At the Limits of Mensural Theory: Tinctoris on Imperfection and Alteration
 Let's be honest: imperfection and alteration are not normally topics to set the heart racing. As notational principles intrinsic to the mature mensural system of the later Middle Ages, they are generally still learnt by students and scholars, for understandably pragmatic, nuts-and-bolts purposes of transcription, through secondary texts, such as those of Apel 1953, Rastall 1983 and Parrish 1959, rather than through primary sources. Indeed, Tinctoris's important treatises on these subjects – his Liber imperfectionum notarum musicalium and the Tractatus alterationum, along with the related essay on dots, Super punctis musicalibus, all from the 1470s – exist only in poor editions and have never been published in English, or, I think, any other translation. As a result, their detailed contents are still largely unfamiliar to all but a few highly specialist scholars of notation.1 I have recently been re-editing these treatises, and I have been frankly astonished at times at the breathtaking sophistication they display in pushing the metrically longer-term, syncopational potential of these rhythmical principles, especially imperfection, to their limits. Even though we often associate such notational sophistry with the rather earlier, circumscribed repertories of the so-called ars subtilior, it is clear that central pedagogical writing of the late 15th century could also eagerly encompass such complexities.
 At the technical extremes of Tinctoris's exposition, any superficial query regarding its 'practical application' must give way to an acknowledgement that the unique semantics of the mensural system are being explored as much for their intrinsic intellectual value as for any day-to-day requirements of polyphonic composition. Perhaps 'intrinsic intellectual value' is not quite right, though. Tinctoris's more innovatory perceptions tend to be, frankly, rather en passant: his prime motivation in most of his pedagogical writing is to provide a practically orientated compendium of relatively traditional theory – packaged, however, as far as possible, so as to give comprehensive, working musical examples of notational phenomena where earlier writers were often more content to give purely verbal explanation. Imperfection is a good case in point. The classic 14th-century texts which grapple with this new mensural phenomenon, in particular those by, or surrounding, the figure of Johannes de Muris, are often more interested in its philosophical ramifications for the nature of the whole and part, the logic of division, the continuity of time, and in the opportunity it affords for a totalizing categorization of its permutations, than in its actual musical embodiment. (Dorit Tanay has much of interest to say on these aspects of Ars Nova mensuration in her 1999 book, Noting Music, Marking Culture: The Intellectual Context of Rhythmic Notation 1250–1400 (Tanay 1999) – a rare and valuable attempt to view medieval notation in its wider philosophical and mathematical context.) Perhaps, by contrast, with Tinctoris we see an example of what we might (anachronistically) call active rather than passive music theory: he is not simply giving the reader the tools to interpret what will be found 'out there' in the 'real world' of polyphonic composition; he is also taking pleasure in demonstrating in a practical way what rhythms and combinations of rhythm are possible and available from a full use of the mensural system. In this sense, his texts belong not to the realm of theory at all, but rather to the world of propaedeutics.
 I have time here to look in detail at only one of Tinctoris's many interesting, and taxing, worked examples of complex imperfection in practice. But before this, we need to be absolutely sure of our terminology: I know that we all think we understand imperfection, but I discovered when doing some of the initial work on my new edition of Tinctoris's Liber imperfectionum that even the summary treatment in Apel's Notation book (Apel 1953, esp. 107–12) is a little, shall we say, economical with its detailed logic. The lineage of Tinctoris's thought basically goes back to the Libellus cantus mensurabilis secundum Johannem de Muris, as regards the relationships between the whole and its parts. Let us take an all-perfect maxima as starting-point [Slide 1]. This can, of course, be imperfected at the most immediate level by one of its three constituent longs [Slide 2]: in strict terminology, the maxima is being imperfected quantum ad totum (with respect to the whole) by one of its partes propinque, or neighbouring parts (i.e. the longs, next down in mensural level). Since these neighbouring parts are also themselves perfect, each of the two remaining longs (i.e. after the discounting, as it were, of the imperfecting long) is also available for imperfection by one of its constituent breves [Slide 3]. These breves are partes propinque, neighbouring parts, of the longs, but are partes remote, which I translate here as 'parts at one remove', of the original maxima. The maxima, then, if it continues to be imperfected to the maximum possible extent by these breves, is said to be imperfected quantum ad duas partes propinquas (i.e. with respect to the two longs at one level up), but a duabus partibus remotis (i.e. by two parts at one remove, namely the breves). Again, once this level of maximal imperfection is achieved, each of the four remaining breves (partes remote) is available to be imperfected by one of its own constituent semibreves [Slide 4], which are partes propinque of the breves, partes remote of the longs, and partes remotiores ('parts at two removes') of the original maxima. So at this level, the correct terminology is that the maxima is being imperfected quantum ad quattuor partes remotas (i.e. with respect to the breves, the four remaining parts of the maxima at one remove), but a quattuor partibus remotioribus (i.e. by four parts at two removes, the semibreves). And finally [Slide 5], each of the remaining eight semibreves, after the previous level is exhausted, is available to be imperfected by one of its own constituent minims, which are partes remotissime, or parts of the maxima at three removes. So, again, the correct terminology is that, at this level, the maxima is being imperfected quantum ad octo partes remotiores (i.e. with respect to the eight parts at two removes, the semibreves), but ab octo partibus remotissimis (i.e. by eight parts at three removes). This process is, of course, demonstrating the maximum number of parts, at each level, by which the maxima can be imperfected; and at any stage a smaller degree of partial imperfection is entirely possible, with shorter notes also able to be re-compounded into longer ones, making the eventual number of permutations very large indeed.2 For instance, in this abstract example, the initial imperfection quantum ad totum is not itself a prerequisite for the subsequent levels; so at the next level down, there could be three neighbouring parts (i.e. longs) each yielding a breve. Or alternatively, only one of the longs may be needed to yield up its breve for the required rhythm. And so on down the chain. Nevertheless, it is remarkable that by the end of this whole process the maxima, in its most complete state of imperfection, will be reduced in value by a full 65 minims' worth, from its original value of 81 minims right down to 16 – in other words, to a duration even shorter than a normal imperfect long in this mensuration.
 Now, when the system is set out in a clear, hierarchical, tabular format such as this, it is (at least in principle) relatively easy to follow. But when transferred to an at least slightly more real-world situation on the staff, the successive levels of imperfection do not, of course, necessarily follow each other in such a simple, linear fashion. Book 2, Chapter 3 of Tinctoris's Liber imperfectionum is entitled, hardly encouragingly for the nervous student, 'On the Fifteen Methods of Imperfecting the Maxima'. If one rises to this tempting challenge, the final example, illustrating the fifteenth method, shows us how the maximal imperfection of an all-perfect maxima, such as we have been looking at in the abstract, might work out in practical notation [Slide 6]. Here is the example given, presumably (judging from the clef, etc.) composed by Tinctoris as if designed for some hypothetical Tenor or Contratenor part, as transmitted correctly in the elegant Valencia codex of his treatises, dating from Tinctoris's time in Naples, around the early to mid-1480s.3 (I should add that this is by no means the most complicated of his examples, just one concise enough to use here today. One or two from the imperfection treatise present such a challenge that I have spent several days on each trying to parse a correct interpretation.) Anyhow, I would be willing to bet that most of us (including me), confronted for the first time with this notation, would work on the initial assumption that the dot attached to the long indicates a dot of division, grouping together the initial maxima, the following two (slightly anomalous-looking) minims, and the long, into the space of a perfect maxima. By this line of thinking, the maxima would be construed as being imperfected both quantum ad totum by the long, and quantum ad duas partes remotiores by the two minims [Slide 7].
 However, before we could get too pleased with ourselves for negotiating this little nursery slope, we would realise that something was very wrong, because of the four breves' worth left over, before the arrival point of the final maxima. We might even imagine, if we weren't helped by the accompanying, explanatory text, that some kind of imperfection of the final maxima by these four breves a parte ante was taking place – and, indeed, without the text or any other contrapuntal parts to check against, this could indeed be viewed as a viable interpretation. The dot attached to the long, however, is crucial. If we read Tinctoris's prescriptions regarding the dot, from his essay Super punctis musicalibus – incidentally, a much more interesting and finely nuanced read than you might imagine – we can see that this cannot be, in Tinctoris's eyes, a punctus divisionis at all, since it would be redundant (a punctus asineus or asinine dot)4 if such a grouping of the first four notes were intended. Instead, it must be read as a punctus perfectionis, indicating a perfect long, which nevertheless does imperfect the initial maxima quantum ad totum [Slide 8]. As we saw in our previous, abstract model, the remaining values, including the two 'displaced' minims near the beginning which have not yet been accounted for, are calculated by Tinctoris precisely to enable the further imperfection of the initial maxima to its greatest possible degree. The two longs' worth from the maxima, remaining after its imperfection by the notated perfect long quantum ad totum, releases two breves' worth quantum ad partes propinquas [Slide 9]; in turn, the four remaining perfect breves at this level can release four semibreves quantum ad partes remotas (in relation to the original maxima) [Slide 10]; and the eight remaining perfect semibreves at this level release eight minims quantum ad partes remotiores, a process which enables the two displaced or anomalous minims near the beginning to be accounted correctly with the six minims' worth near the end [Slide 11]. A correct transcription of the example, therefore, is as here [Slide 12], with the first maxima maximally reduced, as we have seen before, by 65 minims from 81 to 16 minims.
 What really interests me about an example of this sort – brief though it is – is that it requires a process of comprehension that is essentially non-linear. Even the most dense of present-day, complexicist notation is usually, at least in principle, readily graspable in terms of its durational requirements, even if its execution is seriously demanding. In Tinctoris's strangely compelling little example, however, the duration of the initial maxima is in no way knowable, even by skilled 15th-century polyphonists, without prior, careful, non-linear contemplation of the mensural make-up of the phrase. The required retention in the mind of the multiple levels of 'deficiency to be made up' raises interesting questions regarding the use of memory, connecting with possible analogues within the 14th- and 15th-century evolution of mathematical computing, the rise in the use of arabic numerals, and the related, hierarchical conception of arithmetical place-value – what Alexander Murray, in his now classic book Reason and Society in the Middle Ages calls 'The Emergence of the Arithmetical Mentality' at this time (Murray 1978, Chapter 7: 162–87). It also seems to me that the relative lateness of Tinctoris's clear interest in these subtleties of imperfection structure might be understood as an unspoken part of the much more fundamental evolution of interest in notions of dissonance and resolution: not harmonic or contrapuntal dissonance, but a kind of metrical or mensural dissonance, which is held in the mind and in the eye rather than the ear, and which demands and achieves its own kind of temporal resolution as the various levels of perfection are completed. Furthermore, the necessary processes of mediation between and among the mensural levels in Tinctoris's examples – a kind of play with mensural perspective, if you like – is suggestive of how his thinking inherits and develops what Dorit Tanay has called 'the intention of fourteenth-century Nominalism to replace subordination with co-ordination, hierarchical opposition with continuous mediation' (Tanay 1999: 267). There is a sort of counterpoint between the local-level, aural phenomena of melodic contour, cadence, and implicitly harmony, with a sense of 'deep metre' and deep-level syncopation that explores the continuities between the world of perception or contingent reality and the world of imagination and possibility.
 As you will appreciate, I am really only at the beginning of exploring these larger, intellectual contexts lurking behind Tinctoris's continuing fascination with such notational issues, even as late as the 1470s. His musical landscape is obviously very different from that of the Ars Nova theorists who started the ball rolling 150 years previously. Intellectually, too, it seems to me that one compelling distinction between Tinctoris's world and that of Johannes de Muris concerns the nature of complexity itself. As Tanay has outlined in her book, the framework of 14th-century sophistical logic, and its ramifications for conceptions of temporal continuity and hence musical theory, tended towards presenting prima facie complexity as a means of elucidating simple, underlying truths (Tanay 1999, esp. Chapter 7); and, as we all know, there are many examples of subtilior notation that present a kind of ludic counterpoint between the sophistry of the material notation and its relatively straightforward outcome in real time. Looking at the notation presented in Tinctoris's imperfection examples, though, the opposite is rather the case, in that the simplicity of the tools (the notes on the page) belies a high degree of underlying, conceptual sophistication.
 I just want to draw your attention briefly to one particular aspect of Tinctoris's short treatise on alteration, the Tractatus alterationum, which, again, I think, hasn't been quite appreciated up to now, and which also links in a curious way to subtilior notation – though it is pretty rare even there, as far as I know. This is the notion of 'syncopated alteration'. Bearing in mind the rarity of the phenomenon, this comes remarkably high on Tinctoris's list of rules for alteration, forming part of his Second General Rule, which states that [Slide 13] 'it does not matter whether two notes found by themselves, of which the last is due to be altered, are placed directly next to one another or in syncopation, since wherever perfection is necessitated in the latter case, just as in the former, alteration occurs as a consequence, as here ...'.5 He then provides an example in which 'normal' alteration, of the second of two notes before a note of the next larger value, is given, along with two further instances where the altered note is, indeed, separated from its partner [Slide 14]. As with the earlier example of imperfection, a dot becomes crucial to the correct interpretation. It would not be at all surprising if we tried to read the dot attached to the maxima at note 6, and the dot attached to the breve near the end, as dots of division. But, again, as with the imperfection example, Tinctoris would argue that this would be redundant, or asinine; so we ought more properly to construe these dots as dots of perfection. The long following this dotted maxima, then, and the semibreve following the dotted breve, need to be read as altered before their following respective maxima and breve. The final test of the accuracy of this interpretation – apart from the obvious fact that this is the very point that Tinctoris is exemplifying here – is that the example resolves itself at the end correctly with the final maxima coinciding with the new unit of major modus. We sometimes tend to think of theoretical examples of this sort ending with a kind of generic long, whose exact duration is neither here nor there. But Tinctoris, with very few exceptions, is absolutely fastidious about this notion of 'deep metre', and he will almost always end an example in these treatises – however brief – on the correct duration for the accurate completion, or perfection, of the deep-level unit of major modus.
 From a practical point of view, you can see that an incorrect interpretation here induces a significant loss of note-value; and many modern editors, I think, confronted with such a passage without awareness of the possibility of syncopated alteration, and perhaps with a looser sense of distinction between the types of punctus, might well be tempted to argue for a corrupt reading in the MS, and make up something to fill the gaps, or otherwise try to emend the notation. This does, then, make me wonder whether there are instances out there – say in the primary sources of Okeghem, Obrecht, De la Rue, for example – where such syncopated alteration, or complex imperfection structures like the one we saw earlier, has been employed but subsequently edited out, either by modern scholars or by 15th-century copyists who were themselves not sufficiently aware or confident of the subtleties. I have recently been examining an English PhD by Ian Darbyshire on lost notational complexities in the Tudor festal masses of Robert Fayrfax (Darbyshire 2004); and coincidentally (or not) some of Darbyshire's suggested original notations there, which have been disguised through later simplification in the surviving sources, also involve very tricky syncopated alteration and imperfection, much more taxing and long-ranging even than Tinctoris's examples. This itself raises interesting questions about the dissemination of mensural theory in early Tudor circles; but, equally, if any of you have come across parallel instances in continental European, polyphonic sources of the late 15th or early 16th centuries, I would be extremely interested to hear about them.
 I'm very aware that, in the current musicological climate, nitty-gritty notational issues are perhaps regarded as a bit unsexy. I think, myself, that this is a shame, and we sometimes need reminding that the mensural notations of the late Middle Ages and their more recent descendants, for all their limitations and problematics, represent some of the great intellectual achievements of Western Europe. Looking again at texts such as those of Tinctoris, it becomes doubly clear to me that something as apparently unpromising as imperfection and alteration can lead us to explore those deeper intellectual connections in fascinating and unpredictable ways.