Part 2 69 [ Chapter 1:] on Mensural Music

Part 2 69 [ Chapter 1:] On Mensural Music

[c7r] I have completed the first part of this skill (ars) and now will approach the second, in which if I will be a little freer and more prolix, let the reader not wonder,70 since the present subject prevails by far over the former both in the extent and importance of the subject and at the same time in sweetness. Indeed, it prevails in the multitude of its notes, the diversity of its signs, the inequality71 of its Tactus72 and Proportions, so that you will find no facet that does not differ from the other in some distinction. For this reason at the outset of this little book so also here I admonish the boys73 that they should diligently inquire into the song (cantilena) presented and thoroughly study its nature, observe the signs of its Degrees74 (gradus) and of Proportions, Augmentation and Diminution, as well as the individual attributes of the notes, lest carelessness and ignorance of these impose upon them, a thing we see happens in the practice of many. But now to the subject at hand. Notated music is the knowledge (scientia)75 of singing musical Degress and Proportions according to certain notated measure. This consists in various and plainly dissimilar shapes of notes, rests and signs. A shape (figura) is the very condition or disposition by which any note or rest can be recognized with certainty. Indeed, some of the notes are called simple, some ligatures, some major and some minor.

69 Listenius provides no chapter heading for the opening of Part 2. 70 School texts for teaching music, especially in the tradition of the Lutheran Latin Schools, were typically divided into two sections, the first devoted mainly to music fundamentals in support of “choral” music (monophonic plainchant), and the second devoted to mensural music in support of polyphony and the complexities of mensural notation. 71 I have chosen consistently to capitalize words that are important terms in the technique of mensural notation and related subjects. I have found that this makes them somehow more emphatic and seems to give the reader something to “hang onto” in the complex and often confusing descriptions to follow. 72 He previously made clear that the equality of rhythmic values is a distinctive character of plainchant. 73 The address to the boys in the third person is a subtle indication that Listenius is talking to the teacher in a teacher’s text, not to the students. 74 “Degree” is the common English term at this period for the different mensural levels of Mood, Tempus and Prolation. 75 Listenius’ syntax here is not quite correct, but the meaning is clear.

On the Number of the Simple Notes (Figura)

The Three last types toghether with their rests are more suitable for instrumental music than for the human voice because of their excessive speed. They are added as a kind of supplement to the upper

35 ones, with which the ancients76 are reported to have been content, not because no more and no more refined can be conceived, but for the sake of a more elegant and sweet melody.

[c8r] Chapter 2: On Rests

A rest is a line drawn through lines and spaces denoting silence. Or it is a shape (figura) indicating an artistic cessation from music (cantus), which indicates to the singers77 that they must briefly abstain from singing (cantus). Musicians,78 then, position rests both for resting the voice and for the sweetness of the song, which [rest] exists both in Fugas and musical types in perfect prolations79 that enter reciprocally, and also in [voices in] harmonies,80 both for cadences and for reintegrating [them] by means of silences or proceeding otherwise. 81 Likewise for avoiding prohibited intervals, such as consecutive perfect intervals,82 concerning which I speak in the third part.83 There are, then, seven rests corresponding to the aforesaid notes, excepting the Maxima, which has no similar one, unless some one would like to attribute to it a general rest.84

76 “Ancient” could refer to those as recently as only two or a few generations previous, though some authors would distinguish these as “veteres.” 77 Listenius uses this exact quote from Gaffurius, Practica Musice, Bk.1, Ch.6, “est figura artificiosam a cantu desistentiam monstrans, quae cantoribus innuit.” Gaffurius himself seems indebted to Tinctoris, Tractatus de notis et pausis, Pt.2, Ch.1 (CS IV, p. 45) 78 Seay’s translation of this passage from “Musicians” to “otherwise” is quite different from my own. The critical point is that it is a coordinate construction connected with the particles “tum . . . tum,” which means that their respective elements must be kept distinct. Thus, it is not possible to take “perfectis speciebus, etc.” from the first member and transfer it to “concordantiis,” in the second member as Seay has done. Listenius has also already given us the term “modus” or alternatively “intervallum” for interval, so Seay’s understanding of speciebus with that sense does not seem justified. Listenius’ Latin here is, frankly, one of the worst spots in the entire book and Seay’s frustration is more than justified. 79Listenius’ reference to fuga as a perfect species is illuminated by Tinctoris on the genera of compositions in Tractatus de regulari valore notarum, Ch. 1, “Porro cantus alii ex omnibus quantitatibus perfectis componuntur.” “Further some [compositions] are composed from all perfect quantities.” This is elaborated in Ch. 16: Maxima = 3 Longs, 9 Breves, 27 Semibreves, 81 Minims, etc. Such a song using all perfect subdivisions would make for very long phrases and have great need of rests for the refreshment of the singers’ voices, as Listenius says in Chapter 4 when considering Major Mood Perfect. I believe, however, that Listenius probably means something more general, as “composition utilizing perfect subdivisions at multiple levels.” 80 It seems that Listenius is making a comparison of how rests are used in two diferent polyphonic textures: strict conunterpoint as in the Fuga as compared to the freer harmonically conceived polyphony, the latter of which he expresses by concordantiis, here “[voices in] harmonies.” 81 This phrase may be an attempt to convert into musical terms Rhau’s comments on rests under his first heading when he speaks of the orator’s use of silence, Enchiridion musicae mensuralis, Chapter 1, G5v. 82 This also deals with purposes found in Rhau’s same discussion. 83 There is no third part nor does he discuss issues of counterpoint and voice leading in this book. Listenius wrote only two books, this and an earlier Rudimenta. The reference must be to one of the less famous “ghosts” that music authors seem often to be predicting to write but never get around to or never live to complete, Praetorius being one of the more famous examples. 84 He explains the general rest below as a bar line through the staff that separates parts of the work.

Nevertheless, observe that this does not happen everywhere, for in Alteration certain notes are altered, namely, because of the composition of the ternary number.85

Indeed, rests of these are not altered, on the contrary.86 In Imperfection are [c8v] imperfected, the rest corresponding to it is not imperfected. In the remaining types, namely Augmentation, Diminution and Proprotions, they are equal to their notes in whatever way we try in its place. There now remain, as I will briefly tell, the subject of modal and general rests. Modal is a line filling three spaces or four lines, equal to a perfect Long, indicating either Major Mood Perfect or Minor Mood Perfect, whose name it has. Major is double, Minor is otherwise. To be sure, not without cause did Franchinus [Gaffurius] say two are placed for Major when we ascribe one to Minor.87 Now the general rest is [1] a stroke drawn across through all the lines in the middle of the music (cantus) and coresponds with cardinal notes or [2] repeat signs.88 A song (cantilena) repeats the part thus set apart and separated from the remaining, even when the others continue,89 as is seen in the mass of Josquin “Quae est ista” and in the song (carmen) of Johann Walther “Non morier.” It concludes the music (cantus) at the end.

85 Listenius discusses imperfection and alteration below. 86 This is a summary of Gaffurius’ own treatment in Practica Musice, Bk.2, Ch.6. 87 As used for this purpose, these are not actual rests but signs of Major or Minor Mood, Gaffurius in Practica Musice, Bk2., Ch.7. 88 Listenius gives no clue as to which notes are “cardinal.” Probably he means the five principal ones that he alludes to as most appropriate to vocal music: Maxima, Long, Breve, Semibreve and Minim. 89 It was a common practice to use a repeat sign in only one voice when its part repeated a section of music and the others did not. The practice was still in use as late as the nineteenth century.

Here Follows an Exercise on the Six Musical Syllables (Vox)

Another Exercise on the Syllables (Vox)

An Exercise on the Eight Individual Notes.

[d2v] Chapter 3: On Ligatures

A ligature is the proper connection of individual notes by means of a stroke on the right or left side. It has two forms, square or oblique. Square [is the one] whose note is drawn with a square body. Oblique [is the one] whose note is writen with an oblique and transverse body. There are four notes only that can be combined in ligatures, namely [Maxima,90 Long, Breve, Semibreve], which change by various connections, forms and quantities, excepting one, the Maxima, which everywhere is similar to itself. They [the notes] are bound in the like manner at the beginning, middle and end, excepting the Long, which is never joined in the middle [of a ligature] on the example and authority of antiquity, although in our own time by certain ones it is done on the contrary, as will be more clear from the verses below. Further, every note that can be bound in ligature is either an initial, a medial or an ultima. Every medial is between the first and last. Now I will recite in song91 the rules of individual ones, by which boys may more easily commit them to memory, it pleases me to describe. Now beginning with the initials.

Rules for Initials

[1] The first92 when it lacks a tail is a Long when the second descends.93 [2] The same one is a Breve when the second ascends. [3] It is a Breve if the left side sends forth a tail. [4] It is called a Semibreve if that [tail] leads upward.

[d3r] There is only one rule for medials

[5] Anything in the middle is a Breve, with one exception.94

90 The Maxima in origin is nothing other than the triplex or duplex Long, the first mention of which Wolf attributes to Franco of Cologne, Handbuch der Notationskunde, 1, 336. This helps explain why authors always remark that Major Mood Perfect is so rare and also helps explain the elaboration of the cumbersome two-mood system of Major and Minor Mood. Wolf also attributes to the Italian theorist Prosdocimus de Beldemandis (d.1428) the first mention of the second level of Mood, the Modus Maximarum, idem, 1,338. 91 Poetry as a mnemonic device is another pedagogical technique that Listenius inherits from the Middle Ages. 92 I have added the numbers in brackets, which do not refer to the line of the poem but to the number of the example below. 93 These rules conform with others found In contemporary sources and also with those found in Apel, Notation of Polyphonic Music 900-1600, though there is one inconsistency, for which I provide an explanatory note in its proper place. 94 The exception is the ligature with an ascending tail, cf. the fourth line above.

On the Ultimas

[6] The last ascending is a Breve in any ligature, [7] Except when it bears a tail on the left. [8,9] The last is a Long when square and depnding from a higher one. 95

On the Positions of Individual Notes

A Maxima can be at the beginning or in middle or end. A Long can never be in the middle but is bound at the end. A Breve properly occupies the beginning and all positions A Semibreve is likewise at the end middle and beginning.

95 If this is meant to include the Semibreve ligature (cum opposita proprietate) in #9, then further explanation is required. Contemporary understanding is that this c.o.p. ligature always indicates that the first two are Semibreves. Morley’s rule creates the same ambiguity. Listenius seems to have overlooked the same problem, that the rules require greater precision, i.e., “A ligature of two notes that begins with an ascending tail is always Semibreve + Semibreve. For a ligature of two notes that does not begin with an ascending tail and all ligatures with ascending tail of more than two notes, the last note is Long if descending.”

An Exercise on the Rules

[d4v] Chapter 4: On the Three Musical Degrees

There are three musical Degrees, namely Mood, Tempus and Prolation.96 Mood is the obligatory mensuration of Longs and Maximas. Indeed, there is among notes (in some way as in other things) a

96 As earlier noted, a Degree is a level of mensuration. The terminology is somewhat befuddled by the fact that the system has been expanded to include four actual levels for the sake of an unhistorical theoretical completeness, yet the terminology has not been similarly expanded. Because of this, Mood has been made to serve double duty for both the upper two levels. These are the Major Mood (Modus Maximarum, Maxima subdivided into Longs) and the Minor Mood (Modus Longarum, Long subdivided into Breves), respectively called in the English authors the Great Mood and Less(er) Mood. Wolf reports that Franco of Cologne was the first to recognize duplex and triplex forms of the Long, which he then says Prosdocimus de Beldemandis (d.1428) was first to recognize as Modus Maximarum, Wolf, Notationskunde, 1,336-338. The mensuration signs were similarly forced into alien service, which has already appeared above in connection with the rests, where ¡3 indicated the Major Mood and

42 manner always to be observed, in regard to the measure of time, which they consider expressed by regular rhythm. Here in the order of Degrees properly first place is allotted [to Mood], because it exercises its power on the first and special notes, namely Maxima and Long.97 Tempus is next because it acts on the next ones, the Breves. Prolation, then, is the last because it concerns the last ones, Semibreves and Minims.98 For antiquity possessed [only] so many notes, to which they attributed these Degrees individually,99 as proper to themselves, so that they might not become actively mixed up in all. [d5r]

Mood Major Mood is the requisite measure of Perfect is when in the Maxima Is Longs in the Maxima and is itself double. there are 3 Longs and in a Long Of Two Types 3 Breves. ¡3 Imperfect is when there are two Longs in a Maxima and 2 Breves in a Long. C3100 Minor is the number of Breves in a Long Perfect is when a Long is worth according to the requisite consideration. It 3 Breves. ¡ 2 is also double Imperfect is when a Long is worth only 2 Breves. C2

¡2 the Minor Mood. Most important to note, the numbers 3 and 2 emphatically do not refer to the subdivision but to Major Mood (3) and Minor Mood (2). This situation is even further confused by the fact that the subdivision of the Semibreve into Minims is sometimes also referred to as Major and Minor. Gaffurius discusses this issue in Practica Musice, Bk.2, Ch.7, stating that Mood is the division of the Long. Then he says “Nonnulli autem dicunt modum esse mensuram brevium in longis & longarum in maximis.” “Some, however, say that Mood is the measure of Breves in a Long and of Longs in a Maxima.” But, then below he further qualifies that statement. “Hanc autem longae consyderationem: modum minorem vocant recentiores: respectu modi in maxima consistentis. Veteres autem illi unicum.s. & maximae & longae posuere modum ac si unus sine altero perfection computata constitui non posset.” “More recent [musicians] call this calculation of longs the Minor Mood, in comparison with the Mood consisting in the Maxima. Earlier [musicians] however posited a single Mood of the Maxima and Long, as if one without the other, when perfection is computed, were not possible for constituting [it].” By the time of Tinctoris the system of two levels of Mood is recognized. Most earlier authors refer only to the Mood as the division of Longs into Breves. Considering that the Maxima was not originally a distinct note but was a duplex or triplex Long, and further that all authors I am aware of say that the Major Mood Perfect (3 Longs = 1 Maxima) is seldom or never found, even Gaffurius’ reservation seems a bit of a historical distortion, probably influenced by his famous friend Tinctoris, whose reputation for over-theorization is famous. Thus, the evidence would seem to indicate that the fully elaborated Two-Mood system is a construct by theorists for the sake of a tidy, but largely non-historical system. 97 Listenius nods and omits the Degree’s name, Mood. 98 By “last ones” Listenius means the last of the notes that he is concerned with, the Maxima, Long, Breve, Semibreve and Minim. He is not concerned with the lesser values of the Semiminim, Fusa and Semifusa, which latter three are always duple (or imperfect) subdivision. 99 There is what can only be a misprint here of sigillatim for singillatim. The former would have to be an adverb coined from sigillatus, “decorated with figures,” which might lead the reader to assume Listenius was awkwardly introducing the issue of mensuration signs. 100 Note that Listenius, as other authors, does not here discuss the possibility of 2 Longs in a Maxima and 3 Breves in a Long, and the equivalent condition at the level of Minor Mood. His following chart does include further possibilities.

Very many songs (carmen) and examples are found of Minor Mood Perfect, as that of Josquin beyond counting, and especially the Missa Pro defunctis of the same man; likewise “Puer natus” Of Johannes Mouton. Indeed, of the Major Mood Perfect, there are almost no extant examples, even because of the annonying and irritating retardation of the notes in this sign.

On Tempus

Tempus is Double101 Perfect is when the Breve is worth 3 Semibreves. Imperfect is when the Breve is worth 2 Its signs are: ¡3, C3 ¡ Semibreves. Its signs are: ¡1, C2, C

Prolation is the proper disposition of Semibeves and is double.

101 Note that four of the signs do double duty In both Mood and Tempus levels without distinguishing the different significances.

[d5v] On Prolation

Prolation Major Prolation is when the Semibreve makes 3 Minor is when the Semibreve is worth 2 Minims.

Minims. ¯ = Its signs are: ¯ = Its signs are: ¡, C

From the things said, Prolation without Tempus and Major Mood without Minor and Minor Mood Minor without Tempus, I have established is not possible in song (cantilena).102

On the extrinsic signs of the aforesaid Degrees I have interspersed in passing with the preceding ones, for in vain are Mood, Tempus and Prolation recognized apart from their signs. The expressions of Degree also are known by means of certain intrinsic signs, namely by the completion of notes, or the disposition of rests.103 Indeed two or three Longs in coloration are an indication of Mood, Breves or two semibreve rests combined [is an indication] of Tempus. Three or more Semibreves or two successive [d6r] sighs (as they are called) of prolation are an indication of Prolation, as written here.

102 Gaffurius says so in greater detail, Practica Musica Bk.2, Ch.7. 103 In other words, he means by characteristics implicit in the music, which he has just said is nearly impossible.

Another Example of the Three Degrees

[d8v] Table in which the Quantity of any Note may be Explained by the Variety of Signs

104 Chapter 5: On Augmentation [The reader is cautioned that it is vital to read this note.]

[e1r] Not inappropriately is Prolation connected with Augmentation, since they make use of the same signs and notes in common, and yet105 it is done more freely in the former [illa, Prolation] than the latter [hac Augmentation]. In this case106 [of Augmentation] [Perfect Prolation is] in only one part of the song (cantilena), [whereas] in [Prolation] it is in all [parts], and this exact perfect Semibreve, or three Minims, is worth a Proportionate Tactus.107 In this way, indeed, the perfect Semibreve is measured with

104 This text is much troubled by the confusion caused with Listenius’ analogy of perfect prolation in Augmentation as compared with regular Perfect Prolation. According to Apel, at this time the signs of Perfect Prolation indicate that the Minim receives the Tactus in songs with all voices in Perfect prolation (pp. 163-164). This is also true in the case of Perfect Prolation in one voice when used for Augmentation. What Listenius never makes explicit is that when Perfect Prolation is used in only one voice for Augmentation, the other voices must be in Imperfect Prolation. To him, no doubt, this seemed implicit—though schoolboys likely would have been dependent on the teacher’s explanation—but his analogy has confused the distinction. Beyond this, once again Listenius exhibits the tendency for his Latin to deteriorate when the subject becomes more complicated. His source is probably Ornithoparchus, Bk.2, Ch.7 [F4r], (Dowland translation, p.47) as demonstrated by the similarity of Listenius’ definition to the third identification of Ornithoparchus. Ornithoparchus says, “Tertium est punctus in signo temporali, circa unam dumtaxat cantilena partem repertus. Unam dico, nam si apud omnes offenditur, non augmentationis sed maioris prolationis erit indicium.” “The third is a point in the sign of tempus found in only one part of the song. One I say, for if it is found in all, then it is not augmentation but an indication of major prolation.” (Dowland p.47). Listenius’ subsequently attempted explanation fairly clearly also derives from Ornithoparchus’ second and third rules, of which he pretty well makes a hash as well. This type of Augmentation was relatively new, unfamiliar and often confused in Listenius’ time. Apel (p.166) says Tinctoris refers to the practice in Proportionale Musices Bk. 3, Ch.2 (CS IV, 171), yet without complete clarity. I have found no mention of it in Gaffurius, at least where it might be expected in Practica Musice, Bk.4, Ch.4 under subtripla, but Gaffurius is really more interested in the mathematical relationships than in their practical realization. Sebald Heyden’s later work, De arte candendi (1540)—Book 2, Chapter 2, Miller translation p.70 ff.—discusses this species of Augmentation, describing three different possible interpretations of it. Of all the possible interpretations apparently in use at the time, Listenius’ own is only one and the simplest. 105 Seay reads “so that it may be done . . . .” The original is “tametsi id liberius sit.” There is no justification in classical Latin, as humanist Latin imitates, to take the concessive particle “tametsi” as equivalent to the subordinating conjunction of result “ut,” “so that.” I am considerably less familiar with medieval Latin but know of no such usage. In any case Listenius’ Latin is humanist, classical. 106 The use of hac connected specifically with “in one part only” makes clear that Listenius’ use of the correlatives hic and ille is traditional as, respectively, “the latter,” “the former.” 107 Listenius later defines the Proportionate Tactus as when some value other than the Semibreve receives the Tactus; in this case the Minim. I believe Seay goes amiss here by not understanding the reference to be to the Proportionate Tactus of Chapter 10. The pedagogical technique of the time was not “progressive,” but what I call synchronic, in which any part of the text might be relevant for understanding any other part of the text despite its actual sequence of presentation in the book. One of the things that makes this passage so confusing is that both Prolation and Augmentation, as Listenius says here, utilize the Proprotionate Tactus, because in both of them the Minim receives the Tactus. The difference is that in in Prolation all the voices are in Perfect Prolation, while in Augmentation only one voice is in Perfect Prolation, while in the other voices the Semibreve receives the Tactus.

48 a whole Tactus108 for each of three Minims. Whence it is clear109 that Augmentation is nothing other than an enlargement of notes.110 It is indicated extrinsically by these signs and in some one part [only] of the song (Cantilena), or [it is] inscribed in a canon111 [such as] “Breve becomes Maxima,” “Semibreve [becomes] Long,” etc., or “it increases by double, triple,” etc.112

108 Listenius’ use of the phrase “integro tactu” does not apparently have the significance of his later definition of Tactus Integralis but means simply the entire Tactus equal to a Semibreve in the other parts for each Minim. In this I agree with Seay’s reading. Listenius thereby further confuses what is probably the most obscure passage in the book. 109 No comment on the irony of this statement. 110 The phrase is “Unde liquet,nihil aliud esse augmentationem quam notularum multiplicationem.” I cannot agree with Seay’s reading of “it is nothing but augmentation rather than a multiplaction of notes.” Nihil aliud . . . quam is a very common, even stereotyped phrase that has appeared many times previously in Listenius’ text with the normal meaning of “nothing other/else . . . than” or “it is only.” “Rather than” is equally common and is “potius quam.” My reading also takes “multiplicatio” with a standard meaning of both the verb and its substantive derivative of “increase,” which I have rendered as “enlargement” for greater clarity in the context. 111 A canon is a descriptive or explanatory word or phrase. 112 These are references to other means of indicating Augmentation of different types.

[e1v] Chapter 6: On Diminution

Diminution, the contrary species to Augmentation, is a certain decrease of notes and measure, differing not at all from semiditie,113 except that the latter [haec = semiditie] concerns perfect signs and the former [illa = diminution] [concerns] imperfect signs. I make the understanding of it threefold.114

Canonic is first, when the quantity of notes or measure is varied, according to the inscription of a canon, so as to describe it in dupla, etc., or “let the Maxima be a Long,” “the Long a Breve,” etc., which I leave to the intelligence and judgment of musicians.

Number is second, when the proportion dupla or tripla, that is a double or triple diminution of the notes, is represented by some number as in this way: ¡2, C2 C3, in order that it may become clear by means of the proportions.

Stroke is third, when a stroke intersects the sign of Tempus or Prolation, in this way:

113 Seay has taken this with the simple meaning of “halving,” but it was also a term for a specific concept of diminution in the seventeenth century at least as early as Ornithoparchus (1517) and known as widely as Morley in England at the end of the century, the latter probably under the influence of the former. Listenius appears to show the influence of Rhau and/or Ornithoparcus here. 114 This otherwise quite lucid passage either involves a most unfortunate and confusing error, or—even worse— possibly indicates that Listenius does not understand his subject. I have expanded the passage in order to reveal the problem. The opposition of the forms of hic and ille, standard since classical times correlates in English as hic = latter, ille = former. Indeed, Listenius used this correlation correctly in an otherwise very confusing passage just in the previous chapter on Augmentation. Thus, Listenius clearly says that semiditie applies to perfect mensuration. Unfortunately, his correlation reverses the correct one. Semiditie, the English form of the word, relates only to imperfect signs, whereas diminution (proper) alludes to perfect mensuration. This is generally accepted and is as stated by Ornithoparchus in his Micrololgus of 1517 Bk.2, Ch.8: “semiditas est primarie notarum mensure medietas, in imperfect tempore tantum locabilis, cuius hec sunt signa . “Semiditie is the halving of the primary measure of notes appearing only in Imperfect Tempus, whose signs are, etc.” (Micrologus H8r). The likely source for Listenius’ discussion is Rhau’s earlier Enchiridion, which gives a shorter form of that found later in Ornithoparchus, “Nihil a semiditate discrepant, nisi quod illa [diminution] in signis perfectis . . . reperitur, haec [semiditas] in imperfectis,” Rhau, Enchiridion musicae mensuralis, Part 2 of Enchiridion utriusque musicae practicae (H8r). Further, I do not find the term used by Gaffurius in his chapter on diminution in Practica Musice of 1496, which otherwise appears to be Listenius’ main source. A last hope, though I fear a forlorn one, is that the pronoun correlation is not absolutely rigid, though I have never met with it used differently. An author may use the hic/ille forms as they relate to emphasis in his own mind, as if hic = most prominent, ille = less prominent, or the like. Following closely upon the difficulty he seems to have felt with Augmentation, I assume that he has confused the two types of diminution. Ironically the issue is virtually a distinction without a difference because both are simply dupla diminutions apart from the distinction of application to either perfection or imperfection. Seay, perhaps wisely, avoids the issue by means of a vague translation with which I cannot quite agree, “in no way different than halving, except that it may happen on the one hand in perfect signs, on the other in imperfect.”