This study began as a paper given at the Thirteenth Annual Conference on Medieval & Renaissance Music, at Nottingham University, in 1985. I didn't publish it at the time because I intended to use it in a book on the analysis of medieval music, but in the event wrote something else instead (Machaut's Mass (Oxford, 1990)). But it greatly influenced my thinking in that book, and the implications of medieval theory for the way we understand compositions have a focal position in the book I am working towards now (Hearing Medieval Music).<>

I've posted this paper, in virtually its original form (I tinkered with the text two years later, when I momentarily thought about reviving it, and I've now added some comments in square brackets and colour), because the whole subject of counterpoint as a basis for composition has become live again through Margaret Bent's search for an approach to the music that lies more completely than mine or Fuller's within the texts of the theorists. Dr Bent's ideas have recently been applied in a fascinating DPhil thesis by Liz Leach (St John's, Oxford). In that context my attempt to cut a path from the theorists to prolongational analysis, although surely overtaken by events, still seems to have some value as an illustration of the variety of routes that can be taken between the same two bodies of material - the treatises and the compositions.

The paper immediately postdates the appearance of my article on Machaut's Rose, lis, and at the time felt like an attempt to ground the approach taken there in medieval precedents. That may seem like putting the horse after the cart, but the theoretical evidence that medieval musicians thought in terms of decorated structures and were comfortable with operating the procedure in reverse - ignoring decoration to reveal structure - seemed to me to provide a powerful justification for this approach. And it still does (although I now think important points in the Rose, lis article are wrong, and I deplore its tone).

DLW, April 1998]

I Contrapunctus

1 Curriculum

Fourteenth-century counterpoint treatises are practical in intent. They are addressed to students - cantores and would-be musici - whom they provide with instruction in counterpoint. They precede in the medieval music curriculum instruction in mensural notation. But they assume that the student has already studied chant and everything relevant to it: the hand, pitch names, the staff, clefs, hexachords, mutation and the modes.⁽¹⁾

2 Concords

The typical counterpoint treatise begins with the naming of intervals and their classification into concords and discords (if discords are included at this stage, which they need not be).⁽²⁾ The concords are classified, in turn, as perfect or imperfect. And this classification is followed by (or occasionally intermixed with) details of permissible progressions from each or through each concordant interval.

A simple example of this approach may be taken from Quilibet affectans, the first section of the compilation ascribed to Johannes de Muris and printed by Coussemaker under the general heading, 'Ars contrapuncti secundum Johannem de Muris'.

Primo enim sciendum est quod supra octavam non est species; sed quidquid fit supra octavam, potest dici reiteratio vel reduplicatio infra quam octavam inclusive sunt sex species: tres perfecte et tres imperfecte. Prima species perfecta scilicet unisonus...et requirit post se naturaliter semiditonum, id est tertiam minorem. Est autem semiditonus: re fa et mi sol, et e converso; potest etiam habere post se aliam speciem perfectam vel imperfectam, et hoc secundum variationem cantus.⁽³⁾

First note that anything above an octave is not an interval, but anything more than an octave can be called a reiteration or reduplication; beneath which octave are six intervals, three perfect and three imperfect. The first perfect interval is the unison .. which normally requires after it a minor third (that is, re-fa or mi-sol and vice versa), although it can have after it other perfect or imperfect intervals, according to the varying characteristics of the cantus [i.e. the tenor].

This formula is repeated for the other two perfect intervals, the fifth and the octave, and for the three imperfect intervals, minor third, major third and major sixth, the wording varying only in order to specify the progressions which normally follow from each. Thus the fifth is normally followed by a major third, the octave by a sixth, the minor third by a unison, the major third by a fifth, and the major sixth by an octave; and in each case allowance is made for exceptions due to 'the variation of the cantus'.

3 General rules of progression

In addition the student is offered a series of general rules of progression, for instance that counterpoint must begin and end with a perfect consonance, parallel perfect consonances are forbidden, but parallel imperfect consonances are permitted. And a number of treatises offer limited discussion of 'ficta musica' (or whatever the author may choose to call it).

4 Notated examples

More extended treatises, such as Volentibus introduci⁽⁴⁾ and Et nota quod septem sunt species,⁽⁵⁾ expand this central core only by adding the minor sixth to the list of imperfect consonances and by providing much fuller discussion of permitted progressions, using notated examples to supplement their text. Ex.1, for instance, transcribes the illustrations from Et nota quod septem sunt species. These show the student how to treat unisons in the tenor, stepwise ascent and descent, ascents and descents in thirds, fourths and fifths, ascending or descending motion followed by stepwise return, and chains of imperfect intervals over a stepwise tenor.

Other authors, including the Berkeley author, Antonio de Leno⁽⁶⁾ and the author of Ad sciendum componere .. cum tribus⁽⁷⁾ (interestingly, all authors who deal with contrapunctus diminutus or with multi-voice polyphony), provide charts showing which pitches are consonant with each possible tenor note, from which the beginner can read off pitches which seem to suit the progress of his added voice(s) (Ex.2). But in every case the author's aim is to lay down as clearly as possible, and as comprehensively as he believes to be necessary, rules which will enable the student to produce acceptable note-against-note counterpoint.

5 Purpose

Why was that so necessary? Because, as the counterpoint treatise attributed to Johannes de Muris explains, 'Contrapunctus .. est fundamentum discantus' - strict counterpoint is the foundation of composition.⁽⁸⁾

Et prius de contrapuncto sit hec prima conclusio: Contrapunctus non est nisi punctum contra punctum ponere vel notam contra notam ponere vel facere, et est fundamentum discantus. Et quia sicut quis non potest edificare, nisi prius faciat fundamentum, sic aliquis non potest discantare, nisi prius sciat contrapunctum.⁽⁹⁾

'Counterpoint is nothing but placing point against point or placing or making note against note, and it is the foundation of discant. And since one cannot build unless one first makes a foundation, so one cannot discant unless one first knows counterpoint'

Klaus-Jürgen Sachs lists several variants for that penultimate word 'sciat', including 'discat' - nisi prius discat contrapunctum, unless one first learns counterpoint - and 'faciat' - unless one first makes counterpoint.⁽¹⁰⁾ It is not clear which is original. Presumably whoever wrote 'faciat' understood the composer of discant to be working upwards from a contrapuntal structure. And, as we'll see, this uncertainty about whether composers actually worked from a note-against-note structure or whether they simply needed to 'know counterpoint' is reflected even in the contrasting approaches of the different treatises.

II Diminutus

The link between contrapunctus and discantus - between strict counterpoint and composition - is provided for the medieval student by tuition in contrapunctus diminutus, diminished counterpoint; as a minimum definition, counterpoint employing more than one note against each tenor note.

1 Treatises

Of the major fourteenth-century counterpoint treatises, four go on from discussing contrapunctus to introduce the student to the principles of contrapunctus diminutus, namely the Compendium de discantu mensurabili of Petrus dictus palma ociosa, dated 1336;⁽¹¹⁾ the De diminutione contrapuncti, perhaps by Johannes de Muris, from after 1340;⁽¹²⁾ the second Berkeley treatise, of around 1375;⁽¹³⁾ and, from late fourteenth-century Italy, the Regule de contrapunto of Antonio de Leno.⁽¹⁴⁾

2 Leno

The Leno treatise, although very interesting because of the systematic way in which the author leads the student through increasingly complex levels of contrapuntal elaboration, I'm leaving aside for the moment, since its examples seem to apply specifically to the Italian dialect of later fourteenth-century polyphony.

3 Berkeley

The Berkeley treatise stands apart from the other three in offering examples not of decorated contrapunctus but simply of the decorations which might be applied to contrapunctus, leaving the student to decide which of the author's uncleffed melodic fragments - or 'verbula' as he calls them - may best be inserted between the upper-voice pitches of the student's contrapuntal structure.⁽¹⁵⁾

The instructions of Petrus and Johannes de Muris, on the other hand, seem both to apply to French ars nova practices, and are unusually clear and comprehensive. What they have to say has important implications for the way we hear and analyse surviving compositions.

4 De diminutione contrapuncti

The so-called 'Ars contrapuncti secundum Johannem de Muris' consists, in its most complete form,⁽¹⁶⁾ of three sections, Quilibet affectans (the simplest and most widely-transmitted of all discussions of contrapunctus), Cum notum sit (a slightly fuller treatment of the same subject) and De diminutione contrapuncti. The text of the latter is entirely concerned with mensuration, each possible relationship between semibreve and minim being illustrated by a music example. In the table reproduced as Ex.3 Sachs has established the first proper text for these examples, and has laid them out by mensuration.⁽¹⁷⁾ It is, of course, immediately clear that mensuration is not their only subject. What the author is primarily demonstrating is how, using different rhythmic figures, a single tenor melody may be decorated by an upper voice, and further, how the upper voice itself may be decorated. Reading down the page, under each mensuration, each line is a decoration, or 'diminution' or 'prolongation' of the line above.

It is clear, therefore, that the music examples carry a subtext. While, like those of Petrus, their text implies that they illustrate only mensuration, they also show how mensuration interacts with melodic shapes to produce characteristic elaborations of simple contrapuntal progressions. Indeed, this is why they appear in a counterpoint treatise. But the text itself does not attempt to describe this interaction. There is an interesting parallel here with their early 13^th-century precursor, the so-called 'Vatican Organum Treatise'.⁽¹⁸⁾ Just as the examples in De diminutione contrapuncti and Petrus purport to illustrate mensuration but additionally offer models of melodic elaboration, so in the Vatican Organum Treatise the examples are described as note-against-note progressions but actually illustrate decorations of them. In all cases the problem is that what is described in the text is described because it can be; but what cannot be set out in simple rules has to be left for the examples 'silently' to illustrate. As Petrus says, referring to the impossibly large number of potential melodic elaborations, 'of innumerability it is not possible to have certainty'.

One other point to note from a first glance at the De diminutione contrapuncti examples is that they do not look much like surviving compositions, not just because compositions tend to use more than one rhythmic figure in their upper voices, but also because of the large number of upper-voice skips introduced in order to avoid passing dissonance. Johannes, then, is offering not so much examples of compositional practice, as models of contrapuntal motion.⁽¹⁹⁾

5 Petrus frater dictus palma ociosa

Petrus dictus palma ociosa, on the other hand, seems concerned to make clear a more continuous connection between his counterpoint teaching and compositional practice.

i Contrapunctus

His first section, on strict contrapunctus, proceeds according to the traditional format, but describes and illustrates many more possible progressions from each interval than is usual. (The illustrations are transcribed as Ex.4.) His general rules for progression also go into unusual details over exceptions. He recommends contrary motion except 'for the sake of beauty of line, or on account of a defect of pitch' (that is, one requiring correction through 'falsa musica') or in the case of necessity, and advises against parallels unless 'adorned with flowers' (that is, as we shall see, unless disguised in diminished counterpoint).⁽²⁰⁾ He concludes his contrapunctus discussion with the refreshingly undogmatic remark, that

Insuper nota, quod, licet omnes species discantus antedicte decentius stant et ordinantur in locis predictis quam in aliis quibuscumque, possunt tamen ordinari et fieri, ubicumque volueris, hoc cautius observato, quod unicuique speciei discantus debitus numerus tonorum et semitonorum observetur modo et forma superius annotatis.⁽²¹⁾

'It is permitted that all the intervals mentioned above, decently standing and ordered in the aforementioned places [i.e., those of Ex.4.1 - 4.20] and in others of whatever kind, may still be ordered and made wherever you wish, provided that .. in each interval the numbers of tones and semitones be observed..'

ii Falsa musica

After an equally broad-minded and comprehensively illustrated discussion of 'falsa musica',⁽²²⁾ (Exx.4-22 to 4-30) Petrus proceeds to treat contrapunctus diminutus or, as he calls it, 'flores musice mensurabilis', the flowers of measured music.

iii Flowers of measured music

This is his definition of this fascinating term:

Dicunt enim flores musice mensurabilis, quando plures voces seu notule, quod idem est, diversimode figurate secundum uniuscuiusque qualitatem ad unam vocem seu notulam simplicem tantum quantitatem illarum vocum continentem iusta proportione reducuntur.⁽²³⁾

'Flowers of measured music are so called when several pitches or notes, which is the same thing, notated variously according to one and the same quality, may be reduced to a single pitch or simple note containing the full quantity of those pitches in just proportion'

Johannes, Leno and the Berkeley author discuss contrapunctus diminutus, only in terms of the placing of more than one upper-voice pitch against each tenor note;⁽²⁴⁾ they present it, in other words, from the counterpoint student's point of view. But it is difficult to see Petrus' definition as other than analytical: flowers of measured music are to be recognised by the fact that they can be reduced to a single pitch having the total duration of the decorations so reduced. They are flowers because they decorate a structure which can be revealed when they are stripped away. And Petrus has already offered an analogy from nature:

Sicut videmus arborem tempore estatis adornatam et decoratam floribus, et animam sanctam hominis virtutibus necnon etiam beatissimam virginem Mariam de incarnatione filii sui unigeniti sine corruptione, sic omnis discantus de floribus musice mensurabilis adornatur et etiam decoratur.⁽²⁵⁾

'Just as we see the tree in summertime adorned and decorated with flowers .. so is all discant decorated and adorned with the flowers of measured music'

In illustration of the flowers of measured music, Petrus provides 12 examples - or 'modes of measured music adorned with flowers', as he calls them. (Ex.4-36 - 4-47) As in De diminutione contrapuncti, they are arranged by mensuration - reminding the student of the essential connection between diminished counterpoint and mensural notation.⁽²⁶⁾ But, unlike Johannes de Muris, Petrus provides copious explanation of his examples.

Quamvis autem nonnulli dicant et affirment flores scientie musicalis fore innumerabiles secundum diversos modos discantus, et de innumerabilibus non valet haberi certitudo, volentes ob hanc causam de floribus huiusmodi aliquam artem componere. Tamen ne iuvenes et alii cupientes in dicta scientia proficere aliquam artem de eadem non habentes ob hoc fiant tepidi et remissi istam scilicet addiscendo, idcirco ego circa capacitatem ingenioli mei XII modos seu manieres de discantu mensurabili floribus adornato compilavi. Qui quidem modi seu manieres, prout cuilibet competit, ordinantur sub modo perfecto et imperfecto et sub tempore perfecto et imperfecto et sub prolatione maiori et minori.⁽²⁷⁾

'Many people say and affirm that the flowers of measured music are innumerable according to different modes of discant [that is, different mensurations]; and of innumerability it is not possible to have certainty... Nevertheless, in order to make known to young people, and others eager for the said knowledge, some of the art resulting from it.. I have.., so far as my ingenuity allows, compiled 12 modes or manners of measured discant adorned with flowers; which 12 modes.. are ordered in perfect and imperfect modus, perfect and imperfect tempus, and major and minor prolation.'

After some elaboration of this point, Petrus moves on to describe their interval usage:

Quod siquidem 12 modi sive manieres ex eisdem speciebus musicalibus, a quibus simplex discantus componitur et ordinatur, et isti similiter ordinati sunt, et nihilominus iste discantus claris, ut dictum est, floribus adornatus una cum speciebus musicalibus ante dictis quandoque descendit et ascendit vicissim per dissonantias, videlicet per semitonium, tonum, diatessaron, tritonum, semitonium cum diapente, [semiditonum cum diapente] et ditonum cum diapente, de quibus dissonantiis per ordinem est videndum.⁽²⁸⁾

'.. which 12 modes or manners [are made] from the same musical intervals from which simple discant is composed and ordered, and are ordered similarly to it. But nonetheless, at the same time, is this discant clearly adorned with flowers by means of the musical intervals mentioned above [that is, consonances] and, when it descends and ascends again, through dissonances .. which dissonances are now to be examined in order.'

This is the first time Petrus has mentioned dissonance, since it has no place in note-against-note counterpoint; so it is only at this point that he defines the dissonant intervals. But dissonance is, of course, the other essential ingredient in diminished counterpoint.⁽²⁹⁾

Petrus also has some general observations on dissonances: that they generate discord - which he defines conventionally as 'a harsh collision of different sounds mixed into one, striking the ears'⁽³⁰⁾ - and that they should be used only briefly and in passing between consonances:

Nota, quod quamvis in istis dissonantiis non debeamus diutius commorari, possumus tamen ascendere et descendere per eas breviter ad omnes alias species sive differentias discantus tam perfectas et medias quam etiam imperfectas.⁽³¹⁾

'Note that although we must not remain for long in these dissonances, we may, however, briefly ascend or descend through them to all other species or intervals of discant...'

iv Petrus' Examples

Petrus now introduces the examples themselves by describing the mensural characteristics of the first (Ex.4-36):

Primus modus discantus mensurabilis floribus adornati constat ex tribus brevibus perfectis ex maiori prolatione. Unde est advertendum, quod ille tres breves perfecte possunt esse in uno solo corpore integro videlicet in longa perfecta. Et ista longa trium temporum potest dividi et diminui in longam imperfectam a tertia parte sui scilicet a brevi precedente vel subsequente vel valore ipsius sive in breves, semibreves et minimas usque ad 27 minimas vel in valorem aliquarum ipsarum brevium, semibrevium et minimarum scilicet in pausas iuxta voluntatem et possibilitatem sive distinctionem discantantis. Verum sicut discantator potest ordinare discantum in illo primo modo de eisdem notulis et pausis et dividere istam longam in partes predictas vel in aliquas earundem partium, ita et eadem ratione potest dividere et diminuere in omnibus modis sive manieriebus sequentibus omnes alias longas, breves, semibreves maioris et minoris prolationis, prout sibi melius videbitur expedire secundum uniuscuiusque notule quantitatem. Que quidem longe, breves et semibreves suis locis debitis cum exemplis, prout quilibet competit, adoptanti plenius exponentur. Quare de divisione et diminutione huiusmodi quoad presens nihil amplius est dicendum.⁽³²⁾

'The first mode..is established from three perfect breves in major prolation..[which] can be combined into a perfect long. And this long..can be divided and diminished..by its equivalent in breves, semibreves or minims up to the value of 27 minims, or by the equivalent..in rests, according to taste and to the potential or characteristic of the discanting. And indeed, just as the discantor can arrange the discant in this first mode from the same notes and rests,..so..he can divide and diminish, in all the modes or manners which follow, all other longs, breves and semibreves..in whatever way seems most appropriate according to each group of notes. Which certain longs, breves and semibreves, according to their appropriate positions, are set out more fully in the examples, selected just as one might compose.'

And one of the first things that strikes one about these examples is just how like they are to real compositions, specifically to the French motet repertory. (This perhaps explains the otherwise superfluous modes 5, 6, 11 and 12, modes where the tenor moves in breves and the upper voices largely in semibreves and minims, as in the rhythmically diminished closing sections of many isorhythmic motets.) The only slightly unusual feature of the upper voice is its frequent use of minim rests on the second minim of a semibreve beat: Mode 2 (Ex.4-37), bb.9-15, for example, or Mode 3 (Ex.4-38), bb.1-10. It is difficult to avoid the impression that these are breathing and therefore phrasing indications, such as do occur in a number of medieval pieces, but which we are perhaps not used to recognising as hints at a performance practice.

6 Towards an analytic method

The fact that these examples of Petrus are so close in style to surviving pieces, and that they provide such an unambiguously intentional link between counterpoint teaching and contrapuntal practice, leads to the central question which I'd like to try to answer: How can we distinguish between decoration and contrapuntal structure, in Petrus' colourful terms, between flowers and branches? If we can make such a distinction, using as a guide the distillations of fourteenth-century harmonic practice presented in the contrapunctus treatises, then the techniques that enable us to do so can presumably be successfully applied to surviving compositions. If so, then we have ourselves an analytic method.

III Prolongation

1 Evidence from Petrus' examples

i Basic assumptions

How can this be done? We can assume for the moment that a contrapuntal structure will consist of a series of consonant intervals, in the case of the Petrus and Johannes examples, with the tenor providing one note of each⁽³³⁾ (usually the lower).⁽³⁴⁾ It follows - for the moment - that dissonances will be decorative (though not, of course, that decoration need be dissonant).

ii Agents of prolongation

We can usefully begin, therefore, by classifying the decorative dissonances according to their function in relation to surrounding consonances. The beginning of Petrus' Mode 2 will provide a convenient example (Ex.6-1). The consonant intervals here are the initial octave, the sixth and fifth in b.2 (the sixth an imperfect consonance only), the fifth and third in b.3 (the third an imperfect consonance), and the fifth in b.4. The e' in b.1 is clearly a passing dissonance, and in b.3 the first c' and the a are clearly auxilliary or neighbour-note dissonances to the d' and b-flat respectively, while the c' passes back up to the d'. We can label these (Ex.6-2) and reduce them out (Ex.6-3). Of the remaining notes, we've already seen that not all are equal. The d' in b.2 and the b-flat in b.3 are only imperfect consonances, related melodically to the main notes by consonant skip (Ex.6-4: the slurs show their dependence). They are also decorative, therefore, and may also be reduced out (Ex.6-5). (The d', of course, also acts as a passing note; and more complicated examples of such double function will appear in a moment.) Also a consonant skip, but more consonant than the last two, is the c', in relation to the f' (Ex.6-6); so, finally, that must also be reduced out (Ex.6-7).

iii Structural levels

In revealing, by stages, the contrapuntal structure of this very simple example, we have invoked the three most important decorative functions - Passing note, Neighbour note and Consonant Skip - which will apply throughout the repertory.⁽³⁵⁾ We've also, inevitably, come face to face with the question of structural levels. In b.3 of Ex.6-1 the a is subordinate to the b-flat, and the b-flat to the d'. In bb.1-2 the e' is subordinate to the f' and the d' (it passes between them) and the d' to the f' and the c', and the c' to the f'. The pitches function hierarchically. They have different structural weights according to their function within their surroundings.

Clearly it is impossible to make an accurate reduction of a piece, to reveal its true contrapuntal structure, until the function of every note has been determined. Hence the value of the notations used in Ex.6. In Ex.7 these devices are set out more formally. The functions of neighbour and passing notes are self-evident. The consonant skip is particularly common over long tenor notes, where the upper voices - in order to fill the space interestingly - must elaborate more than one pitch consonant with the tenor. The consonant skip can also be used to avoid surface parallels, as in Ex.7-3a; though Petrus has a taste for Ex.7-3b, which creates them (although only at a surface level, as we've seen). In practice, of course, these devices are used in more complex contexts, as in Ex.7-4.

iv Further devices

Ex.8 offers an analysis of a complete example, Petrus' Mode 1. Stems are attached to the more important melodic pitches, and figures point up contrapuntal progressions. In addition, the example can illustrate a number of more subtle points.

v Structural dissonance

First, some consonant upper-voice pitches are extremely superficial in terms of structure, for instance the neighbour-note d' in b.21. Not all consonances are important. Conversely, some dissonances can be surprisingly important, the e' in b.12, for example. This is even clearer from Mode 3 (Ex.8, staves 'e' to 'g': the tenor is the same as for Mode 1). Even at the relatively deep level shown in staff 'g', the dissonant sevenths are clearly very important. Structurally important dissonances often provide essential support for a smooth melodic line, even though they are not admitted into the contrapunctus rules. Nowhere is this clearer than in the examples from De diminutione contrapuncti (Ex.3). One of the reasons why these examples are rather unlike real pieces is that, in contrast to Petrus, their author has not been prepared to admit other than superficial dissonances, and has therefore had to resort to an inordinate number of consonant skips in order to provide consonance on each beat that requires a note according to the rhythmic scheme.

vi Double function

A second device found in Petrus' examples and in surviving compositions but not in the contrapunctus treatises is the 'double function' of which we saw a very simple example a moment ago. A more interesting case occurs at the beginning of Petrus' Mode 1 (Ex.8 staff 'a', bb.1-4).

It is clear that according to contrapunctus theory the essential intervals above the f are 8-6-5, following on through parallel fifths to the fifth on g. That is the reading notated on staff 'b', showing the e' in b.3 to be the least important note in the phrase. But a more thoughtful look at that phrase must show that the first e', in b.1, although certainly a neighbour note to f', has some function as a descending passing note to the following d', and that there is similarly a sense in which the e' of b.3, a prominently placed dissonance in relation to its surroundings, can be heard as passing from the f's of b.1 to the d' of b.4. The reduction on staff 'd' attempts to do justice to this double function. The similar opening of Mode 3 (staves 'e', 'f' and 'g') makes a useful comparison. Clearly metrical position has a bearing on the different reductions which result from these melodically very similar openings.

2 Evidence from De diminutione contrapuncti

Further light is shed upon these questions by the examples from De diminutione contrapuncti.

i Influence of metrical position

On the question of metrical position, Ex.9a shows Johannes' voice 19, b.3 and the corresponding voice 13 b.3, which differ only in their mensuration. However, the different stresses clearly require different reductions. By contrast, in b.8 of these two voices (Ex.9b), the analytic reading is identical, despite the different stresses, because of the way in which the fourth, outlined by the first four notes, is confirmed by the consonant re-iteration of the a at the fifth note. The interval a-d' is too strongly stated and confirmed for metrical placement to make any difference to the reduction. In other words, metrical position only determines a reading where melodic shape shows no preference.

ii Rhythmic displacement

Related to this question is that of rhythmic displacement of structural tones in relation to their tenor notes. In b.1 of the voices in circle-dot (Ex.3, voices 1-8) voices 3-7 clearly reduce to voice 2. And so does voice 8, despite its metrical displacement of the structural tones c'-e'-c'.

By contrast, a case where the elaboration does alter the structure occurs in b.4. Voices 1 and 2 may present the progression from which the author started (8-5-6-3), but the final, apparently decorative c' added in voices 3-5 inevitably turns the previously structural b into a neighbour note. Voice 6 restores the original progression, but voices 7 and 8, particularly voice 8 with its recurring c's, confirm this new reading. Diminution of a contrapuntal structure, in other words, may produce a different structure; and we cannot, therefore, rely upon the basic contrapunctus provided by the author as a guide to the correct reduction of all his elaborations of it.

In addition, voice 8 b.4 offers an example from Johannes of double function, relating back to the beginning of Petrus' Mode 1 (Ex.10; cf. Ex.8 staff 'd'). B.4, although correctly reduced to an 8-6 progression (Ex.10 staff 'a') may be more fully represented as in staff 'b', taking account of the passing quality of the first d' and the b.

iii Compound melody

The last examples that need to be taken from the treatises, before we can move on from theory to practice, concern compound melody.

Ex.11 extracts b.6 from Johannes' voice 8; and it's easy to see there how the two approaches to the d' at the end of the extract - from the e' above and from the b below (rising via c') suggest the operation of two real parts.

Ex.12 shows the whole of voice 2 notated as compound melody, with an implied middle voice at the final cadence. Beneath the end of this example I've added the closing bars of voice 16 for the sake of comparison. Here the middle voice-part is lead back up to meet the top voice for the cadence, so that the implied two voices merge back into one at the end.

3 Examples from compositions

We have now assembled a large-enough collection of analytical tools for us to be able to deal provisionally with some real pieces.

i Machaut V33⁽³⁶⁾

The first of the three compositions at which I should like to look - and in some respects the most surprising - is Machaut's monophonic virelai no.33, Dame a vous sans retollir (Ex.13), which offers a remarkable example of a composer's use of compound melody. As the analysis shows, two implied voices descend in thirds, through a long prolongation beneath b-flat, towards the f final. In the second section of the piece, the implied upper voice is projected back up to the fifth scale degree (actually suspended from an octave projection of the f,⁽³⁷⁾ while the lower voice moves down to the second scale degree, g. That g leads back to the a at the return of the first section refrain, while the upper-voice c' (and its neighbour d') continues into the first c' of the refrain, to be lead back towards the final. Because of the structurally unresolved c' near the end of the refrain, the piece is in effect circular. (The relationship between all this and the piece's marvellous use of metrical ambiguity could fill a study on its own.)

ii Machaut R5

Closer in style to the diminution examples (not least because it shows some use of articulation rests) is Machaut's Rondeau no.5, Quant j'ay l'espart (Ex.14).

The question which we've not faced so far, and which this piece raises as clearly as any, is this: At which level of structure should we stop an analytical reduction? One historically-minded answer might be that we should be aiming at a reduction that offered one upper-voice pitch for each tenor note, on the assumption that that is the likely structure that the composer consciously elaborated. But - referring back to Johannes de Muris - did the composer need to 'make' or simply to 'know' counterpoint as a preliminary to making a piece? The examples from De diminutione contrapuncti do proceed from a made contrapunctus; but those more realistic examples of Petrus do not: his elaborations of repeating chant fragments are not identical.

Further, the evidence of surviving compositions shows clearly that all tenor notes are not equal. Some, in the context of the polyphony around them, become purely decorative. In b.3 of this piece, for instance, the sixth g-e' is simply a decorative neighbour-note motion between the more important F-chords. At a deeper level of structure, therefore, the F's are 'prolonged' through that bar into the beginning of b.4.

Nor is counterpoint just a matter of intervals. We have to take account of melodic structure too. Consequently we cannot overlook the fact that bb.1-16 are concerned essentially with an alternation of upper-voice e's and f's (a comparison of staves 'a' and 'b' should make this clear). These pitches are prolonged through sometimes quite elaborate decoration, particularly in bb.13-16 where the tenor skip from e up to g (in preparation for the cadence) is expanded by movement in both voices.

These - in relation to the brevity of the piece as a whole - are already quite extended prolongations; but even so we cannot logically avoid the further reduction shown on staff 'c'.

I don't wish to seek an Ursatz. I doubt that a universal fundamental structure exists in medieval music. But it does seem to be at this 'deep level of middleground' that the internally repeating structures which do appear, again and again in this repertory, are most clearly seen. This particular example is typical of fourteenth-century songs in presenting an interrupted structural descent up to the end of the first section - in this case starting at what is ultimately revealed as the fifth scale degree a', though others begin at the third or occasionally at the octave - and this structure is completed by a full descent to the final in the second half of the piece.

iii Machaut M18

The isorhythmic motet - as in my last example, Ex.15 - tends to work in a similar way, although of course each color-statement supports a complete descent. Here it is from (what the process of the piece establishes as) the third scale degree down to the final.

There are a number of prolongational techniques demonstrated in that analysis that I've not mentioned, in particular several cases of shifting function, where the role of an event appears to change as its larger context unfolds.

IV Conclusion

But although, during the course of this paper I've introduced only some of the more common means of prolongation which appear in this repertory, I hope that enough details have been provided to show the general tenor of my argument.

There is a continuum of approach from the counterpoint treatises to reductional analysis; the latter follows logically from the premises of the former.

The uses of this analytic approach are, I think, at least dual: 1) as a way of analysing individual compositions, of demonstrating the unique relationships which characterise them; and 2) as a way of describing more fully and more usefully than has been possible before, the general language of a place and time, the context within which individual compositions assert their relationships and their 'difference'.

That the technique can be shown to have medieval precedent, an apparent validation in medieval perceptions of musical structure, is a bonus. Its value for us is ultimately measurable only in relation to its success at making consciously perceptible, and thereby enriching, what it is in this music that we - guided by everything we know of medieval perceptions - hear.

Belfast
December 1984, May 1985

NOTES

1. So when 'Johannes de Muris' writes, at the beginning of the treatise Cum notum sit, that plainchant is the origin and foundation of measured music - he is likely to be referring not so much to the use of liturgical melodies in certain types of pieces as to the necessity of acquiring the rudiments of music, through the primary education designed to explain plainchant, before proceeding to study polyphony. (Edmond de Coussemaker, Scriptorum de Musica Medii Aevi, nova series, vol. III, Paris, Durand, 1869 (hereafter CS III), p.60.) A useful introduction to these foundation subjects may be found in Karol Berger, Musica Ficta (Cambridge, 1987).
Return to text

2. The classic study of fourteenth-century contrapuntal teaching is Klaus-Jürgen Sachs, Der Contrapunctus im 14. und 15. Jahrhundert: Untersuchungen zum Terminus, zur Lehre und zu den Quellen, Beihefte zum Archiv für Musikwissenschaft xiii, Wiesbaden, Steiner, 1974. See also Sachs, 'Der Contrapunctus-Lehre im 14. und 15. Jahrhundert', in ed. Frieder Zaminer, Geschichte der Musiktheorie, vol.5: Die mittelalterliche Lehre von der Mehrstimmigkeit (Darmstadt, Wissenschaftliche Buchgesellschaft, 1984).

3. CS III, p.59; Oliver B. Ellsworth, 'The Berkeley Manuscript (olim Phillipps 4450): a compendium of fourteenth-century music theory', Ph.D. dissertation, Berkeley, 1969, UMI 70-13044, p.77.

4. CS III, pp.23-27.

5. CS III, pp.36-41.

6. ed. Albert Seay, Antonio de Leno: Regulae de Contrapunto (Colorado Springs, Colorado College Music Press, 1977) pp.2-3.

7. CS III, pp.93-95.

8. See also Sarah Fuller, 'On sonority in fourteenth-century polyphony', Journal of Music Theory 30 (1986), p. 38. In translating 'discantus' as 'composition' I am happy to include improvised as well as notated composition. [I wrote about this later on in `Improvised and written polyphony' in C Meyer, ed., Les Polyphonies orales dans l'Histoire et dans les Traditions européennes encore vivantes, Royaumont, 1993, pp. 170-82]

9. CS III, p.60; Sachs, op.cit., p.84.

10. Sachs, pp.84-5.

11. Johannes Wolf, 'Ein Beitrag zur Diskantlehre des 14. Jahrhunderts', Sammelbände der

Internationalen Musikgesellschaft xv (1913-14), pp.504-534; Sachs, op.cit., p.104. [DL-W, article on Petrus in the revised New Grove, forthcoming. I owe my first knowledge of Petrus to the enthusiasm of David Fallows.]

12. CS III, pp.62-63; Sachs, op.cit., pp.146-147. For a summary of current views on the authenticity of the ascription, see Fuller, typescript [became 'On Sonority'] n.11. The date is that suggested by Ulrich Michels, Die Musiktraktate des Johannes de Muris, Beihefte zum Archiv für Musikwissenschaft viii (Wiesbaden, Steiner, 1970) p.15.

13. Ellsworth, op.cit., and The Berkeley Manuscript: A new critical text and translation (Lincoln (Nebraska), University of Nebraska Press, 1984) pp.110-147.

14. Seay, op.cit.

15. Followers of the Berkeley approach include the 15^th-century treatises on keyboard elaboration, D-Mbs 7755 (ed. Theodor Göllner, Formen...), D-Mbs 811 (ibid.) and Regensburg 98 (ed. Christian Meyer...).

16. On its transmission, see Sachs, op.cit., pp.181-185.

17. Sachs, op.cit., pp.146-7.

18. Facsimile, edition and English translation in Irving Godt and Benito Rivera, `The Vatican Organum Treatise - a colour reproduction, transcription and translation into English' in ed. Irving Godt and Hans Tischler, Gordon Athol Anderson - In memoriam (Henryville, 1984), vol. II, pp. 264₁-345₁₁.

19. I take this to be the function of all contrapunctus teaching. Contrapunctus diminutus may either be presented as a further elaboration of this principle, directed towards composition, as seems to be the case in De diminutione contrapuncti, or, as perhaps in Petrus' treatise, it may be presented as a distillation of compositional practice.

20. Wolf, op.cit., p.507.

21. Wolf, op.cit., p.512.

22. Wolf, op.cit., pp.513-6.

23. Wolf, op.cit., pp.516-7.

24. CS III, p.62; Seay, op.cit., pp.14, 19 & 27; Ellsworth, op.cit. (1984), pp.118-121.

25. Wolf, op.cit., p.516. Petrus' word 'discantus' refers to what the other writers label 'contrapunctus'. See also Sachs, op.cit., p.47.

26. Ex.5 shows the sources for Petrus' tenors in two Sanctus chants. Did Petrus write the treatise in order to teach how to make simple, ars nova style settings for the Mass? [See also my Revised NG article: 'The purpose of Petrus's treatise is indicated by its contents. The examples are strikingly like surviving ars nova motets in their melodic and rhythmic style. Nine of the twelve are based on Sanctus chants; they are all two-voice and perhaps improvisable. Cistercian houses did not normally favour decorated polyphony; Petrus therefore seems to have been providing a practical manual on polyphonic mass setting in a simple but up-to-date style for use outside his own order. His sophisticated understanding of ars nova music can only have been acquired from study of recent motets by Vitry and his immediate followers.']

27. Wolf, op.cit., p.517.

28. Wolf, ibid.

29. As the author of Volentibus introduci says, 'dissonances are not used in contrapunctus, but may well be used in cantus fractabilis [his term for diminished counterpoint] in small notes ..' (Et propter earum discordantiam ipsis non utimur in contrapuncto, sed bene eis utimur in cantu fractabili in minoribus notis.) (CS III, p.27.) Sachs sees the whole purpose of the theorists' diminution examples as being to provide instruction in dissonance usage. (Sachs, op.cit., p.142.)

30. Est autem discordantia diversorum sonorum sibimet permixtorum ad aures pervenientium dura collisio. (Wolf, op.cit., p.518.)

31. Wolf, ibid.

32. Wolf, op.cit., pp.518-9.

33. David Fallows has suggested (in conversation) that the counterpoints 'as if in the position of' a triplum and a motetus provided for the Kyrie fons bonitatis chant in Petrus' Ex.21 may amount to just such a contrapunctus foundation for a composition. Certainly the assumption that (at least) upper-voice elaboration is lacking helps to explain the otherwise abrupt semitones in both the added voices. Since no such setting is known, however, it is not possible at present to say that it represents a reduction of a composed piece.

34. For a medieval discussion of the reverse case, where the tenor lies above the discanting voice, see CS III Anonymous I, pp.360-1, quoted in Fuller 'On sonority', p.66, n.18.

35. For a model exposition of these devices in later contexts, see Allen Forte and Steven E. Gilbert, Introduction to Schenkerian Analysis, New York, Norton, 1982. [The CS obviously needs comment in a 14^th-century context. Were I to write this up properly I'd want to bring in a lot of the work on monophonic song (and chant) that points up the importance of third-chains in medieval melodic writing, and extend that into polyphony, perhaps via Machaut's monophonic virelais.]

36. [I got fond of this example, and it turns up again in `Not Just a Pretty Tune: structuring devices in four Machaut virelais', Sonus xii (1991) 27-30, and in `The well-formed virelai' in ed. Patrizia dalla Vecchia & Donatella Restani, Trent'anni di Ricerche musicologiche: Studi in onore di F. Alberto Gallo (Rome, Edizioni Torre d'Orfeo, 1996) 132-3.]

37. Octave projection is another important device which the theorists can be used to introduce.