20

this is then followed by a longa rest of three durational nits. This pattern may be repeated over and over again. The different versions have thus been demonstrated by their descriptions; in this fashion also, we are able to discuss the perfections, etc. All this can be understood in another way: The longa rest is used for the two breves, unless these durational values are preferred; in such a case, there will be a different cyclic kind of diminution, as has been shown above for the fourth modes.
     The first ordo of the third imperfect mode, which has been reduced by three notes from its perfect form, is realized in the following manner: a single longa with a brevis rest of one durational unit; then a brevis of two durational units with a longa rest of three durational units; then a brevis of one durational unit with a brevis rest of two durational units. This pattern may be repeated over and over again, either with the same pitches or with different ones (id est, repetition of the tenor). This ordo is called a solitary ordo, because there is no second ordo to it, as had been the case in the modes discussed previously, unless such an irregular pattern be made by the addition of one of the above-mentioned modes.
     The first ordo of the fourth imperfect mode, which has been reduced by three notes from its perfect form, is realized in the following manner: two ligated notes cum proprietate and sine perfectione in form with a longa rest of three durational units. This may be repeated over and over again.
     The second ordo of the fourth imperfect mode, which has been reduced by three notes from its perfect form, does not exist, unless it results from the splitting of a longa. It does not have a second ordo, since it is called a solitary ordo.
     There is also another solitary ordo, which differs essentially from this in that it is reduced by four notes from its perfect form. This ordo is realized in the following manner: one brevis of one durational unit, one brevis rest of two durational units, one longa of three durational units, one brevis of one durational unit, one brevis of two durational units, one longa rest of three durational units. This pattern may be repeated over and over again, as said previously. These solitary ordines may also be found in the opposite modes (exempli gratia, third instead of fourth, fourth instead of third), as has been stated above and as will be shown below fully.
     The principle underlying the fifth perfect mode is that it consists entirely of longæ of three durational units never with a rest in the position of odd numbers of notes; the first perfect ordo consists of three notes with a longa rest, etc.; the second ordo has five notes, the third has seven, etc., always with a longa rest.
     The fifth imperfect mode is the only mode to consist exclusively of even numbers of longæ.
     The first ordo of the fifth perfect mode consists of: longa, longa, and longa with a rest of three durational units; then, longa, longa, and longa with a rest of three durational units. This may be repeated over and over again as desired, either with the same pitches or with different ones. It should be noted, however, that each longa contains three durational

Coussemaker p. 333a18 to p. 333b28

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units, and thus, each longa is the equivalent of a longa and a brevis of the first mode, or a brevis and a longa of the second mode, or both together, if the harmonies are well conceived, which is difficult in such sciences, unless they are changed from long durational units to more accommodating ones¹.
     Certain older writers, formerly, were accustomed to notate these three longæ in a ligature followed by its longa rest, whereby they used three ligated notes to designate three longæ. Nevertheless, it was impossible to distinguish this ligature from other ligatures of three notes used in modes, which have been or will be discussed. This ligature could only be distinguished by the harmonic considerations of the upper-part adapted to it, or by the progression of the upper-voice in longa brevis, longa brevis, and longa brevis with a rest in accordance with the principle of the first mode, or brevis longa, brevis longa, and brevis longa with a rest in accordance with the principle of the second mode; in the third mode, however, one finds a longa and two breves, longa and two breves, etc., in accordance with the afore-mentioned ordines.
     Thus, this use of three notes, as indicated above, is a method of notating the lower-voices in ligatures, primarily the tenores. Otherwise, it will be disjunct in lower-parts (id est, if the ordo contains more or fewer than three notes) and in all upper-parts. This has been the case, ever since people began to recognize these differences, that is from the time of Perotinus the Great, and from the time of his predecessors. The further back one goes from this time, the less this was recognized, and the more one considered the relationship between the lower and the upper voice (id est, note against note style of writing), or the upper and the lower-voice; there was a total of six harmonic intervals, and that was sufficient for that time. One should not wonder at that, because they used very few of the different kinds of ordines, which have or will be discussed sufficiently.
     The second ordo of the fifth perfect mode consists of three and two disjunct notes followed by a longa rest. This may be repeated over and over again either with the same succession of pitches or a different one.
     The third ordo of the fifth perfect mode consists of three, two and two notes all disjunct, followed by a longa rest of three durational units. This may be repeated over and over again. This afore-mentioned fifth mode is found in many places in different volumes of organa, especially in Laqueus conteritus, which another theorist (scilicet Iohannes de Garlandia) has used for an example of the fifth of the afore-mentioned modes; this same person, however, used Regnat for an example of the third and fourth of the afore-mentioned modes, and exemplified the first and second modes with Latus, as shown in the

Coussemaker p. 333b29 to p. 334a34

1)     The significance of this passage is not quite clear; it would appear that an allusion is made to the transformation of modes, as also in Coussemaker p. 350_a33. The harmonies (id est, the consonances) might be difficult to realize in the fifth mode, if all of the notes were required to form consonances with all of the notes of the upper-part, if in our transcriptions single measures of 3/4 instead of double measures of 6/4 were used. This would, of course, be obviated if two melodies, each in the fifth mode, were reduced to the first or second mode, where only the even-numbered notes would have to form consonances with each other.

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treatise: Habito de ipsa plana musica que immensurabilis dicitur, in which a great number of examples is shown¹.
     The principle underlying the sixth perfect mode can be seen and shown in the example Laqueus; the sixth mode consists of four ligated notes and three and three ligated notes to the extent desired, before the rest, but the perfection will always end with the next-to-the-last note of each ligature. The first ordo of the sixth perfect mode, and all other ordines of this mode may be formed in two ways; either the last note before the rest is a longa or it isn't (id est, it is a brevis). If, as in the first case, it is a longa, the mode is formed as follows: four ligated notes, of which the last note is a longa of two durational units, whereby the four notes consist of three breves and a longa having the same durational value as in the parallel melodic voice (id est, first mode). This is then succeeded by a brevis rest of one durational unit. The four notes with their longa and subsequent brevis rest may be repeated over and over again. You will recognize the perfections, because three breves constitute a perfection, and the other perfection will consist of the longa of two durational units with its brevis rest. This may be repeated over and over again with a longa and a brevis rest for each perfection, and in this manner the perfections may be distinguished. You should know that four notes with their rest may either be reduced to the first mode or to the second. There will be the same number of notes in this mode as in the third mode above, although this mode is quite different from that one The difference, however, may be seen in their formal organization, because four ligated notes are found before a rest in the one case (id est, the sixth mode), and a single one followed by three ligated notes in the other.
     The second ordo of the sixth perfect mode consists of four and three ligated notes, of which the last note is a longa with a brevis rest of one durational unit. This may be repeated over and over again. The perfections, however, always end after the group of three notes, or with the longa and brevis rest as in the first ordo above, which has already been treated fully.

Coussemaker p. 334a35 to p. 334b35

1)     Iohannes de Garlandia, whose treatise is identified here by its incipit, does use the succession of notes derived from the beginning of the psalm verse Vs Laoueus conteritus from the Graduals Anima nostra (M 7) to exemplify the fifth perfect and imperfect modes, and the notes of Regnat from the Alleluia Vs Hodie Maria (M 34) for various forms of the third and fourth modes. Latus (M 14), however, is used in the treatise embedded in the compendium of Jerome of Moravia only as an example of the manner in which a tenor may be fashioned from a plain song (Coussemaker p. 98ff., Cserba p. 196ff.). In the independent treatise of Iohannes de Garlandia (Coussemaker p. 176ff.), Latus is used to exemplify the first mode, but not the second as mentioned in this passage. The first imperfect mode is exemplified in the first treatise by Audi filia (M 37) among others, and this has the same succession of notes as Laqueus (Coussemaker p. 102, Cserba p. 202). Latus does appear again to exemplify the use of rests in the first treatise only (Coussemaker p. 103, Cserba p. 205).

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     The third ordo of the sixth perfect mode consists of four, three and three ligated notes with a brevis rest. This pattern may be repeated over and over again. The different ordines simply add groups of three notes, one to the other. The first group after the first four notes is made up of three notes, as explained above, and the perfections always end with the next-to-the-last note in the groups of three, three and three The final perfection consists of a longa and a brevis rest.
     It is also possible to conceive of these three afore-mentioned ordines in the following manner: four ligated, the last note of which is a brevis with a longa rest of two durational units; in the second ordo there are four ligated and three ligated notes, in which the last note is a brevis followed by a longa rest. The third ordo has four, three, and three ligated notes with a longa rest of two durational units. You should always recognize that the perfections end with the next-to-the-last note of the ligatures, and the (final) perfection consists of a brevis and a longa rest. These modes are perfect modes, just like the others, but by reducing the number of notes, they may be changed into other modes, such as the first or second. It follows from this, that the second form of the sixth mode, as discussed above, may be reduced to the second mode, and the first form to the first mode. Enough has h en stated that this may be understood.
     The first ordo of the fifth imperfect mode, which has been reduced by one note from its perfect form, is realized by having two disjunct longæ with a longa rest of three durational units. This may be repeated over and over again. It should be noted, however, that an entire perfection is realized by every longa of the fifth perfect or imperfect mode, since this value is capable of being reduced to a perfection of the first or second mode.
     The second ordo of the fifth imperfect mode consists of two and two disjunct notes with a longa rest of three durational units. This pattern may be repeated over and over again, either with the same or a different succession of pitches. Accordingly, you will be able to recognize this in each statement, as stated above and below. You can observe that the perfection is the same as in the first ordo of the fifth imperfect mode.
     The third ordo of the fifth imperfect mode consists of two, two, and two notes, always disjunct and followed by a longa rest. This pattern may be repeated as often as desired, as has been stated.
     The first ordo of the sixth imperfect mode, which has been reduced by one note from its perfect form, like the first ordo of the sixth perfect mode, is heard quite often, and is realized in the following way: three ligated breves with a longa rest of two durational units; then, three disjunct breves, that is a single brevis and two ligated breves with a brevis rest of one durational unit. This pattern may be repeated over and over again. The perfections all consist of three durational units. One such perfection consists of a longa rest with a following brevis; the other, two ligated notes sine perfectione with a brevis rest.

Coussemaker p. 334b36 to p.335b8 

24

     The second ordo of the sixth imperfect mode consists of the following: four ligated notes and two ligated notes sine perfectione with a longa rest of two durational units, then a single brevis, four ligated notes, the last note of which appears to be a longa by its form (id est, the ligature is cum perfectione) but which is actually a brevis in durational value¹. This continues for the number of times required by the cyclic process. The perfections may be distinguished as three and three notes or a longa rest of two durational units and a brevis. Of the four ligated notes, three are reckoned for the perfection, whereas the fourth together with the single brevis and the brevis rest total another perfection.
     If the seventh note, however, is a longa (id est, instead of a rest) then it together with its rest is added to the brevis and the following brevis to form a perfection. This is, however, quite difficult to express, since one perfection is thereby caused to overflow into the other perfection. This, indeed, is found in many songs, where, however, one does not know precisely how to define the situation. In many localities, however, this is not used, unless a breath-mark is added, which looks exactly like a rest, but which does not take into account what follows.
     The third ordo of the sixth imperfect mode consists of four, three and two ligated notes, the final group being sine perfectione, and a longa rest of two durational units. This continues in the same fashion, as has been described, for the six notes (id est, second ordo) of this mode for as many cycles as desired, etc. It should be noted that this third ordo may proceed more smoothly in the following manner, namely: In the first ordo, the fourth note becomes a longa, counting the note and the rests. None of these rests may be omitted, as stated above, since otherwise one perfection would overflow into the other; something like this, however, can easily be expressed in the middle, and the continuation is not difficult to recognize. The same is also found in the second ordo in seventh position, whether this be a note or a rest. In the third ordo the tenth symbol may be a longa note or a rest. This may be continued over and over again until the cyclic process is complete. Thus, you will see that such an ordo of notes will not cause the perfections to cross, since whenever it occurs, there will be

Coussemaker p. 335b9 to p. 336a10

1)     Niemann p. 77 suggests an emendation of this passage in Coussemaker's text, which I do not believe is quite justified. The point is, that the four-note ligature of the second group of the second ordo of the sixth imperfect mode is to be notated cum perfectione, whereas the actual note value is a brevis, which would demand a ligature sine perfectione, and is not a longa. Thus, the passage should read: .....quatuor coniunctim, ultima longa secundum materiam, vel secundum actum una brevis, addendo unam brevem, et tunc una brevis pausatio altera (= further).

25

a distinct succession of perfections; thus, disjunctiveness in the middle of an ordo will cause conjunctiveness, or vice versa¹.

     A different form of this mode can be understood more easily. There would be four ligated breves, that is a series of notes with a brevis rest; after this, there are three notes, of which the first two are ligated and the last one is single, followed by a brevis rest; thus, there are three notes of which two are ligated and one stands alone and is followed by a brevis rest. You can recognize that the three notes are always breves, but there will also always be three for each perfection, three for each perfection; these may be reduced to either the first or the second mode. It does not matter if there is a brevis rest at the beginning in the middle or at the end. Thus, you will be able to compute the afore-mentioned ordines, reduced by one note from their perfect form, if you so desire. You may also reckon the ordines which are reduced by two or three notes from their perfect form in the same way as in the case of the third mode above. If, however, you refer these ordines to the fourth mode, then you must consider them as being reduced by four notes².

Coussemaker p. 336a11 to p. 336a30

1)     As Niemann p. 79-80 quite correctly pointed out, this involves a mixing of perfect and imperfect modes, in that the longa rest may be supplanted by a longa note, a breath-mark being added after this. This will, of course, cause the two-note ligature sine perfectione to become a three-note ligature cum perfectione. The difficulty in the description of the perfections (pedes) results from the fact that the second group of notes begins with a brevis, which still forms part of the old perfection, but which belongs to the new group, as if it were an upbeat. This apparently caused some confusion in certain circles, and even our theorist was not clear on this point, so that this is not used unless a breath-mark is indicated. Evidently, since there is no actual rest involved, our theorist considers this to be one long ordo, where the longa occurs in the middle.
2)     Since the first ordo of the fourth perfect mode contains five notes, it will always necessitate the reduction by one extra note to realize the same number of notes over that required for the imperfect forms of the third mode, where the perfect form has only four notes. 

26

     What has just been stated together with that which follows in the second chapter suffices for an understanding of the sixth mode. Since it is impossible to understand this completely without having considered the forms of the notes, all possible computations have been made, such as those which involve number, perfection, figure, conjunction, and disjunction. Quite often, there is a succession of notes of a certain number and quite often there are fewer. Sometimes it is better to have an ordo, sometimes not, according to the rhythmic arrangement.
     The first ordo, which is called the solitary ordo of the fifth imperfect mode, reduced by two notes from its perfect form is realized by having a longa and a longa rest of three durational units. This may be repeated over and over again.
     The solitary ordo of the sixth imperfect mode, just as in the third mode, when reduced by three notes, may be considered in two ways, depending on whether the rest precedes or succeeds.
     Thus, the solitary ordo of this mode may proceed in two ways, according to the manner in which the rests are divided in the perfection, or according to the form of the note. Thus, the note may have the form of a longa, as shall be treated at great length in the second chapter. Note that quite often there is a difference in the number of notes in the ordo; thus, sometimes the ordo is quite changeable, sometimes not. Sometimes it is arranged in only one fashion, sometimes in different ways. All of the above will be treated in the second chapter.

Part 2

Concerning Smaller Values and the Splitting of the Modes

     The second part of the first chapter is presented in the following manner. The first part concerned itself with the six rhythmic modes of melodies, namely the way in which sounds or melodies are perceived by themselves, or how they are or could be combined with other modes or ordines, as shown above. The second part concerns itself with he breaking or splitting of durational values, that is with the splitting or combining (of parts) of any brevis or longa.
     The first mode consists of longa brevis, longa etc. The principle inherent in this mode involves the distinguishing of groups of three, two, and two ligated notes without a rest. It is known, however, that this mode may be presented in the following manner: two notes for the longa (id est, two breves) and a brevis, two notes for the longa and a brevis, etc., as has been sufficiently demonstrated above in the discussion of the sixth mode.
     Again, this mode may be described in this manner: three notes occupying the durational value of the longa brevis etc. without a rest. he note forms, however, are differentiated in the following manner: First, place a note, which formally looks like a longa, and add two notæ currentes, which will split the afore-mentioned longa into smaller note values. Thus, the note shaped like a longa together with its additions will have the durational value of a longa. If the melodic direction of the notes is completely descending, there will be a good result; similarly, if they all move in an ascending direction. Notes ascending and descending seldom occur. This can be accomplished, although this does not cause especially beautiful note shapes. We are, however, not concerned at the present moment with questions of beauty, but rather with questions of the significance of notes.

Coussemaker p. 336a31 to p. 337a14

27

     After this, put a brevis and a longa each having its respective note form, and split the longa with two notæ currentes, as stated above. Then use two ligated notes, and split the latter one with two notæ currentes etc., and thus, we place or feel three breves for a longa etc. Thus, we have the principle of the first perfect mode with a splitting or combining of three notes for the longa, as has been described above

.

     If you wish to combine five notes into a ligature, you will have three notes for the longa, the next-to-the-last will be a brevis, and the final note will be a longa. You can split this longa with notæ currentes, as has been stated above, using two ligated notes with notæ currentes etc., and this will be the same as described above.

     The first ordo of this mode with an integral (id est, not split) longa, before the rest, is constituted in the following manner: five notes ligated cum perfectione with a brevis rest, five with a brevis rest, five with a brevis rest, etc. This is similar to three ligated notes with a brevis rest. Another form of this first ordo, in which there is no integral longa, consists of the following: five conjunct or ligated notes, of which the final longa is split by two notæ currentes, then a brevis rest. This may be repeated with another five notes with two notæ currentes and a brevis rest. Thus, you will recognize three notes for a longa with a brevis rest, three for a longa with a brevis rest etc. In this manner, you will be able to distinguish the subsequent ordines, those with integral longæ at the end from those without. Two ligated notes may be added after this and they will indicate a brevis and a longa, or a brevis and a longa, namely where the longa has been split by two notæ currentes.
     There is another way of splitting and combining the brevis and the longa, namely: three notes for the longa, and two for the brevis. The principle which underlies this is namely: the longa is formed with the addition of two notæ currentes and three notes are added to this, two for the brevis, and the final note is a longa with two notæ currentes, again three notes with two notæ currentes in a similar fashion, and you will combine three notes for the longa and two for the brevis, etc.

Coussemaker p. 337a15 to p. 337b14

28

     The first ordo of this is formed in the following manner: Take an integral longa, that is a note which is identical with a longa in form, and add two notæ currentes which split it. Then, place three ligated notes cum opposita proprietate, of which the first two will occupy the durational value of the brevis, and the final one that of the longa, which is then completed by a brevis rest. This may be repeated over and over again.

     Another form of this is found when there is no integral note. For this, you take a longa and add two notæ currentes, then three ligated notes as above (id est, cum opposita proprietate), two for the brevis and the final note is a longa with two notæ currentes and a brevis rest. This may be repeated over and over again.

     Another form of this is found when you have three notes for the longa and three for the brevis. Thus at the beginning, you will find a note in the form of a longa with two notæ currentes, then four ligated notes cum opposita proprietate and cum perfectione, of which three notes are equal in durational value to the brevis, and the final note, to which two notæ currentes have been added, is a longa, again four notes with two notæ currentes, as stated above etc. Thus, three notes are combined for the longa, three for the brevis, three for the longa, etc.¹.

     The first ordo of this mode with an integral longa at the end consists of: three notes for the longa at the beginning, three of the four ligated notes for the brevis and the final one for the longa with a brevis rest. This may be repeated over and over again.

Coussemaker p. 337b15 to p. 337b37

1)     It is noteworthy, that in the four-note ligature cum opposita proprietate, the first three notes are considered semibreves, not just the first two as otherwise held.

29

     In order to form the second and third ordines of these modes, add two notæ currentes and four ligated notes with a brevis rest, etc., as often as you wish.

     There is another form of the first perfect mode, namely: four notes for the longa brevis at the beginning, then six ligated notes, of which the next-to-the-last will be a brevis, and the last a longa. The preceding four notes will occupy the durational value of a longa, and the ligature will be cum proprietate and cum perfectione. The last note has three notæ currentes, then two ligated notes cum proprietate and cum perfectione with three notæ currentes, etc. Thus, you combine four notes for a longa and then a brevis, four for a longa and then a brevis, etc. You can thus either have an integral or a split longa at the end of the first, second, or third ordo.

     There is another way of forming the beginning of this mode: four ligated notes cum (opposita¹) proprietate and sine perfectione, then three ligated notes cum opposita proprietate and cum perfectione, to which three notæ currentes have been added, then three ligated notes with three notæ currentes, and three ligated notes with three notæ currentes, in the same manner, etc. Thus, you will combine four notes for a longa, two for the brevis, as often as you please. You may always form the ordines either with an integral longa (at the end), or with a split one.

     Similarly, if you wish to have five notes for the longa before the three ligated notes, use a ligature of five sine perfectione, if six, then six sine perfectione, etc.¹. Do this always at the beginning of the first ordo, and not elsewhere; the final three notes may be split by the addition of notæ currentes, as often as you please, and the first longa may be split, if that is also your pleasure. Thus, we say, put four notes in place of the three in the manner described, that is cum opposita proprietate, and thus, you have three for a brevis, and the last note is split by notæ currentes, as stated above. It is customarily quite rare to split notes beyond this; thus, four notes are not used in place of a brevis in vocal music, although this is frequently done in

Coussemaker p. 337b38 to p. 338a34

1)    The addition of opposita is justified by the following paragraph.
2)     Splitting the longa into five or six notes would mean a subdivision of the two breves into three semibreves each, or three and two. The prohibition of four notes equal to a brevis means that there was no further division beyond the semibrevis in vocal music.

30

instrumental music with good results. Thus, the first mode with its finely woven pattern can still be reduced easily to an integral perfect or imperfect mode with a brevis or longa rest, as long as the total durational values are equal. That which has been and which will be stated will suffice for the understanding of this example.
     It should be noted, that we can recognize the second perfect and imperfect modes in the same manner as the first. For example, take two ligated notes cum proprietate and cum perfectione at the beginning with two notæ currentes, and continue in the same manner, etc. Thus, we will be able to combine three breves (id est, two semibreves and a brevis) for a longa and three breves for a longa, etc.; in this manner, you can form the first, second, and third ordines, each in its own way, in the same fashion as we have formed the first ordo, since there are just as many ways of doing this as in the first mode, as stated above. The difference between the two modes, however, is this: In this case, the first note is a brevis, etc., and thus, a longa follows, as given in the definition of this mode. Therefore, the second mode is just the opposite of the first mode, which consists of a longa brevis. nevertheless, there is no difference in the ways in which note-values may be split by notæ currentes, whether the longa or the brevis be involved, etc.
     The third mode consists of the following: the first longa contains three durational units, and is the equivalent of a brevis longa, or of a longa brevis, the first brevis which follows contains one durational unit; the second brevis contains two durational units, since these two breves are equal in durational value to the first longa. Thus, if this mode is reducible to the second mode, it is split or divided according to the arrangement or divisions of that mode; if this mode is reducible to the first mode, accordingly then in the pattern of that mode. From what has been stated, we can gather, that the brevis of one durational unit may be split into three or four notes, and the second brevis into six or eight notes, and the initial longa into nine or twelve notes¹; this is found in some localities, and others not. The note values are to be split according to the circumstances of vocal music, and according to that which fits better according to the nature of the second or first of the afore-mentioned modes, and according to their note shapes, as shall be demonstrated sufficiently in the second chapter below.
     We can recognize the same thing in the fourth mode, which is just the opposite of the third in long and short note-values. We don't, however, speak of this mode as the opposite form of the third mode in the sense that there is an alteration of the place of the longa and the brevis in the perfection.
     Split the fifth mode in the manner listed above, and this shall suffice for that.
     The sixth mode may be split according to the first or second mode, as shall be shown at great length in the second chapter. In addition, if what has been said above has been demonstrated to too limited in extent, it will be discussed in the following six chapters sufficiently and satisfactorily together with the triple perfections, that is: full, perfect, and final. In this context, we might add to what has been said before: The knowledge of all propriety that can be perceived is attributable to the full and final perfection of the knowledge of truth.

Coussemaker p. 338a35 to p. 339a6

1)     Evidently the splitting into larger numbers of small values is intended only for instrumental music.

31

Chapter II

Concerning Notes and Note Shapes

     The second chapter will present the following material: 1. discuss the forms of musical notes, as they are depicted in books, and the manner in which they express melodies, and 2. treat the durational units in two ways. According to the first method, the durational values are presented simply as absolutes without a descriptive passage being added; according to the other method, they are described.
     Musical notes, as absolutes, can be classified in two ways: They are either single notes, that is notes which have nothing added to themselves, and are used in combination with other such notes, or they are combined in ligatures.
     Certain single notes are used in chant books, and certain ones are used in books containing polyphonic music; in the latter case, the use of these notes differs according to the various volumes (id est, according to the different fascicles, and consequently according to the different types of compositions). Thus, they are used in certain of our books of different categories, as they are also used for different kinds of music for all sorts of instruments, etc., as shall be shown in the following.
     The musical shapes, referred to above, have been called notes by various persons, whereby one musical shape is called a note. Others call them figures, since a note may also be called a figure. Still others term them notæ simplices. In this fashion, a musical tone may be formalized with a musical symbol.
     There are three kinds of simplex figures found in chant books; the first kind stands upright with a short stem drawn from the right side; the second stands upright without the afore-mentioned stem. The third kind retains the form of the elmuahym¹.

Coussemaker p. 339a7 to p. 339b1

1)     According to its first description, the elmuahym is a note form identical with the plicated longæ and breves, videlicet In this form, it may be protracted obliquely upwards or downwards to form, what Robert de Handlo (Coussemaker p. 394) calls an obliquitas, scilicet. These latter forms have this in common with the form of the semibrevis, they are both drawn obliquely. The elmuarifa, which could be considered irregular, is the normal form of the first note of the coniunctura in England, id est, In the sense of the shape of the semibrevis, the term is used first to designate the notæ currentes, which are drawn like oblique figures. Finally there is the elnuahym, that others call a semibrevis and which is used in conjunction with another similar note or which forms the third durational part of a brevis; such notes, however, can only be used in connection with themselves, as if they were notæ currentes. It is not clear from the context, as Waite p. 84 seems to feel, that two semibreves are held to be equal in durational value Incidentally, whereas the continental form of the semibrevis is derived from the continental form of the coniunctura, the English form is derived from its typical form. For the Arabic terms used here, confer the footnote by J. Handschin, Zeitschrift für Musikwissenschaft, XIV 1931-1932, p. 321.

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Ii is a figure which is called a plica, and stands upright with one stem or with two, either in an ascending or descending direction, etc. Good notators depict notes in the afore-mentioned fashion. Certain writers make square shapes with or without stems, as was just stated; others do not make square shapes, but form their notes like a quadrangle or like quadrangles. Thus, since such notes rest on the long side, they are extended or protracted in length, and may have or lack a stem. The elmuahym is often protracted in an oblique direction, and others protract the elmuahym in a similar manner.
     Certain elmuarifa, which could be considered irregular, have a stem drawn downwards on the left, such as the English depict and notate, but they are identical in sound (id est, meaning).
     Musical shapes or notes may be combined together into groups of two, three, four, or more notes, as found in liturgical and extra-liturgical books (id est, troparia).
     Two notes may be combined in a ligature in the following manner: one square may rest directly above another square, and these are called two ascending notes, if they are joined by a stem on the right side; the higher note, however, can appear a little smaller than the lower. It should be noted that no ascending figure used for musical melodies may have a stem on the left side drawn downwards from the first note. Two descending ligated notes are recognized in another fashion: The first symbol has a stem on the left side drawn downwards from the first note, and another stem on the right side attached to the next note. The second figure, either a square or a quadrangle, is, however, not immediately below the first (id est, it is to the right). It may be noted, that if these two ligated notes move in an ascending direction, they are called perfect, when the second one rests immediately above the first. The two notes are thereby designated as two ligated notes cum perfectione. When the first note does not have a stem, as stated above, they are said furthermore to be cum proprietate and perfect or cum perfectione. In the case, however, of two ascending notes, where the second note is joined to the first, but does not rest above the first, its perfection is determined by this fact; since, also, the first note may have a stem drawn downwards on the left side of the first note, such a ligature of two notes may be said to be sine proprietate and imperfect or sine perfectione.
     Three ligated notes which are all in an ascending direction can be distinguished in the following manner: Cause one square to be joined to another square, note to note or corner to corner protracted laterally; then take another square connected with the second one and placed immediately above it, as stated in the description of ligatures of two notes. Accordingly, this ligature may be considered cum proprietate and cum perfectione following the considerations mentioned above, as explained at length in the case of ligatures of two notes. We may look at this in another way; take the two ligated notes, that we described above, and add a quadrangle below them so that the uppermost tone is connected on the right side with the lower square; this square in turn is connected on the left with the lowermost tone, and the result again is three notes similar to the three described above.

Coussemaker p. 339b2 to p. 340a18

33

     Three notes which are found exclusively in a descending direction are formed in the following manner: Take two descending ligated notes, as stated above, and add a square to the latter figure, no matter how the second note was joined to the first one. It is, however, not possible to join this final note to a higher one, unless the stem be removed, which in practice is not possible or done. (id est, not ).
     Three ligated notes may be formed in another fashion: Take two notes, such as the two former notes of the three ascending ones, as stated above, and add a square below the second one. The highest tone is connected on the left to the first lower note and on the right en the second one. The connection with this part may either have an intermediary or itself be immediate; it is immediate, if it has no stem, etc. ; it has an intermediary if it has a stem. These notes, therefore, proceed in an ascending or in an ascending direction, and may be said to be cum proprietate and cum perfectione, as stated above.
     Groups of descending and ascending ligated notes can be formed in two fashions, notably: Draw a descending stem, and from the stem protract a short or long elmuahym (oblique figure). Above the latter note, put an upright square, and join the right side of the one to the right side of the other. The second way of forming this, is to take two descending ligated notes, and put a square over the latter note, and join them, as stated above; this figure will now be cum proprietate and cum perfectione (id est, however, this is not the usual form).
     Thus, we have the rule that: Every figure which begins in a descending direction, if the first note has a stem on the left, is said to be cum proprietate. If this ligature begins in an ascending direction, and the first note has no stem, it is also said to be cum proprietate. The perfection of a figure is marked by the relative position of the final notes. If the ligature ascends at the end, and the final note rests above the next-to-the-last, then the ligature is said to be perfect. If, however, the final part of the ligature moves in a descending direction, then, whenever the final note is a square and connected with the others, it is cum perfectione. If the notes are not entirely squares, but are joined in a descending direction obliquely, the perfection is designated by this fact (id est, the ligature is sine perfectione). You can regard four ligated notes similarly to the extent that their perfection is determined by the end of the ligature; the middle notes are distributed, however, as stated above. This is the extent to which the ecclesiastical melodies use notational signs. When these are used with notæ currentes, they take the following form: Draw a square with one stem on the right side. Draw next to this, in an oblique direction, two, three or four elmuahym, and these are called notæ currentes, just like the others which will be discussed exhaustively below. In certain other books, however, five, six, or seven such notæ currentes are used and are combined with other musical symbols also.

Coussemaker p. 340a19 to p. 340b28 

34

     Performers call some of the figures (notæ) simplices and others ligatures; as far as the notæ simplices are concerned, some are called longæ and others breves.
     There are many kinds of notes having the form of the longa; for example, there is the longa which represents two durational units, as in the first, second, and sixth modes, as explained above; and this is formed with an upright quadrangle or square with a stem drawn downwards. there is another longa simplex, which expresses three durational units, as in the (third,) fourth, and fifth modes, as above in the first chapter, and all of this has been mentioned before.
     Again, there is another longa which may represent either the value of two or three durational units, as stated above, and this one is plicated either in an upward or downward direction, as others have said. It is an ascending longa, if it has a stem drawn upwards, or if it has two upward-drawn stems on the right and left side. The stem on the right side must be longer than the other to indicate that this is a longa, according to the larger or smaller durational values of notes, or according to the differences of the modes, as stated above. Again, there is another longa which is called a longa duplex, and which contains six durational units, or not fewer than five (id est, when it is made imperfect by a single brevis). This is a note, which is drawn out at great length, as you will see; it is almost as if two or three longæ have been put together into a square to which a stem, drawn downwards, has been added on the-right side.
     Again, there is a certain note, which is like a longa simplex in form, and which precedes notæ currentes. It is divided into as many parts as there are notæ currentes following them in the same fashion (id est, it looks like a longa but actually is a note of short durational value). There are also other longæ, and they express the larger and smaller durational values, as occur quite often in books of organum purum¹.
     There are also many interpretations for the note which has the form of the brevis. One of these forms presents, for example, an upright square without a stem and is said to be a proper brevis in form, because it depicts a brevis recta of one durational unit, as found in the first, second, or sixth modes, as discussed at length above.
     There is another brevis which contains two durational units, because it designates the durational value of two units, such as the second brevis of the third and fourth of the above-mentioned modes; and this note has the same form as mentioned above.
     Again, there is another brevis which is a square, as mentioned above, but which has two stems drawn upwards or downwards; the stem on the right, however, will always be shorter than the one on the left, such a note will either have the durational value of one unit or two, as explained above.

Coussemaker p. 340b29 to p. 341a33

1)     Certain English compositions, as exempli gratia those in the Worcester Fragments Nos. 44 and 47 (confer my edition: The Worcester Fragments, 1957), use an alternation of longæ to express the rhythm of the composition (id est, a longa duplex is a long note value, a longa simplex a short one).

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     There is also a notational figure which is called an elmuahym, or similar to itself; this figure always lies obliquely in a certain fashion, although it designates different things. Often, it is called a semibrevis if it either follows or precedes a similar figure-; in other cases, it forms the third part of a brevis, and this is the case when three of these notes are placed together as if they were notæ currentes; in this case, three such notes constitute a brevis. Similarly, there might be four notæ currentes taking the place of a brevis, but that rarely occurs and never in vocal music, but may be found in music for stringed instruments.
     It should be noted that one often finds a stem on the left side of the first note of such a series to distinguish it from what precedes, (id est, as the English use).
     It should also be noted that the afore-mentioned figures can be recognized in different ways as occasion demands according to the different modes, either when they are actual musical sounds or when they are rests. Some of the ligatures are called cum proprietate according to the nature of their initial parts, others sine proprietate, and finally others cum opposite proprietate, as described in the opusculum or treatise, the beginning of which reads: Habito de ipsa plana musica, (id est, the treatise of Iohannes de Garlandia).
     It should also be noted that certain musical figures are used when there is no text, and some when there is text. If there is no text, one should ligate notes as much as possible. If there is a text, one should ligate on occasion (id est, when no change of syllable interferes). These ligatures designate the length and shortness of the durational values, as already noted above in the consideration of the modes, etc., and as shall be explained at length below.
     Since we have already spoken of the notæ simplices above, we shall limit ourselves here to a discussion of the ligatures and the methods of determining their significance. The first of these ligatures is fashioned cum proprietate and cum perfectione, and is like those ligatures which are written normally. All ligated figures which are cum proprietate and cum perfectione are interpreted in the following manner: The next-to-the-last note is a brevis; the final note is a longa. If there is a preceding note or notes, it combines with the next-to-the-last note to form a longa¹ . Every figure which is sine proprietate and sine perfectione has just the opposite succession of durational values; the next-to-the-last note is a longa, the final one a brevis.

Coussemaker p. 341a34 to p. 341 b33

1)     This rule is more flexible than the Franconian rule, which states that the first note of a ligature cum proprietate is a brevis, since it permits the three-note ligature at the beginning of the first mode to be interpreted as longa brevis, longa. The ligature sine proprietate and sine perfectione under consideration here apparently has only two notes, although a three-note form sine proprietate but cum perfectione is used by Garlandia (Coussemaker p. 101 and 179, Cserba p. 201) to notate the first ordo of the second perfect mode, scilicet brevis longa, brevis.

36

     These rules were used in many older books; this was so during and after the time of Perotinus the Great. Nevertheless, they did not know how to distinguish these notes from those which will be presented shortly. This was so even since the time of Leo¹, because two ligated notes were put for the durational value of a brevis longa, and in a similar manner three ligated notes were quite often used for a longa brevis, longa. It should be noted that Master Leoninus, about whom it has been said that he was the greatest composer of polyphonic music, created a great book of polyphonic settings of the (solo parts of the) Graduale and the Antiphonale, with many different settings for Divine service; this was in use until the time of Perotinus the Great, who shortened this book, and who composed many very fine clausulæ or musical sections, whereas he became the greatest composer of discantile compositions; and he was a greater composer than Leoninus, although there is no question about the subtilities (id est, rhythmic complexities) in Leoninus's polyphonic compositions.
     Master Perotinus composed certain excellent four-voiced polyphonic settings, such as his Viderunt, and Sederunt, using an abundance of coloration of the harmonic art. In addition, there were renowned three-voiced polyphonic settings, such as his Alleluia Vs Posui adiutorium, (M 5) and Vs Nativitas (M 38), etc. 2). He also composed conductus for three voices, like the Salvatoris hodie, and conductus for two voices, like the Dum sigillum summi patris, and conductus for one voice, among others his Beata viscera, etc.³.
     The book or the books of Master Perotinus were used until the time of Master Robert de Sabilone in the choir of the Marian Church in Paris⁴; and they were similarly continued in use from that time until the present day. The method of notation

Coussemaker p. 341b34 to p. 342a26

1)    There are apparently two distinct personages: Leoninus the original composer of the Magnus liber, and the theorist Leo, whose treatise is reconstructed in my A Central Source of Notre-Dame Polyphony, p. 89-93, and who is mentioned at the beginning of the present treatise. It would appear that Leo might have been the first of the mensural theorists, and this passage may allude to him, in which case it refers to the formulation of a rule, which took place after the Perotinian epoch. Waite p. 59 apparently feels that this Leo is identical with the Leoninus of the following paragraph, which is also quite possible. The corrections in the Latin text are from Waite, loco citato.
2)     These numbers refer to the designations of these compositions in Friedrich Ludwig's, Repertorium organorum recentioris et motetorum vetustissimi stili, 1910
3)     For these sources, confer Groninger, p. 90, 126 and Ludwig, Repertorium, p. 124. Iustitia is an error in Coussemaker's edition p. 342; confer F. Ludwig, Archiv für Musikwissenschaft, V 1923, p. 187, footnote 2.
4)     For the question of whether reference is made to the Cathedral of Notre Dame as we know it today or to the Église de Sainte-Marie (de la Vierge), consult Niemann pp. 5-6, footnote 2. It is quite possible, as Niemann suggests, that Robert de Sabilone was the first director of music in the Cathedral.

37

was perpetuated by Petrus, the greatest notator, and Iohannes called the Primarius (distinguished) and others for the most part until the time of Master Franco the First and the other Franco of Cologne, who began to notate certain things differently; for this reason, there are certain rules which are peculiar to this tradition (exempli gratia, as opposed to the ideas of Lambertus = Pseudo Aristoteles).
     Accordingly, there were certain persons with hindsight, for whom the above-mentioned rules were not sufficient, and they began to use the sign of opposite propriety, as mentioned above. They maintained that in every figure which was cum opposita proprietate, that the final note was a longa, and that the preceding ones had the durational value of a brevis. It follows, therefrom, that there must be fewer than three ligated notes either in actuality or in concept cum opposita proprietate, because if only two notes were to be cum opposita proprietate, they would equal the first two notes of the ligature with the three afore-mentioned notes of the same propriety. Thus, if a longa were to follow, it, together with the other two notes, would have the same durational value as the three notes, as stated above. If there are four, five or six notes in such a ligature, they have the same total durational value as the three.
     There is another imperfect figure, which is drawn or formed in a way different from what has been said above. There is no rule for this, unless we discuss it in terms of perfections. As far as this reduction into perfections is concerned, it quite often occurs that sounds repeat the same pitch, whereby every ligature is said to be ascending or descending, as stated above. Here, however, where the tones have the same pitch, and are neither ascending nor descending, it is consequently not possible to fashion a formal ligature out of these, although it is possible to ligate them in one's mind quite easily by rendering the equivalents and reducing them into longæ and breves. Therefore, two such ligated notes with a disjunct longa following will form the equivalence of three notes, as stated above. Consequently, there will be two notes for the brevis and one for the longa, just as if there had been three ligated notes, where there are two notes for the brevis and one for the longa.
     The situation would be analogous, if there are four notes similar to these; then there would be three notes for the brevis, and the final one would be a longa, etc. All of this would total one durational unit if the figure sine perfectione were notated cum opposita proprietate. If the figure is sine perfectione and cum proprietate, as said previously, the ligature is made perfect by the first longa following, or by its equivalence according to the arrangement of longæ and breves of its mode. In such a case, the entire group is assessed just as if it were cum proprietate and cum perfectione, as stated above. If it should happen to be sine proprietate and sine perfectione, then it is made perfect by the first brevis following, according to the universal rule (judging such cases). Each case is to be assessed according to the rules stated above. Note, however, that there is a difference when only four notes are presented, from the case of more than four. If there are only four notes, they proceed according to the method outlined above: The next-to-the-last note is a longa, the final one is a brevis, and the preceding two occupy the durational value of a brevis. Others explain this in a different fashion; accordingly, the next-to-the-last note is a split longa, which is easier to comprehend, and may be analyzed in this manner. They call the last note as well as the first one a brevis, and the middle two notes together constitute a longa. Thus, these two notes are actually equal in durational value to three notes, and either are or are not reducible to a perfection. This is true, furthermore, for many notes which can not be ligated formally, but which are conceived of as being ligated.

Coussemaker p. 342a27 to p. 343a5

38

     It should be noted that every ligature which has more than three notes can or must be reduced to three ligated notes of the same kind. Thus, it is that every ligature with more than three notes, that is cum proprietate and cum perfectione, is reducible to a ligature of three notes cum proprietate and cum perfectione in durational values. In a similar manner, we understand that those which are sine proprietate and cum perfectione are reducible to three ligated notes sine proprietate and cum perfectione. In a similar manner, the figures cum opposita proprietate and cum perfectione may be reduced to like forms. From what has been said, the different combinations mentioned above may easily be applied to others. Any lack of completeness in what has been said will be compensated for by copious exemplification below.
     Figures with text or above a text are treated as notæ simplices; if they are well drawn or notated, there will be no ambiguity in their interpretation, as was true in the case of the many composers of polyphonic music who gathered all sorts of settings for Divine service in their volumes. Pleasurable Divine efficacies are treated according to their uses in different parts of the world, in accordance with Sacred Scripture, where it is said: Laudate Dominum in timpano et choro, etc. and continuation (Psalm 150). It should be noted, that the difference between texted and textless parts is that when there is no text, you ligate two, three or four notes, or as many as possible, according to the best possible arrangement of notes, as shall be shown below. When there is a text, one sometimes ligates notes, sometimes not; for the main part, however, one has disjunct rather than ligated notes. This yields the following rule: Every notational symbol, as it reveals itself by its name, may be elongated or abbreviated. The division of the notational symbols follows the syllabification of the text, since every syllable will have a nota simplex with further notes attached to it. Others explain this, that by virtue of having a different syllable, every note is said to be an absolute entity, since it can not be combined to form a ligature. If it is to be combined, one note alone can not be combined with itself, but must have another or other notational symbols, with which it can be combined; for example, a brevis recta with one syllable and a longa recta connected to another syllable would form the equivalent of two ligated notes cum proprietate and cum perfectione, according to the afore-mentioned rules, namely that every figure may be notated cum proprietate and cum perfectione, etc. (and will be measured as brevis longa). Another case, if there is a longa on one syllable, and a brevis on another, and a longa for a third syllable, they will form the equivalence of three ligated notes cum proprietate and cum perfectione¹. Again, if there is a longa on one syllable, and two ligated notes cum proprietate and sine perfectione to another syllable, if a brevis and longa were to follow, they would form the equivalence of three ligated notes cum proprietate and cum perfectione. The initial longa could not be

Coussemaker p. 343a6 to p. 343b18

1)    Cum proprietate is correct according to the present treatise, but Franco of Cologne would have required a ligature sine proprietate and cum perfectione to establish a ligature with both an initial and a final longa. Such a ligature is avoided by our theorist, because it might indicate an ordo of the second mode, which has the irregular notation of a three-note ligature sine proprietate and cum perfectione.