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Proportion and Form in the Continental Isorhythmic Motet c. 1385-1450 Author(s): Charles Turner Source: Music Analysis, Vol. 10, No. 1/2 (Mar. - Jul., 1991), pp. 89-124Published by: WileyStable URL: http://www.jstor.org/stable/854000Accessed: 23-09-2015 21:48 UTC

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CHARLES TURNER

PROPORTION AND FORM IN THE CONTINENTAL ISORHYTHMIC MOTET c.1385-1450

The isorhythmic motet was one of the musical glories of the Ars Nova and one of the most celebrated genres passed on to composers of the early Renaissance. Several otherwise well-known composers of the late fourteenth and early fifteenth centuries, however, have left us no motets at all, isorhythmic or otherwise, and, in general, motet composition then became far less popular than before, if we are to judge from the surviving manuscripts. Only a few composers from the period c. 1385-1450 have left more than a handful of motets, and many of these are not isorhythmic. The isorhythmic motet in this period came to be considered a traditional and conservative vehicle suitable for events of state (weddings, dedications, treaties, funerals, etc.) rather than a genre of current compositional practice. It remained for Guillaume Dufay to sum up the era of the isorhythmic motet, at least as practised on the continent, with no fewer than thirteen compositions. In Dufay's isorhythmic motets, and to some degree in those of his contemporaries, it is possible to trace certain aspects of isorhythmic procedure employed in the closing decades of this genre's effective and sometimes symbolic musical life.

It is during this final period of interest in isorhythm that some of the most remarkable examples were created. Several useful commentaries on isorhythmic construction have previously been restricted to individual composers (Mixter 1969, Bent 1981, Bent 1985). In general, these studies, while admirable, have not shown relationships among isorhythmic motets by a number of composers. (For a notable exception, see Fallows 1982.) The subject is both abstract and analytical but has wider applications nonetheless, as proportion in the isorhythmic motet has an important and inherent relationship to performance practice. And because this is an area which has been neglected and occasionally misunderstood entirely, it seems to me that a clarification of the matter would be beneficial to both scholars and performers. The discussion that follows will attempt to do the following:

MUSIC ANALYSIS 10:1-2, 1991 89



CHARLES TURNER

1) review the analytical techniques of isorhythmic analysis employed in musicological literature; 2) define the types of proportion which must be taken into account in both analysis and performance; 3) introduce a new method for analysis, employing a linear representation of musical time and space; 4) examine specific problematical motets previously analysed with only limited success; 5) summarize isorhythmic structure and procedure in thirty-four representative works from continental manuscripts.

The motets considered are cited in Table 1, which also provides reference to individual manuscript sources and scholarly editions. Some of the findings offered here, however, differ from the views of previous editors.

Table 1 Compositions Considered in This Study

Composer Title No. of Sources Edition voices/ (fols.) texts*

Dufay Apostolo glorioso 5 (4) 2 BL 241'-242 Besseler 1966: 33-8 Cum tua doctrina

[Andreas Christi]

Dufay Balsamus et munda 41 BL 191'-192 Besseler 1966: 54-8

[Isti sunt agnm] BL 192'-193

Dufay Ecclesie militantis 53 Tr87 95'-96 Besseler 1966: 46-53 Sanctorum arbitrio Tr87 85'-86 Bella canunt gentes

[Gabriel] [Ecce nomen Domini]

Dufay Fulgens iubar 42 ModB 121'-123 Besseler 1966: 80-8

Puerpera pura parens [ Virgo post partum]

Dufay Magnanime gentis 32 ModB 63'-64 Besseler 1966: 76-80 Nexus amicitie musa

[Hec est vera]

* The number of texts is given in superscript. An alternative number of voices is given in parentheses if the motet is provided with a solus tenor. This does not affect the number of texts. Cantus firmus incipits, when present, are enclosed within brackets.

90 MUSIC ANALYSIS 10:1-2, 1991



PROPORTION AND FORM IN THE ISORHYTHMIC MOTET

Dufay Moribus et genere 42 ModB 74'-76 Besseler 1966: 88-98 Virgo virga virens

[ Virgo est electus]

Dufay Nuper rosarum flores 4' ModB 67'-68' Besseler 1966: 70-5 [Terribilis est locus] Tr92 21'-23

Cop

Dufay 0 sancte Sebastiane 43 BL 218'-220 Besseler 1966: 24-9 0 martyr Sebastiane Ox 31'-32 O quam mira refulsit [Gloria et honore]

Dufay O gemma lux 42 BL 263'-264 Besseler 1966: 29-33 Sacer pastor Ox 130'-131

[Beatus Nicolaus]

Dufay Rite maiorem 4(3)2 BL 196'-197 Besseler 1966: 38-45 Artibus summis

[ Ora pro nobis]

Dufay Salve flos Tusce 42 ModB 64'-65 Besseler 1966: 64-9 Vos nunc Etruscorum

[ Viri mendaces]

Dufay Supremum est 3' BL 190'-191 Besseler 1966: 59-63 BU 56-57 ModB 66'-67 Tr92 32'-33 Mu0 107'-108

Dufay Vasilissa ergo 41 BL 247'-248 Besseler 1966: 21-4 [Concupivit rex] Ox 132'-133

Tr87 57'-58

Ciconia Albane misse celitus 42 BL 271'-272 Bent 1985: 85-8 Albane doctor maxime

Ciconia Doctorum principem 42 BL 270'-271 Bent 1985: 89-93 Melodia suavissima

[ Vir mitis]

Ciconia Petrum Marcello 42 BL 248'-249 Bent 1985: 94-102 Petre antiste




CHARLES TURNER

Ciconia Ut te per omnes 42 BL 260'-261 Bent 1985: 103-7

Ingens alumnus Ox 119'-120

Brassart Ave Maria 42 BL 234'-235 Mixter 1971: 13-15 0 Maria Tr87 51

Brassart Magne decus 42 BL 253'-254 Mixter 1971: 24-6 Genus regale

Brassart 0 rex Fridrice 41 Ao 262'-264 Mixter 1971: 27-31

[In tuo adventu]

Brassart Romanorum rex 41 Ao 267'-268 Mixter 1971: 44-7

[Requiem]

Grenon Ad honorem 42 Ox 127'-128 Borren 1932: 203-4 Celorum regnum BL 216'-218

Grenon Ave virtus virtutum 42 Ox 120'-121 Borren 1932: 194-202

Prophetarum fulti [Infelix]

Grenon Nova vobis gaudia 31 BL 198'-199 Marix 1937: 233-6

Grenon Plasmatoris humani 42 BL 230'-231 Turner 1986: 197-210

Verbigine mater

Carmen Pontifici decori 41 Ox 26'-27 Reaney 1955: 54-61

Carmen Salve pater creator 42 BL 249'-250 Reaney 1955: 48-54 Felix et beata

Carmen Venite adoremus 4(3)2 BL 224'-225 Reaney 1955: 39-48

Salve sancta eterna BL 311'-312 Ox 138'-139

Cesaris A virtutis ignitio 42 Ox 116'-117 Reaney 1955: 32-9

Ergo beata nascio

[Benedicta filio]

Tapissier Eya dulcis 42 Ox 139'-140 Reaney 1955: 72-8 Vale placens

Franchois Ave virgo lux 5(4)' BL 240'-241 Ficker 1933: 19-21





Anonymous 0 Maria virgo 4' Ao 169'-170 de Van 1948: 60-5

Anonymous Argi vices 42 Ao 4'-7 Turner 1986: 211-20 Cum Pilemon

Anonymous Cuius fructus 42 Ao 186'-187 Turner 1986: 221-5 Te Maria BL 235'-236

[Ante thronum] Dij 4'

Manuscript Sigla Cited in Table 1

Ao Aosta, Biblioteca del Seminario, MS Al D19 BL Bologna, Civico Museo Bibliografico Musicale, Codex Q 15 BU Bologna, Biblioteca Universitaria, MS 2216

Cop Copenhagen, Koneglige Bibliothek, Fragment 598

Dij Dijon, Bibliotheque publique, MS 2837 ModB Modena, Biblioteca Estense, MS XX. 1.11 MuO Munich, Bayerische Staatsbibliothek, MS 3232a (olim lat. 14274) Ox Oxford, Bodleian Library, Canonici Misc. 213 Tr87 Trent, Castello del Buon Consiglio, MS 87 Tr92 Trent, Castello del Buon Consiglio, MS 92

Let us first turn to a few specific examples of isorhythmic analysis. Diagram 1 reproduces Gilbert Reaney's analysis (Reaney 1955: xviii) of Carmen's Salve pater. Reaney's method was originated by Heinrich Besseler (Besseler 1925) and is a technique for analysis that has been adopted by several scholars (see de Van 1947, GUinther 1965, among others). In this system, the symbols < and > are used to show increased or decreased temporal dimensions of the taleae. The sign + indicates that those preceding and following it are equal in length. The denominator of each fraction represents a statement of the tenor talea, with the mensural characteristics given in the following order: number of longa, level of modus (M = perfect; N = imperfect) and mensuration sign. The numerator of each fraction gives the isorhythmic configuration of the upper voices (relative to the tenor), with a separate letter assigned to each distinct repetitive pattern. In this case, four patterns are present, represented by letters A-D. It is clear from Reaney's analysis that the first tenor talea (see Diagram 1, denominator of fraction 1) corresponds to two identical taleae (2A) in the upper voices (Diagram 1, numerator of fraction 1), and that this relationship of 2 taleae in the upper voices to 1 in the tenor recurs in the third of the tenor taleae (see Diagram 1, fraction 3). Reaney's analysis also reveals a remarkable, perhaps unique, aspect of the structure, the fact that the two statements of talea B and of talea D in the upper voices are not consecutive, but interspersed (see Diagram 1, numerators of fractions 2, 4, 5 and 6).




CHARLES TURNER

Diagram 1 Carmen, Salve pater: Isorhythmic analysis by Reaney

2A B 2C D B D N + > M >Mo 7Nc

6NO 5Mo 6Mc 7Nc 5Mo 7Nc

There are several shortcomings to this analysis, however. Although it shows increased or decreased duration of the isorhythmic taleae, it does not reveal the extent to which they are augmented or diminished. That is, the system does not allow for the indication of the specific proportions to which the values are subjected. Reaney's analysis reveals the number of taleae (rhythmic repetitions) but not the number of statements of the cantus firmus itself (colores). It does not show the relationship between color and talea, and, finally, the system does not allow for providing any information concerning the mensural configurations of voices other than the tenor.'

Besseler's original idea, to make isorhythmic structure clear 'at a glance', was sound in principle, but in actual practice considerably more than a glance is required to make meaningful use of this analytical system. It is unlikely that the reader can profit from such analyses without access to transcriptions of the works (if not facsimiles of the primary sources) and considerable calculation of his own. Willi Apel (1959: 142) noted the need for a simpler and clearer method many years ago, when, in addressing an international colloquium in W6gimont, he said:

Most of you are, no doubt, familiar with the symbols that have been used by Besseler and, more recently by de Van in his edition of the isorhythmic motets by Dufay. They are rather complex and forbidding and most musicologists will probably refuse to try to decipher their meaning... I have grown up as a mathematician, and I am fully aware of the importance, and susceptible to the charm, of a good mathematical formula. However, a mathematical formula, in order to be good, has to be clear and essential, and it seems to me that Besseler and even more de Van, by trying to include too many details, have defeated their purpose.

This scholar's observations are characteristically astute, and Apel proposed a system of analysis which summarizes the structure in terms of the tenor only, indicating the relationship between talea and color and stating the number of times the tenor is given. Apel's method is exceedingly practical and easy to interpret. A statement of a tenor cantus firmus, or color, is represented by the letter c; a talea by the letter t. A tenor, if repeated, is followed by a multiplication sign and a number representing the total number of statements. Proportional treatment of the tenor statements, when present, is represented by numbers within





parentheses. When applicable, non-isorhythmic preludes and postludes are represented by the letters V and F, respectively. Apel's system of isorhythmic analysis was adopted by Samuel E. Brown, who analysed Dufay's Fulgens iubar as follows: c/2t x 3 (3:2:1) + F (Brown 1962: 277). This means that a single statement of the color (c) corresponds to two taleae (2t), that this pattern (c/2t) is repeated twice, and that the resulting proportions of the three tenor statements are 3:2:1. Thus we are given, in clear and succinct form, some, but not all, essential information about this isorhythmic motet.

Although Apel's method calculates the specific proportions involved (at least those of the tenor), it does not analyse mensural usage (not even that of the tenor), account for the activity of the upper voices (even if they too are isorhythmic) or, most importantly, show the details which led the analyst to the proportions he cites. In short, a truly interested reader of such an analysis will still have considerable research to do if he or she desires an accurate, albeit summary, understanding of the form and structure of the composition.

To cite one further example: Gustave Reese (1959: 22) describes Grenon's isorhythmic motet Ave virtus as follows:

The isorhythmic principle is applied in various ways. Thus, the three upper parts of Grenon's four-part Ave virtus-Prophetarum-Infelix begin by each singing a long talea and repeating it once. They then go through this procedure three more times, each time with new material. Meanwhile, the lowest part, or tenor, which is entrusted with the final fragment of the popular medieval sequence Laetabundus, performs its melody once during the two appearances of talea-group A; then it repeats the melody once, in halved time-values, during the two appearances of talea-group B; then it repeats the melody twice, with the values further halved, once during each appearance of talea-group C; finally, with the values still further halved, it does the same thing during each appearance of talea-group D. The tenor, therefore, has six colores (melodic repetitions); as is normal, there is no color in the upper parts. The piece ends with a brief coda. The way in which the complete tenor is to be evolved from the fragment, which is written out but once in the MS, is indicated by means of a verbal canon [my emphases].

Reese's accounting of events, in which the tenor is progressively diminished by half, would yield the tacet proportions 8:4:2:2:1:1. In fact, the specific reduction Reese describes (half the tenor's previous duration) occurs only once, not three times, for the correct proportions are 8:6(not 4):2:2:1:1. If one wishes to describe the structure as Reese does, in terms of the temporal relationship each tenor statement has to the previous, the actual levels of reduction are 1/4, 2/3 and /2 (these are fractions of the whole




CHARLES TURNER

that each tenor statement removes from the one immediately preceding). This is an odd way to proceed, however. The verbal canon does not specify the proportional relationships of the taleae to each other, but rather the proportions between the notated values of the tenor (integer valor) and those values realized in performance (that is, the degree of augmentation of each tenor statement, relative to integer valor rather than to a previous statement). The sounding proportions between the taleae result from the application of proportions to the notated values of the tenor. Thus the proportions of the isorhythmic taleae are a consequence of the verbal canon and not the content of it, as might be inferred from Reese's description. (The entire canon is transcribed and translated as No. 13 of the Appendix).

A more accurate narrative description of the motet might read:

The first tenor statement occupies eight times the duration of integer valor (canon: 'crescit in octuplo'); the second statement requires six times the duration of integer valor (canon: 'fac in sextuplo'); each of the third and fourth statements occupies twice the time of integer valor (canon: 'canitur in duplo'); and, finally, each of the last two statements of the tenor is performed as notated in the manuscript, or 'as it rests here' (canon: 'concinitur velut hic requiescit'). The application of successively smaller proportions to the integer valor causes the successive statements of the tenor to be diminished, such that the 'real-time' dimensions of the six statements of the cantus firmus are 8:6:2:2:1:1.

This description distinguishes between two varieties of proportions, those of the sounding temporal dimensions of the tenor statements, and the relationship between what is written and what sounds. This rather complex situation demonstrates a semantic problem which will be addressed shortly. Another remarkable aspect of the structure Reese does not mention: since the upper voices have two taleae for each of the first two in the tenor, they may be described in terms of the numbers 4:4:3:3:2:2:1:1, numbers of considerable importance in Pythagorean terms. Perhaps the clearest way to illustrate the structure of a motet such as this is by more graphic means:

Diagram 2 Grenon, Ave virtus: Summary of proportions

Other voices: 4 4 3 3 1 2 2 1111 Tenor: 8 6 2 2 11111

* integer valor





The preceding examples of isorhythmic analysis represent various disparate attempts to represent isorhythmic structure in logical, clear and reduced form. They also demonstrate the difficulties inherent in the subject. Certainly, it is hard to convey isorhythmic structure in a meaningful way by means of narrative description alone, since such descriptions tend by nature to be excessively dense or abstract. Thus the present study will attempt to avoid verbal descriptions of individual pieces altogether, reserving the text for matters of comparison and synthesis. This necessitates a method of analysis which is clearer and more easily grasped than the formulaic and mathematically-derived system introduced by Besseler and one that is more graphic and more comprehensive in its detail than that proposed by Apel. Before arriving at such a system, however, we must consider the very nature of proportions in isorhythmic motets.

One type of proportion, which I designate Type 1, measures the ratio between contiguous isorhythmic taleae. In addition to the proportions formed by contiguous taleae in one and the same voice, additional proportions are sometimes revealed by close analysis. These result from several instances in which two non-coinciding patterns of repetition come into play. Dufay's Ecclesie militantis and Brassart's Romanorum rex, two of the most complex motets of this type, may serve as examples. In Dufay's Ecclesie, the contratenor is divided into three taleae governed by the relationship 3:2:3, while the tenors have two taleae with the ratio 2:1 for each talea of the contratenor. Thus three 2:1 proportions are governed by the overall ratio of 3:2:3. In Diagram 3 we may readily observe the proportional relationships 3:2 and 2:1, but it is also clear that the second 2:1 proportion (labelled B) is not equivalent in duration to the first and third (A, C).

Diagram 3 Dufay, Ecclesie militantis: Summary of proportions

Contratenor: I 3 I 2 I 3 I Tenor land 2: I 2 I 1 I 2 1 2 I 1 I

A B C

We may also observe that the number 2 is used in the contratenor (3:2:3) to represent the totality of the second 2:1 proportion (B). In order to determine the true proportions that result from the superimposition of the patterns, we must convert this complex set of relationships to ratios that employ a common and compatible system of measurement. Let us again examine a line drawing of the isorhythmic proportions of this motet. We find that Diagram 4 reveals the true number of proportions present to be not two but seven:




CHARLES TURNER

Diagram 4 Dufay, Ecclesie militantis: Further summary of proportions

Contratenor: I 9 ' 1 6 9

Tenor 1 and 2: 6~-3-- 4 2---- 3 4

The resulting proportions are the following: 1:2 (3:6, 2:4), 1:3 (2:6), 2:3 (6:9), 3:4, *1:1 (6:6), *2:9, *4:9. Note that the last three ratios (marked by asterisks) result from a combination of taleae from each of the two individual patterns of repetition. They are nevertheless contiguous and structurally significant.

A similar set of conditions presents itself in Brassart's motet Romanorum rex (See Diagram 5). In this case the first tenor statement (letter A in the diagram) is divided into four taleae with the proportions 4:3:2:1, while the second statement (B) has the same pattern in retrograde, 1:2:3:4. The third statement (C) has four taleae of equal length (each is labelled z in the diagram); in terms of duration they are equivalent not only to each other but also to the final talea of tenor statement one and the second talea of tenor statement two (each is also labelled z). The isorhythmic configuration of the other voices, represented by proportions and symbols above the line, divides the temporal duration of each tenor statement (the totality of its four taleae) into equal halves, such that a = b, c = d and e = f. The diagram reveals that the two distinct isorhythmic patterns coincide only at the beginning of each tenor statement and midway through the last of them (A, B, C, f), that is, where the vertical strokes dissect the line:

Diagram 5 Brassart, Romanorum rex: Summary of proportions

a b c d e f I I I I I I I

Tenor I 4 3 1 2 1111121 31 4 1 11 1 11 only: z z z z z z

A B C

The proportions represented by the numerical series 4:3:2:1 (sesquitertia, sesquialtera and dupla) are of symbolic importance in Pythagorean terms. Here the proportions are given twice, with the second presentation in simultaneous diminution and retrograde. However, the proportions between the two tenor statements (the distance from A to B, and from B to C) is 2:1, and thus the proportions of the taleae may be more accurately represented in the first case (A) as 8:6:4:2 and in the second (B) as 1:2:3:4. In the same manner as with Dufay's Ecclesie militantis, we may assign numbers to each talea that represent its duration relative both to its own isorhythmic pattern and to that of the other,





noncoinciding, isorhythmic voices:

Diagram 6 Brassart, Romanorum rex: Further summary of proportions

10 ,

10 5 5-

4 4

Tenor I I I I only: 8 6 4 2 1 2---3---?-4 2 2 2 2

Thus we have in this remarkable motet the following sounding proportions: 1:1 (10:10, 5:5, 4:4, 2:2), 1:2 (5:10, 2:4), 2:3 (4:6), 3:4 (6:8), 4:5, *2:5, *1:10.

As in the previous example, the occasional coincidence of the two independent patterns produces contiguous proportional relationships that are not immediately apparent. In Table 2 (see pp.108-110 below), proportions thus produced by a combination of isorhythmic voices are distinguished by asterisks from those found within a single voice. Determination of proportions such as those disclosed above is essential to a complete understanding of isorhythmic motet structure. Given the numerical basis of musical consonance, the long and continuing influence of Pythagoreanism in the fifteenth century and the general medieval fascination with proportional relationships, there can be little doubt that such considerations were consciously taken into account during the compositional process. According to Sandresky (1979: 108),

A basic function of the new quattrocento humanistic spirit consisted of a transformation of Platonic-Pythagoreanism from a purely speculative concept into an active practical system. The harmonious ratios of the musical intervals became the basis for a new idea of beauty. They symbolized for Renaissance man the artistic truth of nature as represented by natural law, and they provided artists a means for solving the problem of proportion and perspective.

The architecture historian and scholar Rudolf Wittkower (1971: 110) quotes the fifteenth-century architect Leon Battista Alberti as follows:

The numbers by means of which the agreement of sounds affects our ears with delight, are the same which please our eyes and our minds. We shall therefore borrow all our rules for harmonic relations from the musicians to whom this kind of numbers is extremely well known.

In a well-known and controversial article (Warren 1973), Charles Warren speculates that the proportions of Dufay's motet Nuper rosarum were deliberately intended to reflect the specific proportions of Brunelleschi's duomo for the Florentine cathedral Santa Maria del Fiore. The motet was




CHARLES TURNER

specifically composed in 1436 for the dedication of the duomo, at which Pope Eugene himself was present. The evidence would be more convincing if both the cathedral and the motet had correspondingly unique features, which they do not. The presence of two tenors may indeed symbolize the inner and outer domes, but that of two tenors in several of Dufay's other isorhythmic motets tends to soften this argument. The proportions of the taleae themselves (6:4:2:3), while corresponding generally to those of the nave and transept, are different from those of Dufay's Magnanime only because of the augmentation of values in the first talea (12:4:2:3). The proportions of Nuper rosarum are essentially 2:3 (at two levels) and 1:2, the most common proportions employed in the motet repertory under examination. They are present in several other isorhythmic motets by Dufay. Salve flos, for example, has a structure which includes the same proportions as Nuper rosarum, but in inverted order and with each repeated: 6:6:3:3:4:4:2:2. In the preface to his Proportionale musices, Tinctoris praises the proportions of the chapel of King Ferdinando of Sicily, to whom his treatise is dedicated, writing: '... tandemque ad honorem tue proportionatissimae capellae cui similem in orbe non faciliter esse crediderim' (quoted in Coussemaker 1876: 153b-155a).3 This comment seems to support the growing view that, because of the parallels between musical and architectural form, correspondences in structure would seem to be virtually inevitable.

In contrast to the numerical relationships just examined, proportions of a second category, which I designate Type 2, pertain only to certain tenors. In these works, a verbal canon prescribes that the temporal values in performance must have a proportional relationship to the actual values written in the manuscripts, the so-called integer valor. Consequently, the duration of the tenor in performance may be thought of as having a proportional relationship to a hypothetical performance at integer valor. These proportions between written and sounding note values reflect the dimensions of the taleae indirectly, and are often unaccounted for in analysis because the ratios are those of inaudible values (those notated) to audible ones (those performed). However, these proportions were quite tangible to the composers of isorhythmic motets and to the musicians who performed them. Although in verbal form, a proportion indicated by canon has the same effect as any of the more commonly encountered arithmetical ones. The Latin proportion name in the canon must first be translated into numerical form, and then applied to the integer valor.

Representing the effect of a proportion prescribed by verbal canon in linear analysis is sometimes problematical. Diagram 7 below represents the notation of a tenor of 30 breves, and illustrates the manner by which its temporal dimensions may be modified in performance by verbal canon. If the canon calls in one instance for dupla proportion, the ratio between the speed, or tempo, of the sounding note values (C-D) and that of the written integer valor (A-B) will be 2:1. If, for the next statement, tripla





Diagram 7 Temporal relationships between integer valor, dupla proportione and tripla proportione

A B integer valor

30

15 : 10 C D E I dupla I tripla I

proportion is prescribed, a 3:1 relationship between the tempo of the sounding values (D-E) and that of the written values (A-B) will result. Such proportion names as these deal, in terms of numerical ratios, with relative speed (tempo) rather than duration. The relationship between speed and duration is an inverse one. Thus the two performances of the notated tenor (C-D, D-E) will have a 3:2 (15:10) proportion. The first statement of the tenor will occupy the same amount of time as only 15 breves of the music as it is notated; the second statement will occupy the equivalent of 10 breves. However, both statements of the tenor are complete; the numbers in Diagram 7 represent only the total duration relative to integer valor, not the part of the tenor they actually perform. Analysis must avoid the erroneous implication that the two performances of the tenor are incomplete. Since the entire tenor is performed twice, the number of breves in each section must be 30, and the temporal relationship shown in the abstract, as in Diagram 8:

Diagram 8 Temporal relationship between sounding statements of tenor at dupla and tripla proportione (integer valor not shown, since it is not performed)

3:2

30 I 30

Obviously, the proportions which the performers apply to the written values (2:1 and 3:1) must be described in different terms from those of the resulting music (3:2). Hence the distinction between proportions of Type 2 (inaudible proportions prescribed by canon) and those of Type 1 (audible dimensions of contiguous taleae) is essential. The canons to all isorhythmic motets considered in this study, with translations and commentary, are given in the Appendix below. These canons demonstrate that musicians




CHARLES TURNER

indeed distinguished between the varieties of proportion we have observed. When a composer wished to express the tenor's duration relative to a previous statement of it (proportions of Type 1), he either used a fraction name, usually 'semi' or 'tercium' (see Appendix, Nos 1, 2, 3, 9, 12), or prescribed a pattern of mensurations in which the relationship would be implicit (see Appendix, Nos 4, 6, 7, 14, 15, 16). When a composer wished to express the tenor's duration, relative to the written integer valor (proportions of Type 2), he used a proportion name preceded by 'crescit in' (see Appendix, Nos 4, 13) or, less often, 'diminuintur' (see Appendix, No. 8). The composer used proportion names only when he wished to express the speed of the note values in performance, relative to the integer valor (see Appendix, Nos 10, 11). Reese's narrative description of the proportions in Grenon's Ave virtus, cited previously, makes references to proportions of Type 1, contiguous taleae in actual sound. The verbal canon, however, clearly prescribes the degree to which the tenor is to be augmented in each statement (see Appendix, No. 13). These are proportions of Type 2, measuring actual sound against integer valor, an inaudible relationship.

Only four canons in the present repertory describe the second type of proportion (inaudible ratio of integer valor to actual sound): Grenon's Ave virtus, Brassart's O rex Fridrice and Romanorum rex, and Dufay's Magnanime. In the case of Magnanime, the canon (Appendix, No. 4) prescribes that the first tenor statement grow in duple proportion ('crescit in duplo') relative to what is notated. This ratio is one of duration. In order for the tenor to last twice as long as notated, the sounding values must be half as fast. Because of the inverse relationship between speed and duration, another way to express the same relationship would be for the canon to prescribed the speed of the tenor statement, which would have to be given in the inverted form of the proportion name, 'subdupla proportione'. In fact Brassart does precisely this in both O rex Fridrice and Romanorum rex (see Appendix, Nos 10, 11).

Figures 1-4 are linear analyses of the Grenon, Dufay, Carmen and Brassart motets we have previously discussed in some detail. Because of the non-coinciding nature of the isorhythmic taleae in the various voices, these are among the most complex isorhythmic motets from the period c. 1385-1450. We shall return to Carmen's Salve pater.

Table 2 (pp.108-110) summarizes all proportions of contiguous taleae (Type 1) in the thirty-four motets examined. The proportions are presented in order of frequency of occurrence, and in each case all other proportions of the same type are also cited for each motet. Clearly, proportions formed by manipulation of the Pythagorean series (2:1, 3:2, 4:3, 3:1) predominate, with all others represented by a single example.

The subject of systematic, often symbolic, use of contiguous proportions in extended compositions has received considerable attention in recent years. (For a fascinating hypothesis involving proportional treatment of the sections of an entire Mass ordinary, see Taruskin 1986.)





There can be no doubt that composers strove to apply to larger musical forms the same rational, that is, numerical, relationships that governed the basic working materials of their craft, musical intervals themselves. The generation of specific temporal durations is, of course, dependent upon the application of specific proportions to individual notes, most often at clearly articulated points of structural division, and usually indicated by a mensural/proportional sign. The proportions in Table 2 have been calculated with consideration given to the valuable work of Alejandro Enrique Planchart (Planchart 1981), Eunice Schroeder (Schroeder 1982) and Anna Maria Busse Berger (Busse Berger 1990), who has also provided a virtuoso analysis of the history of proportion signs (Busse Berger 1988).

(text continued on p. 110)

KEY TO FIGS 1-4

Proportions between the taleae are measured in real time and are represented in the diagrams by the relative dimensions of the sections. ISORHYTHMIC AND NON-ISORHYTHMIC SECTIONS: Solid horizontal lines define the isorhythmic sections, dotted lines the non-isorhythmic ones. Within the isorhythmic section (solid line) the dimensions of the taleae are bounded by vertical lines, which may either dissect the line (in which case the taleae apply to all isorhythmic voices) or form tangents with it (indicating that the taleae thus bounded are restricted to certain voices). COLORES, TALEAE: Statements of a repeated melody are represented by capital letters, with successive letters assigned to each repetition of the pitch series. Statements of a talea (recurring pattern of durational values), when occurring in tandem with a color (repeated melody), are enumerated with Roman numerals. Unlike the cantus-firmus bearing tenors, the upper voices of motets in this repertory do not employ melodic repetition and normally employ a series of distinctive talea patterns. Talea repetitions in these voices are therefore not enumerated, but indicated by lower case letters, with a different letter assigned to each distinctive pattern. The lengths of all taleae, as measured in breves (p = perfect; i = imperfect), are given in combination with the mensuration signs employed in their respective voices. Mensural usage of each voice is analysed, whether it is isorhythmic or not. Signs in brackets are absent from one or more of the manuscript sources.




0

CI

O

??

to

d V)

Fig. 1 Isorhythmic analysis

Grenon, Ave virtus virtutum (4 vv) Prophetarum fulti [Infelix]

Isorhythmic voices: all

C1/C2 C 36i 36i C. 18i 18i O 12p 12p 212p12p 8p

Ct C 36i 36i C 18i 18i O 12p 12p C 6i 6i 4i

T [C] 6i 6i 6i 6i 6i 6i I 4i

C1/C2/Ct: a a b b c c d d

4 4 3 3 2 2 1 1

8 6 2 2 1 1 Tenor only: AI B.II C.III D.IV E.V EVI

C)

H tB tM



C3 0

z

0

v,

?? 7" to

o


Dufay, Ecclesie militantes (5 vv) Sanctorum arbitrio Bella canunt gentes [Gabriel] [Ecce nomen dominm]

Isorhythmic voices: Tenors 1 and 2, Contratenor

Tr/Mo [] 48p 72p 72i 72i 24p 6p

Ct 72p 03 72i 72p 16p

Tl /T2 C 24i 24i C 24i d24i 0 24p 24p 6p

Ct: A.I B.II C.III 9 6 9

6 9

3 4 2 6 3 T1 / T2: A.I B.II C.III D.IV E.V F.VI

N.B. Symbols and dimensions above the line apply to the contratenor, those below to Tenor 1 and Tenor 2.

0 0

0 z z

tI1

0

C)

0

tCj H

H

z:

?:

tj

?r



oS

5:

o ?? 7"

to


Carmen, Salve pater creator (4 vv) Felix et beata


Tr / Mo 0 6p 6p O 15p C 9i 9i C 14i O 15p C 14i

T/Ct 0 12p O 15p [c] 18i [C]14i [O] 15p [c]14i

Tr / Mo: a a b c c d b d 3 1 3 5 3 1 3 3 5 3

6 5 6 3 5 3 T / Ct: A.I II B.III IV C.V VI

C)

H ta tm



d n

> ?

o9 oo

to?

o


Brassart, Romanorum rex (4 vv) [Requiem]


[O] 30p 30p D 30i 30i O 12p 12p 7p

[O] 30p 30p D 30i 30i (

36i 36i 21i

TO 6p 6p 6p 16p 6pl 6p 6p 6p 6p 6p I 6p 6p 07p

90p 90p 60i 60i 03 36p 36p 21p

a a b b c c 10 I 10 5 1 5 4 4

8 1 6 1 4 12 112 31 4 21 2 122 A.I II III IV BV VI VII VIII C IX X XI XII

N.B. Symbols and dimensions below the line apply to the tenor only; those above it, to all other voices.

0 0

0 z

z

H

0

H

H



CHARLES TURNER

Table 2 Proportions of Type 1 (Contiguous Taleae)

Proportion Composer/Motet Other proportions present

1:1 Dufay Apostolo 1:3

Dufay Balsamus 1:2

*Dufay Ecclesie 1:2, 1:3, 2:3, 3:4, *2:9, *4:9

Dufay Fulgens 2:3, 3:4

Dufay Moribus 1:2, 2:3

Dufay 0 sancte Sebastiane 1:3

Dufay O gemma 2:3

Dufay Rite maiorem 10:11

Dufay Salve 1:2, 3:4

Dufay Supremum 2:3

Dufay Vasilissa Ciconia Albane Ciconia Doctorum 2:3 Ciconia Petrum 1:3, *1:4, *3:4 Ciconia Ut te Brassart Ave Maria Brassart Magne decus 1:3 Brassart Romanorum rex 1:2, 2:3, 3:4, *2:5, 4:5, *1:10 Grenon Ad honorem 1:2 Grenon Ave virtus 1:2, 1:3, 2:3, 3:4, *3:8 Grenon Nova vobis Grenon Plasmatoris 1:3 Carmen Pontifici Carmen Salve pater 1:2, 3:5, 5:6 Carmen Venite 2:3 Cesaris A virtutis

Tapissier Eya dulcis Franchois Ave virgo 2:3

Anonymous O Maria

Anonymous Argi vices

Anonymous Cuius fructus

2:3 Dufay Ecclesie *1:1, 1:2, 1:3, 3:4, *2:9, *4:9 Dufay Fulgens 1:1, 3:4

Dufay Magnanime 1:2, 1:3


Dufay Nuper rosarum 1:2

Dufay O gemma 1:1

Dufay Supremum 1:1

* indicates proportions formed by a combination of isorhythmic voices





Ciconia Doctorum 1:1 Brassart 0 rex 1:2 Brassart Romanorum 1:1, 1:2, 3:4, *2:5, 4:5, *1:10 Grenon Ave virtus 1:1, 1:2, 1:3, 3:4, *3:8 Carmen Venite 1:1 Franchois Ave virgo 1:1

1:2 Dufay Balsamus 1:1

Dufay Ecclesie *1:1, 1:3, 2:3, 3:4, *2:9, *4:9



Dufay Nuper rosarum 2:3

Dufay Salve flos 1:1, 3:4 Brassart 0 rex 2:3 Brassart Romanorum 1:1, 2:3, 3:4, *2:5, 4:5, *1:10 Grenon Ad honorem 1:1 Grenon Ave virtus 1:1, 1:3, 2:3, 3:4, *3:8 Carmen Salve pater 1:1, 3:5, 5:6

1:3 Dufay Apostolo 1:1

Dufay Ecclesie *1:1, 1:2, 2:3, 3:4, *2:9, *4:9


Dufay 0 sancte Sebastiane 1:1 Ciconia Petrum 1:1, *1:4, *3:4 Brassart Magne decus 1:1 Grenon Ave virtus 1:1, 1:2, 2:3, 3:4, *3:8 Grenon Plasmatoris 1:1

3:4 Dufay Ecclesie *1:1, 1:2, 1:3, 2:3, *2:9, *4:9 Dufay Fulgens 1:1, 2:3

Dufay Salve flos 1:1, 1:2 *Ciconia Petrun 1:1, 1:3, *1:4 Brassart Romanorum 1:1, 1:2, 2:3, *2:5, 4:5, *1:10 Grenon Ave virtus 1:1, 1:2, 1:3, 2:3,*3:8

*1:4 Ciconia Petrum 1:1, 1:3, *3:4

*1:10 Brassart Romanorum 1:1, 1:2, 2:3, 3:4, *2:5, 4:5

*2:5 Brassart Romanorum 1:1, 1:2, 2:3, 3:4, 4:5, *1:10

*2:9 Dufay Ecclesie *1:1, 1:2, 1:3, 2:3, 3:4, *4:9

3:5 Carmen Salve pater 1:1, 1:2, 5:6




CHARLES TURNER

*3:8 Grenon Ave virtus 1:1, 1:2, 1:3, 2:3, 3:4

4:5 Brassart Romanorum 1:1, 1:2, 2:3, *2:5, 3:4, *1:10

*4:9 Dufay Ecclesie "1:1, 1:2, 1:3, 2:3, 3:4, *2:9

5:6 Carmen Salve pater 1:1, 1:2, 3:5

10:11 Dufay Rite maiorem 1:1

We have investigated the formal aspects of isorhythmic motets from the point of view of proportional usage. There are, of course, important considerations other than these. We may summarize isorhythmic composition in another way, in terms of the procedures employed by composers. A study of proportions discloses a considerable variety, but a comparable wealth of imagination comes to light when one approaches the same repertory from a different point of view. Tables 3-5 include proportional distinctions, but essentially summarize construction of the motets with regard to 1) the relationship of tenor color to talea, and the number of tenor statements, 2) the number and relationship of the isorhythmic voices, and 3) the presence of non-isorhythmic sections. Table 4 shows that the most often-followed procedure was clearly to establish a reiterated pattern of durational values that recurs at least once within the cantus firmus itself. Twenty-seven motets adhere to this pattern; only seven do not. Considerable variety was employed in the number of tenor statements, ranging from one to six. Twenty-three of the thirty-four motets vary the durations of the taleae proportionally; only eleven do not.

Table 3 Classification of Tenors

TENOR DIVIDED INTO TALEAE (cantus firmus containing repetition of temporal values):

A Two statements of tenor with proportional diminution (8 motets)

Dufay Apostolo 3:3:1:1

Dufay O sancte 3:3:3:1:1:1

Dufay Rite maiorem 11:11:10:10

Dufay Supremum est 3:3:3:2:2:2 Brassart Magne decus 3:3:3:1:1:1 Grenon Ad honorem 2:2:1:1 Carmen Venite 3:3:3:3:2:2 (N.B. Proportions of c.f.: 9:7) Franchois Ave virgo 3:3:2:2





B One tenor statement only (7 motets)

Dufay Vasilissa 1:1 Ciconia Albane 1:1 Ciconia Ut te 1:1 Carmen Pontifici 1:1:1:1:1 Cesaris A virtutis 1:1:1:1 Tapissier Eya dulcis 1:1:1 Anonymous Argi vices 1:1:1:1

C One statement of the entire tenor and repetition of the first half (2 motets)

Dufay O gemma 3:3:2

Anonymous Cuiusfructus 1:1:1

D Two statements of the tenor (2 motets)

Grenon Nova vobis 1:1:1:1 Anonymous O Maria 1:1:1:1

E Three statements of tenor with proportional diminution (2 motets)

Dufay Fulgens 6:6:4:4:3:3 Dufay Moribus 3:3:2:2:1:1

F Three statements of the tenor; each talea is subjected to a different proportion (2 motets)

Brassart Romanorum 8:6:4:2:1:2:3:4:2:2:2:2 Carmen Salve pater 6:5:6:3:5:3 (N.B. Proportions of c.f.: 11:9:8)

G One tenor statement only; two different, successive, patterns of repetition (1 motet)

Grenon Plasmatoris 3:3:1:1

H One statement of the entire tenor; the melody of the first talea is given twice in succession (1 motet)

Brassart Ave Maria 1:1:1:1:1:1

I Two statements of tenor with proportional diminution; alternating taleae give the first half of color in retrograde (1 motet)

Dufay Balsamus 2:2:1:1




CHARLES TURNER

J Four statements of tenor with alternating diminution and augmentation (1 motet)

Dufay Salve flos 6:6:3:3:4:4:2:2

TENOR NOT DIVIDED INTO TALEAE (cantus firmus containing no repetition of temporal values):

K Tenor has two different colores: each color is repeated in diminution (1 motet)

Ciconia Petrum 3:1:3:1

L Three statements of tenor with proportional diminution (1 motet)

Ciconia Doctorum 3:3:2

M Four statements of tenor with proportional diminution and augmentation (2 motets)

Dufay Magnanime 12:4:2:3

Dufay Nuper rosarum 6:4:2:3

N Four statements of tenor with proportional diminution (1 motet)

Brassart O rex 6:4:2:(1) (The last ratio is inexact because the canon specifies that rests are to be omitted)

O Six statements of tenor with proportional diminution (1 motet)

Grenon Ave virtus 8:6:2:2:1:1

P Six statements of tenor with alternating proportional diminution and augmentation (1 motet)

Dufay Ecclesie 6:3:4:2:6:3

Table 4 Classification of Isorhythmic Voices

A All voices isorhythmic; taleae coinciding (22 motets)

Dufay Apostolo 3:3:1:1

Dufay Balsamus 2:2:1:1

Dufay Fulgens 6:6:4:4:3:3

Dufay Moribus 3:3:2:2:1:1

Dufay O sancte Sebastiane 3:3:3:1:1:1

Dufay O gemma 3:3:2





Dufay Rite maiorem 11:11:10:10

Dufay Salve flos 6:6:3:3:4:4:2:2

Dufay Vasilissa 1:1 Ciconia Albane 1:1 Ciconia Ut te per omnes 1:1 Grenon Ad honorem 2:2:1:1 Grenon Nova vobis 1:1:1:1 Grenon Plasmatoris 3:3:1:1 Carmen Pontifici 1:1:1:1:1 Carmen Venite adoremus 3:3:3:3:2:2 (N.B. Proportions of c.f.: 9:7) Cesaris A virtutis 1:1:1:1

Tapissier Eya dulcis 1:1:1 Franchois Ave virgo 3:3:2:2

Anonymous O Maria 1:1:1:1

Anonymous Argi vices 1:1:1:1

Anonymous Cuiusfructus 1:1:1

B Tenor only isorhythmic voice (5 motets)

Dufay Magnanime 12:4:2:3

Dufay Supremum est 3:3:3:2:2:2 Brassart Ave Maria 1:1:1:1:1:1 Brassart Magne decus 3:3:3:1:1:1 Brassart O rex 6:4:2:(1) (last ratio inexact)

C All voices isorhythmic, taleae non-coinciding (4 motets)

Ciconia Petrum Marcello 4 : 4 3:1:3:1

Brassart Romnanorum rex 10:10 :5:5 : 4 : 4 8:6:4:2:1:2:3:4:2:2:2:2

Grenon Ave virtus 4:4:3:3:2:2:1:1 8 : 6 :2:2:1:1

Carmen Salve pater 3:3:5:3:3:3:5:3 6 :5: 6 :3:5:3 (N.B. Proportions of c.f.: 11:9:8)

D Two isorhythmic voices; taleae coinciding (2 motets)

Dufay Nuper rosarum 6:4:2:3 Ciconia Doctorum 3:3:2




CHARLES TURNER

E Three (but not all) voices isorhythmic; taleae non-coinciding (1 motet)

Dufay Ecclesie 9: 6: 9 6:3:4:2:6:3

Table 5 Classification of Non-isorhythmic Sections (this category applies to 18 of 34 motets)

A Non-isorhythmic prelude - bicinium (9 motets)

Dufay Ecclesie Dufay Magnanime Dufay O sancte Sebastiane

Dufay O gemma lux

Dufay Vasilissa Brassart Magne decus Brassart O rex Fridrice Grenon Ad honorem

Anonymous Argi vices

B Non-isorhythmic postlude. A brief extension in the form of a terminal cadence or additional

longa (notes ultra mensuram) is not considered a postlude (5 motets)

Dufay Magnanime Dufay Supremum est Brassart O rex Fridrice Brassart Romanorum rex

Anonymous O Maria virgo

C Non-isorhythmic prelude - All voices (3 motets)

Dufay Apostolo Dufay Supremum est

Anonymous O Maria virgo

D Non-isorhythmic prelude - tricinium (1 motet)

Franchois Ave virgo

Composers would sometimes take advantage of the non-restrictive nature of the free prelude, or introitus, and compose this music either in a generally homophonic or in an imitative style. Occasionally they would employ strict canon, as in Dufay's O sancte Sebastiane, 0 gemma and Vasilissa, and in the anonymous Argi vices. In the introitus to Supremum est,





Dufay employs two notated homophonic voices expanded to three by fauxbourdon. In Apostolo, Dufay has an imitative four-voice introitus consisting of a canon in the upper voices and a second, melodically distinct, canon shared by the two contratenors. In all, seven of Dufay's thirteen isorhythmic motets are preceded by non-isorhythmic sections. The introitus to Johannes Franchois's Ave virgo is unique both in its exceptional length, nearly that of the motet itself, and in its style, which resembles the Italian caccia: two upper voices in strict canon with a more slowly-moving tenor, designated 'Trompetta introitus'. Although isorhythmic composition may seem by nature essentially incompatible with imitation, this is not necessarily so. In Carmen's Pontifici, there is no introitus outside the isorhythmic scheme but rather canon in the upper voices throughout the isorhythmic taleae themselves. In twenty-six of the thirty-four motets examined, all voices employ isorhythm. In twenty-two of these twenty-six, the taleae of all voices coincide; in other words, the motets may be said to be 'isoperiodic' (see Table 4).

Johannes Carmen's Salve pater employs a formal construction which is truly remarkable and merits an exposition of its own. The linear analysis of this motet (Fig. 3) reveals a structure that is unique in several ways. The problem the composer put before himself was, in part, to present the tenor cantus firmus three times, in such a way as to correspond to eight taleae of the upper voices. Carmen's structural organization is highly imaginative. A verbal canon instructs the performers to give the tenor three times 'de prolatione sui moteti', that is, to apply the mensurations of the upper voices to the corresponding (coinciding) taleae of the tenor. This causes two of the signs to recur. The resulting sequence of mensurations has a logic of its own, for it gives (in this order) both varieties of tempus perfectum, tempus imperfectum and prolatio minor: 0 0 c 0 c

The three tenor statements are divided into six taleae, the totality of which correspond to eight taleae, or four pairs, in the upper voices. Six of the eight taleae in the upper voices coincide with all those of the tenor, but two do not. Figure 3 shows that the taleae of the upper voices have the proportions 3:3:5:3:3:3:5:3; those of the tenor have 6:5:6:3:5:3. To summarize the elements of form:

1) the upper voices present four pairs of taleae, but with the normal succession altered (aabccdbd); 2) the tenor melody is stated three times (ABC); 3) each statement of the tenor consists of two unequal taleae; 4) each talea of the tenor has a change of mensuration, corresponding to those of the upper voices; 5) the tenor melody as a whole is subjected to progressive diminution.

With regard to this last observation, the progressive diminution of the tenor employs an interesting combination of proportional relationships, not




CHARLES TURNER

encountered previously: superbipartiente novem (11:9) and sesquioctava (9:8).

melody 11 : 9 : 8 Tenor <

rhythm 6 : 5 : 6 : 3 : 5 : 3

We are not recognizing here proportions of Type 1 (these have been fully accounted for in various tables) but those generated by melodic repetition (color). The unique construction of Salve pater brings us to a final aspect of isorhythmic design not previously investigated. This study has described proportional relationships of taleae (proportions of Type 1), but not those of colores, or repetitions of tenors in their entirety. The reason for this approach is simply that these dimensions are normally dependent upon and reflected by those of the taleae. Generally speaking, the duration of the tenor in an isorhythmic motet will either be the same as the talea (if the melody contains no internal repetitions of rhythm but is itself repeated) or be a multiple of the taleae (if there is repetition of rhythm within the melody).

The unusual proportions of Salve pater confound analysis, for 1) each statement of the tenor consists of unequal taleae, 2) the tenor statements have only an additive relationship to their taleae, and 3) not one of the three tenor statements is an easily calculable fraction or multiple of another statement. These phenomenona, difficult to reduce to analysis, add an imaginative dimension to isorhythmic composition. Compared to Carmen's Salve pater, the same composer's Venite adoremus is relatively simple in its design: 6 taleae divided according to mensural usage into 3 pairs, but according to melodic repetition into 2 colores of 3 taleae each. The resulting taleae proportions are 3:3:3:3:2:2, but those of the cantus firmus statements are 9:7 (3+3+3 : 3+2+2). These two motets by Carmen are the only ones in the present repertory in which consideration of the color proportions adds a dimension otherwise unaccounted for in analysis.

It is particularly appropriate that this study should conclude with Salve pater, for it began with Gilbert Reaney's analysis of this same motet. It is not surprising that such compositions are imperfectly understood today and that performances are often disappointing. It is perhaps inevitable that methods of isorhythmic analysis cannot be adequate, much less ideal, for conveying with ease all the varied possibilities, proportional or otherwise. Indeed, it would be disappointing if any method of analysis could reduce the essence of this music to mere figures or symbols. Nevertheless, an appreciation of the ways in which proportion affects form in both audible and inaudible ways can result in an aesthetically gratifying experience which transcends the resolution of numerical ratios.





NOTES

1. There are a few minor errors in Reaney's analysis. Careful calculation reveals the actual proportions of the taleae to be 6:5:6:3:5:3. Thus the first three tenor taleae are not equal, as the + signs suggest. Further, since the fourth and sixth fractions are identical, they must both be either greater or smaller than fraction five. Reading from fraction four on, Reaney's analysis states first that B is smaller (shorter in duration) than D and then that B is greater (longer in duration) than D when it (D) is repeated, an impossible situation in the context of this motet. The error is easily corrected by inverting the second of the three > signs. The following modification of Reaney's analysis gives the correct symbols:

2A B 2C D B D

6No 5MO 6Mc 7Nc 5Mo 7Nc

For a more detailed, yet more easily interpreted, analysis of Salve pater, see Fig. 3.

2. The proportions 3:2:1 are incorrect, however, for the canon directs the performer to reduce the tenor first by one third and then by half of the original values (see the complete text of the verbal canon in the Appendix, No. 3). It is the first tenor statement to which both of these reductions should apply. Brown's analysis applies the proportions cumulatively, reducing the second tenor statement to produce the third. The actual proportions of the three tenor statements are not 3:2:1, but 3:2:11/2 or 6:4:3, and the analysis should be modified to read as follows: c/2t x 3 (6:4:3) + F. In Dufay's Ecclesie militantis, there are two isorhythmic patterns, one for the contratenor and one that governs both tenors. Brown (1962: 277) analyses the piece as follows:

Ct c/t x 3 ( 1 : 1 : 1 ) V+ +F

T,T2 c/t x 6 (2:1:2:1:2:1)

The isorhythmic patterns are not independent, however, but closely related. While Brown shows the three contratenor taleae to be equal in length, the proportions are not 1:1:1, but 3:2:3. This relationship is inherent in the three statements, which have the mensuration signs 0 (3 1), and these proportions are confirmed by the configuration of the other voices. The proportions of the tenors may be represented as 2:1 in each of the three instances only if they are shown to be governed by the 3:2:3 relationship of the contratenor. Thus the dimensions of the contratenor greatly affect those of the tenors. A more accurate summary of the structure, according to Brown's methodology, would be as follows:




CHARLES TURNER

Ct c/tx3 (3:2:3) V+ +F

T,T2 c/t x 6 (2:1:2:1:2:1) V V V 3:2:3

For a more detailed, yet more easily interpreted, analysis of Ecclesie, see Fig. 2. In fairness, the errors cited in the analyses of both Reaney and Brown are due to individual miscalculation and do not reflect inherent weaknesses in the systems themselves. It must be pointed out, however, that similar errors in the literature of isorhythm are quite common.

3. '...in honor of your most well-proportioned chapel, whose equal I cannot easily believe to exist anywhere in the world' (Strunk 1965: 5).

APPENDIX

Verbal Canons in the Isorhythmic Motets: Transcription, Translation, Commentary

No. 1 Dufay, Apostolo

Et dicitur bis primo de modo perfecto et tempore imperfecto secundo per tercium demptis primis pausis et nota sequente

The tenor is sung twice, first in perfect modus and imperfect tempus, the second time reduced to one third, omitting the rests of the first time and the note following them.

No. 2 Dufay, Balsamus

Canon, BL 191': Tenor dicitur de modo perfecto et tempore imperfecto simili modo retrogradendo

The tenor is sung in perfect mode and imperfect tempus, then in the same manner going backwards.

Canon, BL 192': Tenor dicitur per semi et eodem modo retrograditur accipiendo pro fine nigras

The tenor is sung reduced by half and in the same manner going backwards, accepting the black [notes] before the end. (Two notes in hollow notation provide a kind of 'second ending'.)





No. 3 Dufay, Fulgens

Tenores isti ter dicuntur primo de modo et tempore perfectis minoris secundo tercium tercio cise per semi de primo The tenors are sung three times: first, in perfect modus and tempus with minor prolation; second, reduced by one third; third, in half the time of the first.

Note: The use of tercium seems anomalous here, where it means reduced by one third (or having two thirds of its previous value) rather than reduced to one third, as in Dufay, Apostolo and Brassart, Magne decus.

No. 4 Dufay, Magnanime

Quater dicitur primo crescit in duplo aliis tribus vicibus ut signa notant primas pausas tamen obtinendo

The tenor is sung four times, first increased by duple proportion, the other three times as the [mensuration] signs note, maintaining, however, the rests of the first.

Note: There seems to be an error here. The rests of the first time pertain to the non-isorhythmic introduction by the two upper voices and do not apply to the isorhythmic talea proper. The canon should instruct the singer to omit the rests on subsequent statements of the tenor.

No. 5 Dufay, Moribus

Ut signant > signa

pauseque bases bene

As the signs indicate, [heed] well the note and rest [?].

Note: The meaning of this comment (in the hand of the text scribe) is unclear. It was written beneath the tenor midway through the composition, perhaps as an aid to performance.

No. 6 Dufay, O gemma

Canon qui dicitur sic primo de modo perfecto et tempore imperfecto maioris prolacionis secundo tantummodo primus color resumitur de minori imperfecto, servando modum primis pausis demptis

Canon which is thus: first in perfect mode, imperfect tempus, major prolation; second, in the same manner; the melody is resumed in imperfect tempus, minor prolation, omitting the rests of the first time.




CHARLES TURNER

No. 7 Ciconia, Doctorum principem

Tenor vie [vis?] artis et dicitur imperfecto majori perfecto minori semper ultima semibrevis alteratur imperfecto minori

The tenor (... meaning unclear) and is sung [first] in imperfect tempus, major prolation; [second] in perfect tempus, minor prolation - the last semibreve is always altered; [third] in imperfect tempus, minor prolation.

No. 8 Ciconia, Petrum Marcello

Canon tenores dicuntur sic primo usque ad secundam talliam ut jacent secundo diminuintur resumendo et sic successive alie talie procedant

Canon: The tenors are sung thus: first all the way to the second talea as written. Second, upon resuming, they are diminished, and the other successive taleae continue like this.

No. 9 Brassart, Magne decus

Iste dicitur bis primo de modo et tempore perfectis secundo per tercium sed prime pause non dicuntur

The tenor is sung twice, first in perfect mode and perfect tempus; the second time reduced to a third (of the first time), but the rests of the first are not observed.

No. 10 Brassart, O rex Fridrice

Ca Si me cupis promere sic dic quater sine fine in primo cupio me non subsestupla que secundo subquadrupla iunge sed si vis nun ter teno havere sic studeas canere me subdupla sit sine vere pausas omitto ris quarto ut jaceor promor ego

If you want to reveal me, sing me four times thus without stopping: First, I want you to sing me at inverted sextuple proportion (1:6), and it joins a second at inverted quadruple proportion (1:4). But if you want to have a third time, take care to sing me thus, at inverted duple proportion (1:2). The fourth time I omit the rests as I reveal myself as I lie.

Note: There are problems in the wording of the canon, which may be corrupt in places.

No. 11 Brassart Romanorum rex

Tenor iste qui dicitur: primo modo, prima tallea in subdupla superbipartiente





proportione; secunda tallea in subdupla proportione; tertia in subsesquitertia; quarto vero in sesquialtera. Secundo modo, prima tallea dicitur in tripla proportione; secunda in emiola; tercia sicut iacet; quarta in subdiatessaron. Tertio modo dicitur in dyapenthe. Et es notandum quod secunda et tercia vice capitur in dyapenthe. [Punctuation added for clarity]

This is the tenor, which is sung as follows: The first time - first talea in subdupla superbipartiente proportion (3:8 or 5:12; here, 3:8); the second talea in inverted duple proportion (1:2 or, here, 3:6); the third talea in inverted sesquitertia proportion (3:4); the fourth talea in sesquialtera (3:2). The second time - first talea in triple proportion (3:1); the second talea in hemiola (3:2); the third talea in the values that are written (1:1 or, here, 3:3); the fourth talea in inverted diatessaron (3:4). The third time it is sung at the diapenthe (3:2). And it is notated such that the second and third times it is transposed down a fifth.

Note: This is a remarkable canon. The use of talea (tallea) is normal. Emiola, sesquialtera and dyapenthe are all used for 3:2 proportion. The last of these, dyapenthe, is used in two senses, for a 3:2 temporal proportion and for the interval of a perfect fifth. The second and third tenor statements are transposed a fifth lower. Thus we have a tenor whose melody is sung at actual pitch on only one of three statements, and whose notated values are at integer valor in only one of twelve taleae. The use of subdiatessaron is somewhat unusual, but subsesquitertia had already been used, and the author avoided duplicating a term whenever possible. The number three occurring in all proportions represents integer valor. Modus here has its usual meaning of method or manner, and has no specifically musical connotation. Interpretation of subdupla superbipartiente proportione as either 3:8 or 5:12 is accounted for by Tinctoris (Terminorum Musicae Diffinitorium, s.v. dupla superbipartiens) but 3:8 is implicit in the present context.

No. 12 Grenon, Ad honorem

Canon dicitur bis primo ut jacet secundo cise per [semi]

The tenor is sung twice, first as written, second cut in half.

Note: Charles van den Borren (Borren 1932: xl) states that 'ut jacet is inaccurate, since the context requires the augmentation of the values'. To the contrary, the canon is correct and the context requires not augmentation of the tenor but diminution of the upper voices.




CHARLES TURNER

No. 13 Grenon, Ave virtus virtutum

Canon: Crescit in oc Semel hic tenor incipi

> tuplo > endo Fac in sex Crescat semel in decanen Postea bis ca In duplo cum bene cr

> nitur > escit Sed bis conci Tandem velut hic requi

As it begins, this tenor first grows in octuple proportion. Then sing it in sextuple proportion; let it grow at the same time it is diminished. Afterwards it is sung twice, although it grows at duple proportion. But finally it is sung twice the way it is written here.

Note: Although the notation of the canon in parallel columns gives the impression of four pairs of couplets, each column contains half-lines reading continuously across the columns. As shown above, these not only rhyme but share final syllables, written but once in the MS. Crescat semel in decanendo ('Let it grow as it is diminished') describes the paradoxical situation in which the tenor 'grows' at sextuple proportion relative to the notated values, but is at the same time diminished from its previous statement at octuple proportion. Requiescit, where ut iacet or integer valor might be more common, allows the consistent and playful sharing of syllables.

No. 14 Carmen, Salve pater

Tenor dicitur ter de prolatione sui moteti

The tenor is sung three times, according to the prolations of the upper voices

No. 15 Carmen, Venite adoremus

Tibi color cantus bis pronunciandus unde duo colores de maiori in tempore imperfecto ultimus cum primo de minori in tempore perfecto item secundus et ultimus de eadem in tempore imperfecto

The tenor is sung twice. During this there are two statements [of the rhythm] in imperfect tempus, major prolation. The last [third talea of the first statement of the melody] and the first [first talea of the second statement] are in perfect tempus, minor prolation. The second and last [second and third taleae of the second statement] are in imperfect tempus.

Note: Color seems to be employed to indicate repetitions of both pitch and rhythm.




PROPORTION AND FORNM IN THE ISORHYTHMIC MOTET

No. 16 Anonymous, Cuius fructus

Tenor isti dicitur de modo perfecto et tempore perfecto

The tenor is sung in perfect mode and perfect tempus.

REFERENCES

Apel, Willi, 1959: 'Remarks about the Isorhythmic Motet', L'Ars Nova: Colloque international tenu a Wigimont 1955 (Liege: Universit6 de Liege), pp.140-8.

Bent, Margaret, 1981: Dunstaple (London: OUP). Bent, Margaret, and Hallmark, Anne (eds), 1985: The Works of Johannes Ciconia.

Polyphonic Music of the Fourteenth Century, Vol. 24 (Monaco: L'Oiseau- Lyre).

Besseler, Heinrich, 1925, 1926: 'Studien zur Musik des Mittelalters', Archiv far Musikwissenschaft, Vol. 7, pp. 167-252; Vol. 8, pp.137-258. (ed.), 1966: Guillelmi Dufay: Opera Omnia, Vol. 1. Corpus Mensurabilis Musicae, Vol. 1 ([Rome]: American Institute of Musicology).

Borren, Charles van den (ed.), 1932: Polyphonia Sacra: A Continental Miscellany of the Fifteenth Century (Nashdom Abbey).

Brown, Samuel E., 1962: 'The Motets of Ciconia, Dunstable, and Dufay' (Diss., Indiana University).

Busse Berger, Anna Maria, 1988: 'The Origin and Early History of Proportion Signs', Journal of the American Musicological Society, Vol. 41, pp.403-33. 1990: 'The Myth of Diminutio per Tertiam Partem', Journal of Musicology, Vol. 8, No. 3, pp.398-426.

Coussemaker, Edmond de, 1876: Scriptorum de medii aevi nova series, Vol. 4 (Paris).

De Van, Guillaume (ed.), 1947: Guglielmi Dufay: Opera Omnia, Vol. 1. Corpus Mensurabilis Musicae 1 ([Rome]: American Institute of Musicology). 1948: 'A Newly Discovered Source of Early Fifteenth Century Polyphonic Music', Musica Disciplina, Vol. 2, pp. 5-74.

Fallows, David, 1982: Dufay (London: OUP). Ficker, Rudolf von (ed.), 1933: Sieben Trienter Codices VI. Denkmdler der Tonkunst

in Osterreich, Vol. 76 (Vienna: Universal). Giinther, Ursula (ed.), 1965: The Motets of the Manuscripts Chantilly, Musde Cond&,

564 (olim 1047) and Modena, Biblioteca Estense, a.M.5, 24 (olim lat. 568). Corpus Mensurabilis Musicae, Vol. 39 ([Rome]: American Institute of Musicology).

Marix, Jeanne (ed.), 1937: Les Musiciens de la cour de Bourgogne (Paris: L'Oiseau- Lyre).

Mixter, Keith, 1969: 'Isorhythmic Design in the Motets of Johannes Brassart', Studies in Musicology: Essays in Memory of Glen Haydon (Chapel Hill),




CHARLES TURNER

pp. 179ff. (ed.), 1971: Johannes Brassart: Opera Omnia, Vol. 2. Corpus Mensurabilis Musicae 35/2 ([Rome]: American Institute of Musicology).

Planchart, Alejandro Enrique, 1981: 'The Relative Speed of Tempora in the Period of Dufay', Royal Musical Association Research Chronicle, Vol. 17, pp.33-51.

Reaney, Gilbert (ed.), 1955: Early Fifteenth Century Music, Vol. 1. Corpus Mensurabilis Musicae 11/1 ([Rome]: American Institute of Musicology).

Reese, Gustave, 1959: Music in the Renaissance (New York: Norton). Sandresky, Margaret Vardell, 1979: 'The Continuing Concept of the Platonic-

Pythagorean System and its Application to the Analysis of Fifteenth-Century Music', Music Theory Spectrum, Vol. 1, pp. 107-20.

Schroeder, Eunice, 1982: 'The Stroke Comes Full Circle: ( and C in Writings on Music, ca. 1450-1540', Musica Disciplina, Vol. 36, pp. 119-66.

Strunk, Oliver, 1965: Source Readings in Music History (New York: Norton). Taruskin, Richard, 1986: 'Antoine Busnoys and the L'Homme arm6 Tradition',

Journal of the American Musicological Society, Vol. 39, pp. 255-93. Turner, Charles L., 1986: 'The Isorhythmic Motet in Continental Europe, ca.

1385-1450' (Mus. D. Treatise, Indiana University). Warren, Charles, 1973: 'Brunelleschi's Dome and Dufay's Motet', Musical

Quarterly, Vol. 59, pp. 93ff. Wittkower, Rudolf, 1971: Architectural Principles in the Age of Humanism, 4th edn

(New York: Norton).




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