythagorean framework as, for example, when she states that, construed as the eternal paradigm, the Decad ‘describes a universe adhering to the pattern of emanation from one and return to unity’ (p. 35). Despite the insistence on formulations of the sort ‘the triad is the return of the monad to itself’ (p. 37), it should be kept in mind that such expressions are altogether absent from the Timaeus.