Blachly-From Black to Red to White.htm<title>  <style>  </style> </head> <body bgcolor=white lang=EN-US style='tab-interval:.5in'> <div class=Section1> <p class=MsoNormal align=center style='text-align:center'><span style='font-size:14.0pt'>From Black to Red to White:<o:p></o:p></span></p> <p class=MsoNormal align=center style='text-align:center'><span style='font-size:14.0pt'>The Notation of Musical Proportions at the Dawn of the Renaissance<o:p></o:p></span></p> <p class=MsoHeader align=center style='text-align:center;tab-stops:.5in 135.0pt 3.5in'>by Alexander Blachly</p> <p class=MsoNormal align=center style='text-align:center'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span>The music of the late-fourteenth century has justly earned notoriety in modern times for its extraordinary complexities of rhythm. Some of these are produced by syncopation, but most involve proportional relationships between the durations of notes in different voice parts--or between notes in different passages in the same voice part. This essay will describe the evolution of mathematical proportions in European polyphony at the dawn of the Renaissance: from the period when composers first discovered a way to notate simple proportional durations ca. 1200 up to and beyond the point at which such proportions reached their most bewildering complexity roughly two centuries later, in the 1380s and '90s.</p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span>A fascination with musical proportions related to the notes of the scale was nothing new in the thirteenth and fourteenth centuries. It already had a history extending back to Pythagoras in the sixth century B.C.E., and even further back to the Babylonians and Egyptians, from whom Pythagoras presumably learned much of what he was later said to have discovered.<a style='mso-footnote-id:ftn1' href="#_ftn1" name="_ftnref1" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[1]<![endif]></span></span></a> Since the Pythagoreans recognized that music has a demonstrably numerical aspect, in as much as the notes of the scale can be shown to relate to one another in numerical ratios,<a style='mso-footnote-id:ftn2' href="#_ftn2" name="_ftnref2" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[2]<![endif]></span></span></a> they saw music, by analogy, as a key to how everything else is structured. Music, then, embodying numerical proportions in sound, mirrored the universe in microcosm. Strictly speaking, this recognition did not apply to musical pieces but only to the physics of sound--how the notes in the scale relate to one another in numeric proportions.</p> <h1>Number in musical composition</h1> <p class=MsoNormal><span style='mso-tab-count:1'> </span>The decisive advance made by musicians in the later Middle Ages was to construct musical compositions that expressed numerical proportions between the durations of their various parts and/or with an overall numerical structure based on durations. What began as simple 2 : 1 or 3 : 1 relationships between note values ca. 1200, when the first Gothic cathedrals were being built, evolved rapidly through the work of French and then Italian musicians in the thirteenth and fourteenth centuries,<a style='mso-footnote-id:ftn3' href="#_ftn3" name="_ftnref3" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[3]<![endif]></span></span></a> until by later in the fourteenth century composers were notating vertical and horizontal durational relationships which, when performed, produced such complex rhythmical effects that their like would not to be heard again until the twentieth century. It is clear that the works found in such sources as <i>Ch, Mod</i><span style='font-style:normal'>, </span><i>It</i><span style='font-style:normal'>, </span><i>TuB</i><span style='font-style:normal'>, and <I>OH</I> are the products of composers who, finding the exploration of the notational possibilities available to them a source of intellectual excitement, exploited every new technical discovery and invention to push their field forward--and in so doing challenged singers to ever greater feats of virtuosity.</span></p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span>The process of writing music proportionally in terms of durational values, then, can be said to have begun in its most general sense in the late twelfth and early thirteenth centuries, when, by means of the so-called "rhythmic modes," composers were able to specify that a given note was either a short (Brevis), a long (Longa, twice the duration of a Brevis), or greater than that ("beyond measure," worth three times the duration of a Brevis). Soon the note three times the length of a breve came to be recognized as a "perfect" long, as opposed to the long only twice the length of a breve, which was seen to be "imperfect." From early in the thirteenth century, in the two-voice clausulae of the Notre Dame school, one observes a fascination with numerical proportions expanding beyond simple 2 : 1 and 3 : 1 relationships between individual notes to encompass the construction of entire musical structures. Example 1 shows the clausula <i>Hec dies</i><span style='font-style: normal'> from the manuscript W<sub>1</sub>, where the tenor melody (known as the </span><i>color</i><span style='font-style:normal'>) has been taken from the first 18 notes of the Easter chant gradual. (The tenor is the lower of the two voices.) Relationships between the durations of single notes are still limited here to 2 : 1 but 3 : 1, but the overall structure exhibits a more sophisticated numerical<span style="mso-spacerun: yes"> </span>form:</span></p> <BR> <center> <IMG SRC="W1ClausulaHecDies.jpg" width=80%> </center> <p class=Figuredescription align=center style='text-align:center'>Ex. 1. Clausula <i>Hec dies</i><span style='font-style:normal'> from W<sub>1</sub></span></p> <p class=MsoEndnoteText><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span>The unknown composer of this miniature repeats the rhythmic pattern of the tenor (three long notes followed by a long rest) twelve times (12 x 4L) with no variation. (Each such repeating pattern was known as a <i>talea</i><span style='font-style:normal'>.) The upper voice, meanwhile, has phrases of 8 longs, 4 longs, 4 longs, 8 longs, and 24 longs (counting, as in the tenor, a phrase as consisting of a series of sounding notes and the rest following them). The entire piece, then, can be represented schematically,<a style='mso-footnote-id:ftn4' href="#_ftn4" name="_ftnref4" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[4]<![endif]></span></span></a> with respect to phrase length, as</span></p> <p class=MsoEndnoteText style='tab-stops:.5in 135.0pt 2.0in 3.5in'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoEndnoteText style='tab-stops:2.25in 3.5in'><span style='mso-tab-count: 1'> </span>8L+ 4L + 4L + 8L + 24L = 48L</p> <p class=MsoEndnoteText style='tab-stops:2.25in 3.5in'><span style='mso-tab-count: 1'> </span>12 x 4L = 48L<span style='font-size:5.0pt'><o:p></o:p></span></p> <p class=Figuredescription align=center style='text-align:center'>Fig. 1. The number scheme of the clausula <i>Hec dies</i><span style='font-style:normal'> from W<sub>1</sub></span></p> <p class=MsoEndnoteText style='tab-stops:.5in 135.0pt 2.0in 3.5in'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal>A further numerical feature is the use of the 18-note <i>color</i><span style='font-style:normal'> once repeated, for a total of 36 notes in all, with the result that the numbers 4, 8, 12, 18, 24, 36, and 48 interact to order time through sound (with the number 6, ostensibly absent, present as the mean between the first two and a common factor of the next five). The piece overall, therefore, has a "Pythagorean" structure, in as much as it reflects in various ways the proportions that exist among the integers of the Pythagorean </span><i>tetraktys</i><span style='font-style:normal'> (1, 2, 3, 4) and among numbers built on multiples of these.</span></p> <p class=MsoFootnoteText style='margin-top:0in;line-height:150%'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <h1>Black notation</h1> <p class=MsoNormal><span style='mso-tab-count:1'> </span>The "rhythmic modes" define durations by the placement of notes within sequences of neumes borrowed from Gregorian chant, whereby a single three-note neume (a <I>ternaria</I>) followed by serveral two-note neumes (<I>binariae</I>) would express the rhythm of "first mode": long, short, long, short, long, short, long.... Groups consisting of three three-note neumes followed by a rest, on the other hand, would express the rhythm of "fifth mode": long, long, long, rest, long, long, long, rest....<a style='mso-footnote-id:ftn5' href="#_ftn5" name="_ftnref5" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[5]<![endif]></span></span></a> (The <I>Hec dies</I> clausula exhibits first rhythmic mode with several <I>extensiones modi</I>--individual longs not followed by shorts--in the upper voice; the tenor is in fifth rhythmic mode.)</span></p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <BR> <center> <IMG SRC="W1HecDiesDETAIL.jpg" width=80%> </center> <p class=Figuredescription align=center style='text-align:center'>Plate 1. The clausula <i>Hec dies</i><span style='font-style:normal'> in W<sub>1</sub>, showing the notation of first and fifth rhythmic modes<a style='mso-footnote-id: ftn6' href="#_ftn6" name="_ftnref6" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[6]<![endif]></span></span></a><span style='mso-tab-count:1'> </span></span></p> <p class=Figuredescription align=center style='text-align:center'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=Figuredescription align=center style='text-align:center'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span>Since the square or quadratic form of Gregorian chant that became ubiquitous in the Romanic (non-Germanic) lands during the thirteenth century is written in solid black neumes, the manuscripts preserving the so-called Notre Dame polyphony of Leoninus, Perotinus, and their contemporaries in Paris ca. 1200, most of which sources date from the second half of the thirteenth century, likewise use square black notational symbols. Precisely measured music,<a style='mso-footnote-id: ftn7' href="#_ftn7" name="_ftnref7" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[7]<![endif]></span></span></a> therefore, begins its notated life as solid black notes, and it would stay black until the advent of white notation well into the fifteenth century. </p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span>In the earliest forms of mensural notation,<a style='mso-footnote-id:ftn8' href="#_ftn8" name="_ftnref8" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[8]<![endif]></span></span></a> which employed primarily longs and breves and some semibreves, the shapes of individual notes--being direct descendants of the Gregorian single-note neumes virga, punctum, and punctum inclinatum--retained the appearance of their Gregorian ancestors both in shape and in color. </p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <center> <IMG SRC="FranconianLongaUp.jpg" width=3%> <IMG SRC="FranconianLongaDown.jpg" width=3%> <IMG SRC="FranconianBrevis.jpg" width=3%> <IMG SRC="FranconianSemibrevis.jpg" width=3%> </center> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=Figuredescription align=center style='text-align:center'>Fig. 2. The (unligated) note shapes of Franconian notation: Longae, Brevis, Semibrevis</p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal>Ligated notes (tied together in multi-note groups) also took their form from solid black Gregorian neumes, the prime two-note ascending (podatus) and descending (clivis) forms always being interpreted as Brevis-Longa. </p> <BR> <center> <IMG SRC="Prime2NoteAscending.jpg" width=2%> <IMG SRC="Prime2NoteDescending.jpg" Width=4%> </center> <BR> <p class=Figuredescription align=center style='text-align:center'>Fig. 3. The podatus and clivis</p> <p class=TextTranslation><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal>When minims and shorter notes were added to the system by the Ars nova composers of the fourteenth century, these too continued to be written as solid black shapes. Thus, we speak of thirteenth- and fourteenth-century mensural notation as "black" notation.</p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <center> <IMG SRC="ArsNovaMinima.jpg" width=3%> </center> <p class=TextTranslation><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=Figuredescription align=center style='text-align:center'>Fig. 4. The Minima of Ars nova notation</p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span>It is from the name of the treatise <I>Ars nova</I> by Philippe de Vitry (1291-1361) that the era of fourteenth-century music that ends with the death of Guillaume de Machaut (ca. 1300-1377) takes its name. The Ars nova motet <i>Tuba sacre/In arboris/Virgo sum</i><span style='font-style:normal'> survives intact in a single source (</span><i>Iv</i><span style='font-style: normal'>). Though </span><i>Iv</i><span style='font-style:normal'> dates from ca. 1370, </span><i>In arboris</i><span style='font-style:normal'> must date from ca. 1320, since Philippe de Vitry names the work in his treatise. </span><i>In arboris</i><span style='font-style:normal'> exhibits, in the structure of its tenor, fully-fledged "isorhythm"<a style='mso-footnote-id:ftn9' href="#_ftn9" name="_ftnref9" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[9]<![endif]></span></span></a>--i.e., a complex </span><i>talea</i><span style='font-style:normal'>, conceptually similar to the simple pattern in the tenor of </span><i>Hec dies</i><span style='font-style:normal'> but more varied in the durations of its pitches. The tenor </span><i>color</i><span style='font-style:normal'>, consisting of 24 notes, repeats its eight-note </span><i>talea</i><span style='font-style:normal'> three times (it takes three </span><i>taleae</i><span style='font-style:normal'> to complete one </span><i>color</i><span style='font-style:normal'>); then, on the repeat of the </span><i>color</i><span style='font-style:normal'>, repeats the same </span><i>talea</i><span style='font-style:normal'> another three times but with all values halved.</span></p> <p class=MsoEndnoteText style='tab-stops:.5in 135.0pt 2.0in 3.5in'><span style='font-size:4.0pt'><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p> <BR> <center> <IMG SRC="VitryInArborisTenor.jpg" width=80%> </center> <p class=Figuredescription align=center style='text-align:center'>Fig. 5. The tenor of <i>Tuba sacre/In arboris/Virgo sum<o:p></o:p></i>, with A indicating the first phrase of the <I>talea</I>,<BR> B the second phrase, a the first phrase in diminution, b the second phrase in diminution.</p> <p class=Figuredescription align=center style='text-align:center'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoEndnoteText><span style='font-size:4.0pt'><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span>Not only does the rhythmic pattern of <i>In arboris</i><span style='font-style: normal'> tenor show more varied note values than the tenor of </span><i>Hec dies</i><span style='font-style:normal'>, but the repeat of the tenor melody in halved values introduces the concept of diminution (and its converse, augmentation), where note values may be halved or quartered or reduced or augmented in some other ratio.<a style='mso-footnote-id:ftn10' href="#_ftn10" name="_ftnref10" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[10]<![endif]></span></span></a> In this work, the halved values are actually written out; in later motets we encounter cases where a tenor will be notated a single time but with some indication--either a verbal canon or some set of symbols--or else the context alone that specifies increasing diminution on each repetition.<a style='mso-footnote-id:ftn11' href="#_ftn11" name="_ftnref11" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[11]<![endif]></span></span></a> Thus, the manipulation of musical material in </span><i>In arboris</i><span style='font-style:normal'> goes far beyond the imposition of a simple four-note </span><i>talea</i><span style='font-style:normal'> on a plainchant melody, as was the case in </span><i>Hec dies</i><span style='font-style:normal'>. </span></p> <h1>From black to red</h1> <p class=MsoNormal><span style='mso-tab-count:1'> </span>The composer of <i>In arboris</i><span style='font-style:normal'> (who is thought to Philippe de Vitry himself) has, moreover, created a rhythmic pattern that incorporates a new notational device: the use of colored ink (red) to express a change of mensural value for the notes so colored. (The red notes are indicated in Fig. 5 with small square brackets above the staff; they occur only in the "B" and "b" sections of the <I>talea</I>.) As can be seen in the transcription, the composer includes a rubric to make the meaning of coloration clear: "The black notes are imperfect, and the red ones are perfect." Franco at the end of the thirteenth century defined a perfect note as one containing three of the next smaller value, whereas an imperfect note contained only two of the next smaller value. Red notation appeared for the first time in another motet thought to be by Vitry, </span><i>Garrit gallus/In nova fert/[Tenor],</i><span style='font-style: normal'> which survives in the copy of the </span><i>Roman de Fauvel</i><span style='font-style:normal'> with musical interpolations (Paris, Bibliothèque nationale, fond fr. MS 146=</span><i>Fauv</i><span style='font-style:normal'>) and in Bibliothèque nationale, Collection de Picardie, MS 67 (=</span><i>Pic</i><span style='font-style:normal'>); in this piece, as it survives in </span><i>Fauv,</i><span style='font-style:normal'> the longa rests in black notation lasting three breves and the longa rests in red notation lasting two breves show that black notes are perfect, red notes imperfect. In </span><i>Pic</i><span style='font-style:normal'>, the red notes are written as hollow black.<a style='mso-footnote-id:ftn12' href="#_ftn12" name="_ftnref12" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[12]<![endif]></span></span></a> </span></p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span>As we would expect, the number scheme of <i>In arboris</i><span style='font-style: normal'> shows an advance in sophistication over that of </span><i>Hec dies</i><span style='font-style:normal'>. The clear-cut pattern of the tenor also reveals another feature of Ars nova</span><i> </i><span style='font-style:normal'>practice that distinguishes it from Ars antiqua works (those dating from before the Ars nova, thus, from the thirteenth century): the final note of the piece needs to be understood to be longer than it appears, i.e., a perfect long three breves in length, although it is written as a black long, the very note that the rubric identifies as imperfect. Final longs are typically indeterminate in length, a feature of mensural notation still found as late as Lassus at the end of the sixteenth century, where one voice reaching its final note before the others will still have that final note written as a long, the same as the final note of the other voices.<a style='mso-footnote-id:ftn13' href="#_ftn13" name="_ftnref13" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[13]<![endif]></span></span></a> To make </span><i>In aboris</i><span style='font-style:normal'>'s three </span><i>taleae</i><span style='font-style:normal'> in diminution match the three </span><i>taleae</i><span style='font-style:normal'> in undiminished values that precede them, then, not only must the final note of the tenor be considered the equivalent of a perfect long, but the final note in each of the upper voices likewise needs to be lengthened. Another new feature of </span><i>In arboris</i><span style='font-style:normal'> is a twelve-breve </span><i>Introitus</i><span style='font-style:normal'> for the upper voices before the tenor enters, during which the tenor has rests (indicated in the scheme below by an underlined number; the Triplum, as can be seen, also rests six breves before entering):</span></p> <p class=MsoEndnoteText style='tab-stops:.5in 103.5pt right 369.0pt 387.0pt 405.0pt 423.0pt 441.0pt'><span style='mso-tab-count:1'> </span></p> <p class=MsoEndnoteText style='margin-left:.5in;tab-stops:.5in 103.5pt right 369.0pt 387.0pt 405.0pt 423.0pt 441.0pt'>Triplum:<span style='mso-tab-count:1'> </span><u>6</u> + (20 + 2 + 2) + (20 + 2 + 2) + (20 + 2 + 2) + <span style='mso-tab-count:1'> </span>11 + (12 + 12) + 7 = 120B</p> <p class=MsoEndnoteText style='tab-stops:.5in 99.0pt right 369.0pt 387.0pt 405.0pt 423.0pt 441.0pt'><span style='mso-tab-count:1'> </span>Motetus:<span style='mso-tab-count:1'> </span> 27 + (2 + 22) + (2 + 22) + 2 + 14 + 1 + 1 + 10 + 1<span style='mso-tab-count:1'> </span> + 1 + 15 = 120B</p> <p class=MsoEndnoteText style='tab-stops:.5in 99.0pt right 369.0pt 387.0pt 405.0pt 423.0pt 441.0pt'><span style='mso-tab-count:1'> </span>Tenor<span style='mso-tab-count:1'> </span><u>12</u> + 12 + 12 + 12 + 12 + 12 + 12 + 6 + 6 + 6 + 6 + 6 + 6 = 120B</p> <p class=MsoEndnoteText style='tab-stops:63.0pt right 135.0pt 153.0pt 171.0pt 189.0pt 207.0pt 225.0pt 243.0pt left 3.5in right 261.0pt 279.0pt 297.0pt 315.0pt 333.0pt 351.0pt 369.0pt 387.0pt 405.0pt 423.0pt 441.0pt'><span style='font-size:3.0pt'><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p> <p class=Figuredescription align=center style='text-align:center'>Fig. 6. The number scheme of <i>In arboris</i><span style='font-style:normal'>, counting breves</span></p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span>What are we to make of this scheme? It has obvious elements of order, especially in the tenor, yet seems lacking in an obvious overall theme, though one can easily see that the scheme involves a play of 12's, 6's, 24's, and various "spellings" of these, together with some "irregular" numbers (e.g., 11, 7, 27, 15), all of which add up to 120 breves in each voice.<a style='mso-footnote-id:ftn14' href="#_ftn14" name="_ftnref14" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[14]<![endif]></span></span></a> </p> <p class=MsoNormal><span style='font-size:7.0pt'><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span>As the fourteenth century progressed, composers settled on a single meaning for coloration: that red notes would signify <i>reduction</i><span style='font-style:normal'> in value by a third, thereby turning perfect notes into imperfect ones--the opposite of the procedure specified in </span><i>In arboris</i><span style='font-style:normal'>.<a style='mso-footnote-id:ftn15' href="#_ftn15" name="_ftnref15" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[15]<![endif]></span></span></a> This meaning for coloration can be seen in Machaut's isorhythmic motet </span><i>Felix virgo/Inviolata/Ad te suspiramus</i><span style='font-style:normal'>, which, like the composer of </span><i>In</i><span style='font-style:normal'> </span><i>arboris</i><span style='font-style:normal'>, Machaut has supplied with a rubric for interpreting the red notes' meaning: </span><i>Nigre sunt perfecte et Rubee imperfecte</i><span style='font-style:normal'>.<a style='mso-footnote-id:ftn16' href="#_ftn16" name="_ftnref16" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[16]<![endif]></span></span></a><span style="mso-spacerun: yes"> </span></span></p> <span style='font-size:12.0pt;font-family:Palatino;mso-ansi-language:EN-US'><br clear=ALL style='page-break-before:always'> </span> <BR> <center> <IMG SRC="MachautFelixTENORSbeg.jpg" width=80%> </center> <p class=Figuredescription align=center style='text-align:center'>Fig. 7. The opening notes of the contratenor and tenor of Machaut's <i>Felix/Inviolata/Ad te suspiramus<o:p></o:p></i></p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span>Machaut's <i>oeuvre</i><span style='font-style:normal'>, it should be noted, contains only a handful of works with coloration, and in all of these red notes occur only in the tenor; but in the next generation, that of the Ars subtilior (so-called because of its notational complexities), the use of red notes and other devices for altering durational values became increasingly common in all voices. As coloration migrated to the upper voices in the later fourteenth century and was applied to faster notes, it created a new effect: that of hemiola cutting across the </span><i>mensura</i><span style='font-style:normal'> (what we would now call the beat), a phenomenon we could characterize as rhythmic friction with the underlying mensuration. Trebor's </span><i>Passerose de beauté</i><span style='font-style:normal'> (</span><i>Ch</i><span style='font-style:normal'>, fol. 21)<a style='mso-footnote-id:ftn17' href="#_ftn17" name="_ftnref17" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[17]<![endif]></span></span></a> provides numerous examples of this effect, as in m. 29 and m. 32 at the approach to the medial cadence:</span></p> <BR> <center> <IMG SRC="Trebor - Passerose1.jpg" width=65%> </center> <p class=Figuredescription align=center style='text-align:center'>Ex. 2. Coloration of semibreves approaching the medial cadence of Trebor's <i>Passerose de beauté</i></p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span>Here, the prevailing mensuration calls for a beat on the perfect semibreve (dotted quarter in the transcription). When coloration of semibreves occurs in the top voice (m. 29) or middle voice (m. 32), hemiola cuts across the beat to create the type of effect commonly understood today as a proportion.<a style='mso-footnote-id:ftn18' href="#_ftn18" name="_ftnref18" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[18]<![endif]></span></span></a> This particular relationship, that of hemiola (three against two) at the semibreve level, would remain the most common "proportion" in the period under discussion, closely followed by hemiola at the breve level--this caused by colored breves replacing uncolored ones (in the transcription below, breves equal half notes):</p> <BR> <center> <IMG SRC="Solage - Fumeux fume3.jpg" width=80%> </center> <p class=Figuredescription align=center style='text-align:center'>Ex. 3. Coloration at the breve level in Solage's <i>Fumeux fume</i><span style='font-style:normal'> (</span><i>Ch</i><span style='font-style:normal'>, fol. 59)</span></p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal>In the case of perfect mensurations (those in which at least one level of value is perfect), coloration affected only the value of the perfect notes, changing their value to imperfect, thus reducing their value by a third. The consequence of coloration in <i>tempus imperfectum cum prolatione maiore</i><span style='font-style:normal'> was almost always to create hemiola at the semibreve level (three imperfect red semibreves in the place of two perfect black ones), as in Ex. 2. Similarly, coloration in </span><i>tempus perfectum</i><span style='font-style:normal'> was most commonly employed to create hemiola at the breve level, as in Ex. 3. In works such as Trebor's </span><i>Passerose</i><span style='font-style:normal'> and Solage's </span><i>Fumeux fume</i><span style='font-style:normal'>, with their complexity and strangeness, we witness some of the hallmarks of the Ars subtilior--referring primarily to the 1380s and '90s--the "golden age" of musical proportions.</span></p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span>In certain pieces in the Chantilly manuscript (<i>Ch</i><span style='font-style: normal'>)--which dates from ca. 1415<a style='mso-footnote-id:ftn19' href="#_ftn19" name="_ftnref19" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[19]<![endif]></span></span></a> but includes the core Ars subtilior repertoire--such as Johannes Vaillant's </span><i>Dame doucement trait/Doulz amis</i><span style='font-style: normal'> (fol. 26v) or the anonymous </span><i>Se je cudoie tous iou[r]s</i><span style='font-style:normal'> (fol. 27v), there are nearly as many red notes in every voice as black ones, testifying to an extremely high level of rhythmic friction with the underlying mensuration. Galiot's (Senleches's?) </span><i>En attendant, Esperance</i><span style='font-style:normal'> (fol. 44r) actually has more red and red void than black notes in the top voice and nearly equal red and red void to black notes in the tenor.<a style='mso-footnote-id:ftn20' href="#_ftn20" name="_ftnref20" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[20]<![endif]></span></span></a> We will look more closely at such pieces shortly.</span></p> <h1>Proportional structures</h1> <p class=MsoNormal><span style='mso-tab-count:1'> </span>Returning now to motets and numbers, Margaret Bent nearly two decades ago analyzed Machaut's motet <i>Amours qui a le pouoir/Faus Samblant m'a dece¸/Vidi dominum</i><span style='font-style:normal'> to reveal number operating somewhat differently than in the works cited earlier--namely, in a subtle collaboration with the text. Taking her cue from the theme of deception explored in the poems sung by the upper voices, Bent showed that some structural measurements appear to refer symbolically to the deceit of Peter's threefold denial and Judas's thirty pieces of silver ("[The motet] is 120 breves long. The tenor consists of one </span><i>color</i><span style='font-style:normal'>...disposed in four equal </span><i>taleae...</i><span style='font-style:normal'>each of ten perfect longs.... 10 longs x 3 = 30 breves; 4 </span><i>taleae</i><span style='font-style:normal'> x 30 =120"). Elsewhere, the importance of significant musical and textual events is marked by their occurrence at places that represent Golden Section dividing points of the whole.<a style='mso-footnote-id:ftn21' href="#_ftn21" name="_ftnref21" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[21]<![endif]></span></span></a></span> In this work one sees structural proportions taking on a symbolic role.</p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span>A more famous case of structural/symbolic proportions occurs in Guillaume Du Fay's motet <i>Nuper rosarum flores</i><span style='font-style:normal'>. Here the two tenors are written once but are to be sounded four times in four different mensurations, causing the piece as a whole to express the proportions 6 : 4 : 2 : 3.<a style='mso-footnote-id:ftn22' href="#_ftn22" name="_ftnref22" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[22]<![endif]></span></span></a> As Wright has shown, these are the very proportions of King Solomon's temple in I Kings, the model for many churches in Christendom, including the Duomo in Florence--for the dedication of which in 1436 </span><i>Nuper</i><span style='font-style:normal'> was composed. There are many other numbers at work here as well. Each tenor </span><i>color</i><span style='font-style:normal'> consists of 14 notes, the two tenors together sounding 28 notes in each </span><i>color</i><span style='font-style:normal'>. Each section of the motet begins with 84 breves in which the two upper voices function as a duet, with the tenors silent, followed by another 84 breves in which all four voices sound together. 84 = 28 x 3. It can be no accident that after the first 28 words in the text, the next word is Eugenius, the Pope who presided over the dedication; and that preceding the final 28 words in the text is the word Florentiae, the city in which the Duomo is located.<a style='mso-footnote-id:ftn23' href="#_ftn23" name="_ftnref23" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[23]<![endif]></span></span></a> Wright clarifies the symbolic meaning of still other structural numbers.<a style='mso-footnote-id:ftn24' href="#_ftn24" name="_ftnref24" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[24]<![endif]></span></span></a> Of greatest interest to the subject of the present study is that Du Fay notated his tenors a single time but preceded them with four different mensuration signs, one to govern the tenors' notes in each of the work's four sections.<a style='mso-footnote-id:ftn25' href="#_ftn25" name="_ftnref25" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[25]<![endif]></span></span></a> It is from these four signs, O, C, [cut-C], [cut circle], that the tenor notes change in value in 6 : 4 : 2 : 3 proportions and from which the work overall derives its proportional design.</p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span>Sanders mentions two works from <i>Ch</i><span style='font-style:normal'> and one from </span><i>Ox</i><span style='font-style:normal'> that are laid out in similar proportions: Egidius de Puiex's </span><i>Yda capillorum/Portio nature/Ante thronum </i><span style='font-style:normal'>(</span><i>Ch</i><span style='font-style:normal'>, fol. 61v-62), with a 6 : 4 : 3 : 2 structure, J. Alani's </span><i>Sub Arturo plebs/Fons citharizantium/In omnem</i><span style='font-style:normal'> (</span><i>Ch</i><span style='font-style:normal'>, fol. 70v-71), with a 9: 6: 4 structure, and Grenon's </span><i>Ave virtus/Prophetarum/Infelix</i><span style='font-style:normal'> (</span><i>Ox</i><span style='font-style:normal'>, fol. 120v-121), with a 4 : 3 : 2 : 1 structure.<a style='mso-footnote-id: ftn26' href="#_ftn26" name="_ftnref26" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[26]<![endif]></span></span></a> </span><i>Ave virtus</i><span style='font-style:normal'> achieves its proportions in a manner related to that of </span><i>Nuper rosarum:</i><span style='font-style:normal'> the tenor is notated once but supplied with a canon that specifies that it should grow </span><i>in octuplo</i><span style='font-style:normal'>, </span><i>in sextuplo</i><span style='font-style: normal'>, twice </span><i>in duplo</i><span style='font-style:normal'> and twice "as it rests," i.e., as written.<a style='mso-footnote-id:ftn27' href="#_ftn27" name="_ftnref27" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[27]<![endif]></span></span></a></span></p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span>For sheer virtuosity in controlling the overall structure of a piece by numeric proportions, few works can rival Biteryng's isorhythmic motet <i>En Katerine solennia/Virginalis concio/Sponsus amat sponsam</i><span style='font-style: normal'> in <I>OH</I> (fol. 110v-111), with a 6 : 3 : 2 structural design.<a style='mso-footnote-id:ftn28' href="#_ftn28" name="_ftnref28" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[28]<![endif]></span></span></a> Here isorhythm governs all three voices, each of which has its own </span><i>taleae</i><span style='font-style:normal'> in each of the motet's three major sections. Moreover, </span><i>En Katerine</i><span style='font-style:normal'>, as a so-called "mensuration motet," requires the tenor, written a single time, to be read first "as it stands," then repeated twice more in increasingly faster notes with the arrival of each new mensuration, although no indications of the proportional relationships of the three tenor statements are provided. The underlying beat of the work changes twice, from </span><i>tempus perfectum</i><span style='font-style:normal'> in section I, to </span><i>prolatio maior</i><span style='font-style:normal'> in section II, and back to </span><i>tempus perfectum</i><span style='font-style:normal'> in section III, now with the tenor values reduced by diminution to a third of their original value. The tenor is written with some notes in red ink, which come into increasing conflict with the </span><i>mensura</i><span style='font-style:normal'> as the tenor statements accelerate. At the end of each half of section III the conflict equals a proportion of 3 : 4 with minims in the upper voices (it is a 3 : 2 relationship with respect to the </span><i>mensura</i><span style='font-style:normal'>). (In the transcription below, the upper voices' minims equal quarter notes. The tenor notes are triply diminished from their written values, such that the first note of the </span><i>talea</i><span style='font-style:normal'> has a duration of three beats rather than nine, as in Section I. The 3 : 4 relationship between tenor notes and upper voices occurs at mm. 80-81 and 89-90.)</span></p> <BR> <center> <IMG SRC="BiteryngEnKat4.jpg" width=80%> </center> <p class=Figuredescription align=center style='text-align:center'>Ex. 4. Sections IIIa and IIIb of Biteryng's <i>En Katerine solennia</i><span style='font-style:normal'> (<I>OH</I>, fol. 110v-111)</span></p> <h1>Melodic proportions</h1> <p class=MsoNormal><span style='mso-tab-count:1'> </span>It would be misleading to propose that the importance of number in music was restricted to motets or that the evolution of musical proportions occurred only in such works. The evolution in question is possibly even more indebted to composers' wish to free melody from the constraints of modal rhythm, this by way of exploring in melodies, with medieval thoroughness, every possible durational relationship the developing mensural system could express. This exploration took place primarily in the polyphonic <i>formes fixes</i><span style='font-style:normal'> of the later fourteenth century. In the words of the anonymous author of the late fourteenth-century </span><i>Tractatus figurarum</i><span style='font-style:normal'>, "those coming after" had the task of accomplishing "greater subtleties" by "reforming and perfecting" that which was "left behind by the first masters."<a style='mso-footnote-id:ftn29' href="#_ftn29" name="_ftnref29" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[29]<![endif]></span></span></a> Franco of Cologne, when codifying the first phase of mensural notation in the late thirteenth century,<a style='mso-footnote-id:ftn30' href="#_ftn30" name="_ftnref30" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[30]<![endif]></span></span></a> had shown no interest in subtleties as such: he wished only to define note shapes with specific durational values for the purpose writing more clearly the type of works conceived originally in modal rhythm. (When clausulae, in becoming motets, acquired texts, the many syllables broke up the clausulae's ligatures into individual notes, and it became impossible to read their rhythm without the aid of the untexted clausulae as a point of reference, where the ligatures were still intact; thus the need for notation like Franco's, which assigned specific values to individual note shapes.)<a style='mso-footnote-id: ftn31' href="#_ftn31" name="_ftnref31" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[31]<![endif]></span></span></a> Works written in Franconian notation, while no longer dependent on the arrangement of ligatures to clarify their rhythms, still express a style of melody inherited from music written in the rhythmic modes. </span></p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span>For Philippe de Vitry and the other Ars nova composers, the motivation for extending the note shapes and mensurations of Franco's system can be understood directly from the types of music they composed with the new tools of duple meter and the new note they added to the system, the minim ("smallest note")--soon joined by the semiminim, fusa, and other short-note forms. The tunefulness of the upper voices of Vitry's <i>In arboris</i><span style='font-style:normal'> shows the direction in which </span><i>Ars nova</i><span style='font-style:normal'> melodic experiments would progress.</span></p> <BR> <center> <IMG SRC="VitryInArborisEX.jpg" width=80%> </center> <p class=Figuredescription align=center style='text-align:center'>Fig. 8. Opening of the motetus of Vitry's<i> Tuba sacre/In arboris/Virgo sum<o:p></o:p></i></p> <p class=MsoNormal><i><![if !supportEmptyParas]> <![endif]><o:p></o:p></i></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span>A further step away from the melodic patterns of the Ars antiqua can be seen in Machaut's syncopated style, well represented by his intricate puzzle chanson, <i>Ma fin est mon commencement</i><span style='font-style:normal'>. This work, written entirely in duple meter (</span><i>tempus imperfectum cum prolatione minore</i><span style='font-style:normal'>), is most famous for its design--as a rondeau in which the upper voice is simultaneously to be sung both forwards (as written) and backwards (in retrograde) by two different singers, while the tenor is written as a palindrome that reaches its midpoint at the medial cadence, such that the melody and rhythm of the A section of the tenor creates a mirror image of its B section.<a style='mso-footnote-id:ftn32' href="#_ftn32" name="_ftnref32" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[32]<![endif]></span></span></a> <BR> <center> <IMG SRC="Machaut - Ma finEX2.jpg" width=80%> </center> <p class=Figuredescription align=center style='text-align:center'>Ex. 4a. Measures 23-29 of Machaut's <i>Ma fin est mon commencement.<o:p></o:p></i>, showin the<BR> medial cadence and the palindromic nature of the counterpoint</p> <BR> <p class=MsoNormal><span style='mso-tab-count:1'> </span> Our interest here focuses not on the extraordinary contrivances of counterpoint </span><i>Ma fin</i><span style='font-style:normal'> represents but on the nature of its melodies, which feature a high degree of syncopation at the minim, semibreve, and breve levels. Rhythms such as these, which mark an important innovation in musical style, could not have been notated in Franconian notation, even had thirteenth-century composers been able to imagine them. True, syncopation as a concept may be said to be already implicit in the second, third, and fourth rhythmic modes, with their </span><i>ordines</i><span style='font-style:normal'> featuring short-long motion; but where the rhythmic modes rely on repeating patterns, Machaut's melodies incorporate true rhythmic freedom. Indeed, it is their wide range of rhythmic motion, coupled with rhythmic intricacy, that endows these melodies with their appeal. </span></p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <BR> <center> <IMG SRC="Machaut - Ma finEX.jpg" width=80%> </center> <p class=Figuredescription align=center style='text-align:center'>Fig. 9. Opening of the upper voice of Machaut's <i>Ma fin est mon commencement.<o:p></o:p></i></p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span>Despite this work's characteristic overlapping layers of syncopation between voices, which create passages of intense rhythmic density, Machaut's music overall retains a degree of formal articulation that would be abandoned in the next generation. Specifically, Machaut's music has "hallmark" moments (usually one measure in a transcription, equal to a breve) where all motion stops, only to be resumed at full throttle in the next measure. Both the measures of suspended motion and those in which motion resumes tend to organize the counterpoint for the listener in a way that endows the music with formal clarity, a feature that skilled performers can accentuate in their phrasing.</p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span>While Machaut employed red notation in the tenors of a few works, as noted above, he did not write proportions of the type that cut across the beat. The sophistication of his style lies rather in his syncopations, which endow the music with a palpable rhythmic energy (and some of which do cut across the beat, as in m. 2 in Fig. 9 above). In the next generation that would change dramatically. We have noted already Trebor's hemiolas in the ballade <i>Passerose de beauté</i><span style='font-style:normal'> (Ex. 2). In fact, the melodic rhythms in that piece are considerably more complex than the few measures preceding the medial cadence might suggest, for Trebor uses not just coloration to achieve semibreve hemiola; he also makes extensive use in the contratenor of a note-shape inherited from Italian notation: </span></p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=bullettext style='line-height:normal'><span style='font-size:10.0pt'>The dragma, looking like a minim but with stems both ascending and descending. These notes produce duplets against the minims of <i>prolatio maior</i></span><span style='font-size:10.0pt'> elsewhere in the piece.<o:p></o:p></span></p> <p class=MsoFootnoteText style='margin-top:0in;line-height:150%'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <BR> <center> <IMG SRC="Trebor - Passerose.jpg" width=80%> </center> <p class=Figuredescription align=center style='text-align:center'>Ex. 5. Opening of Trebor's <i>Passerose de beauté, </i><span style='font-style: normal'>showing dragma duplets in the contratenor</span></p> <p class=MsoHeader style='tab-stops:.5in 135.0pt 3.5in'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span>With Jaquemin de Senleches's <i>La harpe de melodie</i><span style='font-style:normal'>, we encounter an Ars subtilior work even more removed from Machaut's style in its elaborate small-note ornamentation, carefully notated as strict mathematical proportions. This work also introduces a proportional reading of its black and white (hollow) notes by way of a verbal canon written in the form of a poetic rondeau.<a style='mso-footnote-id:ftn33' href="#_ftn33" name="_ftnref33" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[33]<![endif]></span></span></a> The canon, as it appears in Newberry 54.1, reads as follows, with Richard Hoppin's English translation to the right:<a style='mso-footnote-id:ftn34' href="#_ftn34" name="_ftnref34" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[34]<![endif]></span></span></a></span></p> <p class=TextTranslation><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=TextTranslation style='tab-stops:.5in 135.0pt 3.25in 3.5in'><span style='font-size:10.0pt'><span style='mso-tab-count:1'> </span>Se tu me veulz proprement pronuncier<span style='mso-tab-count:1'> </span>If you would perform me properly,<o:p></o:p></span></p> <p class=TextTranslation style='tab-stops:.5in 135.0pt 3.25in 3.5in'><span style='font-size:10.0pt'><span style='mso-tab-count:1'> </span>Sus la tenur pour miex estre d'acort<span style='mso-tab-count:1'> </span>you should begin a fifth above the tenor<o:p></o:p></span></p> <p class=TextTranslation style='tab-stops:.5in 135.0pt 3.25in 3.5in'><span style='font-size:10.0pt'><span style='mso-tab-count:1'> </span>Diapenthe te convient comencier,<span style='mso-tab-count:1'> </span>to be in better accord,<o:p></o:p></span></p> <p class=TextTranslation style='tab-stops:.5in 135.0pt 3.25in 3.5in'><span style='font-size:10.0pt'><span style='mso-tab-count:1'> </span>Ou autrement tu seras en discort.<span style='mso-tab-count:1'> </span>or otherwise you will be in discord.<o:p></o:p></span></p> <p class=TextTranslation style='tab-stops:.5in 135.0pt 3.25in 3.5in'><span style='font-size:10.0pt'><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p> <p class=TextTranslation style='tab-stops:.5in 135.0pt 3.25in 3.5in'><span style='font-size:10.0pt'><span style='mso-tab-count:1'> </span>Pars blanc et noir per mi sans oublier<span style='mso-tab-count:1'> </span>Let the black and white parts (notes) sound by half,<o:p></o:p></span></p> <p class=TextTranslation style='tab-stops:.5in 135.0pt 3.25in 3.5in'><span style='font-size:10.0pt'><span style='mso-tab-count:1'> </span>Lay le tonant, ou tu li feras tort.<span style='mso-tab-count:1'> </span>without forgetting, or you will do them wrong.<o:p></o:p></span></p> <p class=TextTranslation style='tab-stops:.5in 135.0pt 3.25in 3.5in'><span style='font-size:10.0pt'><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p> <p class=TextTranslation style='tab-stops:.5in 135.0pt 3.25in 3.5in'><span style='font-size:10.0pt'><span style='mso-tab-count:1'> </span>Se tu me veulz etc.<span style='mso-tab-count:2'> </span>If you would perform etc.<o:p></o:p></span></p> <p class=TextTranslation style='tab-stops:.5in 135.0pt 3.25in 3.5in'><span style='font-size:10.0pt'><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p> <p class=TextTranslation style='margin-right:-9.0pt;tab-stops:.5in 135.0pt 3.25in 3.5in'><span style='font-size:10.0pt'><span style='mso-tab-count:1'> </span>Puis va cassant duz temps sanz forvoier,<span style='mso-tab-count:1'> </span>Then follow (chasing) at two units of tempus, <o:p></o:p></span></p> <p class=TextTranslation style='tab-stops:.5in 135.0pt 3.25in 3.5in'><span style='font-size:10.0pt'><span style='mso-tab-count:4'> </span>without straying;<o:p></o:p></span></p> <p class=TextTranslation style='tab-stops:.5in 135.0pt 3.25in 3.5in'><span style='font-size:10.0pt'><span style='mso-tab-count:1'> </span>Premiere note en .d. prent son ressort;<span style='mso-tab-count:1'> </span>the first note takes its spring from <i>d</i></span><span style='font-size:10.0pt'>;<span style='mso-tab-count:1'> </span><o:p></o:p></span></p> <p class=TextTranslation style='margin-right:-9.0pt;tab-stops:.5in 135.0pt 3.25in 3.5in'><span style='font-size:10.0pt'><span style='mso-tab-count:1'> </span>Harpe toudis sans espasse blechier,<span style='mso-tab-count:1'> </span>harp always without touching (wounding) the spaces;<o:p></o:p></span></p> <p class=TextTranslation style='margin-right:-9.0pt;tab-stops:.5in 135.0pt 3.25in 3.5in'><span style='font-size:10.0pt'><span style='mso-tab-count:1'> </span>Par sentement me puis douner confort.<span style='mso-tab-count:1'> </span>with feeling you can give me comfort (satisfaction).<o:p></o:p></span></p> <p class=TextTranslation style='margin-right:-9.0pt;tab-stops:.5in 135.0pt 3.25in 3.5in'><span style='font-size:10.0pt'><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p> <p class=TextTranslation style='margin-right:-9.0pt;tab-stops:.5in 135.0pt 3.25in 3.5in'><span style='font-size:10.0pt'><span style='mso-tab-count:1'> </span>Se tu me veulz etc.<span style='mso-tab-count:2'> </span>If you would perform etc.<o:p></o:p></span></p> <p class=TextTranslation><span style='font-size:10.0pt'><span style='mso-tab-count: 1'> </span><o:p></o:p></span></p> <BR> <center> <IMG SRC="LaHarpeDeMelodie3.jpg" width=95%> </center> <p class=Figuredescription align=center style='text-align:center'>Plate 2. Senleches's <i>La harpe de melodie</i><span style='font-style:normal'> in </span><i>ChiC, </i><span style='font-style:normal'>rotated to make the staves horizontal</span></p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal><span style='font-size:10.0pt'><span style='mso-tab-count: 1'> </span></span>In addition to demonstrating that the upper voice is to be read as a musical canon, with a second voice chasing the first one at the interval of two breves ("Puis va cassant duz temps"), the verbal canon's instruction of "Pars blanc et noir per mi sans oublier/Lay le tonant" indicates that black and hollow notes (in the upper voice only) are to be read in diminution (at half their normal value). Senleches also employs several novel note-forms in this piece, some of which may have been borrowed from Italian notation: </p> <p class=TextTranslation><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=bullettext style='line-height:normal'><span style='font-size:10.0pt'>Minim with descending rather than ascending stem. This note has twice the value of a normally written minim with ascending stem.<o:p></o:p></span></p> <p class=bullettext style='line-height:normal'><span style='font-size:10.0pt'>Dragma (minim with both ascending and descending stems). Two of these equal three minims with descending stems.<o:p></o:p></span></p> <p class=bullettext style='line-height:normal'><span style='font-size:10.0pt'>White (hollow) dragma with hooked lower stem. Three of these make a triplet equal to one tenor minim.<a style='mso-footnote-id:ftn35' href="#_ftn35" name="_ftnref35" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[35]<![endif]></span></span></a><o:p></o:p></span></p> <p class=bullettext style='line-height:normal'><span style='font-size:10.0pt'>White (hollow) breves. Three of these equal one tenor breve.<o:p></o:p></span></p> <p class=TextTranslation><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal>The result is an upper-voice melody largely composed of intricate, rapid ornaments, carefully notated for precise durations. The proportions encountered in this single voice are:</p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=bullettext style='line-height:normal'><span style='font-size:10.0pt'>Hemiola. Three red semibreves in the upper voice equal two black semibreves in the tenor; also, three white breves in the upper voice equal two black semibreves in the tenor.<o:p></o:p></span></p> <p class=bullettext style='line-height:normal'><span style='font-size:10.0pt'>Reverse hemiola. Two black dragmas in the upper voice equal three black minims in the tenor.<o:p></o:p></span></p> <p class=bullettext style='line-height:normal'><span style='font-size:10.0pt'>Duple proportion. All of the black and white upper-voice notes to be read at one half their normal value (the value of like-shaped notes in the tenor).<o:p></o:p></span></p> <p class=bullettext style='line-height:normal'><span style='font-size:10.0pt'>Triple proportion. Three white upper-voice minims equal one black minim in the tenor.<o:p></o:p></span></p> <p class=bullettext style='line-height:normal'><span style='font-size:10.0pt'>Sesquialtera. Three white upper-voice dragmas equal two black minims in the tenor; <o:p></o:p></span></p> <p class=bullettext style='line-height:normal'><span style='font-size:10.0pt'>Sesquitertia. Four black upper-voice dragmas equal three minims in the tenor.<o:p></o:p></span></p> <p class=bullettext style='line-height:normal'><span style='font-size:10.0pt'><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p> <p class=MsoFootnoteText style='margin-top:0in;line-height:150%'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <BR> <center> <IMG SRC="Senleches - La harpe1.jpg" width=80%> </center> <p class=Figuredescription align=center style='text-align:center'>Ex. 6. Opening of Senleches's <i>La harpe de melodie, </i><span style='font-style: normal'>after transcription by Hoppin 1978b: 169-72</span></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span>When performed with precision and feeling,<a style='mso-footnote-id:ftn36' href="#_ftn36" name="_ftnref36" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[36]<![endif]></span></span></a> the Senleches's virelai fulfills in sound the sentiments of the text: "The harp of melody, made for pleasure without melancholy, should delight everyone to hear and see its harmony resound." In other words, this piece is not just charming to hear but also to <i>see </i><span style='font-style:normal'>(its inscription in a harp in </span><i>ChiC</i><span style='font-style:normal'> evidently being the reference). The virtuosity of its "harmony" has been matched by the virtuosity of its visual appearance. </span></p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <h1>Further notational contrivances</h1> <p class=MsoNormal><span style='mso-tab-count:1'> </span>Virtuosity, then, may be a key to explaining the redundancies of the notation. (Because of contrivances such as these, which are<span style="mso-spacerun: yes"> </span>typical of the Ars subtilior in general, the notation of this period has been characterized as "mannered"--i.e., highly artificial.)<a style='mso-footnote-id: ftn37' href="#_ftn37" name="_ftnref37" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[37]<![endif]></span></span></a> Why else than for the exploitation of contrivance for its own sake would Senleches have hemiola in the upper voice against tenor semibreves notated in two different ways: first in red semibreves, read the normal way, such that three imperfect red semibreves equal two perfect black ones in the tenor; but then also in white breves, read in diminution, such that three white breves also equal two perfect black semibreves in the tenor? Is it really necessary to have white breves just to be able to ligate three in a row (only two red semibreves at a time can be ligated)? It seems more likely that Senleches has taken the opportunity to write the same values in as many different ways as he can devise. Thus, he doesn't represent the value of a black minim in the tenor by a similar note in the upper voice but by an inverted minim with descending stem, now worth two minims (except that the verbal canon instructs that both black and white notes in the upper voice are to be diminished to half their value, thus bringing the inverted minim back to the value of a regular minim). Not only does the writing of a black minim as a black inverted minim to be read in diminution represent redundancy, it poses a hurdle for the singers of the upper voices, since they must continually switch from undiminished notes (written in red ink) to diminished ones (written black or hollow).</p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span>Fig. 10 shows another sort of redundancy:</p> <BR> <center> <IMG SRC="Senleches - La harpeDETAIL1.jpg" width=80%> </center> <p class=Figuredescription align=center style='text-align:center'>Fig. 10. Mm. 23-28 of the upper voice of Senleches's <i>La harpe de melodie</i></p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal style='text-indent:.5in'>In measure 27 Senleches has written the note equal to a dotted quarter in the transcription as a dotted red semibreve--a redundancy in itself, since the color red imperfects the semibreve, while the dot of addition adds to it half again of its value to achieve the duration of a perfect black semibreve in the tenor. Senleches wrote the value of a dotted red semibreve earlier in the upper voice as a perfect black breve (diminished by half its value). In this case not even the negligible advantage of ligating a third note (as might be argued justified white breves instead of red semibreves) accrues to using one shape rather than the other. Similarly, in measure 28 Senleches has written the first three eighth notes in the transcription as inverted black minims (the doubled value of which is undone by the verbal canon specifying diminution). He could easily have written the next eighth note and the quarter following it as one more inverted black minim followed by a black breve but instead chose to change color and write a red minim followed by red semibreve. (If nothing else, such notational riddles virtually prove that music such as this could not be read at sight but would require a certain degree of study--or better, deciphering--and undoubtedly considerable rehearsal.)</p> <p class=MsoNormal style='text-indent:.5in'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal style='text-indent:.5in'>To what end, then, was the introduction of the redundancies just cited? We can perhaps do no better than to recall Craig Wright's memorable characterization of the European world-view in the period under discussion:<a style='mso-footnote-id:ftn38' href="#_ftn38" name="_ftnref38" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[38]<![endif]></span></span></a></p> <p class=MsoNormal style='text-indent:.5in;line-height:normal'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoBlockText style='line-height:normal'><span style='font-size:10.0pt'>Sobriquets, anagrams, riddle canons, and the "mannered" notation were all part of the musical scene of the late-fourteenth century, in the same way that blazons, mottoes and symbolic and emblematic devices pervaded the secular courts and knightly orders of the period; the more medieval man tried quixotically to recapture the essence and meaning of the archaic code of chivalry in the late Middle Ages, the more importance he placed on artifical devices, on factitious values, and on contrived forms.<o:p></o:p></span></p> <p class=MsoBlockText><span style='font-size:1.0pt'><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p> <h1>Syncopation in conjunction with melodic proportions</h1> <p class=MsoNormal style='text-indent:.5in'>Just as earlier we saw composers such as Vitry and Machaut restrict their use of coloration in Ars nova works to slow-moving tenor voices, with Machaut seeking rhythmic complexity and liveliness by way of syncopation, some composers in the Ars subtilior advanced the cause of intricate melody by combining coloration (as in Senleches's <i>La Harpe de melodie</i><span style='font-style:normal'>) with an extreme form of syncopation produced by what might be termed an enhanced use of the mensural rule of "Similis ante similem," first formulated by Franco.<a style='mso-footnote-id: ftn39' href="#_ftn39" name="_ftnref39" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[39]<![endif]></span></span></a> According to this rule, a note could not be imperfected if it was followed by a note of the same level. At first there was no way around the "Similis ante similem" injunction, making the writing of the rhythm perfect long, breve, imperfect long, perfect long (to use Franconian notes) a counterintuitive procedure: this because the injunction prevented the note that was to be equal to an imperfect long in the example from being written as a long.<a style='mso-footnote-id:ftn40' href="#_ftn40" name="_ftnref40" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[40]<![endif]></span></span></a> </span></p> <p class=MsoNormal style='text-indent:.5in'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal style='text-indent:.5in'>With the advent of coloration, it became possible to write a red long before a perfect black long to achieve the desired rhythm of imperfect long followed by another long (or an imperfected note of any value followed by a note of the same level). Ars subtilior composers, however, now had a new use for "Similis ante similem." They had discovered that the dot of division, which originally separated one perfection from another (at the barline, so to speak, in terms of a modern transcription), could now be put virtually anywhere to force a new perfection to begin in the middle of an incomplete one, with the interrupted perfection then to be "completed" following whatever perfections were produced by the off-set notes.<a style='mso-footnote-id:ftn41' href="#_ftn41" name="_ftnref41" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[41]<![endif]></span></span></a> The opening of the anonymous ballade <i>Medee fu</i><span style='font-style: normal'> from </span><i>Ch</i><span style='font-style:normal'> shows how this procedure could be used in a piece with perfect semibreves:</span></p> <p class=TextTranslation><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <BR> <center> <IMG SRC="Anon - Medee fuDETAIL1.jpg" width=60%> </center> <p class=Figuredescription align=center style='text-align:center'>Fig. 11. Mm. 1-5 of the upper voice of<span style="mso-spacerun: yes"> </span><i>Medee fu </i><span style='font-style:normal'>(</span><i>Ch</i><span style='font-style: normal'>, fol. 24v)</span></p> <p class=MsoNormal style='text-indent:.5in'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal style='text-indent:.5in'>Here the first minim imperfects the breve it follows (a necessary imperfection because of the dot following the minim, which<span style="mso-spacerun: yes"> </span>indicates the end of a perfection, i.e., the second half of the breve; such imperfection of a note by one "two (or more) degrees removed from it in value" (Apel 1953: 111) was called <i>imperfectio ad partem</i><span style='font-style:normal'>). The minim following the dot of division (which also functions as a </span><i>punctum syncopationis</i><span style='font-style:normal'>) would normally imperfect the semibreve it precedes, but since this semibreve precedes another one, the first semibreve of the ligature must be perfect, initiating what will be a long syncopation. And since the second semibreve of the ligature also precedes a semibreve, it too must be perfect, as must be both semibreves in the next ligature. Only with the single semibreve just before the three minims do we arrive at a semibreve that can be imperfected--by the minim that follows it. That leaves the final two minims to "complete" the semibreve begun by the minim following the dot but "interrupted" by the two ligatures. </span></p> <p class=MsoNormal style='text-indent:.5in'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal style='text-indent:.5in'>A chain of syncopations such as those in Example 7 cannot technically be called a proportion, but syncopations of this type serve the same function as true proportions elsewhere in the piece: to explore the ramifications of applying devices for altering normal values in unexpected and "intellectual" ways--such that the writing of melody has now become an exercise in creating a puzzle (and the technique required to sing it has become dependent on the skill needed to solve the puzzle). Examination of the entire complex of voices in the first nineteen measures of <i>Medee fu</i><span style='font-style:normal'> reveals that the rhythmic intricacies of the first five measures actually pale in comparison with the ornamental triplets and cross rhythms between the top voice and the contratenor in mm. 9-18: </span></p> <span style='font-size:10.0pt;font-family:Palatino;mso-ansi-language:EN-US'><br clear=ALL style='page-break-before:always'> </span> <BR> <center> <IMG SRC="Anon - Medee fu.jpg" width=80%> </center> <p class=Style1><span style='font-size:1.0pt'><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p> <p class=Figuredescription align=center style='text-align:center'>Ex. 7. Section A of the ballade <i>Medee fu </i><span style='font-style:normal'>(</span><i>Ch</i><span style='font-style:normal'>, fol. 24v) after Hoppin 1978b, No. 68</span></p> <p class=Figuredescription align=center style='text-align:center'><span style='font-size:12.0pt'><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p> <p class=MsoNormal><i><span style='mso-tab-count:1'> </span>Medee fu</i><span style='font-style:normal'> shows several innovations in melodic decoration: 1) the use of the number 3 to indicate that three minims following are to be performed in the time of two previously, 2) the use of the number 2 to indicate that two (imperfect) semibreves are to be performed in the time of one (perfect one) previously, such that four minims governed by 2 equal three minims of the prevailing mensuration with perfect semibreves. In this work, mensuration signs make one of their earliest appearances, e.g., top voice, m. 12, where the sign </span><span style='font-family:PPalatino-Roman'>O</span> signals that the 6/8 that had been in force up to then now yields to 3/4. The tenor has the sign <span style='font-family:PPalatino-Roman'>[dotted C] </span>prior to its initial clef, to indicate that its mensuration throughout will be <i>prolatio maior</i><span style='font-style:normal'> (perfect semibreves). The contratenor has </span><span style='font-family:PPalatino-Roman'>O</span> alternating at various times with <span style='font-family:PPalatino-Roman'>[dotted C], not to mention the use of 2 for three passages. In addition to O, the top voice uses the sign [dotted C] and the numbers 3 (as defined above) and 4 (four minims in the place of three).</span></p> <p class=MsoNormal align=center style='text-align:center'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <center> <IMG SRC="Mod32r-2.jpg" width=95%> </center> <p class=MsoNormal align=center style='text-align:center'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=Figuredescription align=center style='text-align:center'>Plate 3. Matheus's <i>Le greygnour</i><span style='font-style:normal'> in </span><i>Pit</i><span style='font-style:normal'>, fol. 32r, showing such exotic notational forms as hollow note heads and a variety of mensuration signs.</span></p> <p class=MsoHeader style='tab-stops:.5in 135.0pt 3.5in'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal style='text-indent:.5in;line-height:150%'>Matheus de Perusio's <i>Le greygnour</i><span style='font-style:normal'> approaches a limit of rhythmic complexity beyond which it is doubtful that listeners could comprehend the logic of the melodic motion or that performers could execute the intricate proportions and syncopations. Yet the substantial repertoire of pieces in this style bears witness to the existence of virtuoso performers living in the era of the Ars subtilior capable of performing such works and of patrons who were willing to subsidize their composition and performance. This is all the more remarkable when we consider that Franco had standardized a way to notate simple rhythms with discrete note shapes only a century earlier. In </span><i>Le greygnour</i><span style='font-style:normal'>, Matheus, as an Italian, has juxtaposed the French </span><i>prolacions</i><span style='font-style:normal'> of [C] (outer voices) and </span><span style='font-family:PPalatino-Roman'>[dotted-C]</span> (Contre teneur) as though they were Italian <i>divisiones</i><span style='font-style:normal'> sharing a constant breve.<a style='mso-footnote-id:ftn1' href="#_ftn42" name="_ftnref42" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[42]<![endif]></span></span></a> The result is that two imperfect semibreves in the outer voices (in [C]) are forced to equal two perfect semibreves in the contratenor (in </span><span style='font-family:PPalatino-Roman'>[dotted C]</span>), with a sesquialtera proportion (3 : 2) at the minim level persisting for nearly the entire piece (except where it is eclipsed by even more difficult proportions).</p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal align=center style='text-align:center'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <center> <IMG SRC="Matteo-LeGreygnour1.jpg" width=80%> </center> <p class=MsoNormal align=center style='text-align:center'><span style='font-size:10.0pt'>Ex. 8. Matheus de Perusio, <i>Le greygnour</i></span><span style='font-size:10.0pt'> in <i>Pit</i></span><span style='font-size:10.0pt'>, fol. 32r<o:p></o:p></span></p> <p class=MsoNormal align=center style='text-align:center'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal style='text-indent:.5in;line-height:150%'>The composer has not made things easier by writing values consistently; instead, like Senleches in <i>La harpe de melodie</i><span style='font-style:normal'>, Matheus chooses redundant notations for the same effects, e.g., a white semibreve at the beginning of the top voice to equal a value notated as a dotted semibreve in the contratenor in the passage governed by </span><span style='font-family: PPalatino-Roman'>£ (m. 24, second note). Admittedly, these two note-forms that equal a dotted semibreve occur in different mensurations: [C] </span>in the top voice at the opening of the piece, <span style='font-family:PPalatino-Roman'>£</span><span style="mso-spacerun: yes"> </span>in the contratenor at m. 24. Nevertheless, it is an inconsistency at the beginning of the piece to follow the Italian principle of avoiding dots of addition, using instead a special note form (white semibreve and later white minims), and then later to abandon the Italian approach and use a French-style dot of addition in the contratenor at m. 24. </p> <p class=MsoNormal style='line-height:150%'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal style='line-height:150%'><span style='mso-tab-count:1'> </span>Notational redundancies, however, are not the most difficult aspect of this piece. Matheus saves the <i>coup de grace</i><span style='font-style:normal'> of "coloring" the coloration (red notes yielding to hollow red ones) for the ballade's third section, where red hollow notes appear in the top voice following a passage of extended syncopation involving notes of irregular values, almost none of which occurs on the beat (see Example 9, showing measures 51-64). With the arrival of proportions caused by coloration and by coloration of coloration (top voice, mm. 59-61), i.e., large triplets consisting of three faster triplets (resulting in nine fast minims per measure), one might have expected the melodic motion ot become more flowing. It doesn't. The syncopations continue on, with scarcely any of the longer notes coinciding with the beat. (It should be noted that the fifth red-hollow note, which has the appearance of a semibreve in <I>It</I>, is in actuality a minim with its tail either missing or too faded to see. There are no other sources of this work to serve as a check, but the context alone demonstrates the necessity for the note in question to be a red-hollow minim.)<a style='mso-footnote-id: ftn43' href="#_ftn43" name="_ftnref43" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[43]<![endif]></span></span></a> </span></p> <p class=MsoNormal style='line-height:150%'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <center> <IMG SRC="Matteo-LeGreygnourDET2.jpg" width=80%> </center> <p class=MsoNormal align=center style='text-align:center'><span style='font-size:10.0pt'>Ex. 9. Matheus de Perusio, <i>Le greygnour</i>,</span><span style='font-size:10.0pt'> mm. 51-64<o:p></o:p></span></p> <p class=MsoNormal style='line-height:150%'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal style='line-height:150%'><span style='mso-tab-count:1'> </span>In addition to the coloration of coloration, it should be noted that Matheus writes three bi-colored ligatures: a c.o.p.<a style='mso-footnote-id:ftn44' href="#_ftn44" name="_ftnref44" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[44]<![endif]></span></span></a> in the contratenor (governed by <span style='font-family:PPalatino-Roman'>[dotted C]</span>, making semibreves perfect) in which the first half is red (to imperfect the semibreve that "similis ante similem" would require to be perfect), and a c.o.p. in the tenor (governed by C, in which semibreves are imperfect) in which the first half is white (hollow), causing it to increase by half its value (like the first note of the top voice). (In normal Ars nova French notation, such a ligature would have been written with a dot after the first note.) There are two such ligatures in the tenor, one ascending (m. 16) and one descending (m. 60). The third bi-colored ligature also occurs in the tenor, at mm. 51-53. In this case a porrectus with ascending initial stem (c.o.p., making the first two notes semibreves) has its third note flipped to the right, which would normally make this note a breve. However, the first half of this breve is written as white, the second half as black, such that the white half gains half again of its value, i.e., dotted semibreve plus undotted semibreve. The result is a note that in modern notation would be written as a dotted quarter note tied to a quarter--or, more likely, in a transcription with bar lines, a quarter note tied to a dotted quarter.</p> <h1>From black to red to white</h1> <p class=MsoNormal><span style='mso-tab-count:1'> </span>We end the part of our survey concerned with Ars subtilior notation by examining a well known work that probably dates from ca. 1420, though no precise dating has yet proved possible: the famous "Heart Song" by Baude Cordier, <i>Belle, bonne, sage</i><span style='font-style:normal'>.<a style='mso-footnote-id:ftn45' href="#_ftn46" name="_ftnref46" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[46]<![endif]></span></span></a> In this work, notated in part in the "modern" mensuration of [cut-circle], we see all three colors of the title of our study: black notation, in which are inserted red notes, plus two brief passages in "white" notation--that is, black hollow notes, which have the same appearance as the "white" notation of </span><i>Ox</i><span style='font-style:normal'>, dating from the 1430s (the very notation that would eventually cause the abandonment of black notation altogether).<a style='mso-footnote-id:ftn47' href="#_ftn47" name="_ftnref47" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[47]<![endif]></span></span></a> As in Matheus's </span><i>Le greygnour</i><span style='font-style:normal'>, in </span><i>Belle, bonne, sage</i><span style='font-style:normal'> the white notes are not an alternate form of black notation, as they are in </span><i>Ox</i><span style='font-style:normal'>, but rather a means of indicating a sesquitertial proportion (4 : 3) to the notes just preceding them. <a style='mso-footnote-id:ftn48' href="#_ftn48" name="_ftnref48" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[48]<![endif]></span></span></a></span></p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <BR> <center> <IMG SRC="CordierBelleBonne.jpg" width=80%> </center> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=Figuredescription align=center style='text-align:center'>Plate 4. Baude Cordier's rondeau, <i>Belle, bonne, sage</i><span style='font-style:normal'>, (</span><i>Ch</i><span style='font-style:normal'>, fol. 11v)</span></p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal>As can be seen in Ex. 10 (or in the source itself, Plate 4, which has been reproduced countless times in books, journals, advertisements, and on tote-bags and t-shirts), Cordier introduces one proportion after another, beginning in the contratenor, which after just four notes changes from the initial mensuration (not indicated) to <span style='font-family:PPalatino-Roman'>[dotted C] for the rest of the piece</span>. The contrapuntal context prescribes that a semibreve preceding <span style='font-family:PPalatino-Roman'>[dotted C]</span> will now be equaled by a minim following <span style='font-family:PPalatino-Roman'>[dotted C]</span>, confirming that <span style='font-family:PPalatino-Roman'>[dotted C]</span> is in a <i>proportio dupla</i><span style='font-style:normal'> to the opening mensuration. What might the opening mensuration be? The only concrete clue is to be found well along in the top voice, following the second group of notes in white notation, where Cordier writes the sign <![if !vml]><img width=12 height=12 src="Blachly-FromBlackToRedWEB_files/image043.jpg" v:shapes="_x0000_i1046"><![endif]>, with semibreves equal to semibreves in the tenor (which equal minims in </span><span style='font-family:PPalatino-Roman'>[dotted C]</span> in the contratenor). This seems to imply that the notes of the tenor, the opening notes of the contratenor, and all the notes of the top voice not governed by <span style='font-family:PPalatino-Roman'>[dotted C] </span>or some other sign are to be understood to be in [<![if !vml]><img width=12 height=12 src="Blachly-FromBlackToRedWEB_files/image044.jpg" v:shapes="_x0000_i1047"><![endif]>]. Is this sign the same as the signs [diagonal cut circle] or [vertical cut circle], which are found in many sources dating from the 1430s and later? It would seem so, for the relationship of [diagonal cut circle] or [vertical cut circle] to [dotted-C] would soon become common in vertical contexts, where it always signifies a semibreve in cut circle equal to a minim in [dotted C] <a style='mso-footnote-id:ftn49' href="#_ftn49" name="_ftnref49" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[49]<![endif]></span></span></a> (just as <![if !vml]><img width=12 height=12 src="Blachly-FromBlackToRedWEB_files/image045.jpg" v:shapes="_x0000_i1048"><![endif]><span style="mso-spacerun: yes"> </span>does here in its horizontal and vertical relationships with [dotted-C]). The angle of the stroke through the circle, in other words, seems not to affect the sign's meaning:<a style='mso-footnote-id:ftn50' href="#_ftn50" name="_ftnref50" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[50]<![endif]></span></span></a> cut circle is cut circle whatever the angle of the stroke.</p> <BR> <center> <IMG SRC="Cordier - Belle, bonne1.jpg" width=80%> </center> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=Figuredescription align=center style='text-align:center'>Ex. 10. Opening of Baude Cordier's rondeau, <i>Belle, bonne, sage </i></p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span>It appears, then, that both the tenor and contratenor progress essentially in [<![if !vml]><img width=12 height=12 src="Blachly-FromBlackToRedWEB_files/image046.jpg" v:shapes="_x0000_i1049"><![endif]>] (or its equivalent), with red breve hemiola in the tenor and red semibreve hemiola in the contratenor (equal in duration to the breve hemiola in the tenor). These two lower voices provide a stable, uncomplicated foundation for the proportional ornaments of the top voice. The top voice, which is evidently, as we have seen, in [<![if !vml]><img width=12 height=12 src="Blachly-FromBlackToRedWEB_files/image047.jpg" v:shapes="_x0000_i1050"><![endif]>], has six minims to the breve (the semibreve is imperfect). The first red notes--a group of semibreves and minims--are equal to nine red minims in all, and therefore are to be interpreted as minim triplets, with three red minims equaling two of the previous black ones. Following the red triplets Cordier has written the sign <span style='font-family:PPalatino-Roman'>[dotted C], where, just as in the contratenor, minim in [dotted C] equals semibreve in </span>[<![if !vml]><img width=12 height=12 src="Blachly-FromBlackToRedWEB_files/image048.jpg" v:shapes="_x0000_i1051"><![endif]>]. The three white semibreves and two white minims in the group of white notes that comes next are equal to eight white minims altogether, suggesting that four white minims will replace three black minims under [dotted C] in the passage just discussed, to create a sesquitertia proportion. This is confirmed by the contrapuntal context. In the long passage in <![if !vml]><img width=12 height=12 src="Blachly-FromBlackToRedWEB_files/image049.jpg" v:shapes="_x0000_i1052"><![endif]><span style="mso-spacerun: yes"> </span>that follows the second group of white notes, the only proportion is one hemiola at the breve level, notated as three red breves toward the beginning of the second staff. About three-quarters of the way through the second staff, there occurs the sign <span style='font-family: PPalatino-Roman'>[dotted C]</span>, its minim, minim rest, and two semiminims corresponding in value to like shapes in the section of the contratenor governed by <span style='font-family:PPalatino-Roman'>[dotted C]</span> (that is, all of it, except for the first four notes). Following this one semibreve's worth of notes, there occurs the number 3 followed by three semibreve-minim groups (=nine minims), such that each group is equal to a minim just preceding, governed by <span style='font-family:PPalatino-Roman'>[dotted C]</span>. The penultimate "measure" of the piece, coming just before the final note, has an 8 over 9 written just before a group of notes that equals eight minims in all: three minims, a c.o.p., and another minim. These eight minims as a group are to equal the preceding nine minims governed by 3. Though difficult to perform precisely against the three beats in this measure sounding in the lower voices, the fact that the eight minims occur in the penultimate measure, where one might expect a slight slowing down in the approach to the final cadence, makes them sound more like part of a Chopinesque rubato than anything comparable to the extreme effects of syncopation and proportional friction in Matheus's <i>Le greygnour</i><span style='font-style:normal'>.</span></p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span>Baude Cordier's two most famous works--namely, <i>Belle, bonne, sage</i><span style='font-style:normal'>, notated on staves in the shape of a heart, and </span><i>Tout par compas</i><span style='font-style:normal'> (the Circle Canon notated on staves drawn as two concentric circles)--have in recent years become emblems of the Ars subtilior aesthetic.<a style='mso-footnote-id:ftn51' href="#_ftn51" name="_ftnref51" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[51]<![endif]></span></span></a> Musically, though, they usher in a new era of tunefulness, even if both pieces have scarcely a measure that doesn't fleetingly recall the now-receding Ars subtilior with one or another of that period's characteristically intricate proportions. </span><i>Belle, bonne, sage</i><span style='font-style:normal'> announces its novel sound, significantly, not by a proportion but by exact imitation in all three voices--an approach to composition that would not be explored again until the chansons of Antoine Busnoys in the 1450s and '60s. The presence of seven additional chansons by Cordier (but not either <I>Belle, bonne, sage</I> or <I>Tout par compas</I>) in <I>Ox</I>, a source that contains music written as late as the 1430s (see below, note 53), argues further for Cordier's place as a transitional figure at the end of the Ars subtilior and the beginning of the Binchois-Du Fay era.<a style='mso-footnote-id:ftn52' href="#_ftn52" name="_ftnref52" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[52]<![endif]></span></span></a></span></p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span>In fact, Cordier's most famous pieces prefigure a new aesthetic of flowing melody, sonorous counterpoint, and balanced phrases--a turn away from the barely performable complexities and extreme effects made possible by mannered notation. Though difficult proportions would sporadically appear during the rest of the fifteenth century--witness the passage of 9 against 4 at "Qui cum Patre" in the Credo of Du Fay's <i>Missa Sancti Anthonii de Padua</i><span style='font-style:normal'> ([circle over 3]</span><span style='font-family: PPalatino-Roman'> against [cut C]), </span>or the rhythmic complexities in the Credo of the same composer's <i>Missa L'homme armé</i><span style='font-style:normal'>, or the proportional friction in his five-voice motet </span><i>Ecclesie militantis</i><span style='font-style:normal'>, not to mention certain passages in the sacred works of Busnoys, Josquin, and others--the heyday of such complexities was now over. Composers would not show an interest in exploring similarly arcane rhythmic effects for another half millenium.</span></p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <h1>The writing of proportions in white notation </h1> <p class=MsoNormal><span style='mso-tab-count:1'> </span> Yet this is not to say that composers had lost all interest in musical proportions. We have mentioned already such early fifteenth-century works with overall proportional structures as Grenon's <I>Ave virtutum/Prophetarum</I> (8 : 6 : 4 : 2) and Du Fay's <I>Nuper rosarum flores</I> (6 : 4 : 2 : 3). Johannes de Sarto's three-voice Benedicamus Domino trope, <I>Verbum patris hodie</I> (<I>Ox</I>, fol. 12v-13), probably dating from the mid-1430s, <a style='mso-footnote-id:ftn53' href="#_ftn53" name="_ftnref53" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[53]<![endif]></span></span></a> exhibits formidable notational challenges based on a variety of proportional relationships, although its relatively simple musical style poses no hurdles for performers who can decipher the accompanying rubrics; indeed, it serves as a good example of the new "flowing" and simplified style of melody.</span></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span> <p class=MsoNormal><span style='mso-tab-count:1'> </span> Before we examine <I>Verbum patris hodie</I> more closely, a word is in order concerning some of the differences between black and white notation. Aside from a handful of motets by Ciconia, Zacara, Antonio Romano, and Huberto de Salinis, all of which are in black notation, the remainder of <I>Ox</I> is in white notation, with black coloration. <I>Ox</I> shows the standard way scribes transformed a work originally notated in black notation with red coloration into white notation: the black notes became white (black-void) and the red notes became solid black. <a style='mso-footnote-id:ftn54' href="#_ftn54" name="_ftnref54" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[54]<![endif]></span></span></a> One drawback to white notation was that minims in coloration, now written as black minims, were indistinguishable from black semiminims elsewhere. Only context could establish the correct reading. The reason for changing from black to white notation in the first place is examined more fully elsewhere in this volume by Graeme Boone.<a style='mso-footnote-id:ftn55' href="#_ftn55" name="_ftnref55" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[55]<![endif]></span></span></a> Suffice it to say here that with solid black notes limited to those in coloration, there was less ink in white notation to eat through the paper now being used for musical manuscripts such as <I>Ox</I> (replacing the much more expensive choice of parchment, which had been used exclusively for musical manuscripts prior to the 1430s). The contemporary paper manuscript Bologna Q 15, now full of holes from the acidic ink of the black notation, provides a sobering lesson in the danger of not changing to white notation when notating music on paper. </span></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span> <p class=MsoNormal><span style='mso-tab-count:1'> </span> Returning now to <I>Verbum patris hodie</I>, we observe that the first of the work's two <I>partes</I> juxtaposes a top voice in O against a contratenor in [dotted-C]. The rubric accompanying the contratenor, though of somewhat questionable Latin, clarifies the relationship: "Prima pars huius incrementum in proportione dupla" ("the first part of this [voice-part] is augmented in duple proportion [with respect to the cantus]"). Thus, a minim of [dotted-C] in the contratenor equals a semibreve of O in the cantus. </span></p> <BR> <center> <IMG SRC="SartoVerbum1.jpg" width=80%> </center> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=Figuredescription align=center style='text-align:center'>Ex. 11. Opening of Johannes de Sarto's Benedicamus trope, <i>Verbum patris hodie </i></p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> </span></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span> The tenor is supplied with arabic numbers over mensuration signs at the beginning of both the prima and the secunda pars. It, too, has an explanatory rubric. As Bonnie Blackburn has deciphered the abbreviations, it reads, "Canitur per 9/3. figure allegorismi [recte algorismi] ponuntur pro modis necnon circuli pro temporibus,"<a style='mso-footnote-id:ftn56' href="#_ftn56" name="_ftnref56" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[56]<![endif]></span></span></a> which translates roughly as "It is to be sung in 9 over 3, with the arabic numbers denoting the modes and with the circles, in like manner, denoting the tempora." Thus, in the 9 over 3 over O at the beginning of the prima pars, the 9 indicates that the tenor opens in perfect major modus, the 3 shows that minor modus is perfect, and the circle shows that <I>tempus</I> is perfect; thus, the maxima is perfect, the long is perfect, and the breve is perfect. What the rubric fails to indicate is that the tenor notes must be diminished with respect to the cantus by a duple proportion, with a perfect long in the tenor equaling a perfect breve in the cantus (and equaling a perfect semibreve in the contratenor). </span></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span> <p class=MsoNormal><span style='mso-tab-count:1'> </span> For the <I>secunda pars</I> of <I>Verbum patris hodie</I>, the cantus changes from O to [cut-circle], while the contratenor changes from [dotted-C] to C. The tenor has the numeral 2 written over the mensuration sign [dotted-circle], which indicates imperfect minor <I>modus</I>, perfect <I>tempus</I>, and major prolation, i.e., that the long is imperfect, the breve is perfect, and the semibreve is perfect. </span></p> <BR> <center> <IMG SRC="SartoVerbum2.jpg" width=80%> </center> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=Figuredescription align=center style='text-align:center'>Ex. 12. End of <I>prima pars</I> and beginning of <I>secunda pars</I> of Sarto's <i>Verbum patris hodie </i> </p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span> The relationships are: imperfect long in the tenor equals two perfect breves in the cantus governed by [cut-circle] (thus, tenor breve equals cantus breve). In the contratenor, governed by C, a semibreve plus a minim equals a cantus perfect breve. In fact, the contratenor shares the same rhythmic profile as the cantus and tenor (i.e., what looks like modern 3/2 in the transcription), but is written in C so as to nullify the "similis ante similem" rule at m. 48 and elsewhere. The juxtaposition of [cut-circle] in the cantus against C in the contratenor seems to require either that the contratenor be augmented in duple proportion with respect to the cantus or that the cantus be diminished in duple proportion with respect to the contratenor, such that contratenor minim equals cantus semibreve. In fact, all that is specified is that a semibreve in [cut-circle] should equal a minim in C. How fast the <I>secunda pars</I> should move with respect to the <I>prima pars</I> (the relationship of [cut-circle] to O) is a separate question. </span></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span> <p class=MsoNormal><span style='mso-tab-count:1'> </span> However, now that we have established the vertical proportional relationship of [cut-circle] to C, let us address the issue of tempo, that is, the horizontal relationship of [cut-circle] to O. While the French fourteenth-century principle of equal minims among the prime mensuration signs would require that a semibreve in the cantus's O in the <I>prima pars</I> equal a semibreve in the contratenor's C in the <I>secunda pars</I> (resulting in a doubling of tempo in the cantus from O in the <I>prima pars</I> to [cut-circle] in the <I>secunda pars</I>), the flowing, straightforward nature of the music, which hardly changes from <I>prima pars</I> to <I>secunda pars</I>, indicates that some other relationship must obtain, since a duple proportion between O and [cut-circle] in a piece in such a simple musical style--virtually the same in both the <I>prima</I> and <I>secunda partes</I>--would lead to a musical absurdity.<a style='mso-footnote-id:ftn57' href="#_ftn57" name="_ftnref57" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[57]<![endif]></span></span></a> We can hardly consider absurdly slow or fast music to be an appropriate effect for the musical finale of a Christmas service.<a style='mso-footnote-id:ftn58' href="#_ftn58" name="_ftnref58" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[58]<![endif]></span></span></a> Charles Hamm's suggestion that in such music a change from O to [cut-circle] might mean nothing more than a slight increase of tempo (which Rob Wegman would later aptly characterize as a <I>più mosso</I>)<a style='mso-footnote-id:ftn59' href="#_ftn59" name="_ftnref59" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[59]<![endif]></span></span></a> seems the most likely intended meaning here, horizontal duple proportion having been ruled out. In that case, the sign [cut-circle] has a double function: it causes a vertical duple proportion relationship with C in the contratenor, but it causes only a slight change of tempo with the O of the cantus in the <I>prima pars</I>. In this interpretation, the sign [cut-circle] causes the notes in the contratenor in C to adjust to it: [cut-circle] represents the determining sign, to which the others must adjust. Thus, the stroke through the sign [cut-circle], acting as a marker of tempo, controls the interpretation of prime signs with which it is in a proportional relationship.<a style='mso-footnote-id:ftn60' href="#_ftn60" name="_ftnref60" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[60]<![endif]></span></span></a> </span></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span> <p class=MsoNormal><span style='mso-tab-count:1'> </span> It must be said that the stroke through a O or C produced considerable confusion among fifteenth-century theorists, most of whom shied away from discussing the issue, and among sixteenth-century theorists as well, many of whom wished to have the stroke considered solely as a sign of proportion (e.g., Sebald Heyden in his <I>De arte canendi</I> of 1540).<a style='mso-footnote-id:ftn61' href="#_ftn61" name="_ftnref61" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[61]<![endif]></span></span></a> Tinctoris, though, in an aside in a discussion of proportion signs in his treatise <I>Proportionale musices</I>,<a style='mso-footnote-id:ftn62' href="#_ftn62" name="_ftnref62" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[62]<![endif]></span></span></a> clarified the situation when he defined the stroke as a sign of "acceleratio mensure," (acceleration of the beat), insisting repeatedly that it was not to be considered a sign of proportion except when it stood in a vertical relationship with a prime sign. Pieces like <I>Verbum patris hodie</I> confirm the accuracy of Tinctoris's description. As full as this piece is of proportional relationships, the one relationship which cannot be proportional is the horizontal one between the cantus's O in the prima pars and the cantus's [cut-circle] in the <I>secunda pars</I>; for this relationship, only a <I>più mosso</I> produces a satisfactory result. </span></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span> <p class=MsoNormal><span style='mso-tab-count:1'> </span> What obtains for the horizontal relationship between O and [cut-circle] in <I>Verbum patris hodie</I> also obtains for many horizontal relationships between O and [cut-circle] or between O and [cut-C] later in the fifteenth century (e.g., for Busnoys's <I>In hydraulis</I> of ca. 1464, with a <I>prima pars</I> in O and a <I>secunda pars</I> in [cut-C], and in innumerable Mass movements in more than one section that are signed similarly). To notate true proportions, fifteenth-century composers normally employed mensuration signs in vertical relationships or used signs specifically designed to indicate precise mathematical relationships, such as [reverse-C], which had been inherited from the fourteenth century. [Reverse-C] always indicated sesquitertia (4 : 3) whenever it followed any type of triple mensuration, but whether at the level of the minim or the semibreve had to be determined from the context. </span></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span> <p class=MsoNormal><span style='mso-tab-count:1'> </span> Our survey ends with an early work by Guillaume Du Fay, in which he hit upon an ingenious way to indicate a mathematically exact horizontal proportion in his second setting of the Marian antiphon <I>Alma redemptoris mater</I> (<I>ModB</I>, fol. lvii'-lviii).<a style='mso-footnote-id:ftn63' href="#_ftn63" name="_ftnref63" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[63]<![endif]></span></span></a> The notation includes no mensuration or proportion signs at all. Instead, Du Fay has written the central section entirely in coloration in all three voices, implying that the opening and closing sections, which are unsigned, must be in either O or [cut-circle]--probably the latter, because the melodic motion suggests a fast-flowing tempo, with one beat per breve. The middle section displays perfect <I>modus</I> (longs are perfect), as can be determined from various dots of division and altered breves in the three voices. Coloration, therefore, affects only the breve, thereby defining breves in the central section that move one-third faster than the white breves in the first and last sections. </span></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span> <BR> <center> <IMG SRC="Du Fay - Alma II-3.jpg" width=80%> </center> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=Figuredescription align=center style='text-align:center'>Ex. 13. End of the first section, entire second section,<BR> beginning of third section of the Du Fay's <i>Alma redemptoris mater II </i></p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span> Using only coloration, Du Fay has notated an unusual relationship: a horizontal rather than a vertical hemiola, in which three black breves in the middle section of the piece equal two white breves in the sections before and after. The black notes in the middle section do not cut across a beat felt elsewhere in other voices sounding at the same time. Instead the beat itself has changed. A change of (the speed of) the beat from one section of a piece to another, here notated unequivocally, seems implied as well in other works by Du Fay from roughly the same time, the prime case in point being Du Fay's three-voice setting of Petrarch's canzona <I>Vergene bella</I> (<I>Ox</I>, fol 133v-134). This latter work opens in [[cut-circle]], changes to O for the middle section, and closes in [cut-circle], evidently the implied sign of the opening section. Though the mensurations change from section to section in a manner similar to what happens in <I>Alma redemptoris mater II</I>, it seems unlikely that Du Fay had a similarly hemiola-like tempo relationship in mind, because the motion of the first section of <I>Vergene bella</I> seems to require a beat on the semibreve rather than the breve. Thus, the middle section of <I>Vergene bella</I> receives a slower beat than the first and last sections. In <I>Alma redemptoris mater II</I> the reverse is the case: the beat on the black breve in the middle section is faster than the beat on the white breve in the first and last sections. In response to the question of why Du Fay didn't write the middle passage of <I>Alma redemptoris II</I> in O while maintaining white notation, one answer might be that he wanted a different relationship than in <I>Vergene bella</I>: a true horizontal proportion rather a <I>meno mosso</I> or <I>più mosso</I> produced by a horizontal relationship between cut and uncut signatures. </span></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span> <p class=MsoNormal style='line-height:normal'>Alexander Blachly</p> <p class=MsoHeader style='line-height:normal;tab-stops:.5in 135.0pt 3.5in'>University of Notre Dame</p> <span style='font-size:12.0pt;font-family:Palatino;mso-ansi-language:EN-US'><br clear=ALL style='page-break-before:always'> </span> <p class=MsoNormal align=center style='text-align:center'>Sources</p> <p class=Bibliography align=center style='text-align:center'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=Bibliography><i>Ca11<span style='mso-tab-count:1'> </span></i><span style='font-style:normal'>Cambrai, Médiathèque Municipale, MS 11</span></p> <p class=Bibliography><i><![if !supportEmptyParas]> <![endif]><o:p></o:p></i></p> <p class=Bibliography><i>Ch</i><span style='font-style:normal'><span style='mso-tab-count:1'> </span>Codex Chantilly, Bibliothèque du château de Chantilly, MS 564 (</span><i>olim</i><span style='font-style:normal'> 1047)</span></p> <p class=Bibliography style='margin-left:0in;text-indent:0in'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=Bibliography><i>ChiC</i><span style='font-style:normal'><span style='mso-tab-count:1'> </span>Chicago, Newberry Library, MS 54.1</span></p> <p class=Bibliography><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=Bibliography><i>Fauv</i><span style='font-style:normal'><span style='mso-tab-count:1'> </span>Paris, Bibliothèque nationale, fond fr. 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Warren, "Brunelleschi's Dome and Dufay's Motet," <i>Musical Quarterly</i><span style='font-style:normal'> 59 (1973), 92-102</span></p> <p class=Bibliography><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=Bibliography>Wegman 1992<span style='mso-tab-count:1'> </span>Rob Wegman, "What Is 'acceleratio mensurae'?" <i>Music and Letters</i><span style='font-style:normal'> 73 (1992), 515-524</span></p> <p class=Bibliography><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=Bibliography>Wolf 1919<span style='mso-tab-count:1'> </span>Johannes Wolf, <i>Handbuch der Notationskunde</i><span style='font-style:normal'>, Vol. 1 (Leipzig, 1919)</span></p> <p class=Bibliography><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=Bibliography>Wright 1979<span style='mso-tab-count:1'> </span>Craig Wright, <i>Music at the Court of Burgundy 1364-1419: A Documentary History</i><span style='font-style:normal'> (Henryville, Ottawa, Binningen: Institute of Mediaeval Music, Ltd., 1979)</span></p> <p class=Bibliography><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=Bibliography>Wright 1986<span style='mso-tab-count:1'> </span>Craig Wright, "Leoninus, Poet and Musician," <i>Journal of the American Musicological Society</i><span style='font-style:normal'> 39 (1986), 1-35</span></p> <p class=Bibliography><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=Bibliography>Wright 1994<span style='mso-tab-count:1'> </span>Craig Wright, "Dufay's <i>Nuper rosarum flores</i><span style='font-style:normal'>, King Solomon's Temple, and the Veneration of the Virgin," </span><i>Journal of the American Musicological Society</i><span style='font-style:normal'> 47 (1994), 395-441</span></p> <p class=MsoNormal style='margin-left:117.0pt;text-indent:-117.0pt;line-height: normal;tab-stops:135.0pt 3.5in'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal style='margin-left:117.0pt;text-indent:-117.0pt;line-height: normal;tab-stops:135.0pt 3.5in'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <h1>Postscript: challenges for the performer</h1> <p class=TextTranslation style='line-height:150%;tab-stops:.5in 135.0pt 3.5in'><span style='mso-tab-count:1'> </span>The following analysis will demonstrate the sorts of difficulties a piece like Biteryng's <i>En Katerine</i><span style='font-style:normal'> poses for its performers, while simultaneously clarifying the important role of number in every aspect of the musical conception. The tenor, as mentioned in the earlier discussion, is to be sung three times--with its notes in each new statement of the </span><i>color </i><span style='font-style:normal'>accelerating in speed. The upper voices change mensuration twice as well; while they operate somewhat independently of the tenor, their changes of mensuration affect the interpretation of the tenor's notes. The upper voices each show mensuration signs--C at the beginning for the voice that sings the "En Katerine" text, O at the beginning for the voice that sings the "Virginalis concio" text--but no such signs are provided for the Tenor, which must determine its relationship to the upper voices on the basis of context. The only notational devices in the tenor beyond the shapes of notes and rests, the clef, the signature of one flat, and a custos at the end (showing that the repeat of the melody will begin on the note on which it ends), are the line through the whole staff to indicate the end of the full </span><i>color</i><span style='font-style:normal'>, and, following that, a triple-repeat mark, :|||: , from which we learn that the tenor is to be sung three times (see Plate 5, below).</span></p> <p class=TextTranslation style='line-height:150%;tab-stops:.5in 135.0pt 3.5in'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=TextTranslation style='line-height:150%;tab-stops:.5in 135.0pt 3.5in'><span style='mso-tab-count:1'> </span>The tenor begins with three breve rests stacked on top of one another, indicating that the long is perfect. With the two upper voices in this first section of the work sounding O against C simultaneously, a first guess as to the implied mensuration of the tenor might be [dotted-C], but this proves to be incorrect, for the opening tenor note F will then enter after its initial rests in the middle of a "measure" (with reference to a transcription); and, as the piece progresses, it will become increasingly clear, because of the resulting dissonance, that [dotted-C] is not the right choice. The correct implied mensuration turns out to be [O]. Since the implied sign of [O] denotes a perfect breve worth 3 semibreves, this will require each perfect long worth 3 breves to have a total value of 9 semibreves.</p> <p class=TextTranslation style='line-height:150%;tab-stops:.5in 135.0pt 3.5in'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=TextTranslation style='line-height:150%;tab-stops:.5in 135.0pt 3.5in'><span style='mso-tab-count:1'> </span>In order to make sense of the tenor's sequence of note shapes, we need to look at the notation globally, and not simply proceed note by note. What we see in Plate 5 is: perfect long rest, perfect long, breve, breve rest, perfect long,<a style='mso-footnote-id:ftn64' href="#_ftn64" name="_ftnref64" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[64]<![endif]></span></span></a> perfect long rest, a ligature consisting of breve, altered breve,<a style='mso-footnote-id:ftn65' href="#_ftn65" name="_ftnref65" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[65]<![endif]></span></span></a> perfect long, and then a single breve. At this point a passage in coloration is introduced, suggesting that we have reached the end of a perfection. The sequence of black notes and rests thus contains six perfect longs, plus two isolated breves "completed" by the last uncolored note--the "missing" breve--for a total of seven. The notation shows great sophistication and economy. The first breve, which is followed by a breve rest, has no dot between it and the breve rest. Had there been such a dot, the first breve would have imperfected the long it follows. The breve rest, though, would still not have been able to imperfect the long it precedes, because that long in turn precedes another long and thus must be perfect because of "similis ante similem." Lacking a dot of division, two breves between two perfect longs would normally require the second breve to be altered, but this is prevented here because rests have a constant and absolute value. Since rests cannot be altered, we are forced to displace all following perfect longs until the first breve and the breve rest that follows it can be "completed" as a perfect long by the single breve that is written just before the red notes.</p> <p class=TextTranslation style='line-height:150%;tab-stops:.5in 135.0pt 3.5in'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=TextTranslation style='line-height:150%;tab-stops:.5in 135.0pt 3.5in'><span style='mso-tab-count:1'> </span>The next five notes are written in red ink, indicating that they lose a certain amount of value. Since the black notes in the tenor are pefect at the level of both long and breve, coloration (red ink) suggests several possibilities, e.g., imperfection at the level of the long, the breve, or both. The dot of division, however, dividing the first two red notes from the three following, indicates that longs are still perfect; by default, therefore, the coloration affects only the breves. The values in coloration are long, breve, dot of division, ligature consisting of breve, altered breve, long. Since all the red breves are imperfected by coloration, colored longs have a value of 6 semibreves. Note that the implied semibreves in both black and red notation remain equal to one another ("implied semibreves," because the tenor has no notes smaller than a breve) (see Plate 5).</p> <p class=TextTranslation style='line-height:150%;tab-stops:.5in 135.0pt 3.5in'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=TextTranslation style='line-height:150%;tab-stops:.5in 135.0pt 3.5in'><span style='mso-tab-count:1'> </span>The tenor now repeats all the values of notes and rests thus far, first with black notes, then with red ones, thereby showing that each tenor statement (<i>color</i><span style='font-style:normal'>) comprises two </span><i>taleae</i><span style='font-style:normal'>. The Gregorian chant serving as the source of the tenor's pitches, a fragment labeled "Sponsus amat sponsam,"<a style='mso-footnote-id: ftn66' href="#_ftn66" name="_ftnref66" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[66]<![endif]></span></span></a> extends, as a single 24-note melody with no internal repetitions of pitch groups, over the entire length of each tenor statement. </span></p> <p class=TextTranslation style='line-height:150%;tab-stops:.5in 135.0pt 3.5in'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=TextTranslation style='line-height:150%;tab-stops:.5in 135.0pt 3.5in'><span style='mso-tab-count:1'> </span>While the "displacement" in the black notes pointed out above--when two isolated breves need to wait to be completed by a single breve following four perfect longs, with those four perfect longs straddling barlines--causes a slight feeling of asymmetrical dislocation, both the displacement and the ensuing coloration of five notes in each talea are easily negotiated in this first section of the motet, where the Tenor is governed by [O]. In this section, the mensurations O in the middle voice and C in the top voice both imply a beat on the semibreve. Assuming that this beat was transmitted by touch, as the singers stood with arms around their neighbors' shoulders and tapped to keep time (or, alternatively, as a singer marked time by tapping the book from which they sang),<a style='mso-footnote-id:ftn67' href="#_ftn67" name="_ftnref67" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[67]<![endif]></span></span></a> the upper voices would understand each tap to correspond to the value of one semibreve. This would be the understanding of the singer(s) of the tenor as well, who would accordingly rest for nine taps before first entering on the note F.</p> <p class=TextTranslation style='line-height:150%;tab-stops:.5in 135.0pt 3.5in'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=TextTranslation style='line-height:150%;tab-stops:.5in 135.0pt 3.5in'><span style='mso-tab-count:1'> </span>Counting taps, the tenor singer(s) would understand each <i>talea</i><span style='font-style:normal'> in their part to have the following values:</span></p> <p class=TextTranslation style='line-height:150%;tab-stops:.5in 135.0pt 3.5in'><span style='font-size:5.0pt'><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p> <p class=TextTranslation style='line-height:normal;tab-stops:.5in 49.5pt 67.5pt 81.0pt 94.5pt 1.5in 121.5pt 130.5pt 135.0pt 139.5pt 153.0pt 166.5pt 175.5pt 189.0pt 2.75in 207.0pt 3.25in 243.0pt 3.5in'><span style='mso-tab-count:1'> </span>9<span style="mso-spacerun: yes"> </span><span style='mso-tab-count:1'> </span>9<span style='mso-tab-count:1'> </span>3<span style='mso-tab-count: 1'> </span>3<span style='mso-tab-count:1'> </span>9<span style='mso-tab-count:1'> </span>9<span style='mso-tab-count:1'> </span>3<span style='mso-tab-count:1'> </span>6<span style='mso-tab-count:1'> </span>9<span style='mso-tab-count:1'> </span>3<span style='mso-tab-count:1'> </span><span style='color:red'>4<span style='mso-tab-count:1'> </span>2<span style='mso-tab-count:1'> </span>2<span style='mso-tab-count:1'> </span>4<span style='mso-tab-count:1'> </span>6<span style="mso-spacerun: yes"> </span></span><span style='font-size:9.0pt'>(these values to be repeated in the second half of the color)<o:p></o:p></span></p> <center> <IMG SRC="SponsusTenor.jpg" width=80%> </center> <p class=Figuredescription align=center style='text-align:center'>Plate 5. Tenor of Biteryng's <i>En Katerine solennia</i><span style='font-style:normal'> with values of notes indicated above.</span></p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal>Note that the first black long is not imperfected by the breve following for the reasons mentioned above, whereas the first red long is imperfected by the red breve that follows it. Both the black and the red three-note ligatures require their second note to be altered. The total number of beats in each <i>talea</i><span style='font-style:normal'> is, thus, 63 for the black notes plus 18 for the red ones, for a total of 81.</span></p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span>At the end of the first full statement of the tenor, both upper voices change mensurations, suggesting that the tenor should change as well, as it prepares to sound the second statement of its <i>color</i><span style='font-style:normal'>. The top voice, which had been in C, changes to [dotted-C], while the middle voice, which had been in O, changes to [dotted-C]. What might this suggest for the tenor? Just at the time <i>En Katerine</i><span style='font-style:normal'> was written, we begin to see other pieces with one voice in major prolation ([dotted-C]) sounding against another voice in minor prolation (O). Often when this happens, the minim in major prolation equals a semibreve in minor prolation, as follows, [dotted-C] [white minim] = O [white semibreve] .<a style='mso-footnote-id:ftn68' href="#_ftn68" name="_ftnref68" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[68]<![endif]></span></span></a> The counterpoint proves that this is the relationship that obtains between upper voices and tenor in the second section of <i>En Katerine</i><span style='font-style:normal'>, as well. The question still to be resolved is, how fast should the minim of [dotted-C] be? According to fourteenth-century French mensural theory, the four prime mensurations of [dotted-circle], [dotted-C], O , and C all shared a minim of equal value.<a style='mso-footnote-id:ftn69' href="#_ftn69" name="_ftnref69" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[69]<![endif]></span></span></a> This rule turns out not to obtain between tenor and upper voices in <i>En Katerine</i><span style='font-style:normal'>'s middle section, as we have just seen, where the counterpoint demands a 2:1 relationship, but we might want to invoke it to establish the relationship within each of the upper voices between the first section of the motet and the second, i.e., to keep the same minim in the upper voices as Section I makes its transition into Section II. In that case, the semibreves will grow longer in the upper voice as we move from imperfect semibreves in Section I to perfect semibreves in Section II, but the tenor semibreves of Section II will be twice as fast as they were in Section I.</span></p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span>If we have now correctly determined the speed of the minim in the upper voices and the resulting speed of the tenor, we still need to consider how the tenor singer(s) are to calculate the values of their notes. We have seen that in Section I this was done by counting semibreves. If we continue to count tenor semibreves in Section II, they will have to be twice as fast as in Section I, as just demonstrated. Thus, in Section II we see the tenor, counting 9 (fast) beats for its first rest and 9 (fast) beats for its first note, etc., moving twice as fast as it did in Section I. The implied mensuration of the tenor may still be [O], but the new mensurations in the upper voices have in effect caused a doubling of the tenor's speed. If the total number of semibreve beats in Section I was 81 x 2 = 162, in Section II, with beats moving twice as fast, the total number of equivalent semibreve beats will be half of 162, that is, 81. Also, at the faster speed, the hemiola in the tenor produced by the red notes begins to sound like slow-motion triplets. But our calculations are not yet over, for fast beats on minims in the upper voices do not serve well as guides to singers keeping time. From a performer's point of view, a slower beat on the major semibreve in the upper voices would be much easier to follow than a very fast beat on the minim. And the slower beat works well for the tenor, too, moving in perfect breves that have the same duration as upper-voice major semibreves. Things look promising at this point. Perfect semibreve beats in the upper voices translate to three per perfect long in the tenor, and the tenor's red notes will be in slow hemiola to these beats, i.e. three red semibreves per two upper-voice beats.</p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span>In Section III, the upper voices change mensurations once again: the top voice from [dotted-C] back to the C of the opening section, the middle voice from<span style="mso-spacerun: yes"> </span> [dotted-circle] back to the O of the opening section. There is, however, a significant difference between Section III and Section I: the final section of <i>En Katerine</i><span style='font-style:normal'> lasts only 54 semibreves (as measured by semibreves in O or C), whereas the first section lasted 162 semibreves. If the entire section is only one-third as long as Section I, then the tenor will need to move three times faster than in its first statement. How may this be accomplished? We saw that in Section II a doubling of the tenor's [O] was brought about by its juxtaposition against major prolation in the upper voices. Now, however, upper voices and tenor appear to share the same prolation as they did in Section I. The solution is to change the tenor note receiving the beat from the semibreve to the breve. Not the beat of Section II, which we determined should be on the upper voices' major semibreve, but the same beat as in Section I, where it is on the upper voices' minor semibreve. In English sources of the time, a similar change in a tenor part was sometimes signaled by the rubric "per medium," meaning "by half." But that won't serve the purpose here, which requires "by a third." Instead, we might imagine the Continental sign [cut circle], which we find as early as Baude Cordier's "Heart Song"<a style='mso-footnote-id:ftn70' href="#_ftn70" name="_ftnref70" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[70]<![endif]></span></span></a> and "Circle Canon" and in such famous later pieces as Du Fay's <i>Nuper rosarum</i><span style='font-style:normal'>, discussed earlier. While the meaning of </span> [cut circle] is inconsistent,<a style='mso-footnote-id:ftn71' href="#_ftn71" name="_ftnref71" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[71]<![endif]></span></span></a> one possible interpretation is "beat on the breve," which is just what is needed. Let us imagine, then, the understood sign to be [cut circle], with a beat on the breve.<a style='mso-footnote-id:ftn72' href="#_ftn72" name="_ftnref72" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[72]<![endif]></span></span></a> This solves the tempo issues, but the performance of the tenor still requires additional mental gymnastics: for the singers of that part will find it easiest to keep track of what is happening musically--because of the beat falling on upper-voice imperfect semibreves--if they imagine that their black breves (in this third iteration of the <i>color</i><span style='font-style:normal'> only) to be imperfect, too, with the red notes functioning as breve triplets in hemiola to the beat (three red breves in the time of two beats, as seen in Example 8).</span></p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span>It is only fair to point out, however, that mental gymnastics are not restricted to those singing the tenor of this piece. The singers of the upper voices have challenges too, not least being their irregular, non-coincident syncopations and overlapping, non-coincident semibreve hemiolas.</p> <p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> <p class=MsoNormal><span style='mso-tab-count:1'> </span>Looked at from the point of view of overall structure, <i>En Katerine</i><span style='font-style:normal'> exhibits the following prominent numbers. It is in three sections, which accelerate in the proportions 6 : 3 : 2. Section I lasts 162 units, Section II lasts the eqivalent of 81 such units, and Section III lasts the equivalent of 54 such units, for a total of 297 (= 27 x 11 = 99 x 3). The most significant numbers, in terms of symbolism, would be the 12 pitches in each tenor </span><i>talea</i><span style='font-style:normal'>, consisting of 7 black and 5 red notes, more than likely symbolic markers appropriate to the occasion for which </span><i>En Katerine</i><span style='font-style:normal'> is thought to have been composed: the wedding of Henry V to Catherine Valois, whom he married in 1420.<a style='mso-footnote-id:ftn73' href="#_ftn73" name="_ftnref73" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[73]<![endif]></span></span></a> (Seven served traditionally as a number representing virginity, whereas five was one of two numbers symbolic of marriage.)<a style='mso-footnote-id:ftn74' href="#_ftn74" name="_ftnref74" title=""><span class=MsoFootnoteReference><span style='mso-special-character:footnote'><![if !supportFootnotes]>[74]<![endif]></span></span></a> With respect to the focus of the present study, the significance of </span><i>En Katerine</i><span style='font-style:normal'> lies in its demonstration of how a numerically controlled piece can generate in the course of its proportional acceleration complex durational relationships between voice parts, climaxing in the 3 : 4 relationship between red tenor breves and upper-voice minims at the end of each half of Section III.</span></p> </div> <div style='mso-element:footnote-list'><![if !supportFootnotes]><br clear=all> <hr align=left size=1 width="33%"> <![endif]> <div style='mso-element:footnote' id=ftn1> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn1' href="#_ftnref1" name="_ftn1" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[1]<![endif]></span></span></a> For writers in the Middle Ages, though, Pythagoras alone was given credit for opening a previously locked gate to understanding how the universe is ordered and where man and his home on earth, at that time considered the center of the cosmos, fit into the greater scheme. This scheme was primarily numerical, for where such pre-Socratics as Heraclitus saw change as the essence of all things, and Empedocles saw all of creation to be composed of earth, water, air, and fire, Pythagoras (or, more accurately, the Pythagoreans, since we have no written documents from Pythagoras himself) argued that the universe was a structure ordered by number. Pythagoras's follower Philolaus stated this view as follows: "All things that are known have number; for without this nothing whatever could possibly be thought of or known" (Huffman 1993: 44B4).</p> </div> <div style='mso-element:footnote' id=ftn2> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn2' href="#_ftnref2" name="_ftn2" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[2]<![endif]></span></span></a> E.g., a vibrating string in a 2:1 relation in length to another vibrating string of similar material, gauge, and tension sounds an octave with the other string; one with a 3:2 relation to such a string sounds a fifth; one with a 4:3 such relation sounds a fourth; one with a 9:8 such relation sounds a whole step.</p> </div> <div style='mso-element:footnote' id=ftn3> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn3' href="#_ftnref3" name="_ftn3" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[3]<![endif]></span></span></a> The English, too, pursued similar goals, developing notational strategies that paralleled those on the Continent; most of their advances had precedent in French practice.</p> </div> <div style='mso-element:footnote' id=ftn4> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn4' href="#_ftnref4" name="_ftn4" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[4]<![endif]></span></span></a> Ernest Sanders is to be credited with developing numeric analyses of thirteenth-century pieces of the type seen in Fig. 2. For further examples of his technique, see Sanders 1977: 518-525</p> </div> <div style='mso-element:footnote' id=ftn5> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn5' href="#_ftnref5" name="_ftn5" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[5]<![endif]></span></span></a> According to most theorists who addressed the issue of rhythmic modes, there were six. Franco of Cologne ca. 1280, however, argued that only five modes were necessary as models for the patterns of thirteenth-century music. See Franco 1974, Chapter 4.</p> </div> <div style='mso-element:footnote' id=ftn6> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn6' href="#_ftnref6" name="_ftn6" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[6]<![endif]></span></span></a> In the detail of fol. 54v shown here, the ending of the 2v. clausula <i>Tamquam</i><span style='font-style:normal'> takes up about two-thirds of the topmost staves, with </span><i>Hec dies</i><span style='font-style:normal'> occupying the last third of the top staves and all of the bottom two.</span></p> </div> <div style='mso-element:footnote' id=ftn7> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn7' href="#_ftnref7" name="_ftn7" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[7]<![endif]></span></span></a> We omit from this category such earlier polyphony as the <i>conducti</i><span style='font-style:normal'> found in the Codex Calixtinus (ca. 1170), where the values of the notes are not clearly specified, since we have no evidence for whether or not they were sung in measured rhythm.<span style="mso-spacerun: yes"> </span></span></p> </div> <div style='mso-element:footnote' id=ftn8> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn8' href="#_ftnref8" name="_ftn8" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[8]<![endif]></span></span></a> Referring here to the notation codified by Franco (see Franco 1974), although there were precursors to his system.</p> </div> <div style='mso-element:footnote' id=ftn9> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn9' href="#_ftnref9" name="_ftn9" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[9]<![endif]></span></span></a> The term "isorhythm" was coined by Ludwig in 1904 to refer to a repeating pattern of rhythms in fourteenth- and fifteenth-century music, e.g., the same phenomenon for which medieval writers used the term <I>talea</I>. See Ludwig 1904: 233.</p> <p class=MsoFootnoteText><![if !supportEmptyParas]> <![endif]><o:p></o:p></p> </div> <div style='mso-element:footnote' id=ftn10> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn10' href="#_ftnref10" name="_ftn10" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[10]<![endif]></span></span></a> Sanders already sees incipient diminution in Perotinus's <i>Alleluia Nativitas</i><span style='font-style:normal'>. See Sanders 1977: 513.</span></p> </div> <div style='mso-element:footnote' id=ftn11> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn11' href="#_ftnref11" name="_ftn11" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[11]<![endif]></span></span></a> E.g., Nicolaus Grenon's motet <i>Ave virtus/Prophetarum/Infelix</i><span style='font-style:normal'> in </span><i>Ox</i><span style='font-style:normal'>, with a verbal canon that instructs those singing the tenor to augment its values by a factor of eight on its first statement, then by a factor of six, then by a factor of two for two statements, and finally to perform it "velut hic requiescit," i.e., as it lies (as it is written) twice more. See below, Notes 26 and 27.</span></p> </div> <div style='mso-element:footnote' id=ftn12> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn12' href="#_ftnref12" name="_ftn12" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[12]<![endif]></span></span></a> According to Theodoricus, the effect of changing perfect to imperfect could be produced by using either hollow notes or red ones ("rubre vel vacue"), just as we see in the two sources of <i>Garrit gallus</i><span style='font-style:normal'>.</span></p> </div> <div style='mso-element:footnote' id=ftn13> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn13' href="#_ftnref13" name="_ftn13" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[13]<![endif]></span></span></a> For a telling example, see Lassus's famous six-voice motet <i>Cum essem parvulus</i><span style='font-style:normal'>, first published in </span><i>Motetta sex vocum</i><span style='font-style:normal'> (Nuremberg: Berg, 1562). At the closing cadence of this work, the final longs in the six voices have four different durations, depending on how soon before the Discantus's final note each of the others arrives. Notes of the same name (longa) here vary in duration from two breves (Discantus), to three breves, four breves, and six breves.</span></p> </div> <div style='mso-element:footnote' id=ftn14> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn14' href="#_ftnref14" name="_ftn14" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[14]<![endif]></span></span></a> The true significance of the numbers in <i>In arboris</i><span style='font-style:normal'> may be connected with the meaning and structure of the texts sung by the upper voices; if so, this remains to be demonstrated.</span></p> </div> <div style='mso-element:footnote' id=ftn15> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn15' href="#_ftnref15" name="_ftn15" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[15]<![endif]></span></span></a> A clue to the rationale behind the evident discrepancy in the meaning of red notes between <i>In arboris</i><span style='font-style:normal'> and </span><i>Garrit gallus/In nova fert </i><span style='font-style:normal'>is provided by the </span><i>Ars perfecta in Musica Magistri Philippocti de Vitriaco</i><span style='font-style: normal'> (CS III, 33f) (not considered an authentic treatise by Vitry): "Breves ponuntur rubee ad differentiam temporis, ita quod si nigre breves fuerint de tempore perfecto, rubee erunt de imperfecto et e contrario. Nisi cum aliqua longa forsitan ordinentur, sicut in moteti tenore qui dicitur: In arboris, vel in tenore de: In nova fert animus, ut hic [example]; Semibreves rubee ponuntur ad differentiam prolationis, ut, si nigre fuerint de maiori prolatione, rubee erunt de minore, et e contrario. Nisi semibreves forsitan cum aliqua brevi ordinentur, quia tunc ponuntur ad differentiam temporis, sicut in tenore de: In arboris invenitur, ut hic [example]" ("Red breves are written to cause a change of </span><i>tempus</i><span style='font-style:normal'>, such that if black breves are in tempus perfectum, red ones will be in imperfect, and vice versa--unless the [breves] are [so colored] because they are in the place of a [red] long, as in the tenor of the motet </span><i>In arboris</i><span style='font-style:normal'> or in the tenor of </span><i>In nova fert animus</i><span style='font-style:normal'>. Red semibreves are written to cause a change of </span><i>prolatio</i><span style='font-style:normal'>, such that if black ones are in major prolation, red ones will be in minor, and vice versa--unless the [semibreves] are [so colored] because they are in the place of a [red] breve") (cited in Schrade 1956-58b: 25). In other words, coloration in its first phase may cause a change from perfect to imperfect or vice versa and is not restricted in meaning to one or the other.</span><i><o:p></o:p></i></p> </div> <div style='mso-element:footnote' id=ftn16> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn16' href="#_ftnref16" name="_ftn16" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[16]<![endif]></span></span></a> For a facsimile of Machaut's tenor and contratenor, with red notes printed in red ink, and a discussion of the notation, see Apel 1953: 360. Parrish 1959: Plates LII and LIII, gives a monochrome photographic reproduction of the entire motet, with the source identified as Paris, BibliothËque nationale, Fr. 22.546 ("Machaut G"). Here the red notes appear as gray.</p> </div> <div style='mso-element:footnote' id=ftn17> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn17' href="#_ftnref17" name="_ftn17" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[17]<![endif]></span></span></a> It has been proposed that <i>Passerose</i><span style='font-style:normal'> was composed for the 1389 wedding of Jean, Duke of Berry, and Jeanne of Boulogne. See Plumley 2001: 709</span></p> </div> <div style='mso-element:footnote' id=ftn18> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn18' href="#_ftnref18" name="_ftn18" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[18]<![endif]></span></span></a> Strictly speaking, of course, any two notes in a precise mathematical relationship to one another are "proportional." Thus, in modern notation, quarter notes are proportional to eighths, which are proportional to sixteenths, etc. The term "proportional," however, is usually reserved today for relationships that lie outside those inherent in the prevailing meter (or, in fourteenth- to sixteenth-century practice, mensuration).</p> </div> <div style='mso-element:footnote' id=ftn19> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn19' href="#_ftnref19" name="_ftn19" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[19]<![endif]></span></span></a> See Plumley/Stone 2008b: 181.</p> </div> <div style='mso-element:footnote' id=ftn20> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn20' href="#_ftnref20" name="_ftn20" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[20]<![endif]></span></span></a> The notational difficulties of this piece are legendary: Koehler 1986 criticized the then recent edition of Gordon K. Greene (1984) as well as the two earlier transcriptions of the work by Willi Apel (1950, 1970). Stoessel 1999 takes this single work as its principal subject.</p> </div> <div style='mso-element:footnote' id=ftn21> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn21' href="#_ftnref21" name="_ftn21" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[21]<![endif]></span></span></a> Bent 1991: 15</p> </div> <div style='mso-element:footnote' id=ftn22> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn22' href="#_ftnref22" name="_ftn22" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[22]<![endif]></span></span></a> Much has been written about this piece in recent years. See the studies by Warren 1973, Ryschawy/Stoll 1988, Wright 1994, Trachtenberg 2001, and Blachly 2003, among others.</p> </div> <div style='mso-element:footnote' id=ftn23> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn23' href="#_ftnref23" name="_ftn23" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[23]<![endif]></span></span></a> Bent 1998: 143.</p> </div> <div style='mso-element:footnote' id=ftn24> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn24' href="#_ftnref24" name="_ftn24" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[24]<![endif]></span></span></a> Wright 1994: 407; 429.</p> </div> <div style='mso-element:footnote' id=ftn25> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn25' href="#_ftnref25" name="_ftn25" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[25]<![endif]></span></span></a> This is how we suppose the tenors looked in the exemplar(s) used by the scribes of Trent 92 and Mod B, though neither scribe has reproduced this effect precisely.</p> </div> <div style='mso-element:footnote' id=ftn26> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn26' href="#_ftnref26" name="_ftn26" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[26]<![endif]></span></span></a> Sanders 1977: 565.</p> </div> <div style='mso-element:footnote' id=ftn27> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn27' href="#_ftnref27" name="_ftn27" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[27]<![endif]></span></span></a> Modern edition in Van den Borren 1962: 194-202. The canon, which is written in compressed form, reads: "Tenor. Canon: Crescit in octuplo Semel hic tenor incipiendo, Fac in sextuplo Crescat semel in decanendo. Postea bis canitur In duplo cum bene crescit. Sed bis concinitur Tandem velut hic requiescit."</p> </div> <div style='mso-element:footnote' id=ftn28> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn28' href="#_ftnref28" name="_ftn28" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[28]<![endif]></span></span></a> Modern edition in Hughes/Bent 1969-73, Vol. I (part 2), 415-418. </p> </div> <div style='mso-element:footnote' id=ftn29> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn29' href="#_ftnref29" name="_ftn29" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[29]<![endif]></span></span></a> "Sic nunc successive venientes, habentes et intellegentes que primi magistri relinquerunt maiores subtilitates per studium sunt confectii ut quod per antecessores imperfectum relictum fuit per successores reformetur" ("Thus, those now coming after, possessing and understanding that which the first masters left behind, have acomplished by study greater subtleties so that which was left imperfect by predecessors might be reformed by those who follow them"). Citation and translation in Stoessel 1999: 136. For the <i>Tractatus figurarum,</i><span style='font-style:normal'> see Schreur 1989.</span></p> </div> <div style='mso-element:footnote' id=ftn30> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn30' href="#_ftnref30" name="_ftn30" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[30]<![endif]></span></span></a> See Franco 1974</p> </div> <div style='mso-element:footnote' id=ftn31> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn31' href="#_ftnref31" name="_ftn31" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[31]<![endif]></span></span></a> See Sanders 1977: 509.</p> </div> <div style='mso-element:footnote' id=ftn32> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn32' href="#_ftnref32" name="_ftn32" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[32]<![endif]></span></span></a> A recent study (Eisenstein 2007) explores the Christological implications of this work, with its suggestions of Alpha and Omega.</p> </div> <div style='mso-element:footnote' id=ftn33> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn33' href="#_ftnref33" name="_ftn33" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[33]<![endif]></span></span></a> In <i>Ch</i><span style='font-style:normal'> the rondeau canon is written beneath the top voice; in </span><i>ChiC</i><span style='font-style:normal'>, which notates the piece pictorially, with the notes placed on the strings of a harp, the canon is written on a ribbon wrapped around the harp's pillar.</span></p> </div> <div style='mso-element:footnote' id=ftn34> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn34' href="#_ftnref34" name="_ftn34" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[34]<![endif]></span></span></a> Hoppin 1978b: 171-72.</p> </div> <div style='mso-element:footnote' id=ftn35> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn35' href="#_ftnref35" name="_ftn35" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[35]<![endif]></span></span></a> The white dragmas with hooked lower stems have been interpreted differently by the various editors of this piece in recent years. While Hoppin transcribes them as defined above, both Gordon K. Greene and Willi Apel treat them inconsistently, first as defined above, then as notes twice as long--this because the seventh through eleventh tenor notes in <i>Ch</i><span style='font-style:normal'> and </span><i>ChiC</i><span style='font-style:normal'> have slightly different values, causing the tenor of </span><i>Ch</i><span style='font-style:normal'> (the source used by Greene and Apel) to last one perfect semibreve longer than the tenor of </span><i>ChiC</i><span style='font-style:normal'> (the source used by Hoppin). Yet, clearly the tenor of </span><i>ChiC</i><span style='font-style:normal'> is correct, since it allows the white dragmas with hooked lower stems to have a consistent meaning--one that produces better counterpoint as well.</span></p> </div> <div style='mso-element:footnote' id=ftn36> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn36' href="#_ftnref36" name="_ftn36" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[36]<![endif]></span></span></a> The Medieval Ensemble of London produced an excellent recording of this piece in 1982, featuring singers Rogers Covey-Crump and Paul Elliott and harpist Peter Davies (L'Oiseau-Lyre 475 9119). Years later Daniel Leech-Wilkinson would praise this performance as "still unsurpassed," an assessment that<span style="mso-spacerun: yes"> </span>holds true to this day.</p> </div> <div style='mso-element:footnote' id=ftn37> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn37' href="#_ftnref37" name="_ftn37" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[37]<![endif]></span></span></a> See, for example, Apel 1953, Chapter 9, entitled "Mannered Notation."</p> </div> <div style='mso-element:footnote' id=ftn38> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn38' href="#_ftnref38" name="_ftn38" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[38]<![endif]></span></span></a> Wright 1979: 133. Cited in Page 1994</p> </div> <div style='mso-element:footnote' id=ftn39> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn39' href="#_ftnref39" name="_ftn39" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[39]<![endif]></span></span></a> In Chapter 5 of the <i>Ars cantus mensurabilis</i><span style='font-style:normal'>, Franco stated "Si longam longa sequatur, tunc prima longa sub uno accentu tribus temporibus mensuratur" ("If a long follows a long, then the first long is measured by one sound equal to three breves"), which was made more generic (applying not only to perfect longs but also to perfect breves and semibreves) in the Ars nova as "Similis ante similem perfecta non potest imperfici" (to quote Prosdocimus's version in his </span><i>Expositiones tractatus pratice cantus mensurabilis magistri Johannis de Muris</i><span style='font-style:normal'> [Prosdocimus 1966]).</span></p> </div> <div style='mso-element:footnote' id=ftn40> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn40' href="#_ftnref40" name="_ftn40" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[40]<![endif]></span></span></a> Franco's rule of<span style="mso-spacerun: yes"> </span>"alteration" provided a way around this dilemma, by allowing a breve written in the place of the imperfect long to be read as equal to it in value: "Duarum autem brevium prima recta, secunda vero altera brevis appellatur. Recta brevis est, quae unum solum tempus continet, altera brevis similis est longae imperfectae in valore, differens tamen in significatione. Nam utraque sub diversa figuratione duobus temporibus mensuratur." ("Of the two types of breve, the first is called 'recta' while the second is called 'altera.' A recta breve contains only one tempus, whereas the altera breve is similar to an imperfect long in value but different in function, for both are measured by two tempora, though they have different note shapes.")</p> </div> <div style='mso-element:footnote' id=ftn41> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn41' href="#_ftnref41" name="_ftn41" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[41]<![endif]></span></span></a> Apel 1953: 414-422 provides a particularly clear summary of this type of syncopation extended through a long phrase. A slow-moving instance of it is to be seen in the tenor of Biteryng's <i>En Katerine solennia</i><span style='font-style: normal'> (see discussion below, in the Postscript, "Challenges for the performer").</span></p> </div> <div style='mso-element:footnote' id=ftn42> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn42' href="#_ftnref42" name="_ftn42" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[42]<![endif]></span></span></a> See note 55 below.</p> </div> <div style='mso-element:footnote' id=ftn43> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn43' href="#_ftnref43" name="_ftn43" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[43]<![endif]></span></span></a> Stoessel 1999: 137 argues that there are no redundancies in Ars subtilior works, but that, instead, each apparently redundant note form has a mensural logic that justifies its use in addition to another or other note forms that produce the same result.</p> </div> <div style='mso-element:footnote' id=ftn44> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn44' href="#_ftnref44" name="_ftn44" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[44]<![endif]></span></span></a> Apel 1950 transcribes the note this way. Despite the difficulties of execution <I>Le greygnour</I> presents performers, there are three recordings currently on the market that make a convincing case for the musical attractiveness of the piece: soprano Catherine Bott with fiddle and lute in the New London Consort directed by Philip Pickett (Linn Records CKD 039); Catherine Bott supported by Pavlo Beznosiuk and Mark Levy, fiddles, on Hyperion (CDA67549); and Mala Punica on Arcana (A21), where soprano Jill Feldman is, like the Hyperion recording, supported by two fiddles.</p> </div> <div style='mso-element:footnote' id=ftn45> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn45' href="#_ftnref45" name="_ftn45" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[44]<![endif]></span></span></a> C.o.p. is an abbreviation for "cum opposita proprietate," referring to a ligature in which the initial stem ascends, causing the first two notes of the ligature to be read as semibreves, regardless of shape. Walter Odington in the fourteenth century was one of many theorists who defined "opposite propriety" in similar ways: "Opposita proprietas est quae habet a primo puncto tractum ascendentem, sive ligatura ascendat sive descendat" (Odington 1970: 136) ("Opposite propriety is that which has an ascending stem at the beginning, whether the ligature is ascending or descending"), by which he understood the stem of propriety (the descending stem on the prime form of the two-note descending ligature) to point in the opposite direction, i.e., up rather than down.</p> </div> <div style='mso-element:footnote' id=ftn46> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn46' href="#_ftnref46" name="_ftn46" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[46]<![endif]></span></span></a> Famous for being notated on staves curved to form the shape of a heart and uniquely preserved as part of a double flyleaf, on the facing recto of which is Baude Cordier's equally famous "Circle Canon," <i>Tout par compas</i><span style='font-style:normal'>, both works now bound right after the index as the first musical pieces in </span><i>Ch</i><span style='font-style:normal'> (fol. 11v-12).</span></p> </div> <div style='mso-element:footnote' id=ftn47> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn7' href="#_ftnref47" name="_ftn47" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[46]<![endif]></span></span></a> Black notation would still be in use, however, in such northern centers as Cambrai as late as the 1440s, witness <i>Ca11</i><span style='font-style:normal'>, written entirely in black notation with red coloration. See Curtis 1992: 13, who dates this source to ca. 1442-45.</span></p> </div> <div style='mso-element:footnote' id=ftn48> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn48' href="#_ftnref48" name="_ftn48" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[48]<![endif]></span></span></a> Stoessel 1999: 143-144 discusses Prosdocimus's complaint about the new use of red or hollow-red notes to indicate sesquitertia, a proportional relationship which he and other theorists at the beginning of the fifteenth century state should properly be indicated by hollow black notes. </p> </div> <div style='mso-element:footnote' id=ftn49> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn49' href="#_ftnref49" name="_ftn49" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[49]<![endif]></span></span></a> See Wegman 1992 and Bent 1996.</p> </div> <div style='mso-element:footnote' id=ftn50> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn50' href="#_ftnref50" name="_ftn50" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[50]<![endif]></span></span></a> Which is not to imply that cut circle had a consistent meaning. Bent 1996 has shown otherwise.</p> </div> <div style='mso-element:footnote' id=ftn51> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn51' href="#_ftnref51" name="_ftn51" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[51]<![endif]></span></span></a> For a recent study of <i>Belle, bonne, sage</i><span style='font-style:normal'>, see Anheim 2005.</span></p> </div> <div style='mso-element:footnote' id=ftn52> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn52' href="#_ftnref52" name="_ftn52" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[52]<![endif]></span></span></a> The seven Cordier songs in <I>Ox</I> continue to make liberal use of a battery of mensuraton and proportion signs, in the same was as do the Heart Song and Circle Canon. </div> <div style='mso-element:footnote' id=ftn53> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn53' href="#_ftnref53" name="_ftn53" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[53]<![endif]></span></span></a> <I>Verbum patris hodie</I> survives uniquely in Gathering I of <I>Ox</I>, the part of the manuscript copied last, the final piece of which gathering was evidently copied in 1436. The works by Sarto, Quadris, Zocholo, Feragut, and Brassart, all copied in Gathering I, show a "stylistic uniformity" and form the most recent of "several coherently organised groups of pieces in [<I>Ox</I>]" (Fallows 1995: 28) </div> <div style='mso-element:footnote' id=ftn54> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn54' href="#_ftnref54" name="_ftn54" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[54]<![endif]></span></span></a> In the few instances where coloration occurs in the <I>Ox</I> pieces in black notation, it is represented by black-void notes, just as <I>Pic</I> indicates the coloration in Vitry's <I>In arboris</I> (see above, note 12). </div> <div style='mso-element:footnote' id=ftn55> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn55' href="#_ftnref55" name="_ftn55" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[55]<![endif]></span></span></a> See "The Origins of White Notation," pp. ____ to ____. </div> <div style='mso-element:footnote' id=ftn56> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn56' href="#_ftnref56" name="_ftn56" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[56]<![endif]></span></span></a> Blackburn 1991: 664. </div> <div style='mso-element:footnote' id=ftn57> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn57' href="#_ftnref57" name="_ftn57" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[57]<![endif]></span></span></a> This because duple proportion would require either that the <I>prima pars</I> move at a lugubrious pace or that the <I>secunda pars</I> race along at a <I>presto</I>, or both. Wegman criticizes the duple proportion interpretation of O followed by [cut-circle] in these words: "There are numerous examples of horizontal relationships between O and [cut-circle] where anything in excess of 'a little faster' would produce musically unacceptable results" (Wegman 1992: 523). For further on this point, see Hamm 1964: 62-63, Blachly 1995: 204 (especially No. 5), and Blachly 1999: 427. Michael Praetorius's observations on a similar issue in his <I>Syntagma musicum</I> of 1619 seem not only sensible but exactly the sort of attitude we might expect from musicians in any age: "Jetziger zeit aber werden diese beyde Signa meistentheils also observieret, dass das C fuernemlich in Madrigalien, das [cut-C] aber in Motetten gebraucht wird. Quia Madrigalia & aliae Cantiones, quae sub signo C, Semiminimis & Fusis abundant, celeriori progrediuntur motu; Motectae autem, quae sub sign [cut-C] Brevibus & Semibrevibus abundant, tardioi: Ideo hic celeriori, illic tardiori opus est Tactu, quò medium inter duo extrema servetur, ne tardior Progressus auditorum auribus pariat fastidium, aut celerior in Pracecipitium ducat, veluti Soli equi Phaëtontem abripuerunt, ubi currus nullas audivit habenas." (Nowadays both signs [C and cut-C] are observed for the most part such that C is used mainly in madrigals and [cut-C] mainly in motets. Because madrigals and other pieces which abound with semiminims and fusae under the sign C proceed in a faster motion, whereas motets, which abound with breves and semibreves under the sign [cut-C], proceed in a slower motion, for that reason, the latter work will be measured by a faster tactus, the former by a slower, which stands as a mean between two extremes, lest a too-slow motion lead to fatigue in the ears of the listeners, or a too-fast motion lead to a head-long fall, just as the horses of the sun ran off with Phaëton, who gave no heed to the reins of the chariot.) Cited in Blachly 1995: 449. </div> <div style='mso-element:footnote' id=ftn58> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn58' href="#_ftnref58" name="_ftn58" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[58]<![endif]></span></span></a> As Fallows 1995: 29 points out, <I>Verbum patris hodie</I> is a Benedicamus trope, i.e., a polyphonic setting of the Benedicamus Domino formula said at the close of most services in the Divine Office and some Masses. The text begins, "The Word of the Father processes today from the Virgin," indicating that this setting is to be sung on Christmas Day, or at least during the Christmas season. </div> <div style='mso-element:footnote' id=ftn59> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn59' href="#_ftnref59" name="_ftn59" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[59]<![endif]></span></span></a> Wegman 1992: 521 </div> <div style='mso-element:footnote' id=ftn60> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn60' href="#_ftnref60" name="_ftn60" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[60]<![endif]></span></span></a> Already forty-five years ago, Hamm 1964: 39 had proposed exactly this interpretation of cut signatures: that as a marker of tempo a cut signature controlled the diminution or augmentation of any voice-parts with uncut signatures that might be in a vertical proportional relationship with it. In other words, a cut signature functioned as the point of reference to which uncut signatures had to adjust. </div> <div style='mso-element:footnote' id=ftn61> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn61' href="#_ftnref61" name="_ftn61" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[61]<![endif]></span></span></a> See Blachly 1995: 300ff. </div> <div style='mso-element:footnote' id=ftn62> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn62' href="#_ftnref62" name="_ftn62" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[63]<![endif]></span></span></a> See Wegman 1992, which is entirely devoted to this issue, and Blachly 1995: 204, which situates "acceleratio mensure" in the context of Tinctoris's overall teachings about mensuration and proportions. </div> <div style='mso-element:footnote' id=ftn63> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn63' href="#_ftnref63" name="_ftn63" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[63]<![endif]></span></span></a> Hamm 1964 assigns Alma redemptoris II to Du Fay's "Group 2, 1423-1429." </div> <div style='mso-element:footnote' id=ftn64> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn64' href="#_ftnref64" name="_ftn64" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[64]<![endif]></span></span></a> Franco specified in the <i>Ars cantus mensurabilis</i><span style='font-style:normal'> that when two breves occur between two longs, the first long is not to be imperfected.</span></p> </div> <div style='mso-element:footnote' id=ftn65> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn65' href="#_ftnref65" name="_ftn65" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[65]<![endif]></span></span></a> See above, notes 40 and 41.</p> </div> <div style='mso-element:footnote' id=ftn66> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn66' href="#_ftnref66" name="_ftn66" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[66]<![endif]></span></span></a> The fragment is the beginning of the verse of a responsory for Matins on the Feast of St. Catherine of Alexandria, Virgin Martyr (November 25). The complete responsory can be found in the <i>Processionale monasticum</i><span style='font-style: normal'>, 214.</span></p> </div> <div style='mso-element:footnote' id=ftn67> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn67' href="#_ftnref67" name="_ftn67" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[67]<![endif]></span></span></a> See Blachly 2001: 429-437.</p> </div> <div style='mso-element:footnote' id=ftn68> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn68' href="#_ftnref68" name="_ftn68" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[68]<![endif]></span></span></a> In fact, we have already seen a similar relationship in <i>Belle, bonne, sage</i><span style='font-style:normal'>, where [dotted C] [minim] = [cut circle] [semibreve].</span></p> </div> <div style='mso-element:footnote' id=ftn69> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn69' href="#_ftnref69" name="_ftn55" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[69]<![endif]></span></span></a> Whereas in Italian fourteenth-century theory, the <i>divisiones</i><span style='font-style: normal'> shared a breve of equal value. In French theory, then, the minim was a constant for all </span><i>prolacions</i><span style='font-style:normal'> (mensurations), with longer notes being aggregates of this constant. In Italian theory, the breve remained constant, while its subdivisions varied in speed, depending on whether there were two of these, three, four, six, eight, nine, or twelve. </span></p> </div> <div style='mso-element:footnote' id=ftn70> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn70' href="#_ftnref70" name="_ftn70" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[70]<![endif]></span></span></a> See the discussion of this piece above.</p> </div> <div style='mso-element:footnote' id=ftn71> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn71' href="#_ftnref71" name="_ftn71" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[71]<![endif]></span></span></a> For recent studies of the meaning of <span style='font-family:PPalatino-Roman'>[cut circle]</span>, see Wegman 1992 and Bent 1996.</p> </div> <div style='mso-element:footnote' id=ftn72> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn72' href="#_ftnref72" name="_ftn72" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[72]<![endif]></span></span></a> Since English musical sources of the time do not use cut signatures<span style='font-family:PPalatino-Roman'>, the proposal to imagine the use of [cut circle] here is only to make the principle of diminution easier to grasp.</span></p> </div> <div style='mso-element:footnote' id=ftn73> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn73' href="#_ftnref73" name="_ftn73" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[73]<![endif]></span></span></a> Bukofzer 1949: 50.</p> </div> <div style='mso-element:footnote' id=ftn74> <p class=MsoFootnoteText><a style='mso-footnote-id:ftn74' href="#_ftnref74" name="_ftn74" title=""><span class=MsoFootnoteReference><span style='mso-special-character: footnote'><![if !supportFootnotes]>[74]<![endif]></span></span></a> As Atlas 1987: 118 succinctly summarizes the situation: "..the number six was one of the two so-called 'marriage' numbers (the other was five...), being the product of the first female number (two) and the first male number (three)." And a page later (Atlas 1987: 119), </span>"...such commentators as Martianus Capella (in Book VII of his <i>De nuptiis philologiae et mercurii</i><span style='font-style:normal'>) and his follower one millenium later, Pietro Bongo (</span><i>Mysticae numerorum...</i><span style='font-style:normal'>, first published in 1585), called the number seven 'virgin' on the grounds that it neither generates nor is generated within the decad (that is, it is not the product of any two smaller numbers, nor can it be multiplied by any number to produce a larger number that does not exceed ten)."</span></p> </div> </div> </body> </html>