MUSICAL THEORISTS IN TRANSLATION

VOLUME 1

ANONYMOUS IV

Translated and Edited

By

LUTHER DITTMER

Institute of Mediæval Music
1270 Lampman Crescent
Ottawa (Ont.) K2C 1P8
Canada

© 1959 by the Institute of Mediæval Music

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     Translations with annotations serve the purpose of making important historical documents more readily available to a larger circle of readers. It should be borne in mind, however, that a translation is not a substitute for an original piece of work, but only a handy reference tool. Whereas every attempt has been made to render the sense and spirit of the Latin original in English, there are passages, the meaning of which is somewhat obscure, or which continue to be the subject of scholarly debate. Musicology is still a young science, and many of the things which today it has concluded, based on present evidence, tomorrow may be subject to review, proceeding semper ad fontes.
     The erudite Edmond de Coussemaker has left us with a basic corpus of theoretical treatises, which today continue to form the fundamental secondary source for editions of musical theorists of the 13th-16th centuries. Actually, Coussemaker's rendition of the texts has been so excellent, that few changes are necessary. In the century, however, that has passed since Coussemaker's epoch-making achievements, our knowledge of the music of the periods in question has advanced immeasurably. It is hoped that this translation will correct some of the minor flaws in Coussemaker's rendition of the present text, as well as of those mistakes which are to be found in the primary sources themselves; to achieve this a list of errata has been appended. Some of the errata are listed by Coussemaker himself, but have been repeated here since quotations from Coussemaker have not always included corrections from the emendendata.
     The treatise of the fourth of the anonymous writers, according to the order of the first volume of Coussemaker's, Scriptorum de musica medii ævi.....l854 (with several subsequent photographic reprints), is the most verbose, informative and sagacious of all of the known essays of the13th century, concerned with the problems of indicating the durational values of the polyphonic music of the time graphically (id est, mensural notation). The particular treatise appears to have been written about 1275 by a didactition, who if himself not a native of Great Britain, at least was thoroughly familiar with its music. It is perhaps incorrect to consider that the treatise in its present form represents either the lecture notes of a student or of a master, although there are many biographical details of other personalities associated with pæedagogical pursuits for us to believe that the author, himself, was not a master and used the material to instruct others. Our theorist, unlike many of the others, does not resort to strong invective, he will frankly admit, after he has stated something, that this procedure is not used. Perhaps the importance of this essay lies in this point: The author is reporting, stating the alternate possibilities, giving limited information about specific compositions and posing his solutions. In short, one can give more credence to his writing, because of the apparent honesty of his manner. Anonymous the Fourth (as he is commonly called) represents the last of the theorists in the tradition of the middle ages to express the rhythm of melodies in notes, rather than durational values in note shapes. Yet, even in his own day, this was obsolescent and the doctrine of the ordines or rhythmic patterns of melodies is formal and complicated and almost meaningless, because it no longer represented the contemporary trend, as other theorists note by rejecting the laborious task of discussing individual rhythms in favour of the simpler expedient of giving rules for the interpretation of durational values under certain circumstances.

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     This author, almost alone, gives us tangible information regarding the important personalities in the world of music at Paris during the century before his stay there. Unfortunately, these personages are unknown for the most part elsewhere, so that the treatise only whets our appetite but does not satisfy it. The musical compositions cited are fortunately preserved and are listed according to Ludwig' s numbers (Repertorium organorum recentioris et motetorum vetustissimi stili, 1910, new completed edition in progress by the present writer), the conductus are cited according to E. Groninger, Repertoire-Untersuchungen zum mehrstimmigen Notre Dame-Conductus. Other works cited include: S. Cserba, Hieronymus de Moravia O. P. Tractatus de Musica, 1935; W. Niemann. Über die abweichende Bedeutung der Ligaturen in der Mensuraltheorie der Zeit vor Johannes de Garlandia, (Beiheft VI, Series 1, of the Publikationen der Internationalen Musikgesellschaft, 1902), and W. Waite, The Rhythm of Twelfth Century Polyphony, its Theory and Practice, 1954.
     The treatise is contained in three manuscripts in the British Museum in London. It is contained in the manuscript Royal 12 c VI ff. 59^r-80^v, provenience Bury St. Edmunds, late 13th century, calligraphically well written. The second source is Cotton Tiberius B IX, provenience Bury St. Edmunds, late 13th century, beginning with f. 215^r; unfortunately, this copy was largely destroyed by fire in the eighteenth century, and only smaller portions may be read. This latter manuscript, however, was the source for the copy made for Dr. Pepusch in the eighteenth century and contained in Additional 4909 ff. 56^v-93^r.
     The pages and lines of the treatise of Anonymous IV in Coussemaker's edition are contained on the following folios of the manuscript Royal 12 C VI.

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     The following corrections should be made in Coussemaker's text:

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5 

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     The technical terms listed below have been retained in their Latin forms and are italicized in the text. In the case of nouns used in the plural, these forms are added. It should be noted that the full form of a technical term is given, even if it is shortened in the text, exempli gratia: notæ currentes instead of currentes. For the sake of consistency, sine perfectione is used throughout instead of imperfecta.

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     The term tenor (tenores), the voice which holds to the succession of pitches of the plain chant on which the composition is based, is used without italics.
     The following English terms are always used to translate the following Latin words:

     All text within parentheses in the translation represents editorial additions.

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Anonymous IV - CONCERNING THE MEASUREMENT OF POLYPHONIC SONG

Chapter I - Part 1

Concerning the Modes and the Ways of Arranging the Modes

     In that we know about the division of melodies according to the eight modes, as used in the services of the Catholic faith, we shall now concern ourselves with questions of the measurement of the durational values, that is the length and brevity of music. This has been recorded by the ancients in treatises; such persons as Master Leo and quite a few others have arranged this material according to ordines and colores, in the following manner:
     The mode, or manner, or the consideration of the durational values of a composition is the recognition of the length and brevity of a melodic tone or sound. There are the usual modes:
     The first consists of longa brevis, longa brevis, longa brevis etc. The second consists of brevis longa, brevis longa, brevis longa etc. The third consists of a longa and two breves, longa and two breves etc. The fourth consists of two breves and a longa, two breves and a longa etc. The fifth consists of longa, longa, longa etc. The sixth consists of brevis, brevis, brevis etc. The first mode, when it is combined with the others or with itself, results in five modes and one. The second mode, when it is combined with the others or with itself, results in five modes and one. The third mode, when it is combined with the others or with itself, results in five modes and one. The fourth mode, when it is combined with the others or with itself, results in five modes and one. The fifth mode, when it is combined with the others or with itself, results in five modes and one. The sixth mode, when it is combined with the other modes or with itself, results in five modes and one. There are no more combinations of modes than these thirty admixtures of different modes and the six of the same mode, and thus there are thirty six modes all told.
     It should be noted that admixtures are quite often formed either with or without the help of rests, as may be seen from case to case. There are other modes, like those mentioned above, which split the brevis or breves into two, three, or four notes, as in instrumental music. We shall have cause to speak of the measurement of the durational values of modes, etc.; we shall present in the course of this treatise many different modes formed by the splitting of durational values.
     There are other modes, which are called unused modes; they are like irregular modes, although they aren't such; these are like those used in parts of England and elsewhere, and they consist of longa, longa brevis; and there are many more such, as shall be demonstrated in the course of this treatise. A longa simplex contains two durational units, whatever position it shares, first, second, or third. A brevis simplex contains one durational unit in any similar position.

Coussemaker p. 327a1 to p. 328a36

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     A tone of one durational unit may be defined as a sound which is not held for the minimum or maximum length of time possible, but which though sufficiently short may yet be split into two, three or four rapid notes; further division is not possible in vocal music, although this is feasible in instrumental music.
     The ordo of a mode is determined by the number of notes occurring before a rest. Thus, an ordo is selected according to the principle underlying its arrangement. The basic principle of organization of the different modes corresponds to the disposition of notes or melodic tones, which contain no rests, and which are ligated in a definite order of musical notes according to the nature of the tropes¹. Thus, any particular (id est, rhythmic) arrangement of notes is independent of the accompanying trope (id est, succession of melodic pitches), as if it were used only as a melodic source¹, as shown in the Graduals antiphon (id est, verse). This was followed extensively by older composers in the books of the Church of the Holy Mother (Église de la Sainte Marie, or perhaps the Cathedral itself is meant), and elsewhere in an even more general manner as musicians have often used in different places.
     Certain modes are perfect, others are imperfect. A mode is considered perfect when it ends with the same durational value as it began. A mode is considered imperfect when it ends with a different durational value than it began. The methods of making the modes perfect or imperfect vary according to the differences of the six modes listed above.
     The first example of the realization of this principle involves the first mode in this form: Take any succession of musical pitches, notably notes, sounds or tones which are thickly ligated (id est. melismatic rather than syllabic) in a gradual. Such an example is found in Latus, which is taken from the Alleluia (Vs Pascha nostrum) immolatus est Christus, Write such examples down on parchment.

     After this is done, make a different disposition of the notes in accordance with the nature of the mode, unless that which you already have is sufficient. If you are going to apply the first mode, you should arrange the notes in the manner shown below in the second chapter. Thus, the notes will be arranged in the following way: three ligated and two, two, two without rests. There are however, two things to be considered in the art of arrangement of these notes, the first concerns itself with tones and melodies, the other with suitable and appropriate notational symbols, that may be written into books, as shall be demonstrated at length in the second chapter.

Coussemaker p. 328a37 to p. 328b39

1)    Tropus est. per quem cognoscimus principium, medium ac finem cuiuslibet meli. Cserba p. 154_24-25. Thus, tropus refers definitely to the succession of pitches, or melodic formulæ of a melody, in contradistinction to the groupings of notes, which give the durational values. This is further confirmed by the use of the term radix, which Iohannes de Garlandia (Cserba p. 196) also uses: Radix est quilibet cantus primo datus, and which we have translated as musical source. The example which follows in both Garlandia's and Anon. IV's treatises involves the conversion of a piece of plain chant into a measured melody (id est, with a definite ordo or succession of notes, which thereby imply a specific rhythm) used as a tenor. 

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     The first ordo of the first mode consists of three notes with a brevis rest, and three notes with a brevis rest, and three notes with a brevis rest, and three notes with a brevis rest, etc.
     The second ordo of the first mode consists of five notes, and five notes before a rest; and these are always divided into groups of three and two.
     The third ordo consists of seven notes, and seven notes, and seven notes, always divided into groups of three and two and two, always without rests; if three notes are left over at the end, they form a first ordo of the first mode.
     The fourth ordo adds two notes to this.
     The fifth ordo adds two notes to the fourth; whenever one wishes, one may add additional groups of two notes to form additional ordines. It therefore follows, from what we have said previously, that all of the ordines of the first perfect mode have an uneven number of notes before a rest; these notes are always found in the order of longa brevis, which notes constitute a pes simplex, no matter how often this relationship is split (id est, into notes of smaller durational value). It should also be understood that the afore-mentioned rests have the durational value of a brevis, according to the arrangement or definition of the first perfect mode.
     The ordines of the second perfect mode are arranged similarly and have the same number of components, but this mode is treated in just the opposite way in the quantitative measurement of its parts. Thus, the second perfect mode begins in this manner: two, two, two, and three ligated notes at the end before a rest. The first ordo of this mode consists of a brevis longa, and a brevis with a longa rest of two durational units; this then is the first ordo. The second ordo consists of five notes and five notes, always divided into groups of two and three.
     The third ordo consists of two, two, and three notes.
     The fourth ordo consists of two, two, two, and three notes; these ordines may be increased by groups of two notes at the beginning of the ordo, never at the end, or, in other words, in just the opposite way f row the first mode.
     The ordines of the first imperfect mode consist of even numbers of notes, namely by the appendage of such groups of notes such a mode begins with two, two, and two notes without rests, but never has three notes at the end.
     The first ordo of this mode consists of a longa brevis, with a longa rest of two durational units, to which is added a brevis longa, followed by a rest of one durational unit.
     The longa brevis, the longa rest and brevis, and the longa brevis rest are all equal in durational value in the same fashion.
     The second ordo of the first imperfect mode proceeds in the same manner, except that there are four notes instead of two. These are grouped as two and two notes with a brevis rest. As many of these may be used as desired, as we have indicated.
     The third ordo consists of six notes, grouped as two, two, and two notes with a longa rest, and two, two, and two notes with a brevis rest. Thus, there are six notes and a longa rest, and six with a brevis rest etc. This may be repeated as often as desired.

Coussemaker p. 328b40 to p. 329b16

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     The principle underlying the second imperfect mode is realized by having two, two and two ligated notes, but there is a difference between the principle underlying this mode and that for the afore-mentioned first imperfect mode. This mode is marked by differences of propriety and perfection of the notational symbols; the first group is sine proprietate but cum perfectione, as shall be explained fully in the second chapter.
     The first ordo consists of two notes with a brevis rest, and four notes with a longa rest, and may be repeated over and over again; it is just the opposite of the first imperfect mode.
     The second ordo consists of four notes and a brevis rest, and four notes and a longa rest, and may be repeated as often as desired.
     The third ordo consists of six notes and a brevis rest, and six and a longa rest, and this may be repeated. Observe, that many such groupings of the same mode may be combined with specific ordines.
     The principle underlying the third perfect mode is realized by having one longa and three, three, and three ligated notes without a rest. The perfection is completed by the next-to-the-last note, whereas the perfection of the first mode ends with a brevis, and the second with a longa, as we have already indicated above. It should be noted, however, that the brevis of the first, second and sixth modes always has one durational unit. The longa of these modes always consists of two durational units. The longa of the third perfect mode, however, contains three durational units, and is the equivalent of a brevis and a longa, according to the second mode, or a longa and a brevis, according to the first mode. It is noteworthy, that the situation is similar when different voices are used together (id est, with contrasting rhythms). The first brevis of the third mode contains one durational unit; the second brevis contains two. This is true, because, whenever there is a succession of breves, the more such notes approach the end of the group, the longer their durational value. This is true of the third and fourth modes, w shall be discussed below, and of every mode in which the durational values are split.
     The first ordo of the third perfect mode contains four notes, followed by a longa rest of three durational units, and these notes are divided into groups of one and three. This is repeated in this fashion over and over again as often as one wishes.
     The second ordo of this mode adds a group of three notes and contains seven, seven, and seven notes, followed by a rest of three durational units. It is known that this rest completes the perfection after the group of seven notes, which are divided into one and three and three.
     The third ordo contains three more notes than the second ordo, as| listed above, and has ten notes, divided as above, followed by a longa rest. In this manner, you may understand the other ordines of the afore-mentioned mode.
     The principle underlying the fourth perfect mode is realized by removing the first longa of the third mode; this mode then contains three, three, three, and two ligated notes with the propriety and perfection as explained below in chapter two (id est,

Coussemaker p. 329b17 to p. 330a32

1)    This passage is discussed at length by Niemann, p. 51-53, who misinterpreted the form of the final two-note ligature of the fourth

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only the final ligature is sine perfectione). Because of these distinctions, we are able to recognize the difference between two such groups and the second or first imperfect mode, etc.
     The second ordo adds three notes to the previous ordo, that is three, three, and two ligated notes with a longa rest, as has been stated; this may be repeated over and over again, always with a longa rest at the end.
     The third ordo of this mode adds three notes to the beginning of the previous ordo, so that there are eleven notes, divided into groups of three, three, three, and two, where the eleventh note is followed by a longa rest. Thus, this last case will contain eleven notes. The question of the propriety and the form of.the notational symbols of all of the musical figures mentioned above and their recognition has been postponed, and will be discussed at length in the second chapter.
     The principle underlying the third imperfect mode is realized by having one longa, and three, three, three, etc. ligated notes with two at the end cum proprietate but sine perfectione. The perfection is always completed by the next-to-the-last note of every group of three, as was shown above in other cases.
     The first ordo of the third imperfect mode is derived from the first ordo of the third perfect mode but has one note less. Thus, there will be three notes or tones with a longa rest of three durational units. After this, there will be three notes of a different kind, again with a rest of three durational units, if it is to occupy the place of the two breves. This may be done differently, however, if the rest only occupies the space of a brevis of one durational unit. In such a case, a further group of three notes follows with a different division, resulting in an irregular mode.

Coussemaker p. 330a33 to p. 330b23

perfect mode. Niemann implies that Garlandia also notates the final two-note ligature of the fourth mode as cum proprietate and cum perfectione, whereas this is actually the case only in the example reprinted by Coussemaker p. 177, but not according to the text and example on p. 101 (= Cserba p. 201). Actually, these theorists are in complete accord with Franco and Odington, since the final ligature is always to be considered as being sine perfectione in this mode. Thus the second ordo of the first perfect mode differs from the first ordo of the fourth perfect mode, only insofar as the final two-note ligature is cum perfectione in the first mode and sine perfectione in the fourth. Incidentally, I had suggested that the terminology sine perfectione for ligatures derives from the fact that the position of the final note of such a ligature gives the impression that one further note is required to restore the ligature to the normal forms inherited from the chorale notation, (confer, Musica Disciplina, IX 1955). I find tacit justification for this explanation in the phrase: sub intentione perfectionis unius longæ in pausatione, Coussemaker p. 346b_18-19.

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     Following this example further, we have a brevis of two durational units, a longa of three, and a brevis of one, with a rest of two durational units; at this point, the entire process is repeated from the beginning, as if it were going around in a circle. If we were to begin with the first part of this, which we have just described, then it would be recognized by its nature as the afore-mentioned ordo. One of the afore-mentioned parts would contain a longa and two breves with a longa rest of three durational units; the next part would have two breves with a longa of three durational units with a brevis rest of one durational unit; a brevis, longa and a brevis with a brevis rest of two durational units are joined then to these; this can be repeated over and over again. It should be noted that the first group of notes is divided into a single note and a ligature of two notes, as if they were to be sung to different syllables; even if they are sung to the same syllable, the are not ligated, for there is no way of conjoining them, although the are understood as ligated. The next three notes are both formed and conceived of as being ligated or combined in one figure. The final three notes are both formed and conceived of as consisting of two ligated and one separate note. This series may be repeated over and over again.
     As already has been stated, a different way of handling this situation is to form and conceive of a longa and two ligated breves with a longa rest of three durational units; the next three notes after this are both formed and conceived of as ligated; these then are followed by a longa rest of three durational units, which supplants the two breves. The distinctions between any of the groups of three notes have been shown by what has just been said. This process may be repeated over and over again.
     The second ordo of the third imperfect mode consists of a longa and three ligated notes cum proprietate but sine perfectione, as has been stated previously;then, there is a longa rest of three durational units; following this, there are three ligated and three ligated notes, followed by a rest of one durational unit; thereupon, there are two ligated notes, and three ligated, and a single note with a rest of two durational units. This process may be repeated over and over again.

     This ordo adds three ligated notes to the first group between the one and two notes; in the following group, it adds three

Coussemaker p. 330b24 to p. 331a21

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notes at the beginning, and in the final group, it adds three notes after the two ligated notes. This continues in a cyclic pattern over and over again, etc., either with the same or a different succession of pitches, or both¹.
     A different version of this consists of a longa, three ligated, and two ligated notes, as shown previously, and followed by a longa rest of three durational units. Then, there will be three ligated notes, followed by another group of three ligated notes with a longa rest of three durational units; finally, there will be two breves and three more notes conjoined; and this may be repeated over and over again.
     The third ordo of this mode consists of a longa, three ligated, and three ligated notes, with two ligated notes followed by a longa rest; after this, there will be three, three, and three ligated notes with a brevis rest of one durational unit; after this, there will be two, three, and three ligated notes with a single note, followed by a brevis rest of two durational units. Consider the mode to be increased by three notes after the first longa, thereafter by three notes before the longa, and finally by three notes after the two. Thus, it will be understood that all the other variants of this mode may have additions made to them in like manner, assuming of course, that the group may be increased or augmented. It is known, that the third mode, which is reduced by one note from its perfect form, is well conceived.
     Another form of this mode is found when there are one longa, three, three, and two ligated notes, only the last of which is sine perfectione, followed by a longa rest of three durational units; after this, three, three, and three notes always ligated, followed by a longa of three durational units, which takes the place of the two breves. This may be repeated over and over again.

     The difference between this ordo and the one described just before, is that this one has only longa rests, whereas the other one also has brevis rests. Another difference is that there are two methods of organizing this mode (id est, only two different rhythmic groupings), whereas the other has three methods. The principle underlying the fourth imperfect mode reduced by one note from its perfect form is realized by having three, three, three etc. ligated notes followed by a single one².

Coussemaker p. 331a22 to p. 331b14

1)    This and subsequent remarks of this nature undoubtedly refer to the fact, that the number of notes of the tenor could (and mostly did) coïncide with the number of notes required for the completion of a cyclic formula, id est, the tenor is repeated with the same rhythmic groupings. There are some compositions, such as those motets whose tenores are constructed in the form of a rondeau or other such composition, where the succession of notes in the tenor is repeated at certain times, but at others the formula has alternate material. This would provide an explanation for the use by the theorist of vel ambo (or both).
2)    This passage is incorrectly emendated by Niemann p. 53, who felt that there should be two ligated notes sine perfectione.

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     The first ordo of the fourth imperfect mode consists of three ligated notes and a single note with a brevis rest of two durational units; after this, there is a longa and three ligated notes, followed by a brevis rest of one durational unit; after this, there are two ligated and two ligated notes sine perfectione and then a longa of three durational units; this may be repeated over and over again.

     Another form of the same mode is found when there are three notes and one with a brevis rest of two durational units; then a longa and three ligated notes with a longa rest of three durational units, which takes the place of the two breves. Then, there will be a longa again with three ligated notes and a longa rest, and a longa and three ligated notes etc. Four more notes beyond this will no longer be in the same mode as the first groups; thus, we see that the fourth mode is converted into the third, etc.

     The second ordo of this mode adds a group of three notes just before the last note; then, three notes are added at the end; three notes are added to the final sequence between the groups of two and two; in each case, the rests that are used correspond to the various ones listed above; this may be repeated over and over again.
     The other form of this ordo of the fourth imperfect mode has a brevis rest (of two durational units) after the first group, and a longa rest after the second group, as was stated above. This is then repeated over and over again leaving the first group aside, and thus, we see that the mode has been converted, as shown above.
     The third ordo adds three notes ahead of the final note in the first group. Thus, there will be three, three, three ligated notes and a single one followed by a brevis rest of two durational units. After this, three notes are added at the end to the second group, so that there are one, three, three, and three notes; finally, three notes are added to the final group immediately before the two ligated notes, and this is followed by a rest of three durational units. This may be repeated over and over again with the same sequence of rests.
     The other form of this ordo of the fourth imperfect mode consists of three, three, three ligated notes, and a single one, followed by a brevis rest of two durational units; then, there follows a longa with three, three, and three ligated notes; finally, we have a longa with three, three, and three notes. Thus, these latter groups proceed uniformly with ten notes, differing only from the first group. Thus, the mode has been converted, as was stated previously.

Coussemaker p. 331b15 to p. 332a10

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     The first ordo of the third imperfect mode, which has been reduced by two notes from its perfect form, is realized in the following manner: one note and one note with a brevis rest of two durational units. Then, the longa and brevis are followed again by a rest of two durational units, and are repeated as many times as desired. For each repetition, there will be a brevis rest (of two durational units) at the end.

     Another¹ form of this ordo has a longa rest of three durational units, followed by a brevis rest of one durational unit. The first note is omitted, but the second and third are ligated and are followed by a brevis rest of one durational unit. This may be repeated with groups of two ligated notes and a brevis rest of one durational unit. This then may be repeated over and over again without the rests found at the beginning.

     It may be noted, that if this preceding ordo of two notes is used in connection with the other ordo of two notes, they will fit well together, whether they coïncide in sound, or whether they are used or arranged with different pitches in other such groupings².
     The second ordo of the third imperfect mode, which has been reduced by two notes from its perfect form, is realized as: a longa, three ligated notes and one note with a brevis rest of two durational units.

Coussemaker p. 332a11 to p. 332a36

1)    Coussemaker's text is repetitious at this point, and the second paragraph expresses this ordo better than the first. Thus, the following paragraph is omitted in translation in the text. Another form of this ordo has a brevis rest at the beginning, followed by two ligated notes with a brevis rest of one durational unit; then, there will be two ligated notes with a brevis rest of one durational unit, repeated as often as one pleases. It is, however, necessary to place a double rest of one longa of three durational units, and one of a brevis of one durational unit at the beginning'. This was understood differently by Niemann p. 41-42, who added a single longa at the beginning of one of the ordines.
2)    They will coïncide in sound, if, as in an hoquetus, they are combined as in Example 7 with the alternate form of Example 8.

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     This can be repeated over and over again. You can recognize the addition of this second ordo over and above the first ordo of the third mode as presented above, because it can be seen that three notes are added to each ordo immediately preceding the last note of each group. If it is necessary to continue, as shown above, begin with two ligated notes, then three ligated with a brevis rest of one durational unit. This may be repeated with two and three ligated notes and a brevis rest, and this pattern may be repeated or varied.

     It should be noted, however, that the alternate version of this ordo, which we mentioned, does not begin in the same way as does this one, since the latter is preceded by a brevis rest of one durational unit and the other a brevis rest of two durational units. It should be noted, that it is better to have a single longa rest of three durational units, in place of two for the two breves. Thus, you will require for the one example, not only a single duplex rest of three durational units, but also one of one durational unit, or vice versa. When the former begins with a longa of three durational units and a brevis of one, there will be a good connection between the two (confer Examples 7 and 8), if they are properly combined. The same will be true, if a note, not part of the pattern, is added to the beginning of the one ordo to occupy the durational value of the rest and to form a consonance with the beginning of the other ordo, as shown above¹.
     The third ordo of the third imperfect mode consists of a longa with three and three ligated notes and a single note of one durational unit with a brevis rest of two durational units; again a longa with three and three ligated notes and a single note and a similar rest; this may be repeated over and over again, but it should be differentiated as stated above.

Coussemaker p. 332a37 to p. 332b24

I)    It would appear, that this could be realized in two ways, as shown below. In the first case the ordo would become identical with the normal third mode without any rests so that the alternation, or hoquetus-like principle would be realized only in the one voice, as in the composition in the Montpellier manuscript (Y. Rokseth, Polyphonies du 13^e siècle, II 1936 No. 137). In the second case, the more likely, the hoquetus would be continued, but the two parts would begin together.

19

     The first ordo of the fourth imperfect mode, which has been reduced by two notes from the perfect form, is realized in the following way: three ligated notes with a brevis rest of one durational unit, then two ligated notes and a single note of one durational unit with a brevis rest of two durational units, then a longa and two ligated notes (sine perfectione) with a longa rest. This pattern may be repeated over and over again. The various versions of this mode have been discussed in the preceding section sufficiently.

     The second ordo of the fourth imperfect mode, which has been reduced by two notes from the perfect form, is realized in the following way: three and three ligated notes with a brevis rest of one durational unit, then two and three ligated notes followed by a single note of one durational unit with a brevis rest of two durational units, then a longa, three ligated notes and two ligated notes sine perfectione with a longa rest of three durational units. This may be repeated over and over again, either with the same musical pitches (id est, repetition of the tenor) or with a different succession. This ordo is formed by the addition of three notes to the first ordo; thus, three ligated notes have been added at the end of the first part of this ordo, and three notes have been added to the second part between the two notes anal the single one, finally for the third part, three notes have been added between the single and the two notes. The various forms of this mode have thus been demonstrated by their description; in this fashion, we can recognize all of the modes previously and subsequently mentioned.

     The third ordo of the fourth imperfect mode, which has been reduced by two notes from its perfect form, is realized in the following way: Add three notes to the second ordo, as shown above, at the end of the first group before the brevis rest of one durational unit; then, add three notes before the last single note, which is found just before the brevis rest of two durational units; then, add three notes to the third group before the two notes sine perfectione and the longa rest of three durational units, in that these groups are arranged as before: three and three and three ligated notes with a brevis rest of one durational unit: then two and three, and three ligated notes with a brevis rest of two durational units; finally, a longa and three, and three, and two notes, the final ligature being cum proprietate and sine perfectione in form;

Coussemaker p. 332b25 to p. 333a17

Anon. IV Part 2