As promised by the title, this book traces a ‘nexus’ of interrelated themes – harmonic space, its division into smaller units according to varied patterns of tones and semitones, attempts to reconcile an inherited tone-system with a repertoire created outside its ambit, and endeavours to endow the melodies with visual shape. As the author acknowledges, the foundations of this book, much of which has to do with tracing the sometimes shifting meaning of terms, had already been laid in previous articles on the polyvalent terminology of medieval music theory. Two of the most important terms were clarified in Atkinson's masterly articles ‘Modus’ and ‘Tonos/tonus’ for the Handwörterbuch der musikalischen Terminologie (1996 and 2005). Large portions of the present book are taken up with similarly astute analyses of musical and grammatical terminology.

The Critical Nexus opens with a long chapter on ‘The Heritage of Antiquity’, the intellectual basis of medieval chant theory. Succeeding chapters are grouped under two comprehensive headings: (I) ‘The Eighth and Ninth Centuries’ and (II) ‘The Synthesis of Ancient Greek Theory and Medieval Practice’. The concluding Epilogue does not merely summarise previous findings but explores new aspects of topics previously introduced. Atkinson eschews an unmanageable survey of all medieval theorists who treated system, tone/mode, and notation, preferring to concentrate instead on those few theorists whose originality and influence carried most weight.

Chapter 1 takes up an examination of ancient Greek music theory as transmitted and interpreted by authors of Late Antiquity, principally Boethius (De institutione musica) and Martianus Capella (De nuptiis Philologiae et Mercurii), substantial works that enjoyed considerable circulation. Atkinson lays out diagrams of the Greater and Lesser Perfect Systems (exx. 1.1 and 1.2), the octave species (ex. 1.7), Greek alphabetic notation (ex. 1.4), the Alypian signs (ex. 1.5) and monochord division (ex. 1.6) as transmitted by Boethius. He points out that ‘modus’ in Boethius is the equivalent of the Greek τόνος or τρόπος, defined as ‘an entire collection of pitches, brought together within the framework of a consonance such as the diapason, diapason-plus-diatesseron, or the bis-diapason’. According to this definition the ‘tones’ or ‘tropes’ are the (fifteen) transposition scales known as Dorian, Phrygian, etc. ‘Tonus’ had yet another meaning in Greek music theory: the 9:8 whole tone, and for Martianus Capella it could signify merely sound.

In a single sentence that was to have wider repercussions than Cassiodorus could ever have intended, he (and following him Isidore) forged a link between music theory and grammar: ‘tonus is a size and difference of the entire harmonic system [the Greek transposition scales], which consists in the accent or inflection of the voice [a grammatical concept]’.Footnote ¹ The first part of the definition (from Gaudentios) pertains to tone-system and mode, the second (from Donatus) to grammar, thus mingling in a single phrase the three topics in the subtitle of The Critical Nexus.

As Atkinson demonstrates, the borderline between the accentus (cf. ‘ad cantus’) of classical prosody and its realisation as a physical sign on the page (both derived from Donatus) was easily broached, and Donatus, the prime medieval source for all things grammatical, linked the accents (acute, grave, circumflex) with notae.Footnote ² In the chapter ‘De accentibus’ of the Ars maior he defined accentus as three kinds of notae – acute, grave and circumflex – with the understanding that these were graphic signs, which he related to the length of syllables in Greek and Latin prosody.Footnote ³ Though ancient grammarians and their epigones describe accent in terms of pitch, Atkinson reminds us that the accent in spoken Latin was a stress accent by the end of the fourth century, the very time when chant repertories were beginning to take shape. Martianus Capella allowed all syllables to receive an accent (‘argiletum’; pp. 61–2), a portent for the future development of musical notation. Tonus occupied a borderline between grammar and harmonics, a fact that consequently ‘proved influential in shaping the development of a medieval tone-system, a theory of mode, and a method of notation for the plainchant of the Western church’ (p. 46).

Chapter 2 attributes revived interest in ancient music theory to successive Carolingian legislative edicts, not only the Admonitio generalis of 789, but also the circular letter De literis colendis, which encouraged the cultivation of all learning, sacred and secular. Following a scenario sketched out by Marie-Elizabeth Duchez, Atkinson elucidates the successive glossings of Donatus, of book IX of De nuptiis, and of Boethius's De institutione musica in order to understand the role these late antique texts played in the genesis of medieval music theory. Martianus's treatment of the accents (acute, grave and circumflex) ‘made a direct connection between prosodic accents and music’ (p. 58), a point not lost on the glossators, who supplied acute and grave accents to some words (exx. 2.9 and 10) and in one case even neumes (fig. 2.3). There were fewer glosses on harmonic theory because, as Atkinson observes, the material was far more difficult.

In addition to explaining the pronouncements of Carolingian authors, Atkinson must very often determine what they did not understand about the sources on which they commented. Remigius of Auxerre and even John Scottus Eriugena did not always get things right in their commentaries on De nuptiis. Atkinson surmises that one of the problems might have been the fact that they were trying to interpret the ancient sources in the light of the system of eight ecclesiastical modes with which they were familiar. In some respects they were not much different from theorists who strove to develop a chant theory with the tools furnished by the ancient Greek tone-system. A Boethius gloss in D-Mbs Clm 14523 (ex. 2.27) introduces for the first time Byzantine terminology for the ‘modes’ (protus authentus, etc.) that dominated early chant theory.Footnote ⁴ The same gloss interprets the ‘modi quos eosdem tropos vel tonos nominant’ not as transposition scales (τόνοι) but as arrangements of tones and semitones.

In Chapter 3 (‘The Heritage of the Church’) many readers will find themselves on somewhat more familiar terrain. Whether or not a theoretical system for Gallican chant existed, Frankish singers who began to sing the new (for them) Roman chant in the latter half of the eighth century sought to devise such a system. Central to the enterprise was determining how the formulae for the singing of psalms were to be related to antiphons. They had at their disposal elements of Byzantine theory (protus, deuterus nomenclature, the intonation syllables) and an exceedingly complex ‘heritage’ derived from Late Antiquity. One early response to the challenge was the creation of tonaries, and Atkinson explores the principles that generated these. Regino of Prüm's (c.900) is the principal focus here, since the earlier tonaries of St Riquier and Metz are unnotated. Two folios from Regino's tonary (fig. 3.1B (Office) and 3.1C (Mass)) exemplify the method of classification by differentia and antiphon incipit – one as important as the other – based on Regino. Aurelian's discursive description of the repertory, his definitions of terms dependent on Cassiodorus and Isidore, is a kind of ‘verbal tonary’. He ignores Greek theory entirely, exemplifying intervals with reference to chant pieces known to his readers.

Atkinson argues that the original conception of Aurelian's Musica disciplina intended the presence of musical notation. In Chapter 19 of the treatise, Aurelian refers several times to ‘notae’, but none are present in any of the manuscripts of Musica disciplina. Aurelian calls the intonation formulae (Noeane) of the protus authentic ‘notarum formula/forma’, and he assumes that the reader will be able to undertake an ‘inspectio notarum’ (CSM 21:119). Atkinson concludes that Aurelian's point of reference must have been what is known as ‘palaeofrankish’ notation, later added to some portions of the oldest complete manuscript of the treatise (F-VAL 148; c.875/85).Footnote ⁵ Several well-chosen melodic comparisons accompanied by two facsimiles demonstrate that Aurelian's verbal descriptions match notation in the practical sources.

Atkinson's brief excursus on palaeofrankish neumes (pp. 106–13) will be welcomed by readers who have come away from Handschin, Jammers, and other authors more perplexed than enlightened. He transcribes the Gloria patri of the protus authentic in square notation following Aurelian's instructions. Posing the question as to whether there was ‘any kind of musical notation in broader use in the ninth century that corresponded to the prosodic accents that Aurelian discusses’, Atkinson offers an affirmative response, while acknowledging that the milieu in which Aurelian worked was broadly speaking ‘a notationless musical culture’. He concludes that the old theory that neumes were derived from the accents was far too simple an explanation: ‘given that the notae for the accents cannot be separated from their character as vocal inflections in Classical sources, it can hardly be argued that they should be so separated here’ (p. 106).

Chapter 3 continues with a discussion of the curious tone-system of Musica enchiriadis, which consists exclusively of disjunct T-ST-T tetrachords (thus producing augmented octaves). Musical examples in the treatise combine daseian notation with equidistant horizontal lines representing pitches on which the syllables of the chant are placed. Transpositions of the short phrase ‘alleluia’ to adjacent pitches exemplify the character of each mode.Footnote ⁶ A passage in Chapter 9 of the treatise (ex. 3.17) distinguishes among potentially reduplicative terms: pthongos, sonus, tonus, epogodos, modus and tropus – terms which make frequent appearances in The Critical Nexus.

A decisive moment in the history of chant theory came with the recognition and justification by the Musica enchiriadis of ‘vitia’ – chromatic (in terms of early medieval tone-systems) notes lurking outside the medieval gamut which had to be accounted for. The presence of these pitches, mainly F sharp and E flat, embodied the confrontation between the oral tradition of chant, a tone-system borrowed from a foreign musical culture, and the practical limits of staff notation. The author of Scolica enchiriadis provides an ingenious solution for the dilemma of chants such as the communion Beatus servus that require both a whole and a half step above the final.Footnote ⁷ On its normal final (D) this would have been impossible, so either it had to be transposed to the confinalis (a), or internal phrases had to be altered in a way that potentially changed the character of the piece; but Scolica Enchiriadis suggests a mental reinterpretation that seems to look ahead to the system of mutations of the musical (‘Guidonian’) hand.

Chapter 3 closes with reflections on the development of notation: ‘one of the most important legacies of the Western church in the eighth century’. The relationship between palaeofrankish script and later neumations is reviewed, followed by observations on liquescence and the litterae significativae of St Gall and Laon notations.

Part II continues the exploration of the ‘nexus’ that binds together tone-system, mode and notation. It begins with a chapter devoted mainly to Hucbald but with a closing section that brings back Regino. The centrality of Hucbald of St Amand (c.850–930) for the conception of musical ‘space’ in the Middle Ages has long been recognised, and Atkinson describes his De harmonica institutione as ‘in many ways a remarkable synthesis of ancient Greek theory and contemporaneous chant theory and practice’ (p. 162). Although well informed about his predecessors (even Eriugena), Hucbald wrote ‘for singers rather than mathematicians’ (p. 150).Footnote ⁸ He interpreted Greek music theory (even the ‘Alypian’ signs) for his contemporaries and supplemented their technical descriptions with more readily accessible concepts, drawing on the familiar chant repertory to illustrate Greek harmonic theory. The interval of a whole tone, for example, is not defined as mathematical proportion but by reference to an antiphon, Ite dicite Iohanni, whose incipit moves up and down by this interval. Adiastematic notation, the only kind available in Hucbald's time, had to be supplemented with alphabetic notation (‘litterulae quas pro notis musicis accipimus’). At this point Atkinson introduces a few comments about the Montpellier tonary-gradual (Bibl. Interuniversitaire, Section Médecine, H 159), which features such a combination with special signs thought to represent quarter-tones. Though Hucbald defined the broad outlines of a music theory that transcended earlier writings and laid the foundations for later schemes, a more satisfactory synthesis had to wait another century until the time of Guido.

Chapter 5 delves into the problematic Alia musica, ‘one of the most complex and perplexing of all medieval writings on music’, as Atkinson says in the opening sentence of the chapter. The treatise was first edited by Martin Gerbert in 1784, but a more ‘critical’ (and rather cumbersome) edition was published by Jacques Chailley in 1963. Atkinson thoroughly revises Chailley's assessment of the treatise based on an array of manuscript sources far exceeding what previous editors had at their disposal.Footnote ⁹ These sources reveal that sections of the complete treatise circulated separately. Atkinson rejects Chailley's proposed division into three ‘treatises’ (table 5.1), substituting for it a more nuanced analysis that identifies five layers and four different authors/commentators (table 5.3). Authors α–γ are identified with Chailley's ‘First Quidam’, ‘Principal Treatise’, and ‘Nova expositio’, respectively.Footnote ¹⁰ The fourth contributor-editor (δ) commented on passages of the Nova expositio (concerning the eight modes of plainchant) and compiled the final version of this multipartite treatise. Detailed comment on the nuances of Atkinson's proposed new arrangement is not possible here, but one can note that Chailley's ‘First Quidam’ is dethroned from first to last place! Atkinson observes that Alia musica took a decisive step towards ‘distinguishing authentic from plagal modes via differences in their octave species’ and ‘in describing plainchant by means of species of fourths and fifths’ (p. 202).

Chapter 6 reviews the contributions of a quartet of theorists: Bern of Reichenau (d. 1048), pseudo-Odo, Guido of Arezzo, and the less familiar pseudo-Bernelinus (c.1100). The brief treatise by the last of these – ‘simple and direct’ as Atkinson observes – explained the modes not as octave species but as combinations of species of diatesseron and diapente, just as Alia musica had done.Footnote ¹¹ Though pitch names are not assigned, the eight ecclesiastical modes are obviously to be assumed.

Bern's Prologus in tonarium (after 1021) confirms the trend of interpreting the church modes as combinations of species of fourths and fifths (here using Greek terminology). This allowed theorists to distinguish between the first and eighth modes, which occupy the same tonal space (D–d). With Bern's synthesis ‘the implications held by Boethius's De institutione harmonica for the theory of mode and tone-system in the Middle Ages were well on their way to being fully realized’ (p. 211). Atkinson illustrates how the flaws in Bern's presentation were later corrected by an interpolator drawing on the treatise Duo semisphaera.Footnote ¹² As Jane Warburton has shown, the latter and Bern's original treatise are linked by their common dependence on the anonymous Cita et vera divisio monochordi.Footnote ¹³

Contemporaneously, the monochord experienced a revival as a practical tool not only for illustrating intervals, but also for reconciling Boethian theory with chant theory, exemplified by the pseudo-Oddonic Dialogus de musica (c.1000). The anonymous author's division of the monochord seems to be original: he begins with Γ (‘rarius in usu’, as he admits) and instructs his pupil to divide the string into nine parts writing ‘A’ at the lowest segment, a process repeated for ‘B’. A fourfold division of the string gives the note ‘C’. This process yields a complete tone-system with both square and round B (exx. 6.9–10). The final is considered the primary modal determinant, but tessitura and starting pitches also share in this determination, particularly when the bounds of the modal octave are exceeded. The Dialogus author distinguishes clearly between ‘modus’ and ‘tonus’. The former signifies ‘all eight of the tones (!) or modes of chant’. For clarity's sake he restricts the latter to ‘the tones in the ninefold disposition and division [of the monochord]’. Pseudo-Odo also tackled the problem of chants that do not fit into the tone-system inherited from the distant past. If a piece will simply not fit into one of the prescribed eight modes, it may be emended to fit with the mode ‘with which it least disagrees’ (‘in quo minus dissonat’). At this juncture Atkinson makes a comment that has probably more than once occurred to the reader: ‘it seems not to have occurred to [pseudo-Odo – or to other authors, for that matter] that perhaps the system itself ought to be changed’ (p. 219).

No survey of this kind would be complete without reference to Guido of Arezzo (c.990–1050), who drew on the heritage of earlier medieval music theory to devise a synthesis that was to make his name famous for centuries.Footnote ¹⁴ Guido extended the gamut beyond the canonical fifteen pitches to ‘dd’. He focused on the modi vocum, the unique arrangement of tones and semitones surrounding each pitch in the gamut. Three of these modi are replicated at the fifth (D at a, E at b, F at c), but a fourth (G) is not. The introduction of soft b alters the modus vocum, making G sound as if it were moving in a protus environment. Imitating Musica enchiriadis, Guido illustrated how the modi vocum differed from each other by writing out at four pitch levels (D–G) the phrase ‘Tu patris sempiternus est filius’ from the Te deum (also the foundation of the organum examples in Musica enchiriadis). This phenomenon, not unknown to earlier theorists, underlies the doctrine of affinities (ex. 6.20; from Micrologus, ch. 7). Atkinson points to Guido's acknowledged central role in musical notation. He eschewed the awkward syllables-on-lines technique for one that combined diastematic neumes with (coloured) lines corresponding to pitches in alphabetic notation. This development – a noteworthy one in the most literal sense – so consolidated the tone-system that ‘the use of scale degrees not in the tone-system became [remained?] a problem’, which subsequent theorists felt obliged to ‘correct’.

The Epilogue to The Critical Nexus, after yet another examination of what might be called the ‘Beatus servus problem’, turns to the De musica of John Cotto/Affligemensis (c.1100). Atkinson identifies three areas in which John staked out new territory: (1) a (not entirely satisfactory) etymological-historical definition of mode, (2) the integration of the psalm tone tenor into the theory of mode, and (3) a theory of the characteristics of each church mode. John further commented on the ambitus of chants, tessitura as a criterion of modal discrimination, the affinitates, and chants that had been ‘corrupted’ by singers.Footnote ¹⁵

The Epilogue closes with a foray into the fourteenth century: Marchetto of Padua's Lucidarium (1317/18) and the first treatise of the Berkeley Anonymous (c.1375). Marchetto took up the traditional themes of chant classification, modelling his treatment on that of pseudo-Odo but refining its specificity.Footnote ¹⁶ He discriminated among chants in (1) tonus perfectus (those that fill out the modal octave), (2) tonus imperfectus (those that do not fill out the modal octave), (3) tonus mixtus (authentics that descend more that one tone below the final and plagals that extend beyond the sixth above the final), and (4) tonus plusquamperfectus (authentics that exceed an octave and plagals that descend more than a fourth below the final). As had so many of his predecessors, Marchetto addressed the problem of Beatus servus and frankly admitted the necessity of C sharp and F sharp, whose introduction produced a tonus acquisitus (ex. E.16). Many years later, the Berkeley Anonymous rationalised a completely chromatic scale, which made use of coniuncta, defined by Atkinson as ‘the attribute, realized in actual singing, of permitting one to make a semitone out of a tone and conversely; … the mental transposition of any property of any hexachord from its own location to another location above or below’ (p. 256). The Berkeley author thus approved the placement of a flat or sharp before any note to lower or raise the pitch. As a ‘coda’ to the Epilogue the communion Beatus servus puts in one last appearance: its tonal vagaries could finally be accommodated by a newly conceived tone-system.

The Critical Nexus supplies a generous number of examples, diagrams, quotations from the sources, and clearly reproduced facsimiles. All original texts are provided with accurate and idiomatic translations, either in the running text or in footnotes at the bottom of the page – more than a courtesy to the reader, since the footnotes are often an integral part of the argument. There is an extensive bibliography of primary and secondary sources, but in the case of the former one must know the identity of the editor.Footnote ¹⁷ The ‘list of manuscripts cited’ (pp. 260–3) is repeated later as part of an index of chants and manuscripts, this time with references to pages where each manuscript is discussed. While the small type of the running text and the even more minute font size used for the footnotes allows a cornucopia of scholarship to be compressed within little more than 250 pages of text, choice of a larger format might have made a challenging book slightly more of an ‘easy read’.