Anna Maria Busse Berger positions her important and provocative study of medieval music as a corrective to the past hundred years or so of musicology, based on the charge (familiar since Joseph Kerman's Musicology, 1985) that the so-called positivistic concerns of understanding sources and establishing musical texts have excluded other important issues. Arguing that musicology's aims have been dominated by Friedrich Ludwig's (1872–1930) ‘narrowly defined questions of fact that can be directly answered by reading the sources’ (p. 10),Footnote ¹ Busse Berger claims that there has been a consequent lack of attention to ‘the role of memory in the composition and transmission of polyphony’ (p. 44). She proposes that Ludwig's younger contemporary Jacques Handschin (1886–1955) provides better working models for evaluating medieval polyphony, since he emphasised the nexus of improvisational skills that connect medieval ‘paraphrase’ (the mimicking of standardised formulae) to the memorisation of theory, and noted the importance of these skills in composition (pp. 36–7). Busse Berger proposes that examining several aspects of composition and transmission in relation to the group of mnemonic practices and techniques (often called the artes memorativae) will bring Handschin's less individualistic view of the compositional process into play. Relying principally on the work of Mary Carruthers, whose two books (1990, 1998) demonstrate the ubiquity of the artes memorativae in medieval learned culture,Footnote ² Busse Berger, in the first half of her study, links these artes (either directly or by analogy) to the creation of a musical ‘memorial archive’. The building blocks for this ‘archive’ include (1) tonaries, which, among their other functions, help in memorisation of chant, (2) mnemonics for the acquisition of basic music theory, and (3) theories of counterpoint (e.g. organum and discant treatises), which are replete with examples to be learned by rote. The second half of her book treats how this ‘archive’ may have been deployed in the creation and transmission of Notre Dame polyphony and may have helped in the visualisation of notated music, and therefore helped in the performance and creation of fourteenth-century isorhythmic motets.

Busse Berger clearly demonstrates some broad analogies between the artes memorativae designed for the manipulation and acquisition of texts and the various music-theoretical sources she describes, and this allows her to give a convincing description of how the music-theoretical sources could have helped the process of memorisation. However, by narrowly focusing on how the material was memorised rather than on ‘what they had to learn’ (p. 6), she glosses over detail that is directly relevant to understanding improvisation, and therefore compositional process. In particular she does not pay sufficient attention to two important differences between memory techniques: those designed for words and those designed for music. First, unlike most verbal composition, musical improvisation often takes the form of a thoroughly social practice, both in its instruction and in its performance. Improvised polyphony was most often taught orally by a master demonstrating its techniques and was ‘composed’ by singers preparing for, and performing, the liturgy, in which it was adjusted to the timing of the liturgical action and the solemnity of the feast. For these reasons, musical memory techniques often differed considerably from those employed in other arts, and focused instead on the creation of mental tools to be used in conjunction with a group of performers rather than on finished pieces that were ‘composed in the mind’. Second, many musical mnemonic devices (e.g. the Guidonian hand, the Gamut, consonance tables, etc.) are designed to produce mental instruments that are often to be used as a replacement for, or in conjunction with, notated monophonic music, in order to create improvised polyphony. Thus while Busse Berger clearly recognises that most polyphonic music throughout the Middle Ages was not composed by individuals at all, but made in rehearsal and performance, her discussion of the specific ways in which the ‘memorial archive’ was deployed is often sketchy and occasionally inaccurate. In this review, we will therefore focus on the points at which her discussion could have benefitted from a closer investigation of music pedagogy and improvisational practices, providing some supplemental bibliography of studies that have treated these issues.

In chapter 1, Busse Berger examines tonaries, comparing them to florilegia, and suggests that both types of book function as ‘promptbooks’: they regulate and ensure the memorisation of the content that each classifies, and they enable the compiler to retrieve ‘a lengthy passage through a short one’ (p. 77). She advances the mnenonic functions of tonaries over the conventional view of them as reference works designed to help cantors ensure the smooth connection of antiphon to psalm tone. However, there is no reason to believe that tonaries could not fufil both mnemonic and reference functions. Although she later quotes Regino of Prüm, who explains that he made his tonary because he had observed the frequent disagreement (dissonantia) in the choir about the correct tone for singing the psalms, Busse Berger argues that a tonary would not be useful if there were uncertainty over the mode, since a cantor would have to scan the entire book, which would be impractical: she concludes that ‘the purpose of the tonary was simply to organize the repertory for memorization’ (p. 60). However, the parties of the dispute described by Regino would each maintain that they did already know which was the right tone (and therefore mode). With Regino's comprehensive tonary, they did not need to scan its entire repertory to resolve this disagreement: they needed only to look at the first level of classification to see whether it confirmed the modal classification championed by one or the other party.

Although tonaries could be used (and may normally have been designed) as reference books, this does not contradict Busse Berger's more important insight that tonaries may have been extremely useful in establishing the memory of the chant more securely and in gaining better access to, and control over, that memory. The analogy that she develops between the process used by the compiler of a florilegium and of a tonary is apt, since it highlights their control over their memorised material through the technique of divisio (the hierarchical classification of a large group into smaller logically organised groups, and subgroups, pp. 51–6 and 77–80).

However, Busse Berger argues more controversially that, since the classification system employed by tonaries changed little over time, ‘the invention of diastemmatic notation resulted in no radical change in the procedures used to commit chant to memory’ and that ‘the basic techniques for memorizing and retrieving the pieces remained similar from ca. 800 to 1500’ (p. 50). While she is correct in claiming that chant continued to be memorised both before and after the creation of diastemmatic antiphoners, the distribution and change in content (though not of the classification system) of tonaries specifically suggests that the chant no longer needed to be memorised using the same procedures. The key studies of tonaries (especially those by Huglo)Footnote ³ demonstrate that early tonaries, in the main, classify the repertory of chants more or less comprehensively, but during the eleventh to the thirteenth centuries, they broadly divide into an Eastern group continuing the older comprehensive model and a Western group, in which only a few representative examples are classified for each mode. The division of the types can be linked both geographically and chronologically to the use of diastemmatic notation. Busse Berger convincingly argues that the chant incipits as listed in the earlier comprehensive type are written in a form that suggests that they are intended to be memorised (pp. 74–7), but it is precisely the comprehensive lists which disappear along with the adoption of diastemmatic notations. The particular technique of memorising lists of antiphons may well have waned because any doubt about the specific issues relating to pitch could now be resolved through consulting the notation. Even though the classification system, modal mnemonics, modal formulae and model antiphons persist in the truncated tonaries, it is by no means clear that they remained intimately connected to any specific method of memorising the repertory. They could, for example, still be used to teach how to recognise the features of the mode, and remained valuable for reasons such as ensuring accuracy of intonation. One suspects that it soon became clear to those who used diastemmatic notation that the most efficient way for a cantor to gain a comprehensive analytical view of the repertory (that would fix the melodic details in his mind) would not be to create a comprehensive tonary, but would instead be to copy out a diastemmatic antiphoner: these later antiphoners supplied details such as psalm intonations and euouae formulae. While Busse Berger is certainly correct in claiming that the most basic way of memorising chant never changed, it is not in the use of the tonary that one encounters the primary analogy that can be made with the artes memorativae. The most basic technique was always to memorise chant by rote, within the strongly articulated structure of the liturgical cursus that is mnemonically classified by hour, day, week, season and year.

Busse Berger's analysis of florilegia in chapter 2 similarly pays little attention to their history and diversity and therefore develops only the most basic analogy (the use of divisio) relevant to thinking about florilegia, tonaries and the artes memorativae in conjunction. However, florilegia, like tonaries, are not used by their compilers alone, and their function often changes when they become books to be memorised in their own right. For example, memorising a florilegium might allow those whose reading was not wide to attain a veneer of classical education, with only a minimal amount of memory work.Footnote ⁴ Similarly, a survey of the types of florilegia and their historical and geographical context might produce evidence of the influence of the artes memorativae on medieval music. Could one, for example, connect the Carolingian prosodic and metrical florilegia (which classify short prose and poetic extracts according to their use of metrical cursus and metre) to tonaries produced in the same region, perhaps even by the same people, since the cantor often had charge of grammar instruction? Might the creation of such florilegia be connected to the cultivation of liturgical tropes written in hexameters in the same period?

Chapter 3 surveys previous work on the well-known mnemonic devices used in standard musical instruction, demonstrating that musical instruction clearly shared in a culture in which rote memorisation flourished and the artes memorativae enabled the flexible recall and manipulation of memorised material. Busse Berger briefly considers tools that help with rote memorisation, such as interval songs and versified treatises (pp. 94–102) and the graphs and diagrams for the classification and recall of intervals, modes, mensuration and proportion (pp. 102–9), concluding with the observation that ‘these methods are not different from those used in other areas of knowledge in the Middle Ages and Renaissance’ (p. 110). More pertinently, in discussing the Guidonian hand and the Gamut (pp. 85–94), she stresses the analogy noted by Karol Berger to rhetorical loci (visual seats which help an orator remember the structure, order and content of a speech). At one level the analogy works well, since the hand provides just such a set of placeholders that can be filled with the content of the medieval pitch system and its intervallic structure. However, such loci were used in functionally different and often precisely opposite ways. For example, in constructing a speech one mentally assigns the sections of the speech to the loci and proceeds through them in fixed order. To compose another speech, one replaces the content, but keeps the loci and the order the same. In contrast, in using the hand, its loci and content are fixed, so that one may move mentally through the loci in any order and have a mental aural access to the pitch that is indicated. This is why medieval authors explaining the hand compare it to a monochord.Footnote ⁵ The hand is not, therefore, simply a tool for recalling the pitch system (which is already memorised), but an instrument that enables one to hear pitches in any order or sequence in relationship to the whole system. Since the content of its loci never changes, one may also use the hand to secure one polyphonic line by placing one's finger on the notes in succession, while singing or thinking another line.

Busse Berger's analysis of the possible use of memorised images such as the Gamut represented in scala (ladder) form leads her to consider that an ‘imagined staff’ may also be used to create mental loci: ‘The lines of the [imagined] staff function as the background grid, while the notes are the ever-changing images placed onto the grid’ (pp. 86, 93–4). Again, the discussion is marred by its insistence that such a tool exists solely in the mind. First, a real staff filled with the content of one of the lines (available to anyone with access to a diastemmatic antiphoner or gradual) would clearly be of more use in mentally constructing polyphony over a cantus firmus, and the practice of singing super librum probably dates back at least to the invention of diastemmatic notation, even if documentation of the practice comes late. Moreover, the scala form of the Gamut may also easily be represented by the hand turned sideways, thinking of the fingers and spaces between the fingers as the loci, a practice described by Bermudo.Footnote ⁶ Using the hand in this way enables a singer to represent the cantus physically while improvising over it, or while having another singer improvise over it.

Chapter 4 completes the survey of the ‘memorial archive’ by comparing methods of instruction implied by grammar treatises and abacus treatises to the teaching of polyphony in organum, discant and counterpoint treatises. Busse Berger correctly notes that elementary grammar and mathematical instruction placed an emphasis on memorisation of examples and on drill and suggests that the same is true of the vast majority of music treatises (pp. 111–19).Footnote ⁷ The emphasis on drill allows her to supplement Steven Immel's work on the Vatican Organum treatise (VT), which clearly associated it with the repertory of Notre Dame.Footnote ⁸ Busse Berger's principal contribution is to call into question Immel's view of the treatise as a kind of manual for writing music in the style of the Magnus Liber. She allows for the possibility that the treatise itself could have been compiled from notated sources, but argues that the purpose of such copying would have been to memorise the formulae for reuse, and that the treatise could therefore serve as a form of oral transmission of the style (pp. 122–30). Although she does not comment on the introduction to the treatise, characterising it as ‘a short theoretical tract’ (p. 119), it could support her view that the repertory was improvised (though not unambiguously). The treatise starts as if it is describing improvised counterpoint practices, naming an organizator who responds to the lead of the cantor; both use the Guidonian hand as an aid. It then describes the problems associated with chromatic alteration (momentary shifts to and from b♭ that might force a change of hexachord) and states that there is a ‘figure’ called synemenon for this. It now sounds as though the treatise were discussing notation rather than the use of the hand, but it could also be describing how a singer could switch from the soft to hard hexachords quickly to create what would be illegal mutations in the normal use of the hand. On representations of the hand (as well as in notation) from the twelfth century on, ‘b’ is signalled by round and hard shapes, i.e. different ‘figures’ similar to flat and natural signs. This interpretation finds support in the two examples cited in the treatise that Immel discovered can be found in the Notre Dame repertory. Ad nutum Domini, from O 18R, has no b♭ in the cantus, but the organal part momentarily introduces one, fitting the situation described in the treatise as ‘when we have added something we ought not to have added or taken away something we ought not to have taken away’. In Iustus germinabit, from M53, the b♭ is in the cantus itself, and the treatise correctly states that ‘one frequently finds this in chant’, and that it creates ‘the greatest difficulty in organum’. The introduction concludes that no organizator should be ignorant of how to adjust the pitch of the hexachord syllables to the hexachord proper for responding to b-fa or b-mi in the chant, and that each ‘b’ should therefore have its own set of solutions (suas facturas). This suggests that the most common problem encountered by organizatores was the presence of b♭ in the cantus itself, implying that the singers were sighting from the hand or the book.

Busse Berger might also have argued that although Immel has properly characterised VT as a copula treatise, he has not given a satisfactory explanation of its rhythmically imprecise notation.Footnote ⁹ Unlike the concordances in the Notre Dame repertory, VT regularly uses conjuncturae instead of standard ligatures, and makes a highly inconsistent use of suspiratio marks instead of carefully indicating perfections with them. Both features obscure the precise indication of rhythmic modes, and while Immel suggests that the writer's haste accounts for these practices and the treatise's numerous mistakes, they are better explained if one accepts Busse Berger's argument that the compiler of VT was copying the examples both to organise his memory of the repertory and to pass on instruction in the style orally. When sung accurately from memory, any imprecision in the notation would, of course, disappear.

Busse Berger then briefly discusses largely fifteenth- and sixteenth-century discant and counterpoint treatises that comprehensively list the allowed consonances over each interval progression within one hexachord or between two hexachords an octave or fifth apart. In presenting such treatises within the context of the artes memorativae, she not only supports the standard view that students were expected to memorise these progressions, but also calls attention to ways in which the material was visually organised for its better retention and recall (pp. 130–50). However, she states that the goal of this memorisation is to ‘put all of these progressions together’ (p. 141), i.e. to have the contrapunctus of an entire composition in one's mind (p. 208). Her warrant for this is the anonymous treatise Iuxta artem conficiendi [compositiones]: ‘Whereby the first rule must be this one: anyone who wishes to compose should proceed in all such discant in such a way that he places the progressions according to the way planned before’ (p. 141).Footnote ¹⁰ However, both the original language and its context in the treatise call this translation into question. The context is a discussion of the proper relationship of the contratenor to the tenor at cadences (clausulas): the contratenor can always move from a fifth below the tenor to an octave except when the tenor is ‘fa–mi’, in which case the tenor should move from a third below to a fifth (creating a plagal cadence in modern terminology), and the text refers to the examples of such cadences in both simple and decorated forms given at the end of the treatise. The author is not speaking about stitching together the various simple and decorated formulas into new progressions, but about learning them as cadences. A translation that takes the context into account would not support Busse Berger's argument: ‘Whereby the first rule must be this: all who wish to compose should so progress in this method in all discant that he should set the several cadences in place according to the pattern mentioned above.’

Busse Berger's discussion of discant treatises treats those that rely on ‘sights’, such as Lyonel Power's, and those that assess consonances strictly in terms of hexachord syllables, like the ‘grado’ treatise of Ugolino of Orvieto (and, one might add, like VT). However, she does not connect these two differing processes with the two forms of the Gamut discussed earlier. ‘Sight’ treatises imply either that the scala form of the Gamut or staff notation is being used, and locate the cantus on the Gamut through using hexachord syllables while expressing the consonances by the numbers of their intervals. ‘Grado’ treatises imply the use of the Gamut as expressed on the hand, where one may keep track of the movement of the tenor by placing one's finger on the appropriate joint, and the consonances are expressed through the hexachord syllables (i.e. the syllables which are always consonant in the same hexachord, or in the same syllables of the hexachords a fifth and octave above). Both types of treatise imply the interaction of memorised material with the physical expression of the cantus, whether on the hand or in notation. This obviates the need to hold a series of consonance progressions in one's memory.

The chapter ends by examining diminished or florid counterpoint treatises and concludes that the examples given in them were not meant to be memorised, since the possible variations are so many that no comprehensive catalogue could be given (p. 152). However, this contradicts the analogies between florilegia and other anthologies of verbal composition mentioned earlier in the book. The extensive self- and intertextual quotation of musical flores suggests that such examples were memorised, though they would not have to be memorised from written treatises. Moreover, although writers like Coclico seem to imply that students proceeded directly from learning simple counterpoint to florid, some theorists describe a number of intervening stages. For example, the early fifteenth-century theorist Antonio de Leno gives rules and copious examples of not only note-against-note counterpoint but also of two notes to one and three notes to one, before going on to florid counterpoint. He mentions that the first and third notes of three notes against one need to be perfect or imperfect consonances (approached either by step or by leap) and the middle note may be a dissonance, but approached by step alone. What is being codified here is the practice already demonstrated by the many flores given in treatises like the VT. The difference is that generalised rules supplement (but do not replace) the copious examples. The fifteenth-century theorist Guillelmus Monachus describes another way of generating two notes to one while controlling dissonances in his rules for, and examples of, syncopation over ascending or descending stepwise movement of the tenor. When memorised, such formulae provide a fuller framework than simple or note-against-note counterpoint, and this framework also could be rhythmically and melodically decorated to create florid counterpoint. For this reason, we suspect that there was a very complex interaction between sighted or notated examples and oral instruction, in which memorised formulae continued to play a crucial role even when teaching and singing florid counterpoint.

The second half of Medieval Music and the Art of Memory addresses how the ‘memorial archive’ may have been deployed in the composition and transmission of polyphony. Chapter 5 argues that the Notre Dame repertory was conceived by deploying memorised formulae, such as those given in VT, and was both performed and transmitted without the aid of notated sources. Busse Berger's argument is based, in part, on the lack of surviving written sources. For example, the singing of tripla and quadrupla was reported in documents dating to 1198 and 1199, but no notated sources exist that can be dated earlier than the 1240s. Moreover, Busse Berger reiterates the argument already made by Craig Wright and Rebecca Baltzer that no cathedral inventory lists polyphonic manuscripts (p. 163). While one should not underemphasise this negative evidence, it should be noted that there is a continuous tradition of cantors being associated with specific liturgical books containing the solo portions of the Mass, and that they may often have compiled these and owned them personally. Not only did they own and compile cantatoria but also tropers, which contained new repertories for soloists (and in the case of the Winchester Troper CCCC 473, notated organa).Footnote ¹¹ Moreover, as Margot Fassler has demonstrated, the duties of the cantor came to be increasingly associated with the maintenance and correction of liturgical books: if anyone knew how to notate, it would be the cantor.Footnote ¹² Since the Notre Dame repertory develops polyphonically the very same material as that contained in cantatoria, and was sung by the local experts in notation, it is simply not likely that notation played little or no role in its transmission. If Wright is correct in his support of Handschin's identification of Perotinus with Petrus Succentor of Notre Dame †1238, there is in fact a fairly brief period between the death of the most famous organista and the appearance of manuscript ‘F’ (perhaps dating before 1250), which conforms to the Parisian chant dialect and calendar precisely and (according to Baltzer and Edward Roesner) is Parisian in origin.

Busse Berger's argument that much of Notre Dame organum is written in a style which could be improvised and performed (at least in part) from memory is much stronger. She supports her argument in two ways. First, she compares the formulae given in VT against one of the examples given at the end of the same treatise, the organum Operibus sanctis (pp. 165–74). Her detailed comparison demonstrates that although the formulae form the backbone of the composition in every instance, they are flexibly deployed and display melodic variations, all of which suggests improvisatory recomposition. Second, her more radical suggestion is that the use of modal rhythm and ordines in measured discant create metrical loci that could help performers retain large segments verbatim in memory (pp. 174–95). In support of this second point, she argues against Treitler's and Rudolf Flotzinger's contention that the study of quantitative metre had little bearing on the evolution of modal rhythm. Although she recognises that William Waite made similar arguments as early as 1954, she dismisses his specific arguments, repeating the claim that Augustine's De Musica was not widely known in the thirteenth century (p. 182). Nevertheless, Augustine remains the only author to write about metrics who has a clear notion of a musical rest, an essential element for establishing metrical rhythm without text, and Patrick Le Boeuf's dissertation, ‘La tradition manuscrite du De Musica de Saint Augustin (et son influence sur la pensée et l'esthétique médiévales)’ (Paris, 1986) has identified about twenty manuscripts dating from the thirteenth century, several of French origin, that document the revival of interest in De Musica at precisely the correct time to be influential.Footnote ¹³ Moreover, Busse Berger's argument about the connection between the study of metrics and the creation of modal rhythm does not take into account a similar and more fully documented argument made by Margot Fassler.Footnote ¹⁴

On the other hand, Busse Berger's emphasis on the mnemonic potential of rhythmic cells and ordines is an important new contribution to the discussion of modal rhythm and may help explain how discant sections could remain relatively stable in the absence of notation. Even so, such organisation has another important function: the organisation of the patterns of strong and weak ‘beats’ in modal perfections allow discanters to create specific sets of flores that avoid any series of strong dissonances and that work over many different cantus movements. Immel has called attention to the repeated-note pattern through a descending fourth (e.g. c–cb–ba–ag) that frequently occurs in the Notre Dame repertory. The reason for the ubiquity of this pattern may be that it can be applied to almost any succession of notes of a cantus organised into an ordo of four perfections (three notes plus one rest). As well as making the discant section easier to memorise, it makes that voice easier to improvise.Footnote ¹⁵

We would propose an alternative view of the compositional process of the Notre Dame organa in which the cantus line is both fixed in memory and physically signalled either in monophonic (not polyphonic) notation or on the hand, or both. The organal line is ‘created’ from the memorised consonance patterns but ‘improvised’ in the sense that these patterns are applied to the chant as it is given in the book or on the hand. The easiest parts to improvise would be measured discant of two voices, since the steady rhythmic pulse of the cantus would make the coordination of cantus to discantus predictable. Less easy to coordinate would be copula and organum purum, but if the tenor singing and signalling the cantus had the formulae memorised, he could cue off the discanter and know when to change to the next cantus note.

Busse Berger supports her argument against the use of ‘written notation’ by citing Craig Wright (p. 163): ‘Taking the absence of polyphonic sources at face value, we must conclude that much of the organum, discant, and counterpoint of the church, whether sung by memory, by improvisation, or by some combination thereof, was performed without the assistance of written notation.’ However, the Notre Dame inventories list many notated graduals as well as an antiphoner given to the ‘right side of the choir’ in 1250. It is only in the early fifteenth century that a full inventory of choir books is given, and we do not know when the practice of chaining two graduals and two antiphoners on each side of the choir began, but one cannot logically argue that an absence of notated polyphonic sources suggests that notated monophonic sources would not be used. Wright's most intriguing reference is also late (1497) but it describes a new gradual with large notes containing repertory not found in the other large books placed on the lectern in the centre of the choir on high feast days, i.e. to be used on the same occasions and at exactly the same location that earlier references establish that organa were performed. The principal point to stress here is that even in the absence of such ‘visual aids’, the tenor responsible for the chant line would still be able to signal the movement of the cantus on his hand, allowing a discanter who had developed a quick sight and who had a set of memorised stock formulae to improvise his part (without needing to hold any memory of the entire ‘composition’ in his mind).

In her final chapter, Busse Berger discusses music that presupposes mensural notation, e.g. isorhythmic motets, written down by ‘a composer in the modern sense of the term, that is . . . by someone who conceived his music not only as something to be heard, but also as something to be seen’ (p. 198). Her focus is therefore on the various ways the artes memorativae might be deployed to build up a mental grid through which the entire composition might be worked out ‘in the mind without a wax tablet or parchment’ (p. 199). She suggests that the visualisation techniques used in English sight treatises may have been adapted for composers. She argues that Pseudo-Chilston expected beginners ‘to visualise on the staff the consonant intervals for the entire piece’, and that therefore accomplished composers could ‘similarly have visualized their note-against-note structures’ of more complex polyphonic music (p. 208). However, as we have already discussed, Pseudo-Chilston's treatise, like the other sight treatises, is concerned only with interval progression and not with memorising lengthy series of interval progressions. The reason for this is that the chant is already present or physically signalled, so the full structure never needs to be memorised. This does not mean that composers did not work out contrapuncti, nor does it call into question Busse Berger's more important, if speculative, insight that imagined notation, memorised and controlled by techniques made famililar in the artes memorativae, could have helped composers work out complicated compositional relationships in their minds. The rest of the chapter examines two specific practices: (1) the creation and manipulation of memorised or notated contrapuncti against which other parts may be conceived, and (2) the organisation of rhythmic patterns that might be visualised in grids and manipulated through applying mensurations. While Busse Berger proposes that such complex manipulations of the tenor are the musical analogue of medieval word games based on the manipulation and permutations of syllables and letters in medieval texts (p. 235), she does not discuss how the texts of motets were created or how the complex verbal relationships among tenor, duplum and motetus could have been arrived at and deployed in the musical setting.

Although Busse Berger stresses that the artes memorativae may have created an ability of composers to retain entire complex compositions in their minds without the need of ‘real’ writing, we would again propose a more complex interaction between notated and visualised music. Motet tenors are relatively short, and even quite complex rhythmic organisations may be notated using very little space on a page. Once the tenor structure is fixed, consonances could easily be sighted against it in two, three or more parts, and different permutations of the consonances worked out for each repetition of the tenor. It would even be possible to work out consonances over the retrograde of the tenor or combine complex proportions among the parts.

We also think it worth mentioning that the emergence of instruments, especially keyboard instruments capable of playing two or more polyphonic lines, should be considered with regard to how composers could retain complex harmonic structures in their minds without needing any real notation other than a chantbook. The organ keyboard (as well as the later keyed monochord and clavichord) is based upon the scala version of the Gamut and provides a ready visualisation (albeit sideways) of the stave. As Busse Berger reports, the use of such real instruments in the teaching and learning of counterpoint and improvisation is documented from the fourteenth century on (p. 113). We believe that the influence of such instruments was just as profound as the use of the Guidonian hand and of the stave, allowing the physical representation of complex harmonic and melodic relationships and thus providing the means for the ‘composer’ to work out even more complex musical relationships, sighting from chant or mensural notation and using the eyes and the hands as well as the mind.Footnote ¹⁶

Anna Maria Busse Berger's focus on the nexus connecting oral and written transmission of music to the artes memorativae suceeds in allowing us to think beyond the ‘naive picture of a written musical culture replacing an oral one’, replacing it with ‘a more complicated picture of a culture in which orality and literacy interacted in many unexpected ways’ (p. 254). We hope that in focusing on the specific means through which singer-composers improvised music during this period we have contributed an additional layer of complexity that nevertheless supports Busse Berger's view that the memory arts were central to musical thought in the Middle Ages.