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     The difference between the smaller minor second and the major second is the larger minor second (3⁷ /2¹¹ or 2187/2048 compared with the smaller minor second, or 2⁸/3⁵ or 256/243), but that is not used. The perfect fifth may have a major third added to it, and it is then called a ditonus cum diapente; the difference between the major seventh and the perfect fifth will be a major third. The perfect fifth may have a minor third added to it, and then it is called a semiditonus cum diapente; the difference between the minor seventh and the perfect fifth will be a minor third. In a similar fashion, all proportions larger than the unison may be added to the single, double and triple octave, and these are called semitonium cum diapason, tonus cum diapason etc. until you reach the largest possible intervals.
     The double octave and other consonances can be called such things as the double sesquitertia, which is the octave and a fourth (the eleventh), or the double sesquialtera, where the larger number is three times the smaller, and this is the octave and a fifth (the twelfth). These consonances differ from each other in most notable ways, You can always compound intervals with double and triple octaves, if it becomes necessary.
     The word diapason is derived from dia; that is de in Latin, and pan that is totus and sonus, as if it contained all sound in itself; this is, that it includes all the harmonic proportions in itself, and it also has the same letter on the monochord when one proceeds through eight letters. The word diapente is derived from dia that is de in Latin, and pente, that is quinque, as when you proceed through five letters or notes. A perfect fifth may be conjunct or disjunct (id est, CDEFG would be conjunct, FGABC would be disjunct, not being contained completely in one hexachord). The word diatessaron is derived from dia, that is de in Latin, and tetra, that is quatuor, as when you proceed through four letters or notes. A perfect fourth may be conjunct or disjunct. The ditonus (major third) is so called, because it contains two tones. The tritonus (augmented fourth) is so called, because it is formed as if it contained three whole tones, but it is not used except in rare instances by organists. The minor third is called a semiditonus, because it is like an imperfect ditonus. The interval which is called the semitonium maius (larger minor second) is not used, except on rare occasions. The minor second is an imperfect major second, because it is not exactly half of a whole tone, since if any proportion is removed from any superparticularis function, that which remains will not be half of it. Tonus is named after tonando, that is sounding.
     If you multiply two times two, you will get four; and if you multiply one times one, you will get one. Thus, the proportion of four to one is the result of twice two, and in this manner, the proportion of two to one is added to two to one. If you multiply two times one, you will get two, which is the geometric mean between four and one.
     Observe the series of terms: 4 : 2 : 1; two to one is a double proportion, and four to two is another double proportion, and this is the way proportions are added to proportions. In the same manner, you can add similar and dissimilar proportions. Again, if by chance, you should be subtracting proportions, take the first term, such as four to one; then, put a two under the four and a one under the one (id est, invert the fraction and multiply). Divide the four by two, and the result will be two, and divide the one by one, and the result will be one. These two numbers represent the result of this operation, and they represent

Coussemaker p. 353b7 to p. 354a30  

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the difference between the double (2/1) and the quadruple (4/l) proportions, which latter is another double proportion; thus, the result will be 2/l. If you multiply the reciprocal, you will get the difference between the relationship of the quadruple (4/1) and the double (2/1), and you will get the other double proportion. Thus, if one method of doing this is not adequate, the other will be. It should be noted, that whenever it is necessary to apply this rule, if the terms of this proportion are multiplied by the same number (id est, in both the numerator and the denominator) that the proportion will remain the same. According to these rules, one can produce all of the afore-mentioned proportions from the double proportion or from equality etc. More of this will be shown in the following.

Part 2

Concerning Polyphonic Compositions

     It is known that the term organum is quite ambiguous; often, we speak of organum purum, such as Iudea et Ierusalem (O l) for two voices, or Descendit de celis (O 2) or Gaude Maria (O 5) etc. Again, we use the word organum to designate the organ as an instrument, as the poet says: in cordis et organis (Psalm 150). Again, it is used in another fashion to designate three-voiced polyphonic compositions, such as Posui adiutorium (M 51). Organum simplex often designates three-voiced polyphonic compositions, like the conductus simplices. The term is often used in a different fashion by ordinary clerics, who apply it to two, three and four-voiced conductus, but this is quite improper. There is also a kind of organum, which was so designated by our forebearers, and this involves the consonance of one sound with another. Of all of them, this one consisted of connected conductus simplices with any kind of mensuration, and any kind of melody. To explain this universal method, we shall begin with two notes. The first has the melody or tenor on low C C.; the organum begins on high c against the afore-mentioned low C. and may end on the same c forming an octave with the afore-mentioned C.; thus, there will be high c c against low C C.; or it could end on low G a fifth above; thus, there would be c G.; or it could end on F. yielding c F.; again, it could move by a major third to e, as some people like, although this is improper, as c e; or it could move to e^b, which is marked by a flat, as c e^b, by a minor third. Again, it could end with the same sound in unison as C c. The rule, that the best composers of polyphonic song follow for every natural harmonic progression, is to begin with an unison, octave, fifth, fourth or either the major or minor third; properly speaking, however, a composition does not end with a major or minor third, although there are some persons who use these improper sounds at the end. Every composition should end with an octave, fifth, fourth or unison. It should be noted, however, that two-voiced compositions rarely end with a fourth, whereas three and four-voiced compositions use this interval in conjunction with other consonances.
     Again, if we take the example of C C in the melody or tenor, the organum may begin with G and end with c an octave above, or with G a fifth, or with the fourth F. or with the major third E. or the minor third E^b, or with a C unison. These may be collected together as: G c; G G; G F; G E; G E^b; and G C.; similarly (if we began with the fourth), we have F c;

Coussemaker p. 354A31 to p. 355a10

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F G; F F; F E; (F E^b); F C. In a similar manner, if we began with the major third E. we would have: E c; E G; E F; E E; and E C. Again, if we began with the minor third E^b, we would have and be able to sing: E^b c; E^b G; E^b F; (E^b E^b); and E^b C.
     Again, we could begin with an unison and end with an octave, as C c. Again, we could end with a fifth, a fourth, a major or minor third or an unison in the following manner: c G; c F; c E; c E^b; and c C. All of this can be done with any measurement that you might wish, although it is better if this doesn't go too quickly, and the consonances will be better, if they proceed slowly.
     Just as we have considered the progressions in relation to C C, so we can now use the progression C D. If C D is in the tenor, the organum can begin on the octave and end on the fifth, fourth, minor or major third, or unison, thus: c high d; c A; c G; c F; (c F^#); c D. Again, it can begin on the fifth and end in the same way, namely: G d; G A; G G; C F; (G F^#); C D. Again, it can begin with a fourth and end in the same fashion, such as: F, d; F, A; F, G; F, F; and F D. Again, it can begin with a minor or major third and end in the same way: E d; E A; E C; E F; (E, F^#); E D (or using E^b). Again, it can begin on the same sound, as: C d; C A; C G; C F; C D.
     If the lower voice has C E. we could begin with an octave and end with an octave, fifth, fourth, major or minor third, or with the same pitch, as c e; c B; c A; c G.; c E. Again, we could begin with the fifth G and end with e; B; A; G; or E. Again, we could begin with the fourth and form the following to the tenor C E; F e; F B; F A; F G; F E; or F C. Again, to C E. we could have E e; E B; E A; E G; E E; or E C.
     To the notes C F we could have c f; c c; c B^b; c A; c A^b; c F; F c. You proceed in the same fashion for all groups of two higher notes.
     Again, we can have lower notes also. Thus, if we have high c c, we can begin at the lower octave, and end in the same way as above, thus: C c; C A; C G; C F; or C C.
     In the same fashion, we can descend from the unison: c c; c A; c G; c F; or c C.
     We can also descend from A, that is from the lower minor third. Accordingly with the upper-part c c, we can begin on the lower minor third, yielding: A c; A A; A G; A F; or A C.
     Again, we could begin below c c with: G c; G A; G G; G F; or G C. Thus, we have given all of the individual examples both above and below.
     We shall proceed in the following manner: Above, we spoke about the consonances formed between two notes according to the six harmonic consonances. It is now our intention to continue with three notes which begin in this manner: C C C in the melody or tenor, and beginning with the octave above. The first two notes will not be different for the present. The final note can be like the first two as c c c. The last note may also form a fifth, fourth, minor third, major third, as in the following: c c G; c c F; c c E; or with a flat c c E^b; or c c C. The second note may also be different from the afore-mentioned notes, as c G c; c G G; c G F; c C E; or c G C. Again, the second note may differ as follows: c F c; c F G; c F F; c F E; or c F G. Another position for the second note is: c E G; c E F; c E E; or c E C. Another form for the second note is: c c C; c c G; c G F; c C E; c C C.

Coussemaker p. 355a11 to p. 356a12

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     That which we have added above a voice, may also be placed below; if the melody or tenor has high c c c, the other voice may begin with low C C C. The last of these notes can be: C C F; C C G; C C A; or C C c. The second note can be different, and the third can be changed in this manner: C F C; C F F; C F G; C F A; or C F c. Similarly: C C C; C G F; C G G; C G A; C G e. And again: C A C; C A F; C A G; C A A; or C A c. Finally, C c C; C c F; C c G; C c A; or C c c. If any particular finesse is desired, you can continue each succession o£ three notes, preserving the above-mentioned arrangement to the extent that they are contained in the consonances of the monochord,

Part 3

Concerning Discant

     In the following, we shall concern ourselves with discant. Discant involves the consonance of otherwise different melodies; it will be necessary to have at least two voices which consonate with each other, for what I shall have to say. We shall consider the relative nature of these voices, which may be fashioned in many different ways both in the upper-part as well as in the tenor.
     The melody of the tenor is the first song to be composed or made. The discant is the second song which is composed or fashioned, and it forms consonances with the tenor. These consonances are subject to the considerations listed below. In the first mode, they observe this pattern: The first note of the discant must form a consonance with the first note of the tenor, and this will be an octave, fifth, fourth, minor third, major third or unison. In the first mode, the first note is a longa in both the discant and the tenor, because they are coëqual. We are little concerned with the brevis which follows, since it may be used indiscriminately for one durational unit against the tenor. The third note is a longa and is equal to the third longa while forming a consonance as stated above, and the fourth brevis is coïncidental with the fourth brevis This is the way you proceed in the first mode, as far as you may desire, from which the following rule follows: All odd numbers of notes in the first mode are longæ and must form consonances with the tenor. The remainder of the notes, the even-numbered ones, may be composed without regard to consonance.
     It should be noted that you have to keep three things in mind: 1) the sound or proportion of the consonances, 2) the durational values, and 3) the total number of durational units contained. Among others, there is one universal rule, which is usually adhered to, and that is: If the melody or tenor repeats any note or maintains an unison, the discant may also retain an unison, descend or ascend to another consonance, whichever is best. This is stated in regard to two notes, as if there were no others, to demonstrate that all final notes must form consonances. Again, you may have a series of notes, and one set will ascend and descend, or descend and ascend etc.

Coussemaker p. 356a13 to p. 356b35

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     The discant must have relative oblique motion in the voice-parts, if it is to be considered natural. The difference between singers of discant and the vagrant singers, who are also discantores, is, that if the tenor rises for the latter group, they sing the upper-part higher also, and if it falls, they also sing the upper-voice lower in pitch. The true discantores, however, do not do this, but follow rather the method outlined above.
     It should be noted that there are three kinds of discantores; one kind consists of charlatans and novices; and these do what I have just said except very rarely, since without recognizing what they are doing, they will on occasion descend or ascend to a consonance other than the one they have been using. The second group uses a different method in following the principle stated above (id est, oblique motion in the various voices); here, we find the real discantores. There are still others who follow the method employed by the first group some of the time, and that of the others the rest of the time, as shall be shown below fully.
     It should be noted that the real discantores have three ways of composing a melody. The first method makes use of the neighboring consonances, that is the lower fourth and fifth. The other method employs the more remote intervals, which include the lower octave along with the others. The third method utilizes the most distant intervals, such as the lower 12th and 15th, or even larger ones.
     Let us begin with the closest intervals, those which do not exceed a fifth from the tenor line, whether they fall between the unison and the major third inclusive, or whether they are formed of more distant ones: those which are smaller than the fifth, but larger than the major third, or both.
     First of all, let us write a clausula or period in the usual fashion, that is in accordance with the paragraph above. The tenor may be notated in the following way, using no rests at first: c c c c d d c c d c; all of these notes are longæ of three durational units, since they equal a longa and brevis of the first mode.
     Again, form the discantus according to the first mode, namely: c, d, e, c, f with a descending plica, d, c, e, d, B, G with an ascending plica, the durational value of this note being a longa and a brevis c, d, e, c, d, B, c. The text of these notes is in French: Jo quiday mes maus celer, me jo n'i puis kamors ne mi lesse, etc. The discant is as mentioned above.

The Motet Je cuiday mes maus celer according to Wolfenbüttel, Helmstad.1099 f. 238a^v

Coussemaker p. 356b36 to p. 357a36

60

 61

     It should be noted that the discant has to consider many things in its relationship to the tenor: there may be a modus rectus (id est, first, second or sixth) in the discant as well as in the tenor, that is when the mode is formed in both cases with the lengths of proper longæ and proper breves, and then it is a right mode.
     There is another kind of mode, which has values in excess of the proper measurement of proper longæ and of proper breves, and this is called, by contrast, a mode beyond measure; or this might also be called an oblique mode transsumptively, and the third mode is one of the oblique modes.
     There are many combinations of proper modes; thus, the first can be used with or contrasted with the first mode, or the second against the second, or the sixth against the sixth; thus, these three modes can be used with themselves.
     In another way, different modes can be used against others; thus, the first mode can be contrasted with the second, the first against the sixth, and the second against the sixth, and there are also other combinations. This is accomplished by employing the second mode in the discant and the first mode in the tenor, or the sixth mode in the discant and the second mode in the tenor.
     Thus, the first mode against the first constitutes group A; the second against the second B.; the sixth against the sixth C.; the first against the second D; the first against the sixth E.; the second against the sixth F.; the sixth against the first G.; the second against the first H. This is said to be the contrast of proper mode against proper mode, because the first, second and sixth modes proceed with proper longæ and breves, as stated above.
     There are certain other modes which are beyond proper measurement, since they have elements which differ from those used in the other modes. These include the third, fourth and fifth modes. They are found in three ways: either combined with other modes or themselves. Thus, we find the third mode against the third, fourth against fourth, fifth against fifth. Contrasting modes include the third against the fourth, the third against the fifth, the fourth against the third, the fourth against the fifth, the fifth against the third, and the fifth against the fourth.
     Finally, there are many ways in which a proper mode may be contrasted with a mode, which is beyond measure or oblique; thus, the first mode may be used against the third, the first against the fourth, the first against the fifth; or again, the second against the third, fourth or fifth; the sixth against the third, fourth or fifth.

Coussemaker p. 357a37 to p. 358a4

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     To complete the possibilities, put the tenor into the place of the discant, just as in the case of all those which are beyond measure; and then put other modes into the place o£ the tenor, and you will have turned upside down or inverted what has just been said. Thus, you have the third mode over the first, second or sixth; or the fourth mode over the first, second or sixth; or the fifth mode over the first, second or sixth.

Chapter Five

Concerning Compositions for Three and Four Voices and the Copula

     In the following, we shall concern ourselves with compositions for three and four voices and with the copula.
     Concerning compositions for three voices: Let us consider a discant of the first mode, which has this mode in both the discant voice as well as In the tenor; the tenor will be: F G F D F with a further brevis rest, F F A G F with a brevis rest; in this fashion, you can recognize the second ordo of the first mode. This is named omnes, because it is taken from Viderunt omnes (M 1). This may be stated twice, thrice, or as many times as the tenor needs it. The discant, or the second melody, begins on the same pitch (sic!), having the range of a fifth above, thus: F, E, F, G, F with a brevis rest with the above-mentioned ordo, c, B, c, B^b (sic!), A, G, A with a brevis rest, and then c, B^b, c, A, A, G, c, c, A, G. A, G brevis etc.
     Then, we add a third melody to the two melodic consonances, and it is called a triplum to distinguish it from the second voice, which is the duplum, and the tenor which is the first one.
     In the triplum then, which begins below in the range of the lower octave, we have: A, B, c, B^b, d, c with a brevis rest in the same fashion as the other voices, c, B. c, B^b, A, G, F three for a longa, A, c, d, c, d, c, d, c, B^b, A, G three for a brevis, A, Bb, C, B^b brevis¹ .

Coussemaker p. 358a5 to p. 358b3

1)     The sequence of notes of the triplum is given twice, at the bottom of Coussemaker pp. 358a and 359a. In the first rendition, measure 2 has c B. d, the second c d; the first version moreover specifies in measure 4 B rotundum in the triplum (B^b) and B natural in the duplum. Note that the tenor is transposed down a fifth from its normal position; this, otherwise, occurs only in motets (confer Cserba p. 253). This motet is apparently not known elsewhere, although Montpellier VII, 288 begins with the same sequence of notes in the second mode. Thus, we can add an additional motet to the known répertoire. 

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     It is to be noted that the uneven-numbered notes of the first mode in the duplum form consonances with the odd-numbered notes in the first voice or tenor in the first mode. These are the perfect consonances of the unison and the octave, the middle consonances of the fourth and fifth, and the imperfect consonances of the minor and major thirds. Other composers, however, count the major and minor thirds among the other types of consonances. Thus, there are excellent composers of polyphonic music in certain places, such as England, in that part which is called the Westcountry, who consider these to be the best possible consonances, since they use them so much.
     There are certain good composers and writers of polyphonic song, who as a rule use discords in place of consonances, even in consideration of what we have just said. To attain a certain refinement, we can, for example, place a whole tone of squares, notes or sounds before a perfection (id est, unison). This will also form a consonance, whether this is the next-to-the-last note of the composition, or is found else-where, since this is a proper rule.
     Every next-to-the-last note before the rest, which is the end of a section, is a longa. If the next-to-the-last note is a whole tone above the tenor, as in organum purum, it will still form a good consonance, although the major second by itself is not a consonance. This method is quite often used by composers of organum purum, and by the Lumbard composers of polyphonic song. The difference, however, is that the composers of polyphonic song add the final note in their books, that is the one right after the next-to-the-last note, and this note either forms an unison or an octave with the tenor. Certain Lumbards, however, sometimes notate the final note, and sometimes they don't, and they draw back down under the sound of the final consonance and then return to the same pitch, this does not correspond with the philosophy of consonances.
     An example may be given of the consonance of the major second in organum purum, as mentioned above. You have G G for a longa florificata, then in the organum purum you have e c, with the proper rest, B^b d B^b d, ligated, then d B^b G

Coussemaker p. 358b4 to p. 359a2 

64

with a rest, B^b d B^b d B. B. c A A touching disjunctly final G. The tenor remains continually on G. and has G at the end in a stabilizing manner, as in the bass part of organum¹.

     This example and similar ones are or were to be found in many places in organum purum, and shall be shown to the fullest possible extent in the seventh chapter below.
     An example involving a discant would have the tenor A, G. F presented two, three or more times, and the duplum of the discant would go c, d, e, f in opposite motion, as has been stated etc.

     Thus, we have shown an example of that vile and loathsome discord which is the sixth, and which is mostly to be avoided. If, however, it is the next-to-the-last note before a perfect consonance, which is the octave, it is the best consonance in this arrangement of notes or sounds, as shown above. In this fashion, you will also understand the other discords, which shall be presented below fully, and in this light also, you will consider the next-to-the-last note.

Coussemaker p. 359a3 to p. 359a21

1)     This passage could conceivably be a corruption of a two-voiced organum purum from a composition contained in the Magnus liber. The identical endings of M 38 and M 49 (confer Waite pp. 202 and 231) bear some similarities to the notes given here and may be compared, since they raise a problem similar to that of the theorist.

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     There are certain people who heap discords on discords before a perfect consonance, such as before the octave, and for this reason, they are at the same time both greatly praised and derided. This may be seen by those who write accordingly, namely d, f, c, B. natural, c, B^b, c in the upper-voice, and D. D. E. F. E. D. C in the lower. Thus, by these means you will be able to recognize all discords, similar to those presented, as well as others that will be shown below.

     If we have assumed a two-voiced setting until now, scilicet duplum and tenor, that is the interrelationship of two voices, we now may add a third voice, which might have: A, B^b, c, d, c with a brevis rest similar to the one stated above, then c, B^b, c, B^b, A, G. F. three notes for a longa, A, c, d, c, d, c, d, c, B^b, A, G three notes for a brevis, A, B, c, B. This may be continued as far as one wishes and will form consonances with what has been stated before¹.
     The rules for the triplum are as follows: In the first mode, all uneven-numbered notes must form consonances with all odd-numbered notes of the duplum. Thus, the first note will form a consonance with the first note, the third with the third, the fifth with the fifth, and so on for all individual notes.
     Then, we also state that every uneven-numbered note must form a consonance with every odd-numbered note of the tenor; the first with the first, the third with the third, the fifth with the fifth, and so on for all individual notes.
     Thus, if we collect items, we find that the duplum will form consonances with the tenor on all uneven-numbered notes, the triplum similarly with the tenor, and the triplum with the duplum; thus, any one note will form a consonance with any other note, and similarly this one with the other two, because all three notes must consonate and form consonances with each other.
     All even-numbered notes in the triplum can be composed without regard to this; they may form consonances or not. Good performers, however, also consider the even-numbered notes and concern themselves with these, and consider their relationship to the duplum and to the tenor. Thus, if the duplum goes up in pitch, the triplum will fall, and vice versa. This is followed for one, two, three or more steps.
     In a similar fashion, they allow the triplum to move with the tenor for one, two, three or more steps. It is, however, regarded as reprehensible, if two voices move upwards excessively in parallel motion with one another, no matter which voices are involved, as shall be shown below.
     In the same manner a quadruplum can be added to the afore-mentioned. If it is also conceived in the first mode, every odd-numbered note must form a consonance with the same uneven-numbered note in the triplum, duplum, and tenor. If it now moves upwards in parallel motion with one of the other voice-parts, it can be fashioned so that it will go well for

Coussemaker p. 359a22 to p. 359b35

1)     Confer footnote 1, page 62.

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two or-three notes, as stated above in regard to the other voices. It may, however, move in the opposite direction, proceeding obliquely now to the triplum, now to the duplum, now to the tenor, or it may move along in unison with one of them. It is understood that the other voices may move in the same fashion, provided they don't do it excessively.
     It should be noted that the quadruplum may occasionally form a discord with the other voices, which is not possible between the triplum and the voices presented below it. Thus, the quadruplum might form a major or minor third with the tenor, or duplum, and the other three might form consonances of an octave, fifth or fourth. Thus, it will be necessary to have a discordant sixth, if a major or minor third is formed above the tenor, and this is quite astonishing. Such things are possible in ways which are mutually compatible without forming discords. T he continuation of this will be presented below. Thus, the fifth chapter ends.

Chapter Six

Concerning the Collections and Different Kinds of Discants

     Now we shall go over to the final task of this kind. We know that there are many diversities and many different varieties of collections, as we have said before, which contain these compositions. One specific collection contains quadruple, such as the Viderunt (M 1) and the Sederunt (M 3), which Perotinus the Great composed, and which contain much beauty and many ornaments. For the most part, this art has been continued to be used in its entirety, together with other similar compositions etc.
     There is another collection which contains great triple, such as the Alleluia Vs Dies sanctificatus (M 2), in which ornaments and beauties are found abundantly. If anyone performs the Divine Service in this manner, (id est, with the intent of using polyphonic compositions), he will have the best collection of this kind in these compositions. We shall discuss this collection below in the present chapter of this treatise.
     There is a third collection of conductus for two voices, which have caudæ, such as the Ave Maria (gratia plena), which was formerly for two voices, and the Pater noster commiserans, and the Hac in die rage nato¹. Similar names of many other conductus are also contained in this collection.
     There is a further collection of conductus for four, three and two voices without caudæ, which is quite often used by lesser singers and others like them.

Coussemaker p. 359bS5 to p. 360b4

1)     It would be most enlightening to discover which version of the Magnus Liber is being referred to by our writer. These two conductus are in sequence in both Wolfenbüttel, Herzog August Bibliothek, Helmstedt 628 (W₁) and Florence, Biblioteca Mediceo-Laurenziana, pluteo 29,1, where the first strophe is a 3, the second, however, a 2. Since in the next paragraph, reference is made to Ave Maria, and Pater noster commiserans, which are in sequence only in W_l, it is probable that this writer is reflecting a répertoire closely related to that source available to him; this would also explain his neglect of motet fascicles in his enumeration of the contents of the Magnus liber, since W_l has no motet fascicle. Confer Groninger pp. 80, 86 and 90.

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     There is a sixth collection of organa for two voices, such as the Iudea et Ierusalem with Vs Constantes (O l); this was never made into a three-voiced composition, just as if it were fashioned according to its own manner, which was held to be extraneous (id est, irregular in relationship) to the others. Because the longæ are excessively long, and the breves excessively short, it may be seen that this contains a mode which is irregular when compared with the above-mentioned modes of this discant, which are regular in themselves. We shall explain this more fully in the seventh chapter.
     We also find many other collections, which have different ways of e arranging their songs or melodies, such as the laïc conductus simplices. There are also countless others, all sorts of which may be found in their books or collections.
     For the present, we intend to consider the larger three-voiced compositions mentioned above. These have three different melodies which go together, and thus there are only small differences between and among them. In this fashion, the first, second and third voices are brought together, so that we may have the good, common distribution mentioned above, as well as others of this kind.
     One type of diversification is found when each of the three voices has a different kind of measurement. This is realized in the same fashion as in the case of the discant,namely, in any two voices or in all three, as in the tenor, the duplum and the triplum, or as is found in organum purum or similar compositions.
     Another kind of diversification is realized when the initial (mode) is not retained for the entire first section, except for two notes with the same or different pitches; the same also applies to half a section. In the duplum and in the triplum, you continue according to the discantile method, ending always with one of the consonances, as stated above.
     The third kind of diversification is found when you have the same tenor, but you have an extraneous (id est, irregular) mode in the duplum and triplum, as when the first note is too long or too short. Thus, such a section will temporarily seem to occupy a middle position between discant and organum. It is then neither discant nor organum, but rejoins the first again at the end, as stated previously.
     It should be noted that the first note of the tenor can and will continue to sound in places in which it agrees to a greater extent with the consonances sounded above it, and it will rest when it would form disagreeable dissonances etc. This results in the best solution.
     A second sort of difference is found in tripla, and it is one of the three kinds of diversification mentioned above. This is found when one voice proceeds according to the discantile manner, as stated above, and this in the most delectable manner. Accordingly, the tempo may be slow, slower, or slowest; fast, faster, or fastest; medium, more medium or most medium. The tempo refers to the first state of mensuration, or the second, or the third, and it may be increased or decreased or made more average.

Coussemaker p. 360b5 to p. 361a24

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     The third difference of tripla is found when you have a most pleasing or delightful note at the beginning, which is presented slowly, resulting in the fact that in the first mode, the second note will have to move quickly (id est,). Thus, this will not quite be the same as any of the discantile modes, but will be different and extraneous to them, as stated above. All of this is found in the Alleluia Vs Posui adiutorium (M 51), the great triplum, and in many other compositions.
     It should be noted that good composers of organum combine notes to produce the kind of pleasure, ornamentation or beauty inherent in this art. For this reason, there are different ways of writing melodies for each part. Thus, some people add a section of organum purum to a section with a better-known ending. For the same reason, they add two or three sections of organum at the beginning, forming good consonances similar to the more usual openings. This section, which precedes the longa rest, is displayed quite ostentatiously, as is usual in organum purum and similar compositions. Sometimes, they even go further, as is shown, for example, in various tripla, or in examples extracted from them.

Chapter Seven

Concerning Irregular Modes

     The seventh chapter of this treatise concerns itself with irregular modes, which are called voluntary modes (id est, they may be chosen ad libitum), and there are many of these. The first of these begins with a longa duplex (of five durational units) followed by a semibrevis (= a brevis recta of one durational unit), or a (longa) minima or diminished longa (longa recta of two durational units), and continues with such breves and longæ. This is shown in the Alleluia Vs Posui adiutorium (M51), where the following succession of notes is found as the copula: f longa duplex, f e ligated, f d ligated, e d (ligated), d f (ligated), g f (ligated) with plica, d c with plica, a, a longa duplex ligated to c; this is the first irregular mode, and this occurs quite often in organum purum.

     The second irregular mode has a brevis parva or brevis minima (of one durational unit) with a longa duplex, or ligated to a longa which occupies an excess of time. Then, you have the same brevis and continue with the same longa etc.

 Coussemaker p. 361a25 to p. 361b24

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     The third irregular mode has a longa minima with two longæ tardes which are irrespective breves; the third brevis is a brevis mediocris, which produces an excellent result when ligated. Then, there is a similar group of three ligated notes proceeding in the same fashion¹.
     There is another kind of irregular mode which exhibits a major degree of divergence, and this has a longa and three ligated notes, as stated above; and then, another three of which the first two are breves minimæ and the third a brevis mediocris, or the final one is a longa (minima), and then you continue with three more similar ones etc.

     Finally, there is another major variant, namely: a longa minima and three (breves) mediocres, as stated above; then three fast ones, except for the last, which is a (brevis) mediocris, then three mediocres and three festinantes.

     The fourth irregular mode consists of three (breves) festinantes, as stated above, although the last note is a longa minima, and then three more (breves) festinantes, continuing this pattern over and over again etc. Again, you could have three (breves) festinantes, as stated above, and three (breves) mediocres, and then you could continue with (breves) festinantes; or you might often mix (breves) festinantes with (breves) mediocres etc.

     There is another arrangement of breves mediocres, whereby the final note of the three is a (longa) minima This finest is understood as yielding identical patterns. The fifth irregular mode has a (longa) minima and many, that is three, four or five longæ mediæ, although the next-to-the-last note will be a (longa) minima. Use as many as you wish, or as many as will fit according to the requirements of any of the above-mentioned modes.

Coussemaker p. 361b25 to p. 362a18

1)     Niemann p. 91ff. has apparently missed the point that the irregular motes follow the pattern set by the regular modes. Thus, the first and second modes are distinguished by the fact that the longæ duplices, which may have five

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     The sixth irregular mode proceeds in this fashion: four ligated notes and three ligated notes, and three ligated notes etc. All of the notes are breves mediocres, although the next-to-the-last note of the four ligated notes is a (longa) minima, and each of the next-to-the-last notes of the three ligated notes is also a (longa) minima. This mode may be diversified in many ways, as has been shown, and according to the seven gifts of the Holy Ghost.

     The seventh mode is the most notable and dignified, and follows any predilection or desire; this is a mixed or common mode, and consists entirely of groups of twos, threes and fours etc., and properly speaking it is identifiable with organum purum et nobile.
     It should be noted, that the irregular modes, which have just been or will be discussed, are sufficient for the understanding of the durational values involved in organum purum. Again, it should be noted, that as far as the method of notating this music is concerned, the explanation given suffices, as shown above. The notation of these modes is identical with that of the regular modes, although occasionally there are differences in some of them.
     Rests can be used quite arbitrarily, according to what the singer or performer finds better. He can shorten, lengthen or make them more moderate. Longæ duplices are rarely found in organum purum.
     The consonances of organum purum are the usual ones: unison and octave, fourth and fifth, minor and major thirds. These are the best consonances, and are called perfect, imperfect and middle ones, as explained in the chapter concerned with consonances. The secondary ones are the five not formed with the unison; they include the double octave, which corresponds harmonically to the quadruple proportion; the 12th, which is the triple proportion; the 11th,which is the double sesquitertia; major and minor 10th, as they were named above in the chapter which concerned itself with these things. Thus, we get six and five consonances, which make eleven.
     Again, there are those consonances which are most distant, such as the minor and major seventeenth, the perfect eighteenth, and the perfect nineteenth, which must be transposed for some, but not for all men. Thus, there are ten and nine strings in the compass of the diatonic genus, whereas the number of consonances is fifteen.

Coussemaker p. 362a19 to p. 362b24

durational units, supplant the normal longæ. The third and forth modes correspond to the regular third and fourth modes, only the theorist has difficulty

71

     Others even beyond this, although this is rare, proceeding to the triple octave, although this is in common use only with the organ as an instrument; other instruments go beyond this, as exempli gratia a number of string and pipe instruments. These instruments sound good with the cymbals, as are kept by good musicians.
     The longæ and the breves are recognized in many ways and manners in organum purum. According to one method, all initial notes form one of the afore-mentioned consonances. Before this, there will be a longa parva, a longa tarda, or a longa media, and this will be found in any ligature of two or three notes etc.
     This, however, is the difference; if these longæ form a consonance, the tenor will be sounded; if they do not form a consonance, the tenor will not be sounded. In this case, the following note will be long, and will form a consonance.
     Again, all notes immediately before a longa rest, as at the end of a period or clausula, will be long.
     Every next-to-the-last note which is similarly conceived of as long according to the mode, will sometimes form a consonance, sometimes not.
     Of two notes, if the first forms a consonance, it is long; if it forms a dissonance, it is short in so far as it is found in a neutral position and not the next-to-the-last note.
     Again, of two notes, if the latter one forms a consonance, it is long; if it forms a dissonance, it is reduced in quantity.
     Again, of every group of three notes, the first note is long, if it forms a consonance; if not, it isn't. Every second note of the three is long, if it forms a consonance, otherwise it is not, and it is long, if it is the next-to-the-last note. Every last note of three is long, if it forms a consonance, short if it forms a discord, except when it is the next- to-the- last note, where it is long.
     If there are four notes together, the first, second, third or fourth note will be long, to the extent that it forms a consonance, short if it forms a dissonance.
     If two notes have the same pitch, they may or may not form consonances, and together they form a longa florata.
     Notæ currentes may also be used with the preceding, as in Viderunt, (Waite p. 67 m. 27-28) where eleven notes follow a certain extraneous pattern. They are not concerned with forming consonances with other notes, or do not all attempt this equally. They move downwards quite rapidly. If the first figure is a nota currens, then the first, second or third note before it will also be one.

Coussemaker p. 362b25 to p. 363a34

in expressing these time values, using terms such as minimæ and festinantes to indicate shorter forms of the respective longæ and breves, and mediocres to indicate a longer form. If, however, the third, fourth, fifth and sixth modes are irregular, then only because of the distribution of durational values. Since these irregular modes are identified with the English (as exenpli gratia Coussemaker p. 346b), it is quite reasonable to assume that they represent modes in binary rhythm, and they have been transcribed accordingly. Perhaps such modes were also represented in organum purum!

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     In this particular descent, the rhythmic pattern will begin not with the previous note, but rather with the note before this one, despite the fact that this is a longa (in form). The first note previous, even though it moves upwards, will also be a nota currens, whereby there are twelve notæ currentes, rather than eleven. The second note preceding is a long note which is found on the consonance of the fifth. If, however, it did not form a consonance, it would still be long, for the tenor could be silent, since every note which directly precedes, or which is two or three before, is of necessity long, as if it were to prepare for it to descend, or for its descent.
     This is notated in a simple fashion in vocal music, but this is not necessary for instruments. We can regard notæ currentes which rise and fall, or fall and rise in pitch etc. together with their antecedent or antecedents in a similar fashion.
     Again, longæ are quite often used for reasons of ornamentation, or beautification of the melody, and sometimes they form consonances, and sometimes not, as is often shown in performance.
     Again, there is a longa duplex florata, and this is found at the beginning: In nomine sanctissimi alpha, and is called the beginning before the beginning and always forms a consonance.
     Again, some persons put two and three ligated notes in the place of one. The first of these can either form a consonance or a discord, and it begins shortly before the tenor does, and the tenor enters with the second note, if it forms a consonance with this, otherwise with the third note. This third note is then drawn out, as we stated before, with a flourish. Others will even put three or four ligated notes before the beginning of the tenor. If the last note forms a consonance, the tenor will begin at this point. If the note forms a discord, then the tenor will begin with the first consonance which follows. Thus, it comes to the same result if there is a modulation of the melody with any sort of appropriate beauty or ornamentation; the same thing may be said of the beginning before the beginning.
     The final section In nomine sanctissimi can be completed in many different fashions. Some persons finish the composition with a single note on the octave, unison or fifth, rarely, however, on the fourth except in string music, and even more rarely on a major or minor third, as done by the men from the West (id est, the Westcuntry in England). All of these consonances, however, may be used at the beginning.
     There are others who sometimes use a consonance, sometimes a dissonance on the next-to-the-last note before an octave.
     There are others, who put one, two, three or more dissonances before the next-to-the-last note, as long as they sound good together in one or several ways, as long as this does not extend too far, and reach to the end.
     There are also certain good composers of polyphonic music, who are happy to put dissonances in penultimate position in preference to consonances etc. Examples of this are found throughout books of organum.
     Whatever occurs between the above-mentioned beginning and ending is considered as mediation.
     In the Name and Honour of the Holiest Mediator of all, who is the True Saviour Iesus Christ, to whose glory we may all come with the most Holy. Explicit.

Coussemaker p. 363a35 to p. 364b10

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Addendum to p. 44 footnote 2

The difference between the third and fourth modes is that in the third mode the longa begins the double perfection, and in the fourth mode the brevis, as in the following examples, cited by Robert de Handlo (Coussemaker p. 401):

Since the consonances will normally fall on this part of the perfection, and since any accents will also normally fall only this part of the perfection, it is quite important to establish a distinction between these two modes. The constant repetition of ordines, however, will finally result in a shift to the third mode, as above in Example 6, page 16, where the longer note becomes coïncidental with the beginning of the double perfection.

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