Among the many interesting features of Jacques de Liège's Speculum musicae (1320s) is its definition of the term cadentia in reference to harmonic progression.Footnote 1 Frederick Hammond's article on the author for the New Grove Dictionary of Music and Musicians signalled the interest of this concept,Footnote 2 referring to it, perhaps misleadingly, as ‘cadence’. Since then several authors have cited it as a medieval witness to the notion of directed musical motion (the ‘directed progression’), by which an imperfect interval (a third or sixth) implies resolution to a perfect interval (a unison, fifth or octave) following.Footnote 3 Such an interpretation appears to follow from Jacques's account of imperfect concordFootnote 4 ‘striving’ towards adjacent perfect concord, which seems very much like a description of harmonic progression directed from one chord to another;Footnote 5 and the linguistic similarities between Jacques's exposition of the concept and later writings seem to attest a continuing tradition of musical thought of which he was a part.Footnote 6 Yet the absence of the word cadentia in this sense from these other writings is odd if Jacques was indeed part of the same tradition as they; and his well-known opposition to the stylistic and theoretical developments of the Ars nova makes him unlikely as a conduit for one of its striking new ideas.
This article challenges the connection that has been made between Jacques's cadentia and the theory of the ‘directed progression’ as this was adumbrated in fourteenth- and fifteenth-century music theory. It argues instead that cadentia served a purpose in Jacques's treatise entirely congruent with his intellectual stance overall. Rather than seeking to explain or describe the seemingly inexorable passage of one chord to another, he presented a solution to the metaphysical problem of imperfection as it concerned the harmonic dimension of polyphonic music. The concept thus related to the polemic of his treatise's famous seventh book. Cadentia comes near the end of Jacques's lengthy discussion of concord that forms the latter part of the fourth book of Speculum musicae. Concord was a concept with an aesthetic grounding; and so cadentia too related to an aesthetic conception of polyphony, one that had developed in the thirteenth century and was distinctly different from that of the new musical order emergent at the beginning of the fourteenth.
THAT JACQUES'S NOTION OF CADENTIA IS NOT A THEORY OF DIRECTED MUSICAL MOTION
There is much in Jacques's chapter that resembles the concept of the directed progression, so it is necessary first of all to establish the respects in which cadentia runs short of this. David Cohen has identified the theory of directed progression as emerging in the fourteenth century from the assimilation of the Aristotelian account of natural motion, via scholastic transmission, into music theoretical discussions of harmonic intervals and their use in polyphony. He proposes five general elements to the idea:Footnote 7
1. a basic distinction between perfect and imperfect intervals;
2. a principle stating that imperfect intervals ‘tend towards’ or ‘seek’ perfect ones;
3. a principle stating that (2) holds because the imperfect intervals are imperfect;
4. a principle stating that (2) holds by nature;
5. a principle (usually implicit) stating that an imperfect interval tends towards a specific perfect interval (its own perfection).Footnote 8
The first of these was widely recognised in the thirteenth century; and the fifth, as a general proposition about the tendency of imperfect things to perfect ones, was extensively cited in thirteenth-century writings outside the sphere of music theory, being taken up there only in the fourteenth century.Footnote 9 Cohen illustrates the coalescence of these with the other three as a coordinated idea through an examination of Marchetto of Padua's Lucidarium, arguably the earliest treatise to expound it. Aspects of the theory in this work are complicated (as will be discussed below), and a neater example to set in direct comparison with Jacques's chapter can be found in Ugolino of Orvieto's Declaratio musicae disciplinae (c. 1430), which is also cited by Cohen. The sixth chapter of the second book follows similar content and plan to Jacques's, beginning with a general statement and then illustrating that with specific cases. Ugolino's general statement is as follows:
By nature, that which is imperfect and incomplete, tending to that from which it is deficient, is compelled to move so that it has perfect form; and as the aforementioned imperfect consonances, or dissonances, are imperfect in the comparison of consonances and do not have their perfection, each longs ardently to go towards it, so that it may be resolved to belong to perfect consonance. Certainly these things have been observed by expert theorists through experience, because if the placement of notes is at a third, sixth or tenth, there is no repose, but rather whichever it may be it is moved compulsively so that it should be joined to its perfection. And hence it is that any mensurally regulated cantus closes with a perfect consonance, although from time to time before the final end one has to close on an imperfect consonance, or dissonance, at which, because the well-ordered ear of the listener is not quieted, the last consonance is adjudged to be the end.Footnote 10
This passage clearly presents all five of Cohen's elements for the concept of directed progression. It goes even a step further by specifically identifying this process as a movement and by recognising that subordinate cadences on imperfect chords are less satisfying to the listener than final cadences on perfect chords, thereby aligning a natural movement from imperfection to perfection with the effect of progression in the music, one in which the end may be sensed before it arrives.
Following this passage, Ugolino exemplifies the progression of imperfect intervals to perfect ones, discussing the minor third passing to unison, the major third to perfect fifth, the major sixth to octave, the tenth to octave or twelfth (depending on whether it is major or minor), the thirteenth to double octave and the seventeenth to double octave or nineteenth (depending again whether it is major or minor). He mentions but rejects progressions of the sixth to fifth and of the thirteenth to twelfth. Each progression that he accepts is, then, unitary, one imperfect interval leading to one perfect interval, with the sole exception of the minor third, which, seemingly as an afterthought and without discussion, he adds in progression by similar motion to the fifth. Ugolino's is, thus, very nearly a doctrine of directed progression. In any case, it is as close as any single theorist came to formulating the idea and may be regarded as a classic statement of it.
Jacques's definition is less straightforward:
Cadentia, as far as it concerns the present matter, seems to refer to a certain order or a natural inclination of a more imperfect concord to a more perfect one. For it seems that an imperfect thing naturally inclines to a more perfect one, as if towards better being, and what is feeble wishes to be sustained by a thing that is stronger and stable. Therefore cadentia is said to be in consonances, when an imperfect concord strives to reach a more perfect concord next to it so that it falls into it and is joined to it below and above, namely by descending or ascending [progression]. (Appendix 1, paragraphs 2 and 3)
The first, second and fourth of Cohen's conceptual elements are present; but the third is not, as imperfect things are said to incline to perfect things not because they are imperfect but because they are weak and unstable.Footnote 11 Nor is there any statement that the imperfect thing seeks its proper perfection, even though this adage was known to Jacques and cited by him later in the treatise;Footnote 12 and while Cohen allows that this principle may be implicit, it is far from clear that it is in this case, given the examples of cadentia offered in the chapter's fourth paragraph.
Jacques illustrates cadentia with a substantial list of concords seeking greater perfection that is much longer and more inclusive than that given by Ugolino. It includes intervals that other fourteenth-century theorists excluded from polyphonic composition (the tone, perfect fourth, minor seventh and the octave compounds of the first two of these). There is a question to be answered of how exactly these descriptions are to be understood, and this is considered in Appendix 2; but even the most restricted interpretation does not illustrate the notion of directed motion, as the progressions of the fourth, fifth and octave to unison, of the ninth to fifth, of the fifth, eleventh and twelfth to octave and of the twelfth to double octave can be realised in several different ways even on an assumption of contrary motion; and alternative continuations are specified for progressions from the fourth, fifth, sixth, seventh, ninth, major tenth, eleventh and twelfth. The examples give the impression that Jacques was attempting to identify all the progressions from more imperfect to more perfect concords that could be foreseen within the contrapuntal style of music at the time, given the constraints upon the melodic progression of the individual voices. He was illustrating the possibilities rather than specifying what should in any instance be the case, as suits the harmonically diverse style of the Ars antiqua motet. It is clear from what he says in the seventh book of the treatise that he thought that the lines of good discant should have independent movement and melodic beauty.Footnote 13 Harmonic diversity is liable to arise from music that emphasises such features.
What is significantly absent from Jacques's chapter but which is more than once present in Ugolino's is the concept of movement. The falling implicit in cadentia itself is a sort of movement, of course, though one of a particular sort.Footnote 14 Unlike Ugolino, Jacques does not speak of nature compelling the imperfect thing to move to the perfect one but rather of its being sustained by the perfect thing.Footnote 15 The idea is quite different. It might be conceived by analogy with a book on a shelf leaning against and thus being propped up by a bookend. There is no movement between the two as they are in a stable equilibrium (the bookend sustains the book in a vertical position); yet the book is ‘falling’ towards the bookend in the sense that if the bookend were moved away, the book would fall. The falling in the book is potential, not actual. So while Ugolino views polyphonic music as being naturally propelled by the progression of imperfect to perfect concords, for Jacques polyphonic music passes through unstable and stable states, the stable ones sustaining the unstable ones. He does not suggest that the passage of the music is motivated by these states.
Although cadentia in its raw sense indicates a kind of movement, and the verb cado had been used since at least the eleventh century to refer to descent in a musical line,Footnote 16 Jacques makes no connection between its falling and effects of musical movement, as is apparent from the awkward juxtaposition at the end of the third paragraph, where he states that cadentia occurs when an imperfect concord ‘falls into [a perfect concord] … by descending or ascending [progression]’. In the examples of the fourth paragraph of the chapter, there are some that use just descending motion, but others that use just ascending motion, and a majority that use both at the same time. In terms of actual musical motion, cadentia meant at least as much ascending as descending motion, and with the double-leading-note ‘cadence’ common in three-voice counterpoint at the time, twice as much.
This difference between the direction of the interval progressions and the descending motion inherent in a notion of falling is not a trivial one. The size and direction of melodic progressions is the subject of a detailed examination in the sixty-ninth chapter of the sixth book, where the concept of musical movement is explicitly introduced through what Jacques presents as a quotation from Guido (though it is, in fact, closer to a statement in the Anonymous Expositio de motu): ‘Music is the motion of notes and consists in quantity, because [the motion is] in the number of notes joined together.’Footnote 17 As presented, the idea of motion is one of changing quantity, specified by the sizes of intervals between consecutive notes in a melody, as assessed by the numbers of notes included within its intervals. It corresponds to the first of the three types of motion that Jacques identifies in the chapter on motion, which forms part of an extended discussion of the nature of sound in the first book of the treatise: quantitative, qualitative and local.Footnote 18 Quantitative motion consists in a change of size, whether an increase (motus augmenti) or decrease (motus detrimenti). So the musical motion is described in terms of the increase or decrease of melodic interval in the melodic line. Implicitly the music is understood as a substance that is getting bigger or smaller through time.Footnote 19
In fact this view is not strictly maintained, as the ensuing discussion of ascending and descending intervals suggests a covert change to an idea of local motion. Such inconsistency may be indicative of uneasiness on Jacques's part with the metaphor of musical movement, if not as a description of musical effect, as an intellectual conception. His adoption of the ‘Guidonian’ definition is rhetorical rather than assertive:
Surely the progression from one note to another is a sort of movement? Surely ascent and descent are sorts of movement? And these relate to unequal [i.e. distinct in pitch] notes, since one ascends from re to mi, or by the reverse descends; and similarly between all conjunctions of unequal notes there is ascent or descent; and since there could be motion which neither ascends nor descends, as that which is in the same space, the progression that is between equal notes is not entirely free from motion.Footnote 20
Jacques does not defend or explain the idea, but merely invites the reader to accept it. Yet the rhetorical mode admits a degree of uncertainty. He was rightly troubled by the problem of unison movement, which does not fit in with the quantitative view.Footnote 21 The concept of melodic motion itself poses problems: what substance is getting bigger or smaller or changing through this motion – or, in the case of local motion, moving from place to place? How could an idea of pitched motion be reconciled with the temporal motion embodied in mensural concepts? The metaphor of musical movement remains difficult to extrapolate in its specifics,Footnote 22 and so we need not be surprised to discover that the problem had not been resolved in the fourteenth century. We can sense in what Jacques says that while he acknowledges the aesthetic experience of musical movement he recognises intellectual difficulties with the concept.
Jacques's chapter on cadentia does not present all the aspects essential to the idea of the ‘directed progression’ as these are contained in the classic articulation by Ugolino of Orvieto. Nor do the interval successions with which he illustrates his concept support the view that he had in mind a kind of harmonic motion in which the second term was specifically implied in the first. Indeed, the idea of musical motion is not overtly mentioned in the chapter, and his comments on it later in the treatise suggest that it was one he was happier to use than to expound. In any case, there is a clear difficulty in reconciling the descending motion implicit in the concept of cadentia with the types of musical motion that he allies to it. Even where the concept of falling is applied to harmonic progression elsewhere in Speculum musicae, it appears that the ‘descending’ invoked applies to some other quality associated with a progression that does not instantiate it in any unambiguous way. Given that Jacques chose and formulated the term cadentia for harmonic progression, there must have been a specific purpose to his choice. What this was will emerge from an exposition of what Jacques intended by cadentia, his motivation for the idea, and the context in aesthetic thought to which it related.
CADENTIA IN MUSIC THEORY
The word cadentia itself is an essential element of Jacques's theory, and a historical assessment of its use in music theory provides a necessary context for an exposition of that theory. In referring to cadentia in Jacques's treatise as ‘cadence’, Frederick Hammond was, of course, correctly marking both the Latin word as the origin of the modern term and Jacques's use of it as the earliest surviving in connection with harmonic progression. There is nothing, however, in Jacques's exposition of the concept that restricts the phenomenon to formal or stylistic instances that we should now think of as cadential. His reference in the sixth paragraph of the chapter to endings of organa and discant is exemplary of particular incidences of cadentia (in his understanding), but he says nothing to indicate that they represent the typical or exclusive case. Moreover, it is important to note the isolation of Jacques's usage of the word. The next surviving text to use it in a similar connection is the Liber musices of Florentius de Faxolis, dated well over a century and a half later, in the years 1485–92.Footnote 23 The final chapter of the second book (18, mislabelled 17 by the scribe, entitled De neuma et cadentia) introduces the term cadentia in relation to polyphony, though the author cites only the superius voice in his examples.Footnote 24Cadentia is owned there as a term borrowed from the vernacular and presented as the mensural synonym for neuma, which is discussed in relation to non-mensural music.Footnote 25 A quarter of a century or so after Faxolis, Aaron presented an extensive discussion of cadentia in the second half of the third part of his De institutione harmonica (1516), and there is no doubt here that the term applies to harmonic formulae in polyphony that we would now recognise as cadential.Footnote 26 The two-voice schemata that he discusses are the sixteenth-century ancestors of the modern cadence. The term cadentia is not defined, which suggests that Aaron assumed his reader was to be already familiar with it, presumably from vernacular use, as Faxolis testified earlier.
Stephano Vanneo's Recanetum de musica aurea from a couple of decades later also presents an extensive discussion of cadentia, somewhat indebted to Aaron's. It does, though, present a definition (the earliest extant) of cadentia, one that is clearly an ancestor of the modern concept of cadence:
And so a cadentia is a small segment of some part of a cantus at the end of which either a general pause or a perfection is obtained. Either cadentia is general [i.e., simultaneous in all voices], or a perfection is attained. Or a cadentia is a certain ending of a part of the song itself, just as in the context of oration there is a medial punctuation and a final punctuation. And learned musical experts strive to ensure that the end of cadentiae occurs where either the part of the oration or its clause finishes.Footnote 27
There is a point of contact with Jacques de Liège in that the attainment of perfection (perfectio reperitur) may be a characteristic of cadentiae. This formulation is not indebted to Jacques, however, and there is no reason to think that Vanneo was acquainted with his theory.
Jacques's use of the word cadentia was both original and not directly related to the later tradition that we have inherited; so in assessing it, no preconceptions born of later usage should be brought to bear. The comparative novelty of the word itself at the time Jacques took it up must be borne in mind. Originally a neuter plural form of the participle (cadens) of cado (I fall), the Latin word cadentia was only adopted as a substantive in the late Middle Ages. It has been found employed in three different ways by writers of the thirteenth century: in the simple sense of falling in physical movement, as a technical term in regard of accidence and inflection in grammar, and as a technical term in regard of rhythm in poetry.Footnote 28 If the modern word ‘cadence’ has lost its connection with the ‘falling’ that is its root meaning, that is unlikely to have been the case in the thirteenth and fourteenth centuries, when it was new.
In fact, Jacques's use of cadentia was not the earliest to occur in music theory, as he himself appears to recognise in saying ‘as far as it concerns the present matter’, implying that it applied to others too. The Tractatus de discantu by Coussemaker's Anonymus II, dated sometime at the end of the thirteenth century or beginning of the fourteenth, refers to ‘bona cadentia dictaminum’ as the fourth of four requirements for good discantus.Footnote 29 The meaning here is ‘good “falling” of the words of the text [in the music setting them]’ (i.e., ‘good word setting’), and the text goes on to specify the alignment of long syllables with long notes and of short syllables with short notes. The expression recurs derivatively in the Tractatus de musica mensurata et musica falsa seu ficta secundum complures scriptores Footnote 30 and towards the end of the fifteenth century in Franchino Gaffurius's Extractus parvus musicae and Guillaume Guerson's Utillissime musicales regule.Footnote 31 These instances attest a continuous tradition of using the term; and as it is nowhere defined or expounded, nor in any of these cases does it merit more than a single mention, its sense must have been obvious in the context, and may have reflected the way in which musicians talked about fitting words to music.Footnote 32
When Jacques appropriated the word, cadentia was a neologism with a variety of current applications, including another in the domain of music theory. He was the first to define the word in a music theoretical context, but his definition appears to have found no adherents, at least as far as surviving sources attest. In any case, cadentia was for Jacques a general concept, not, as it was for late fifteenth- and sixteenth-century writers, a specific musical object.Footnote 33Cadentia in his understanding was manifestly associated with particular kinds of harmonic progression; but his adoption of the term poses the question of what was thought to be ‘falling’ in them.
AN EXPOSITION OF CADENTIA
The two aspects
Jacques's definition of cadentia contains an equivocation: ‘Cadentia … seems to refer to a certain order or a natural inclination of a more imperfect concord to a more perfect one.’ On the face of it, the alternatives here are two quite different aspects of the concept: the first, ‘a certain order’, places concords in sequential relation to each other, whereas the second, ‘a natural inclination’, indicates a condition inhering within the concord itself. Before these two aspects of the idea can be addressed, the underlying concept of concord must be considered.
The chapter on cadentia is the penultimate in a series of twenty-one devoted to the topic of concord and its converse, discord. Unlike many previous writers, Jacques distinguishes concord from consonance.Footnote 34 For him a consonance is simply any combination of two notes taken together at the same time, following the word's literal sense of ‘sounding together’.Footnote 35 It may be distinguished quantitatively, according to the size of the interval between its combined notes, and qualitatively, depending on whether it is concordant or discordant. Jacques defines concord as a mixture of distinct sounds that gives sensory pleasure, and discord as a mixture of distinct sounds that is harsh or displeasing to the senses.Footnote 36 The pleasing combination of sounds in concord creates a blend which he regards as a third element.Footnote 37 The archetype of this blend is represented by the unison, in which distinct voices are indissociably united as one. Concord in general is the harmonious blend that arises between two (or more) distinct sounds standing at certain pitched intervals from one another.
As the unison represents an extreme of concord, the absolute blend of distinct sounds, other concordant consonances manifest such blend to a greater or lesser extent. Jacques ranks these hierarchically according to a scheme similar to those of John of Garland's De mensurabili musica and Lambertus's Tractatus de musica.Footnote 38 Both concords and discords are organised into perfect, middle and imperfect categories. Jacques further divides the perfect concords into subdivisions of ‘good’ or ‘perfect’ (bona; perfecta), ‘better’ or ‘more perfect’ (melior; perfectiora) and ‘best’ or ‘most perfect’ (optima; perfectissima). A complete order of concords emerges from this (Table 1) and it forms the background to the theory of cadentia.
The first aspect of cadentia, as an order of concords, relates directly to this order. In this aspect, cadentia comprises the relationship in which a more imperfect concord stands to a more perfect concord adjacent to it in a musical progression. The progressions that Jacques gives in the fourth paragraph of the chapter fall between the divisions of the order of concords set out in Table 1. In each case the progression is from the middle or imperfect categories to the perfect one, or within the perfect category from good to better, or from good or better to best. Aesthetically, each comprises a passage from less blended to more blended concord. In this sense, cadentia refers to harmonic progressions. Any ‘falling’ conveyed by the term is conceptual: the passage from less to more blended sonority may as well be climbing as falling.
Table 1 Jacques de Liège's order of concords (SM IV. xxxiii–xxxvii)
NB. Fourth, fifth, octave and their compounds are all perfect intervals in the modern sense.
It is unclear why Jacques does not include progressions from imperfect to middle concord (such as a tone to third, and tenth or eleventh to ninth), since the basis of cadentia is the ‘inclination of a more imperfect concord to a more perfect one’, which is to say progression between comparative not absolute qualities of concord. Indeed the function of the category of middle concord is unclear, as Jacques is unable to distinguish it in general terms from imperfect concord:Footnote 39
Middle concord is when notes sounded together are perceived to differ much in the hearing; however, they are pleasing to it. … Imperfect concord is said to occur when notes, sounded together, are perceived to differ much in the hearing, otherwise however they concord.Footnote 40
Cadentia as order, then, concerns progressions between sonorities that are perceived to be the one less and the other more concordant (which means less or more blended). In order to account for this phenomenon, Jacques has erected an unnecessarily complicated conceptual edifice of hierarchically differentiated categories. It is within the terms of this framework that progressions are reckoned to be falling – not within the sound of the music, where in this regard only a contrast of sonorities can be discerned.
The second aspect of cadentia suggests that the more imperfect concord is in a state of ‘falling’ whether or not it abuts a perfect concord. This state is part of what it means for the concord to be imperfect. As was discussed above, Jacques's clarification of the condition of imperfect things (in the chapter's third paragraph) characterises them as weak and needing to be propped up by stronger (perfect) things. Cadentia in its second aspect ties in with this view of imperfection. Imperfect concords have a predisposition to ‘fall’ within themselves irrespective of their context. So cadentia in this sense is an abstract quality of particular concords. It has a perceptible correlate in the distinctness of a concord's constituent notes; and cadentia as a property of an individual concord stands in parallel to the deficiency of its degree of blend by comparison with the complete blend of the unison. As regards cadentia as natural inclination, any ‘falling’ that might be discerned in the incomplete blend of a certain consonance is purely conceptual: the falling of imperfection into perfection. This may serve as a metaphor for instability perceived in concords owing to a lack of blend between their notes.
The two senses of cadentia are synthesised in the final sentence of the opening section, where the imperfect concord is said to strive, conveying the inclination in the imperfect concord itself, to attain the more perfect concord near to it, conveying the order from imperfect to perfect. It is here that Jacques's language comes closest to that of the directed progression: for the suggestion is very strongly that an imperfect concord is falling towards and trying to reach the perfect concord following it; and both motion and direction are expressed in this idea.Footnote 41 It follows from the discussion above, however, that these relate to the conception of the concords as imperfect and perfect and not to their aesthetic effects. There is no musical effect (or experience) of falling or striving in them; that is part of the conceptualisation of the concords as being more or less perfect. When the tone is said to ‘seek’ the unison, this seeking is not within the music itself but within the domain of imperfection, to which the tone is said to belong.Footnote 42 No grounds are given for maintaining a specifically musical effect of directed motion in it.
Yet there is a tension in the concept. For if cadentia is a matter arising in the comparison of concords, what has it to do with falling in the domain of perfection? Indeed, this reveals a tension between the senses and the intellect endemic to Jacques's definition of concord, and for that matter to the entire Pythagorean-Platonic harmonic tradition. The matter is addressed explicitly in the first book of the treatiseFootnote 43 and surfaces again near the beginning of the section on concord in the fourth book:
Although sensation may judge some sounds to concord, others not, and, as regards concording, it may say that some concord more, others less, and similarly in discording, sensation is not, however, the cause of concord in concording [sounds], nor of discord in discording ones, but this stems naturally from the proportion of the miscible notes or in the proportion from the principal parts of such consonances, or by some essential or accidental causes, which even if sensation does not know them, the intellect can perceive.Footnote 44
The cases of the twelfth and the tone reveal how problematic Jacques's position is. Both of these consonances manifest high levels of perfection in the judgement of the intellect, as is demonstrated by the order of consonances that Jacques draws up ‘under the way of perfection’ in Book 2.Footnote 45 His order (which proceeds from left to right according to decreasing perfection) is based on the numerical properties of the ratios representing the consonances:
Greater perfection inheres in multiple proportions (decreasing as the magnitude of the ratio increases), then in superparticular proportions (decreasing as the magnitude of the ratio decreases). Both the twelfth and tone are demoted in the order of concords, however, because both sound less perfect than their ratios seem to the intellect. The twelfth is subordinated to the double octave as a merely good, rather than a better, perfect concord; and the tone is relegated to the imperfect concords, beneath the thirds and ninth (and other concords not in use). If the ratios are the cause of perfection, then why do not the senses reckon these concords more perfect, or at least learn to subordinate impression to knowledge? If the tone is perceived as less concordant than the third, why is that perception not the cause of the imperfection that is thought to arise in that judgement?Footnote 46
The concept of perfection and the sensory perception of concord are two independent systems that have a (perhaps illusory) common origin in the unison. For the one, this is absolute simplicity of proportion; for the other this is complete blend. Cadentia, then, seems to address two dimensions: a conceptual order according to which less perfect ratios fall (or are in the state of falling) towards the absolute perfection of unity; an aesthetic effect of passage from less blended to more blended concord. Inasmuch as it is grounded in the aesthetic concept of concord it is associated with perceptible qualities of progression in polyphonic music, from less to more blended sonority. The metaphor latent in the concept of cadentia does not refer directly to such qualities, however, for nothing is falling in them. Rather, the falling is of an abstract sort.Footnote 47
‘Ea, quae prope sunt, sunt quasi idem’
Paragraph 5 of the chapter on cadentia represents an important development of the argument. It states that an imperfect concord before a perfect one is perfected. Evidently there is some conceptual sleight of hand involved in saying that an imperfect concord is perfected by an adjacent perfect concord, as an imperfect concord is by definition not perfect; and this sleight of hand is apparent in the formula ‘things that are close are more or less the same’ (ea, quae prope sunt, sunt quasi idem – hereafter this is referred to as the ‘formula’).Footnote 48 The primary difficulty of the formula is what is meant by ‘close’.Footnote 49 Fortunately, some commentary on this notion can be gleaned from occurrences of the formula four times elsewhere in the treatise.
Two of these come close together in the forty-ninth chapter of the second book, in connection with numerical reasoning. The chapter investigates which are the smallest numbers that may legitimately be taken to represent the proportion of the comma (‘Qui sint minimi numeri proportionis commatis’), and the ‘formula’ occurs first to justify numerical approximation and then to reject it.
First Jacques examines the claim that the comma is the amount by which the whole tone exceeds two minor semitones, representing the two minor semitones through the proportions between the numbers 65536 (g), 62208 (h) and 59048 (i). Thus ‘g:h’ and ‘h:i’ both represent the proportion 256:243. In fact this claim is not quite true, as while ‘g:h’ is exact (256:243 × 256:256), ‘h:i’ is one out (it should be 256:243 × 243:243 = 62208:59049; he gives 62208:59048). Jacques's numeral ‘i’ is one unit too small.Footnote 50 He justifies the approximation with an exact statement of the ‘formula’: ‘But things, which are near, are more or less the same and those terms [i.e., g and i] are the smallest numbers of the proportion of the minor tone, as will appear below.’Footnote 51 In other words, ‘i’, a unit off from the true number of the proportion, is near enough to the true value to be taken as identical with it. Figure 1 shows the proportions Jacques uses in comparison with their true values. Only the minor semitone ‘g:h’ and the whole tone ‘f:i’ are accurate; all the other intervals are approximated.
Figure 1 Divisions of the tone given by Jacques (SM II. xlviii) compared with the correct values
In fact, Jacques seems to be advancing here an argument propounded by others as a foil for his own view, as he goes on to reject it. Towards the end of the chapter the ‘formula’ returns but now modified to register the difference between true identity and the approximation previously accepted. His rejection of approximation has the fortunate effect of leading him to qualify ‘close’ (prope): ‘And these things, which are close, are not the same, but more or less the same. It does not follow: “This is close to the house, therefore it is in the house or at the house.” For this word “close” is seen to convey a certain exceptional or exclusive force.’Footnote 52 The use of a spatial metaphor to clarify the meaning of prope suggests that Jacques viewed numbers not as discrete quantities but as members of an ordinal series. What makes 59049 and 59048 close is not the similarity of their quantities, but their proximity in the series of natural numbers. A quantitative element cannot be denied to the idea, as 1 and 2 are close by the same reckoning, but he would surely not have regarded them as being ‘more or less the same’ in this context. Although he distances himself from such approximation, the fact that he discusses it at all indicates that it was a tool of reasoning sometimes used at the time; and approximation is at the heart of the next two chapters, which consider representations of the comma by the superparticular ratios on either side of it (75:74 too small and 74:73 too large).
This use of the ‘formula’ gives us a way of understanding cadentia as the assimilation of imperfect concords to perfect concords close to them in the order of concords. This explanation is not entirely satisfactory, however; for while it may account for the relationship of middle concords to perfect ones, which are adjacent in the order, it is unconvincing for imperfect concords, which also progress to perfect concords, but which are not so close in the order: they ought to progress to middle concords.
The other two instances of the ‘formula’ (of which the second is a reformulation of the first) use it in a rather different way. The first of them is found in the first book of the treatise, where the twenty-sixth chapter discusses the differences of sounds (‘sonorum differentiae’):
And it should not be thought that a single sound is generated from a single impulse of a stretched string in the air, but several, as many that is as when a vibrating string in oscillating strikes the air sufficiently to cause sound. But because things that are close are seen more or less not to differ at all, these neighbouring sounds are adjacent and similar. Not discriminating between them, sense judges the sound there to be joined together as one, just as the colour red or green placed on some part of the surface of a hoop, when the hoop is spun rapidly, is seen to affect circularly the whole surface of this hoop, and the same appears in any of the colours of the rainbow.Footnote 53
Jacques here conceives the sound of a vibrating string as comprising many individual vibrations. These many short constituent sounds (the vibrations), because they are temporally very close together and very similar to one another in pitch and tone colour, are assimilated to one another by the senses so that a single sound is perceived. ‘Close’ means here temporal adjacency at a certain threshold of perception.
The comparison with the spinning hoop,Footnote 54 with which Jacques attempts to clarify the conception of closeness, offers another way of understanding cadentia. Although colour is located on just one part of the hoop, as this coloured section spins with the hoop at velocities in excess of a certain minimum, the eye ceases to distinguish the movement of the coloured portion with the spinning of the hoop or the flashing of the colour as it returns repeatedly to the same point in the revolution; it sees instead a continuous line of colour, as if the whole hoop were coloured. Instead of perceiving so many revolutions, one sees a complete circle of colour. In a similar way, in discant, comprising a mixture of perfect and imperfect concords, when it is performed at speed, the whole is rendered perfect because the perfect concords add, so to speak, a sheen of perfection to the whole, as does the splash of colour to the rotating hoop.
There is an air of plausibility to this as an understanding of how discant might sound perfect despite the imperfections in its harmony. However, this way of understanding cadentia is also not entirely satisfactory, for there is no reason in it why the imperfect concords could not be discords, just as it does not matter what colour the hoop is before the paint is added to it.
So Jacques uses the ‘formula’ in the first two books to argue for approximation on the grounds of two types of closeness: ordinal proximity in the case of approximated numbers and temporal proximity in the case of the vibrations of a string. Although neither of these applies in an entirely satisfactory way to cadentia, both seem to shed some light on it. In the first sense, imperfect concords are ordinally closer to perfect concords than are discords. So the use of imperfect concords can be understood on the grounds that they are close enough to perfect ones to be more or less the same as them. In the second sense, the perfection of discant with imperfect concords can be understood on the basis of the temporal proximity of the perfect concords to each other when the music is performed at an appropriate tempo, assuming (as is likely) that Jacques had in mind here the Ars antiqua style, grounded in an alternation of contrasting sonorities.Footnote 55 Jacques offers no help in deciding between these two senses, and the meaning at this crucial point in the chapter is frustratingly unclear. Of course, the two do not conflict with one another, and both may be taken as conditions for the effect of cadentia. A key point emerges, though, that through the agency of cadentia, however exactly this may work, imperfect concord is perfected.
CADENTIA AND THE REDEMPTION OF IMPERFECTION
That the perfection of imperfect concord is the purpose of the concept is borne out by a further occurrence of the term in the seventh book, the term's only recurrence in Speculum musicae. In the eighth chapter Jacques considers the relationship between the consonances of the fourth and the fifth. Arguing that the fourth concords better above the fifth than below it, he proposes that it does so because the high note of the fourth is joined to the high note of the octave arising from the combination of the consonances and because the low note of the fourth is joined to the high note of the fifth. His reasoning stems from a belief that concord derives more from the higher than from the lower of two notes comprising a consonance.Footnote 56 He argues further that the combination of fourth on top of a fifth on top of an octave also concords well and caps these arguments with the statement: ‘Since an imperfect concord before a perfect one is perfected on account of the cadentia into it [i.e. the perfect concord], it seems rational that some concord, for instance a fourth, is enriched from a more perfect one if it comes into contact with that.’Footnote 57 By this he means that the fourth is enhanced from its contact with the octave that results from combining the fourth with the fifth in the same way that imperfect concords are perfected in sequence with perfect ones. The reasoning of the statement does not help his case, since both the fourth and the fifth belong to the same kind of perfect concord in his scheme; accordingly either ought to be enriched by contact with the octave, and thus this constitutes no ground for preferring the placement of fourth above fifth over that of fifth above fourth. The remark is valuable, however, in showing that Jacques regarded cadentia as the agency of perfection, the property of an interval by which it might be bestowed with greater perfection through contact with more perfect intervals.Footnote 58
Underlying Jacques's view of cadentia, then, is what might be termed a ‘redemptive effect’. Imperfect consonances are made perfect and therefore good through it. Comments on the unison and octave bear this out:
This consonance [the unison], because of the ultimate unity that the notes bring to it, makes other bad ones around it better, for as much as they fall into it. …
Also on account of its goodness, [the octave] makes better imperfect consonances beside it, such as a sixth or seventh, so that they strive to fall into it directly, coinciding in this with the unison.Footnote 59
The unison and octave make less perfect intervals beside them better. They redeem them for perfection.
The capacity of cadentia to perfect imperfect concords was an important function for Jacques's theory, as perfection was an essential value for him. An aversion to the lesser degree of perfection in modern music was one of the most important components underlying the polemical stance of the seventh book. A diatribe against the use of imperfect time in the music of the Ars nova is given in chapter 45 and spills over into the three that follow it. It was not imperfect time values themselves that Jacques objected to in modern music, as these were a standard part of the Ars antiqua. Rather it was their autonomy in Ars nova theory that concerned him.Footnote 60 In the theory of the Ars antiqua, imperfect longs frequently occurred, but they were always accompanied by breves (or by semibreves to the value of breves), creating a perfection. Thus imperfect things were subordinated to perfection. In Ars nova music, composers were often content to render temporal values imperfect throughout a composition, with no completion in perfection at the end. In the temporal domain, the music of the Ars antiqua was superior in his view because imperfect values were always resolved in perfection.Footnote 61 This music posed a problem for him in the harmonic domain, however, as it was quite liberally composed with imperfect concords. Yet Jacques maintained that good (that is, perfect) concord was essential to good discant.Footnote 62 He had to find a way in which imperfect concords did not vitiate the perfect concord, and the concept of cadentia was it. Like the concept of perfection in the rhythmic domain, cadentia enabled imperfect elements to become effectively perfect through their contact with perfect things.
The need for perfection in music is reinforced by Jacques in the seventh book in a passage that also offers a synthesis of perfection in the temporal and harmonic elements. The thirtieth chapter adds a theological dimension to the argument by directly relating perfection to the Holy Trinity.Footnote 63 The tone of the chapter is defensive, the bulk of it countering three objections raised against the view.Footnote 64 It seems likely that the stakes of this argument would have been high, in the light of Ars nova developments in the treatment of imperfection. Jacques restates his position boldly:
All perfection proceeds from the highest and first perfection and as a consequence every ternary number, threeness or thirdness by reason of the perfection which it rightly conveys is drawn into the first, highest and most perfect Trinity. … And since song [assembled] from perfect [time values] is drawn into the highest first and most perfect Trinity by reason of threes, it is not so for song [assembled] from imperfections since no imperfection falls back or could fall truly into that highest Trinity.Footnote 65
This much is clear for the rhythmic dimension of music, but what of its harmonic dimension? Should harmony be based on the interval of the twelfth (expressed by the ratio 3:1)? Plainly this would be unrealistic; and Jacques has a solution for the problem of reconciling harmonic reality with theory, which arises as he answers the second of the objections introduced near the beginning of the chapter:
‘In the same way that God is three in persons, just as he is one in substance, natural song should no more be drawn into the divine Trinity than into its unity.’ It should be said that cantus naturally composed from perfections is carried both into the divine Trinity and into its unity: into the Trinity by reason of the perfections and distinctions that the ternary number conveys, into unity itself indeed by reason of the concord which in singing of this type is required, for concord is named from the concord of distinct notes or distinct melodies reduced to one.Footnote 66
If the dual numerical identity of God as one and three seems to undermine the force of Jacques's claims for the triple number and the Trinity as a basis for musical order, he answers this by applying these numbers to two different dimensions of mensural music. The perfections of rhythm are ‘drawn into’ the Trinity; while the unity of concord, the meld of two notes into a single sound, represents God's unity. The agency for this latter, though it is not acknowledged here, is cadentia, by which every more imperfect concord is supported in its falling by a more perfect concord, of which the unison is the most perfect example, the end point of unity into which all imperfections fall.Footnote 67
The stakes underlying the redemption of rhythmic and harmonic imperfection seem to be the object of the chapter's closing paragraphs, though the argument develops tangentially. Jacques addresses the third of the objections, that whether songs are sung in perfect or imperfect time, God remains unchanged as three and one. He accedes to this because God is intrinsically perfect. He goes on to extend the range of the statement by underlining God's immutability in relation to His creation. Yet we should not infer from this that there is no difference between sinful and virtuous behaviour, for God rewards good deeds and punishes bad ones. Quotations from Plato, St John and Augustine establish that sin does not come from God. A further quotation from Boethius states that sin is nothing because God cannot sin. Jacques qualifies this: although sin is not a thing in itself, it is nonetheless the lack of the good moral quality that deeds should have.
The chapter ends at this point without establishing how these reflections on sin relate to the argument about singing in imperfections. Readers are left to make the connection themselves. The intention seems to be to equate singing in imperfect time with sin, implying that although God is unchanged by it, he will punish it. We may infer that Jacques thought a polyphony too much compounded of imperfect concord would incur similar censure.
Part of Jacques's polemic in the seventh book derives from his commitment to a theory of divine reference according to which the order or disposition of elements in music refers back to God. Cadentia is the means by which ‘sinful’, imperfect concords may be redeemed to the perfection which is caused by God. It would have been easier for this theory if every polyphonic composition had actually ended with a unison. In the sixth paragraph of his chapter on cadentia, Jacques acknowledges both the octave and unison as the chief final concords of discant, before admitting the fifth and even the fourth as occurring also. Practice could be refractory from a theoretical perspective. If the fifth and fourth are overlooked, even the octave represented a slight problem for him. If it were allowed to stand as perfect in and by itself (as he states in the fourth paragraph), then this admits perfection to the ratio 2:1, since the ratio of an interval is one of the causes of its perfection;Footnote 68 and if this is admitted, then his objection against imperfect time is weakened. If he required that the octave actually be perfected by the unison (which he allows as a possibility in the fourth paragraph), his theory is in direct opposition with an overwhelming quantity of the actual practice that he endorses.
There is another problem with the theory of cadentia for Jacques's polemical ends. Although Ars nova music often made a greater use of dissonance than that of the Ars antiqua, it nonetheless respected the supremacy of perfection in the harmonic realm: there was always a perfect chord at the end. Cadentia could have been used as a justification of Ars nova harmonic practice just as it was by Jacques for that of the Ars antiqua. Yet Jacques was no less critical of the harmonic practices of the Ars nova than of its rhythmic ones.Footnote 69 It is understandable, then, that the concept of cadentia was tucked away at the end of the fourth book and scarcely acknowledged in the seventh. The completeness of his work required a chapter on concords set against each other in successive progression (cadentia) and simultaneous combination (partitio). Cadentia exceeded this remit, for it addressed other issues, both speculative and aesthetic. Yet its solution to the problems was not complete, and it could not represent a key element in the polemics of the seventh book, even though it was a necessary background to them.
THE RETROSPECTIVE AESTHETIC OF CADENTIA
The redemptive purpose of cadentia addressed a problem of an intellectual sort posed within the conceptualisation of concords as perfect and imperfect. As has been shown, concord was regarded as a primarily aesthetic conception, and the notion of cadentia carried with it an aesthetic dimension, addressing the problem of how music can sound good in which certain of the elements are intrinsically dissatisfying. Surely the easiest course for a composer to choose would be to compose only with perfect concords, as imperfect concords would represent an offence against aesthetic sensibility. A true conservative might have advocated polyphonic composition only in the style of the ‘fifthing’ and ‘fourthing’ mentioned in the tenth chapter of the seventh book, where each instance of the texture would sound well.Footnote 70 Yet composers did compose with imperfect concords, including those admired by Jacques. The theory of cadentia sought also to explain how this could be.
In Jacques's view of polyphony, there is not a drive from imperfect concords to perfect ones; rather the force is in the opposite direction: the perfect concords prop up the imperfect ones. This idea is implicit a few chapters earlier in discussion of the middle concords, where the case of the major and minor thirds is alluded to: ‘For a third pleases much, if it is sung sweetly in its place, namely before unison or fifth.’Footnote 71 Despite not being wholly concordant on its own, a third can sound pleasing if it is put in the right place. The order of concords is of aesthetic consequence not to an understanding or appreciation of musical movement but to an acceptance of the concords themselves. In the right context, concords that might not be acceptable on their own become pleasurable.
Because of the cadentia of more imperfect concords to more perfect ones, the more perfect ones propped up the more imperfect ones and made them effectively perfect. In other words, music in which imperfect concords were used would sound well provided these intervals, less satisfying in themselves, were adjacent to perfect concords that made them sound good retrospectively (or perhaps it should be ‘retro-auditively’).
Jacques was not the first writer to have confronted this problem. The expanding range of intervals increasingly used in polyphonic composition during its development posed a significant problem to theorists given the strong current of Pythagorean ideology in their intellectual tradition.Footnote 72 The simple organum of the Musica Enchiriadis treatises, doubling melody in parallel perfect consonances (or ‘symphonies’), conformed to Pythagorean ideals completely: each instant of the polyphony was perfect and sounded well on its own. The use of non-consonant intervals was hard to defend in theory. How could music that used these other intervals be good? Practical music does not appear to have been much troubled by such questions, and composers quickly took to exploring the other intervallic combinations of two voices and the melodic independence of voices that their use enabled. Theorists then had to find a way of shoring up the gap between practice and theory, and the tradition that developed was one of explaining the phenomenon by the adjacency of concordant and discordant (or imperfect) intervals in polyphonic music.
The first hint of it can be found in the De musica mensurabili of John of Garland. He does not appear to have been concerned by the imperfection of major and minor thirds; but he did feel the need to justify the use in polyphonic practice of consonances that he regarded as discords:
It should be known that every discord [i.e., seconds, tritone, sixths and sevenths] may equal a middle concord [i.e., a fifth or fourth] before a perfect [i.e., unison or octave] or middle concord, and this is rightly employed before unison or octave: [He gives music examples of a tone before a unison and before an octave] and so it is for one [example]. And it is inappropriately employed before a middle [concord]. But this is frequently found in many parts of organum, such as a tone before a fifth, as in this example: [He gives music examples of a semitone before a fifth and of a tone before a fifth and before a fourth] And it should be known that a discord is never put before an imperfect concord, unless there is a cause in the colour or beauty of the music.Footnote 73
Although the case is not explicitly stated, the implication of this passage is that discords become acceptable (because they become equal to middle concords) when put before perfect, or less acceptably middle, concords. A similar idea is behind a passing remark of Franco of Cologne: ‘any imperfect discord [by which he means a tone, major sixth or minor seventh] concords well immediately before a concord’;Footnote 74 and in the same vein, Anonymus IV, referring to an instance in discant, owns:
And thus it appears, that a worthless or offensive discord, which the sixth is, and rejectable by all for the most part, is itself the penultimate before a perfect concord, which the octave is, and becomes the best concord under such ordering and positioning of notes or sounds, as was said before.Footnote 75
And he goes on to say that the same may apply to any discord, and that some musicians even treat several discords in sequence in the same way.
Incidental comments of this sort may have served as a starting point for more elaborate formulations developed by others. Walter Odington adumbrated a theory of how the phenomenon worked: ‘Diaphony is a concording discord of lower notes with higher ones, so called because it does not proceed by concords throughout, but because the following concord removes the displeasure of the preceding discord, and this is commonly called organum.’Footnote 76 Interestingly, Odington puts discord at the forefront of his conception of polyphony.Footnote 77 The function of the concords is to make the discords bearable.Footnote 78 Discord that is made concordant is the essence of this kind of polyphony for him, a view that is borne out by the bold writing of many late thirteenth-century motets.
The Pseudo-Franconian Compendium discantus found a neat way to sidestep the problem by formulating a basic differentiation between consonances that are perfect in themselves and those that gain perfection through their placement:
Three of the consonances are perfect by themselves, namely: unison, diapason and diapente. Three are [perfected] by circumstance, namely: semiditone, ditone proceeding in order to the diapente or unison, or tone with diapente proceeding in order to the diapason; one is perfect, and not perfected by circumstance, namely the fourth.Footnote 79
There is no categorial difference here between perfect and imperfect consonances, so the author does not have to confront the problem that Jacques tried to solve; all consonances used in polyphony are reckoned to be perfect, but it is the way in which they are perfect that differs, and this is their defining characteristic.
Jacques's notion of cadentia is similar to Odington's idea of concord removing the displeasure of discord, but it regards consonances that Odington thought discordant to be imperfect concords. Suiting the speculative nature of the first part of his treatise, Jacques developed the idea, presenting a theory of the agency of this phenomenon. Imperfect concords, which sound unsatisfactory on their own, are relatively weak and in a state of falling towards the perfection of perfect concords, which sound well on their own. In the light of an ensuing perfect concord, the imperfect concord also sounds well. The aesthetic is not the prospective one of the directed progression, but a retrospective one; it is not the aesthetic of the Ars nova but that of the Ars antiqua.
ENVOY
Marchetto of Padua, writing in his Lucidarium in arte musice plane at roughly the same time as Jacques, can be seen to address a similar problem to that of Jacques's cadentia but in importantly different ways and with significantly different effects. He was more modern than Jacques in that the only intervals with which he concerned himself for the purposes of polyphony were the unison, fifth, octave, major and minor thirds and major sixth. In contrast to Jacques, but similar to Franco and Odington, he classified the thirds and major sixth as types of dissonance; so the challenge he faced was to explain how acceptable music could be written with dissonances, which, following Boethius (though crediting Isidore of Seville), he defined as ‘a mixture of two pitches that strikes the ear as rough and unpleasant’.Footnote 80 Clearly, with such a definition in mind, it ought not to be possible to use them. For him, certain dissonances could be made acceptable through voice-leading propinquity to consonances, thus the thirds and major sixth were ‘endurable by the ear and mind’Footnote 81 on account of their proximity to unison, fifth or octave by voice-leading. The reason for this is that dissonant sounds, which he regards as being imperfect, require perfect, that is consonant, ones. ‘The less distant the dissonance lies from the consonance the less distant is it from its perfection and the more it is assimilated to it, and thus the more agreeable it is to the ear, as if it partook more of the nature of the consonance.’Footnote 82
What is also strongly contrasted with Jacques's notion of cadentia is the explicit connection that Marchetto establishes between the accommodation of dissonances and an idea of musical motion. Like Jacques, he cites Boethius' definition of dissonance, stating that each note of a dissonant interval ‘strives to flee’ (gliscit ire) the other, and as thirds and major sixth are for him dissonances, this introduces a dynamic element directly into the substance of polyphonic music. He goes on to say that each note ‘seeks to go to the location where it will produce a pleasant, amicable, sweet mixture, that is, a consonance’.Footnote 83 The statement is then consolidated: ‘When two notes lie in a dissonance compatible to the ear, each, seeking consonance, must be moved so that if one tends upward the other tends downward.’Footnote 84 Marchetto very clearly presents the idea here of resolution of dissonance through the movement of musical progression, specifying contrary motion and voice-leading propinquity. In this respect his thinking is fundamentally different from that of Jacques and relates more closely to the discourse of the directed progression that can be found in later writings, one that related also more closely to contemporaneous polyphonic practice.
There is a characteristically anti-modern strain in Jacques's aesthetics. For while Marchetto and other Ars nova theorists of contrapunctus own a prospective aesthetic, one by which the end may be known from the implications of the events preceding it, an aesthetic sense that has continued into modern tonal theory, Jacques reckons the effect of the music retrospectively. We do not know the following perfect concord from the preceding imperfect one; rather, we look back at the preceding imperfect concord from the vantage point of its perfect successor and discover that it too is, in a sense, perfect, by association. There is a similarity between Jacques's thinking here and Guido d'Arezzo's in his description of the effect of modal colour, in a passage that Jacques himself cites.Footnote 85 Guido does not attribute it to a prospective drive towards a tonal goal (the tonic of common-practice tonality), but to a retrospective comprehension of the colouration in the light of its end-point (the modal final):
And although any chant may come about through all the notes and intervals, the note that ends the chant, however, is the chief one; for it sounds longer and more lingeringly. And the foregoing notes, as appears only to those who have been trained, are thus adapted to it so that in an amazing way they seem to draw a certain appearance of colour from it.Footnote 86
As the Ars nova style simplified harmonic progression, centring more and more on thirds and sixths as the constituent elements of imperfect chords and resolving them predictably to fifths and octaves, theory had to take only a small step to arrive at the view that an imperfect consonance heralded the coming perfect one, and thus to create an aesthetic model based on anticipation.Footnote 87 This is the idea clearly articulated in some later treatises, and it is this that gives rise to the notion of the directed progression. Such is only possible where the goal of a progression is specifically predictable from the first element, and this is the case with the typical progressions of minor third to unison, major third to fifth and major sixth to octave that the other theorists intend. Jacques had these progressions in view; but as advocate of the older style, he had many other progressions in view too. His notion of cadentia was not useful to the musicians of his time; and when a notion of harmonic cadentia finally made its way into the mainstream of music theory, it was in a form unrelated to his conception.
APPENDIX 1
Jacques de Liège, Speculum musicae, Liber 4, Capitulum L
SM, vol. iv, pp. 122–3. In the translation the letters in square brackets refer to Example 1 (see p. 116).
Example 1 Interval progressions manifesting cadentia (SM IV. l)
APPENDIX 2
Interpreting the Interval Progressions Exemplifying Cadentia
The interval progressions listed in the fourth paragraph of the chapter on cadentia and aiming to specify the location of the phenomenon in musical practice are ambiguous, stemming first of all from the question whether there is a requirement of contrary motion in Jacques's statements about cadentia; and secondly, if there is not this requirement, how widely the progressions may then be interpreted. Example 1 will be referred to in the following discussion; it aims to represent each intervallically distinct realisation of the progressions of consonances referred to by Jacques. For reasons of economy, progressions using the melodic intervals of a tritone, diminished fifth, or any interval greater than a perfect fifth other than the octave have been excluded, though a glance through the repertory of Ars antiqua motets will reveal that composers were not in practice limited by restrictions of this sort (and they are not stipulated by theorists of the time).Footnote 88
Sarah Fuller has translated the third paragraph of the chapter as follows: ‘With respect to consonances, cadence occurs when an imperfect concord strives to attain the neighboring more perfect concord so that it falls into it, and is joined to it from below and above, that is in descending and ascending [i.e., contrary motion of the voices].’Footnote 89 On the face of it, this is a plausible translation; but it becomes questionable when set against what Jacques goes on to say. The very first instance that he gives is the progression of a tone to a unison from below or above, a progression that is most easily realised by oblique motion (though it can frequently be found in Ars antiqua motets following similar motion); it cannot be realised by contrary motion. I suggest that vel near the end of this sentence should be translated as ‘or’, indicating that any progression is either a descending or an ascending one, as is clearly the case for the tone progressing to unison. Then secundum sub et supra merely indicates that the progression proceeds by melodic movement of lower and upper voices.
This explanation works for oblique progressions, of which Jacques gives several (as well as the tone to unison there is the fourth to fifth, the major sixth to fifth, the minor seventh to octave and the ninth to octave), but apparently not for progressions by contrary motion. Further light is cast on ‘by descending or ascending [progression]’ by what Jacques says at the end of the fourth paragraph: ‘Now the twelfth … in ascending seeks the fifteenth, that is the double octave, on the part of him that has the high voice, or, in descending, returns to the octave.’ Ascending and descending are again directly referred to here. The progressions could both be oblique, in which case the ascending and descending are simply the melodic movements of the upper voice. It seems more likely, however, that progressions by contrary motion would be included, in which case each progression is regarded as a whole as ascending or descending by reference to the upper voice alone (see Example 1: n). The advantage of this latter interpretation is that it enables Jacques's opening general remarks to be viewed as consistent with all of the examples that he gives, whether the voices move by oblique or by contrary motion – or, as may also be possible, by similar motion – the progression in each case being regarded as ascending when the upper voice rises and descending when it falls.Footnote 90 To maintain that contrary motion was a necessary part of Jacques's conception of cadentia requires that his general statement is contradicted by the specific examples he gives.
In addition to the oblique progressions, Jacques cites others by contrary motion. He gives the ones that are usually associated with the directed progression: minor third to unison (minor tenth to octave), major third to fifth (major tenth to twelfth), and major sixth to octave (Example 1: b, c, fi – also k, li). He also gives some progressions between perfect concords that fourteenth-century theorists tended to ignore: the fifth to unison or octave, the octave to unison, and the twelfth to double octave or octave (Example 1: e, h, n). His list is further expanded by fourteenth-century norms because of his different conceptions of concord;Footnote 91 so he gives: the tone progressing to unison, the perfect fourth to unison or fifth, the minor seventh and ninth to fifth or octave (Example 1: a, d, g, j). While the progressions of the thirds, major sixth and minor seventh all proceed by stepwise contrary motion in both voices, and could thus be regarded as directed, those of the fourth, fifth, octave, ninth and twelfth can proceed variously in two or three different ways, and clearly could not be so regarded; in any case, two possible progressions are given for the major sixth and tenth. Yet, as no requirement for contrary motion is clearly stated, it is not clear that any of the progressions might not also be realised by oblique motion and (with one or two practical exceptions) by similar motion too (as the respective columns of Example 1 show). Jacques's preference for music of the Ars antiqua must be remembered in this connection, as the progressions that he describes may be readily exemplified in the repertory of that style.Footnote 92
