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Text and translation:
Jacobus on _cadentia_
Speculum musicae, Book IV, Chapter 50
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The Latin text of _Book IV_ of the Speculum musicae used here is
available at ,
THESAURUS MUSICARUM LATINARUM, School of Music, Indiana University
Bloomington, IN 47405 Jacobi Leodiensis Speculum musicae, ed. Roger
Bragard, Corpus scriptorum de musica, vol. 3/4 ([Rome]: American
Institute of Musicology, 1963), 1-126 (Used by permission).
Capitulum L.
Concordiarum comparatio quantum ad cadentiam.
Chapter 50
Comparison of the concords as far as concerns _cadentia_
Antequam concordantes, quantum ad cadentiam,
comparentur, quid sit cadentia videatur.
Before the concordant intervals, as far as concerns _cadentia_, are
compared, it should be seen what _cadentia_ is.
Cadentia, quantum ad praesens spectat propositum, videtur dicere
quendam ordinem vel naturalem inclinationem [-123-] imperfectioris
concordiae ad perfectiorem. Imperfectum enim ad perfectionem
naturaliter videtur inclinari, sicut ad melius esse, et quod est
debile per rem fortiorem et stabilem cupit sustentari. Cadentia igitur
in consonantiis dicitur, cum imperfecta concordia perfectiorem
concordiam sibi propinquam attingere nititur ut cadat in illam et illi
iungatur secundum sub et supra, descendendo videlicet vel ascendendo.
_Cadentia_, insofar as it pertains to the present purpose, seems to
speak of a certain order or natural inclination of a more imperfect
concord toward a more perfect. The imperfect indeed naturally seems to
incline toward perfection, as if toward a better existence, and that
which is weak desires to be sustained by something stronger and
stable. Thus _cadentia_ is spoken of in intervals when an imperfect
concord strives to arrive at a more perfect concord near it, so that
it may fall into and be joined to it below and above, namely by
descending or ascending.[1]
Quantum igitur ad cadentiam, una secunda sive tonus sub vel supra
petit unisonum. Et similiter una tertia in semiditono. Sed una tertia
in ditono petit quintum. Et similiter una quarta in diatessaron, aut
petit unisonum, aut quintam. Quinta, scilicet diapente, propter
bonitatem suam, tenet locum suum. Perficitur tamen in unisono aut in
diapason. Sexta, scilicet tonus cum diapente, petit duplum vel
quintam. Septima in semiditono cum diapente petit diapente vel
diapason, ut praecedens. Octava, scilicet dispason, stat in se ipsa,
non perfectibilis per aliam, nisi per unisonum. Nona, idest tonus cum
diapason vel bis diapente, petit diapason vel in diapente
revertitur. Decima in semiditono cum diapason petit diapason. Decima
in ditono cum diapason petit duodecimam, scilicet diapason cum
diapente, vel revertitur in diapason. Et consimiliter est de undecima,
scilicet de diatessaron cum diapason. Duodecima autem, scilicet
diapente cum diapason, quiescit in se ipsa, aut, ascendendo, petit
quintam decimam, idest bis diapason, apud eum qui altam habet vocem,
vel, descendendo, revertitur in diapason.
Therefore, so far as it concerns _cadentia_, a second or tone [major
second, 9:8] below or above seeks the unison.[2]. And similarly a
third in semiditone [minor third, 32:27].[3] But a third in ditone
[major third, 81:64] seeks the fifth.[4] And similarly a fourth in
diatessaron [4:3], either seeks the unison, or the fifth.[5] The
fifth, namely the diapente [3:2], according to its goodness, holds its
place. It is perfected nevertheless in the unison or in the octave
[2:1].[6] A sixth, namely a tone plus fifth [major sixth, 27:16],
seeks the octave [i.e. _duplum_, the 2:1 ratio] or the fifth.[7] A
seventh in minor third plus fifth [minor seventh, 16:9] seeks the
fifth or octave, like the preceding interval.[8] The octave, that is
the diapason, stands in itself, not perfectible by another, unless by
the unison.[9] A ninth, that is the tone plus octave or double fifth
[major ninth, 9:4], seeks the octave or returns to the fifth.[10] A
tenth in minor third plus octave [minor tenth, 64:27] seeks the
octave.[11] A tenth in major third plus octave [major tenth, 81:32]
seeks the twelfth, namely the octave plus fifth, or returns to the
octave.[12] And it is similar with an eleventh, namely the fourth plus
diapason [8:3].[13] The twelfth, however, namely the fifth plus octave
[3:1], rests in itself, or, ascending, seeks the fifteenth, that is
the double octave [4:1], on the part of the singer who has the high
voice, or, descending, returns to the octave.[14]
Omnis igitur concordia non solum summe | [P2, 193v in marg.]
perficitur et quietatur in unisono et in diapason, sed etiam, si
imperfecta sit ante perfectam, perficitur, cum ea, quae prope sunt,
sint quasi idem.
All concords therefore are not only fully perfected and brought to
rest in the unison or in the octave; but even, if an imperfect concord
is before a perfect one, it is perfected, since things which are near,
are as if the same.[15]
Unisonus autem et diapason, quia perfectissimae sunt concordiae, ideo
finis principalis sunt discantuum et organorum. Possunt tamen
discantus in diapente terminari, quia saepe fit ubi unus supra tenorem
est discantus. Quodsi terminetur discantus unus cum tenore in
diatessaron vel in alia a praedictis tribus concordia, hoc rarum est.
The unison and octave, however, because they are the most perfect
concords, therefore are the principal endings of discants and organa.
It is nevertheless possible to end a discant in the fifth, which often
is done where only one discant is above a tenor. But even though a
single discant with the tenor may be ended at the fourth or in some
other concord than the three mentioned above, this is rare.[16]
De concordiis transcendentibus diapente cum diapason, mentionem non
facio, quia propter ipsarum nimiam altitudinem raro vel nunquam in usu
sunt. Habemus multas alias bonas quibus uti possumus et utimur.
About concords beyond the twelfth, I make no mention, since because of
their too high range they are rarely or never in use. We have many
other good concords which we are able to use and do use.[17]
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Notes
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1. Jacobus evidently views _cadentia_ as the tendency of an imperfect
or unstable concord to resolve into a perfect or stable one, with many
of his specified two-voice resolutions involving either stepwise
contrary motion (characteristic of the strongest directed progressions
in the 13th and 14th centuries alike) or stepwise oblique motion.
Although I have preferred to retain the Latin term _cadentia_ in my
translation, "resolution" might be the best modern English equivalent
for the overall concept of Jacobus. An unstable interval "strives" for
resolution, thus creating a sense of anticipation; and the apt arrival
at a perfect concord brings a sense of repose.
The main complication of translating _cadentia_ simply as the English
"cadence" is that the latter term may for many readers imply the end
of a piece or a portion thereof, a complication more generally
applying when the standard resolutions described by theorists of the
14th and early 15th centuries are termed "cadential," which in fact
often occur within phrases as well as at their conclusions. As long as
this point of usage is clear -- e.g. "In some 13th-century motets,
directed cadential progressions occur continually as an unstable
sonority at the end of a rhythmic unit characteristically resolves to
a stable sonority at the beginning of the next" -- I see no problem
with such a translation.
2. The oblique resolution from major second to unison is a classic
gesture much favored by Guido d'Arezzo in his Micrologus (1025 or
1026) as one of the best progressions for the _occursus_ or "coming
together" at the unison in two-voice organum or diaphony at the end of
a phrase, and it retains favor during the 13th century both in
practice and among theorists such as Johannes de Garlandia and
Coussemaker's Anonymous IV. The inclusion of this progression is an
example of how Jacobus is interested in resolutions by oblique as well
as contrary motion.
3. The specific preference that the minor third resolve by stepwise
contrary motion to the unison (e.g. D-F to E-E) rather than the fifth
(e.g. D-F to C-G), and that a third contracting stepwise to a unison
be minor rather than major (e.g. F#-A rather than F-A to G-G), is one
aspect of the pattern of "closest approach" embraced by the modern Ars
Nova theory of the earlier 14th century -- and also by Jacobus.
In such closest approach progressions from an unstable to a stable
interval by stepwise contrary motion, one voice more specifically
moves by a tone (9:8), and the other by a regular or diatonic semitone
(256:243): thus in the resolution from minor third to unison
e.g. F#3-A3 to G3-G3, the upper voice descends by a tone (A3-G3),
while the lower voice ascends by a diatonic semitone (F#3-G3). In the
resolution D4-F4 to E4-E4, there is conversely an ascending step of a
tone (D4-E4) and a descending step of a semitone (F4-E4).
Elsewhere in the _Speculum musicae_, Book II, Chapter 80, Jacobus
notes how singers would rather sing the minor third than the major
third when proceeding by stepwise contrary motion to the unison, and
observes that they will use _musica falsa_ accidentals where needed
for this purpose. That is, singers will go beyond the eight notes of
the standard _musica recta_ gamut -- C, D, E, F, G, A and the two
forms of B (Bb and B-natural) -- and use added inflected steps such as
F# so as to sing the minor third F#-A in approaching a unison on G. In
recognizing and approving this practice, Jacobus seems in agreement
with Ars Nova "moderns" such as Philippe de Vitry, famous for writing
that _musica falsa_ is in fact "true and necessary."
In contrast, 13th-century practice also embraces directed progressions
by stepwise contrary motion where both voices move by a tone: for
example, from major third to unison (C3-E3 to D3-D3), or from minor
third to fifth (e.g. E4-G4 to D4-A4). Thus in embracing the closest
approach principle, Jacobus is in this respect aligning himself with
the "moderns." Manuscript accidentals in pieces by composers of the
later 13th century such as Adam de la Halle and Petrus de Cruce
suggest tendencies toward this preference for closest approach
codified in the early 14th century by Jacobus as well as the Ars Nova
theorists with whom he differs on some other points.
It is also possible to move from a minor third to a unison by oblique
motion, but Jacobus is evidently focusing on the closest approach
progression.
4. Like minor third to unison, major third to fifth is a principal
closest approach resolution. Earlier in Book IV, see Chapter 11,
Jacobus notes that musicians would rather sing a third that will
expand by stepwise contrary to a fifth as major rather than minor
(e.g. D-F# rather than D-F before C-G), and will use what he elsewhere
terms _musica falsa_ if needed in order to do so (see n. 3 above).
5. In this discussion of _cadentia_ as a property of two-voice
intervals, the fourth, although a perfect concord, is addressed only
as an interval seeking resolution to the more conclusive unison or
fifth. The first resolution, 4-1, goes back to one of the first
recorded examples of Western European polyphony, the setting of the
sequence _Rex coeli, Domine_ from the _Musica enchiriadis_ treatise of
around the 9th century, and often involves what might be called
near-conjunct contrary motion (e.g. C3-F3 to D3-D3), with one voice
moving by step and the other by a third. The 4-5 resolution,
reflecting a perception in the 13th century that while the fifth and
fourth are both stable concords, the fifth is more restful and
conclusive (as Jacobus agrees), often involves stepwise oblique
motion.
For Jacobus, the simple fourth may not have seemed conclusive enough
to serve as a goal for standard resolutions in two-voice writing,
which seems the main focus of this chapter. In contrast, 13th-century
composers, as noted by the theorist Johannes de Garlandia, often use
the fourth as a goal of directed or other resolutions for unstable
intervals, in two-voice as well as multi-voice writing.
6. Jacobus is very realistic in noting the restful conclusiveness of
the fifth; he will observe in Book VII, Chapter 3, that in two-voice
improvising or composing the perfect concords are to be mainly used,
and "especially" (_praecipue_) the fifth, although with more voices a
greater variety of concords can be pleasingly used. Here he seems to
be saying that while the 3:2 fifth can in an academic sense "resolve"
to the yet more perfect 1:1 unison or 2:1 octave, _cadentia_ in its
usual practical application means that an unstable interval resolves
to a stable one, with a sense of tension followed by repose. The
resolution that he has just addressed of the perfect but generally
inconclusive concord of the 4:3 fourth resolving to the more
conclusive 3:2 fifth might be an interesting intermediate case.
7. The first alternative, a resolution from major sixth to octave, is
one of the principal closest approach resolutions (e.g. G3-E4 to
F3-F4): it plays a vital role throughout the 13th century, but with
the directed resolution from minor sixth to octave (e.g. A3-F4 to
G3-G4) where both voices move by a tone as equally important. The
preference by around 1300 that a sixth expanding stepwise to an octave
be major, like the min3-1 and Maj3-5 resolutions, provides a
motivation for inflections outside the regular gamut (_musica falsa_
as writers around 1300 often call it, or _musica ficta_, "contrived"
or "invented" music, as it will often also be known).
The second alternative, major sixth to fifth, suggests an oblique
resolution, most often with the upper voice descending by a tone. This
resolution is common in the 13th and 14th centuries, with Machaut for
example using it effectively -- and indeed Dufay in the earlier 15th
century at the close of _Quel fronte signorille_.
It is an interesting touch that here Jacobus for the major sixth
mentions first the resolution to the octave most likely achieved
through a closest approach resolution, and then the resolution to the
fifth most likely involving stepwise oblique motion (although this
might also occur through similar motion, e.g. D3-B4 to F3-C4).
8. Jacobus remarks that the minor seventh generally seeks the
same resolving intervals as the major sixth he has just addressed, but
with a reversal of the order for these intervals of resolution: first
the fifth, and then the octave.
The resolution from minor seventh to fifth by stepwise contrary motion
(e.g. B3-A4 to C4-G4) is another closest approach progression used not
only by 13th-century composers, but prominently by Machaut; and it is
also found in some 14th-century English sources. However, "modern"
14th century theory, unlike Jacobus, regards the minor seventh as too
discordant to be used in such standard directed progressions. To
Jacobus, the minor seventh is an "imperfect concord": that is, an
interval whose voices, "sounded at the same time, are recognized by
the sense of hearing to differ greatly, and nevertheless somehow to
concord" (Book IV, Chapter 37).
The alternative resolution from minor seventh to octave suggests
stepwise oblique motion, also a very common 13th-century resolution of
this interval.
9. Like the fifth, and even more so, the octave is itself a stable and
conclusive interval; but, in theory, it can "resolve" to the yet more
perfectly blending unison.
10. The major ninth at a ratio of 9:4 is here is referred to by both
of its usual formal names: either a _tonus cum diapason_ or "tone plus
octave"; or _bis diapente_, a "double fifth," since it can be formed
from two pure 3:2 fifths (e.g. D3-A3-E4, with string ratios of 9:6:4).
As Jacobus notes, the most common resolution is an oblique one by step
to the octave, found from Perotin to Machaut. The resolution from
major ninth to fifth may also occur by oblique motion, with the upper
voice typically falling by a fifth, as happens in two-voice as well as
multi-voice settings of Machaut. Additionally, the major ninth may
sometimes resolve to a fifth by contrary motion, as in this
three-voice context:
E4 C4
A3 C4
D3 F3
11. The resolution of minor tenth to octave by stepwise contrary
motion (e.g. G3-Bb4 to A3-A4) is another closest approach progression.
12. For the major tenth to seek the twelfth is a closest approach
progression (e.g. G3-B4 to F3-C5) very characteristic of the era
around 1300, as a taste for wider voice ranges in polyphony makes the
twelfth, although certainly in evidence through the 13th century, a
more common interval.
13. In the view of Jacobus, the eleventh is considerably more tense
than the simple fourth; thus it would be even more so an interval
seeking _cadentia_ or resolution rather than serving as a goal for
it. In classic Greek times, the Pythagoreans held that all concords
other than the 1:1 unison must have either multiplex ratios of n:1
(e.g. 2:1 octave, 3:1 twelfth, 4:1 fifteenth or double octave); or
else superparticular ratios of n+1:n (3:2 fifth, 4:3 fourth). While
the fourth has a superparticular ratio of 4:3, the eleventh at 8:3
fails this test. Ptolemy, however, held that adding an octave to an
interval does not change its nature, making the eleventh, like the
fourth, a _symphonia_ or concord.
In the _Musica enchiriadis_ and _Scholica enchiriadis_ around 850-900,
two of the earliest sources on Western European polyphony, the
eleventh is freely accepted as a good concord; and it occurs
prominently, for example, in some 13th-century compositions. However,
one 20th-century perspective (in a journal called _In Theory Only_
that I saw around 1985) holds that in fact the eleventh is somewhat
less acoustically simple than the fourth, suggesting a possible aural
basis for the opinion of Jacobus that it is considerably less
concordant.
Either a resolution to the twelfth (likely by stepwise oblique
motion), or to the octave (as by contrary motion, e.g. E3-A4 to
F3-F4), provides a nearby restful goal.
14. The 3:1 twelfth, like the fifth and octave, is described as
restful in itself, although in theory it might "resolve" either to the
4:1 fifteenth (unlikely in practice, since as Jacobus notes the
twelfth is close to the limit of vertical intervals in use) or the 2:1
octave.
15. This passage might be interpreted in various ways. One reading is
that although the restful and conclusive fifth and twelfth are not as
purely blending as the unison, octave, or fifteenth, yet they are
perfect enough to have a "near" resemblance to these intervals, and so
are "as if the same" in providing unstable intervals with fully
satisfying resolutions. This reading would tie in nicely with the next
passage about how two-voice pieces, although characteristically ending
on a unison or octave, often end in practice on a fifth.
16. It is vitally important to understand that Jacobus is saying that
the simple unison and octave are the "principal endings" specifically
of two-voice improvisations or compositions, a view that Zarlino,
interestingly, takes in 1558, despite all kinds of changes in musical
style and intonational practice over the intervening two centuries and
a bit more. Zarlino explains: "Since [musicians] found that among
consonances no greater perfection could be found than in the octave,
they made it a fixed rule that each composition should terminate on
the octave or unison and no other interval. The rule was ignored only
by a few of poor judgment. If we wish to follow those who instituted
and observed good rules, we will terminate our compositions upon one
of these consonances, because they are more perfect than the others."
See Gioseffo Zarlino, tr. Guy A. Marco and Claude V. Palisca. _The Art
of Counterpoint: Part Three of Le Istitutione harmoniche 1558_
(Norton, 1976), Chapter 39, p. 84.
Here Jacobus writing sometime around the 1320's is actually more
flexible than Zarlino: he notes that even in two-voice writing, which
is what both are addressing, having the upper voice end on a fifth "is
often done." His observation that endings on intervals other than
these three -- the unison, octave, or fifth -- are rare also reflects
13th-century practice. The one piece with which I am familiar having a
part ending on a fourth above the lowest voice is actually for three
voices -- Montpellier 98, _Vilene gent/Honte et dolor/HEC DIES_, in
Yvonne Rokseth, ed., _Polyphonies Du XIIIe Siecle: Le Manuscript H196
de la Faculte/ de Me/decine de Montpellier_ (Editions de L'Oiseau
Lyre, Louise B. M. Dyer: 1936), Tome II, p. 198, mm. 27-28: concluding
with an outer octave, lower fourth, and upper fifth, G3-C4-G4 (with
the octave G3-G4 between tenor and motetus or duplum, and the fourth
at C4 in the triplum). The twelfth might be another candidate in a
two-voice context, but maybe, even around 1300, more characteristic as
a closing interval in writing for three or four voices.
With three or more voices, however, Jacobus makes it clear (Book II,
Chapter 36) that the "best way to end a measured song" is with a
complete sonority of outer octave, lower fifth, and upper fourth,
which his contemporary Johannes Grocheio around 1300 describes as
expressing the "threefold perfection of harmony" (_trina harmoniae
perfectio_). This preference reflects the practice of 13th-century
composers from Perotin on, and Jacobus takes great interest in the
question of why the octave is best arranged in such sonorities with
the fifth below and fourth above (Book VII, Chapters 6-8).
17. Jacobus expresses his view that the twelfth is close to the
largest simultaneous interval in practical use, although his theory
does address wider intervals, sometimes in intriguing ways.
To sum up, Jacobus in addressing _cadentia_ or the resolution of an
unstable concord by a stable one includes in his catalogue of standard
resolutions all the closest approach progressions used, for example,
by Machaut involving arrival at a restful and conclusive perfect
concord: min3-1, Maj3-5, Maj6-8, min7-5, min10-8, and Maj10-12. This
list agrees with Machaut's practice, but differs from Ars Nova
counterpoint and discant theory in recognizing min7-5 as well as the
others involving thirds, sixths, and tenths.
While the inclusion of min7-5 is in accord with traditional
13th-century practice as well as some varieties of 14th-century
practice, there are from a 13th-century perspective some notable
omissions. Directed progressions by stepwise contrary motion involving
motion of a tone by both voices include min2-4, min3-5, Maj3-1,
Maj6-4, min6-8, and Maj7-5, for example. Jacobus may have deemed some
of these directed resolutions unfashionable because of the lack of
closest approach (e.g. min3-5, Maj3-1), but two other factors may have
come into play.
First, the minor second, minor sixth, and major seventh play an
important role in 13th-century resolutions by directed contrary motion
and also oblique motion, but under the theory of Franco and Jacobus
are generally excluded as acute discords, along with the tritone or
augmented fourth (729:512) also, and for Jacobus likewise what he
recognizes as the distinct interval of the semitritonus or diminished
fifth with its slightly less dissonant character (1024:729).
Secondly, Jacobus does not include the 4:3 fourth as a goal of
resolutions in his treatment of _cadentia_, which excludes the closest
approach resolution Maj2-4 (e.g. D4-E4 to C4-F4). In 13th-century
practice, this resolution often occurs in two-voice as well as
multi-voice writing; in the 14th century, it remains relevant
especially in multi-voice writing in styles where progressions such as
G3-D4-E4 to F3-C4-F4 are idiomatic, with a major second resolving to
the upper fourth of a sonority with outer octave and lower fifth.
In addition to closest approach resolutions, Jacobus includes in his
catalogue a number of oblique resolutions: e.g. Maj2-1, 4-5, Maj6-5,
min7-8, Maj9-8, Maj9-5. These are relevant to 13th-century practice,
and also to 14th-century styles like that of Machaut. A fuller
catalogue would also include very useful 13th-century resolutions of
this kind involving strong discords: e.g. min2-1, Aug4-5, min6-5, and
Maj7-8.
While these 13th-century directed resolutions not involving closest
approach (among which should also be mentioned Maj2-5 by near-conjunct
contrary motion), and oblique resolutions involving strong discords,
may not be mentioned in the catalogue of Jacobus, they follow a
similar paradigm: an unstable interval resolves efficiently to a
stable one (unison, fourth, fifth, octave, eleventh, or twelfth).
Thus we can readily expand his concept of _cadentia_ to a broader
survey of 13th-century practice, and also apply it to the directed and
oblique resolutions of Machaut. This concept admirably fits the genius
of the Ars Antiqua, an art which from Perotin to Petrus de Cruce
features some of the most dramatic and telling use of vertical
instability and its satisfying resolution in the history of European
music.
ACKNOWLEDGEMENT: In arriving at this translation of Book IV, Chapter
50, I am much indebted to two previous articles translating this
chapter in whole or part: Jennifer Bain, "Theorizing the Cadence in
the Music of Machaut," _Journal of Music Theory_, Vol. 47, No. 2
(Fall, 2003), ,
pp. 325-362, see especially p. 328 and pp. 357-358 n. 8; and David
Maw, "Redemption and Retrospection in Jacques de Li`ege's Concept of
_Cadentia_," _Early Music History_ Vol. 29 (2010), pp. 79-118, with
complete text and translation of Book IV, Chapter 50, in Appendix 1,
pp. 114-115. Like both of these authors, I find that leaving the Latin
term _cadentia_ untranslated may be the best policy. My warm thanks to
Rob Wegman for calling the article by David Maw to my attention.
-----------------------------------------
Text and translation:
Jacobus on partition
Speculum musicae, Book IV, Chapter 51
-----------------------------------------
The Latin text of _Book IV_ of the Speculum musicae used here is
available at ,
THESAURUS MUSICARUM LATINARUM, School of Music, Indiana University
Bloomington, IN 47405 Jacobi Leodiensis Speculum musicae, ed. Roger
Bragard, Corpus scriptorum de musica, vol. 3/4 ([Rome]: American
Institute of Musicology, 1963), 1-126 (Used by permission).
[-124-] Capitulum LI.
quoad partitionem.
Chapter 51
Comparison of the concords as far as partition
De partitione concordiarum loqui volentes, dicatur prius quid sit talis partitio.
When we wish to speak about the partition of concords, it should first
be said what such partition is.
Est autem concordiae partitio alicuius tertiae vocis mediantis
aliqualiter cum qualibet extremarum vocum partitae concordiae
concordantis acceptio. Patet ex hac descriptione quod nulla concordia
partibilis est nisi divisibilis sit in plures partes aliqualiter
concordantes, et ideo concordia quam importat unisonus impartibilis
est, quia partibus, quae consonantiae sint, caret, nec, inter voces
eius extremas, aliqua tertia vox illis inaequalis mediat. Item
repugnat tono et semiditono partitio, quia non sunt divisibiles in
duas partes aliqualiter secundum se concordantes.
The partition of a concord is, moreover, when by the means of a third
mediating voice in some fashion, as you please, its outer voices are
partitioned into an agreement of concordant concords. It appears from
this description that no concord is partible unless it is divisible
into two or more parts each in some fashion concordant, and therefore
this means that the unison [1:1] is impartible, since it lacks parts
which are intervals, nor, between its outer voices, may any third and
unequal voice mediate. Likewise the tone [9:8] and semiditone [minor
third, 32:27] resist partition, because there are not divisible in any
way into two concordant parts.[1]
Ditonus autem aliqualiter partiri potest cum divisibilis sit in tonos
duos. Findit enim vox media extremas ipsius ditoni voces et illas
dividit in duas aequales partes, ut est cum sunt duo simul cantantes
quorum unus dicit fa, alius sonat la, tertius vero dicit sol (sed tali
partitione non est multum utendum, cum voces ditoni non perfecte
concordent.)
The ditone [major third, 81:64], however, is able to be partitioned in
a certain fashion, since it is divisible into two tones. A middle
voice indeed splits the outer voices of the major third and divides
them into two equal parts, as it is when two are singing at the same
time, of whom one sounds fa, and the other sounds la, a third truly
sounding sol (but such a partition is not much to be used, since the
voices of the ditone do not perfectly concord.)[2]
Partitur autem una quarta ex semiditono et tono cadentibus in unisono vel quinto.
Morever a fourth [4:3] is partitioned into a minor third and tone,
cadencing in the unison or the fifth.[3]
Quinta partitur ex tertia duorum tonorum inferius et semiditono
superius, et dicitur quinta fissa, vel e converso. Sed prima partitio
melior est. Vel partitur ex quarta et secunda cadentibus in unisonum.
The fifth [3:2] is partitioned into a major third below and minor
third above, and called the "split fifth," or conversely. But the
first partition is better. Or it is partitioned by a fourth and
[major] second, cadencing in the unison.[4]
Sexta partitur ex tertia duorum sonorum inferius et quarta superius,
vel e converso, vel ex quinta et tono, superius vel inferius. In
partitione autem ipsius sextae locum non habet semiditonus, | [P1,
165r in marg.] quia pars alia esset tritonus qui, in talibus
partitionibus, non recipitur, ut patet ex descriptione partitionis hic
intentae.
A sixth [i.e. tonus cum diapason or major sixth, 27:16] is partitioned
into a major third below and a fourth above, or the converse, or by a
fifth and tone, above or below. In the partition of this same sixth,
however, the minor third has no place, because the other part would be
a tritone [729:512] which, in such partitions, is not received, as
appears from the description of partition here set out.[5]
Semiditonus cum diapente partitur ex duabus quartis, vel ex quinta et
semiditono ascendendo vel descendendo. Non habet in istius partitione
ditonus locum, quia maneret ex parte altera semitritonus qui inter
discordias numeratur. Ideo a talibus excluditur partitionibus.
The semiditone plus fifth [minor seventh, 16:9] is partitioned into
two fourths, or into a fifth and a minor third, ascending or
descending. The ditone has no place in this partition, since there
would remain in the other part a semitritone [diminished fifth,
1024:729] which is numbered among the discords. Therefore it is
excluded from such partitions.[6]
[-125-] Octava, quae est diapason, partitur in diapente et
diatessaron, et convenientius est ut diapente inferius et diatessaron
ponantur superius. Item, descendendo in semiditonum, tonum,
semiditonum et ditonum.
The octave, which is the diapason [2:1], is partitioned into a fifth
and fourth, and it is more agreeable that the fifth be placed below
and the fourth above. Also, descending, into minor third, tone,
minor third, and major third.[7]
Nona finditur ex tono et diapason, vel ex bis diapente. Habet et alias
partitiones, sed non sic convenientes.
The [major] ninth [9:4] is split into a tone and octave, or into two
fifths. It has also other partitions, but not so suitable.[8]
Semiditonus cum diapason partitur in partibus ex quibus nominatur, et
tunc vox ipsius diapason medians est una tertia respectu alterius
vocis proximae illius consonantiae decimae. Item partiri potest in ter
diatessaron, sed praevalet prima partitio. Non convenit autem hanc
concordiam partiri per diapente, quia manet ex alia parte diesis cum
diapente. Et minus ab hoc convenit haec per ditonum, quia maneret
tetratonus cum tono minore.
The semiditone plus octave [minor tenth, 64:27] is partitioned into
the intervals by which it is named, and then this same mediating voice
is at a [minor] third with respect to the nearer other voice of this
interval of the [minor] tenth. Also it may be partitioned into three
fourths, but the first partition prevails. It is not agreeable however
for this concord to be partitioned by the fifth, since there remains
in the other part a diesis plus fifth [minor sixth, 128:81]. And it is
less fitting yet by a major third, since there would remain a
tetratone [augmented fifth, 6561:4096] plus a minor tone [diminished
third, 65536:59049].[9]
Decima in ditono cum diapason partibilis est in dictas partes, item in
tonum cum bis diapente, item in diatessaron, ditonum et diapente, vel
e converso.
A tenth in major third plus octave [i.e. major tenth, 81:32] is
partible into these said parts, also into a tone with two fifths. Also
into a fourth, major third, and fifth, or the converse.[10]
Undecima in diatessaron et diapason, item in diapente, diatessaron et
diatessaron; item in semiditonum, tonum et diapason.
An eleventh [8:3] into fourth and octave, also into fifth, fourth, and
fourth; also into minor third, tone, and octave.[11]
Duodecima in diapente et diapason, item in bis diapente et
diatessaron, item in ditonum et ter diatessaron. Prima partitio melior
est.
A twelfth [3:1] into fifth and octave, also into two fifths and a
fourth, also into a major third and three fourths. The first partition
is better.[12]
De partitione autem concordiarum quae sequuntur diapente cum diapason,
quia in usu non sunt, in speciali non prosequor; patet satis ex dictis
ipsarum partitio.
Concerning the partition however of concords which follow the diapente
plus diapason [fifth plus octave or twelfth] we do not specifically
pursue this; the partition of these is sufficiently clear from what
has been said.[13]
Est autem intelligendum quod omnes consonantiae simul se compatiuntur,
exclusis discordiis, si debito modo disponantur. Quamvis autem
consonantiae aliquae dissonae partes habeant concordantes, partitio
tamen, de qua loquimur, proprie in eis locum non habet. Semitritonus
enim ex duobus componitur semiditonis et ditonus cum diapente ex
ditono et diapente miscetur, tetratonus ex duobus ditonis.
It should be understood that all intervals agree with each other,
excluding discords, if they are disposed in due manner. And although
certain dissonant intervals have concordant parts, nevertheless the
partition of which we speak properly has no place in them. The
diminished fifth indeed is composed out of two minor thirds, and the
major seventh is combined out of the major third and fifth, and the
tetratone [augmented fifth, 6561:4096] out of two major thirds.[14]
Talibus et consimilibus consonantiis partitionem tactam non
applicamus, quia ipsarum voces extremae, simul mixtae, dissonae
sunt. Nec moveatur aliquis quod eaedem vocum mixtiones consonae
vocentur et dissonae. Utimur enim nomine consonantiae generaliter ad
voces concordantes et ad discordantes, prout supra visum est libro
secundo. Notandum est autem quod, in tactis consonantiarum
partitionibus, si debeant consonantiae partitae suam retinere naturam
convenienter, [-126-] tres cantores requiruntur (duo qui extremas
concordiarum dicant voces, unus qui mediam), si in duas partes fit
partitio, vel plures si in pluribus, ut cum finditur diapente quae est
inter ut ipsius .C. tertiae et sol .G. septimae, dicat unus cantor ut,
alius sol, tertius vero dicat mi ipsius .E. quintae. Et consimiliter
est cum finditur diapason per diapente | [P2, 194r in marg.] et
diatessaron.
To such and similar intervals we do not apply the said partition,
because the outer voices of these, combined at the same time, are
dissonant. Let it not influence anyone that the same mixtures of
voices are called consonant and dissonant. We use indeed the name of
"consonance" [i.e. an interval with both voices sounded at the same
time] generally for concordant and discordant voices, as accordingly
seen above in the Second Book. It should moreover be noted that, in
these partitions of intervals touched upon, if the partitioned
intervals are to retain their fitting nature, three singers are
required (two who sing the outer voices of the concords, and one who
sings the middle voice), if the partition is made in two parts, or
many if in many parts: as when the fifth is split which is between the
same C the third step and sol G the seventh step, one singer sounds
ut, another sol, and a third in truth sounds mi, the fifth step E. And
it is likewise when the octave is split by the fifth and fourth.[15]
Et haec dicta de consonantiarum concordiis sufficiant ad
praesens. Valent, puto, haec ad componendum discantus aliquos. Quid
autem sit discantus, et qualiter sit componendus, et quae eius
species, ad locum alium dicere reservamus.
And let this said about the concordant intervals suffice to the
present. This is of value, I think, for the composing of certain
discants. What, however, is discant, and how it is to be composed, and
what are its species, we reserve to speak of in another place.[16]
Aliae consonantiarum fieri possent comparationes, sed sufficiant quae
tactae sunt, et hic librum hunc quartum determinantes.
Other comparisons of the intervals may be made, but let those touched
upon suffice, delimiting this Fourth Book.
Ad aliam materiam stylus convertatur.
The pen may be turned to other matter.
Explicit liber quartus Speculi musicae.
Here ends the Fourth Book of the Speculum musicae.
-------
Notes
-------
1. To Jacobus, a partition is thus a multi-voice sonority in which a
suitably concordant outer interval is "partitioned" or "split"
(_fissa_) by one or more additional "mediating" or middle voices into
two or more included intervals, all of which are also to one degree or
another suitably concordant. No "discords" are admitted. In
appreciating the very diverse universe of multi-voice sonorities or
partitions endorsed or permitted under these guidelines, we must
consider how Jacobus classifies the intervals, the theme of Book IV as
a whole. Specifically, he expands the scope of the "concords" to
include all intervals deemed by Franco in the 13th century to have
some degree of "compatibility," with the Franconian "imperfect
discords" thus reclassified as "imperfect concords" to emphasize that
they are acceptable when aptly resolved, for example, and sometimes
more generally when used in multi-voice sonorities in association with
more concordant intervals. Here is a comparison Franco's scheme and
that of Jacobus, with my own characterizations of each category. Note
that Jacobus, unlike Franco, draws distinctions between the fourth
(4:3) and what he perceives as the rather less concordant eleventh
(8:3); and between the major second or tone (9:8) and the somewhat
more concordant major ninth (9:4).
-----------------------------------------------------------
Franco Jacobus Possible description
-----------------------------------------------------------
Perfect concord Perfect concord Stable
1 [1:1], 8 [2:1] 1 [1:1], 8 [2:1] Purely blending
-----------------
Middle concord 5 [3:2]. 4 [4:3] Stable
5 [3:2], 4 [4:3] Optimally blending
===========================================================
Imperfect concord Middle concord Mildly unstable
Maj3 [81:64] Maj3 [81:64] Relatively blending
min3 [32:27] min3 [32:27]
Maj9 [9:4]
-----------------------------------------------------------
Imperfect discord Imperfect concord Relatively tense
Maj2 [9:8] Maj2 [9:8] Some compatibility
min7 [16:9] min7 [16:9]
Maj6 [27:16] Maj6 [27:16]
-----------------------------------------------------------
Perfect discord Imperfect discord Acutely tense,
min2 [256:243] dim5 [1024:729] "Incompatible"
Aug4 [729:512] min6 [128:81]
min6 [128:81] Maj7 [243:128]
Maj7 [243:128] ------------------
Middle discord
min9 [512:243]
dim12 [2048:729]
min13 [256:81]
Maj14 [243:64]
------------------
Perfect discord
min2 [256:243]
dim3 [65536:59049]
Aug4 [729:512]
Aug5 [6561:4096]
------------------
Note: Jacobus, _Speculum musicae_, Book IV, Chapters 38-40,
also discusses some rare discords, such as those involving
intervals larger than the double octave or fifteenth [4:1],
not mentioned in his chapter on partition, and not listed
in the table.
----------------------------------------------------------------------
Of special importance are the major second or tone, minor seventh, and
major ninth, prominently used by composers of the 13th century from
Perotin to Adam de la Halle and Petrus de Cruce, and also by Guillaume
de Machaut in the 14th century, and present in various approved
partitions of Jacobus, but excluded as "discords" from the basic
vocabulary of "modern" 14th-century discant and counterpoint theory.
Since the 9:8 major second or tone is the smallest interval with some
degree of concord, it follows that neither the tone nor the minor
third (equal to a 9:8 tone plus a 256:243 diatonic semitone,
e.g. D-E-F or re-mi-fa, or 32:27) can be divided into two parts both
of which would be suitably concordant. Thus, as Jacobus observes,
there are no proper partitions of these intervals -- although, in
13th-century practice, dissonant sonorities like simultaneous D-E-F do
sometimes occur. Thus see _Deo confitemini-Domino_, m. 19 (second
system, m. 5), D4-E4-F4 (here C4 is middle C, with numbering by C-C
octaves), in Archibald T. Davison and Willi Apel, _Historical
Anthology of Music: Oriented, Medieval and Renaissance Music_ (7th
ed., Harvard University Press: 1962), p. 33.
2. Jacobus describes how a major third may be partitioned into a
three-voice sonority with two equal 9:8 major seconds, e.g. in the
three-voice conductus _Praemii dilatio_. m. 77, C4-D4-E4, in W. Thomas
Marocco and Nicholas Sandon, eds., _The Oxford Anthology of Music:
Medieval Music_(Oxford University Press: 1977), pp. 112-114 at 114.
The string ratios for this impressively complex and dense, but not
acutely discordant, sonority would be 81:72:64.
3. This partition of the fourth into a minor third and tone, in either
order, does occur from time to time in 13th-century polyphony. Here
_cadentibus in unisono vel quinto_ might more precisely be translated:
"with unstable or less conclusive intervals resolving to the unison or
fifth." As discussed by Jacobus in the previous chapter on _cadentia_,
Book IV, Chapter 50, a major second tends to resolve to the unison
(typically by oblique motion, an idiom going back to the 10th-11th
centuries); and a fourth, although a perfect concord, often may tend
toward the more conclusive fifth. A fine example of both resolutions
occurs in a three-voice setting of _Haec dies_ in a style like that of
Perotin, where this partition resolves by oblique motion over a
stationary lowest voice, with the two upper voices expanding from
minor third to fifth (a progression not mentioned by Jacobus, but
common in 13th-century practice):
F4 G4
D4 C4
C4
Note how the lower pair of voices resolve Maj2-1, and the outer pair
4-5. See _Hec dies_, mm. 15-16 (second system, mm. 1-2), Davison and
Apel (note 1 above), p. 31. This same piece also includes a version of
this partition of the outer fourth with both unstable intervals
resolving by stepwise contrary motion (Maj2-4, min3-1), arriving at a
relatively stable but less conclusive fourth, see ibid., fourth
system, mm. 5-6, and again at mm. 7-8:
G4 F4
E4 F4
D4 C4
4. Both the _quinta fissa_ or "split fifth" of Jacobus with a major
third below and minor third above (e.g. F3-34-C4), or the converse
with the minor third below (e.g. A3-C4-E4); and also the partition
with a lower fourth and upper major second (e.g. G3-C4-D4), or the
converse (e.g. G3-A3-D4), are very common in 13th-century writing.
The conductus _Praemii dilatio_ (n. 2 above), for example, has
beautiful examples of both types of sonorities with an outer fifth.
The words _cadentibus in unisonum_ for the partition into fourth and
major second, which I read as "cadencing in the unison," may refer to
the resolution of the unstable tone or major second into a unison with
one of the outer voices, often brought about by oblique motion of the
middle or mediating voice. This variety of resolution occurs, for
example, at the close of Montpellier Codex piece #125, the motet Je ne
puis/Amors me tienent jolis/VERITATEM, m. 44, in Yvonne Rokseth, ed.,
_Polyphonies Du XIIIe Siecle: Le Manuscript H196 de la Faculte/ de
Me/decine de Montpellier_ (Editions de L'Oiseau Lyre, Louise
B. M. Dyer: 1936), Tome II, pp. 240-241 at 241:
1 2 3 |
D4
A3 G3
G3
There is a momentary partition of the final fifth into a lower major
second and upper fourth (G3-A3-D4, string ratios 9:8:6), with the
middle voice resolving from major second to unison with the lowest
voice in a classic gesture of two-voice polyphony already noted and
approved by Guido d'Arezzo in the earlier 11th century, and also
recommended as a standard resolution by Jacobus (see text and
translation above of Book IV, Chapter 50).
A taste for three-voice combinations with fifth, fourth, and major
second (the latter two arranged in either order) may be a trait shared
by 13th-century Western European polyphony and such other world
musical traditions as Georgian three-voice polyphony evidently dating
back to at least the 12th century; certain Russian polyphony of the
16th-17th centuries; and Albanian polyphony. This is not necessarily
to imply either genetic relationships or diffusion, but possibly to
point to convergent evolution in some polyphonic traditions involving
three or more voices with a preference for fifths and fourths.
5. The major sixth thus may be partitioned either into a major third
and a fourth in either order (e.g. G3-B4-E4 or G3-C4-E4); or into a
fifth and major second (e.g. G3-D4-E4 or G3-A3-E4). In 13th-century
practice, the sonorities of M6|M3_4 (e.g. G3-B3-E4) and M6|5_M2
(e.g. G3-D3-E4) are common in directed progressions, since either can
expand stepwise to a complete 8|5-4 sonority (e.g. F3-C4-F4). Thus:
E3 F4 E4 F4
B3 C4 D4 C4
G3 F3 or G3 F3
In the 14th century, the newer rules of discant and counterpoint would
exclude, at least from note-against-note writing, vertical sonorities
with a major second; but this progression can be found in Machaut, and
also in the Old Hall Manuscript from the later 14th to early 15th
centuries. Thus see Richard H. Hoppin, _Medieval Music_ (W. W. Norton
and Co.: 1978), p. 511, Example XX-1, "Sanctus in English Discant
Style by Lambe," with a cadence of G3-D4-E4 to F3-C4-F4.
The preference of Jacobus to avoid a division of the major sixth into
a minor third and a discordant augmented fourth or tritone
(e.g. D3-F3-B3, or G3-Bb3-E4) was evidently not always shared by
Perotin and other 13th-century composers who use these combinations,
it sometimes being an open question whether performers might have sung
G3-Bb3-E4 or G3-B3-E4.
Note also that Jacobus does not list any partitions of the minor sixth
[128:81], to him as to Franco one of the outright discords, although
such sonorities seem very common in much 13th-century practice.
6. The partition of the minor seventh into two fourths, e.g. G3-C4-F4,
invites a classic resolution by oblique motion where the fourth
proceeds to the fifth and the minor seventh to the octave (two
resolutions listed by Jacobus in Book IV, Chapter 50), featured in
Perotin and other composers as a final or sectional cadence:
F4 G4
C4 D4
G3
The partition into a fifth and minor third, in either order
(e.g. G3-D4-F4 or G3-Bb3-F4), often invites a directed progression
involving two resolutions by stepwise contrary motion listed by
Jacobus, ibid.: minor seventh to fifth and minor third to unison, e.g.
F4 E4 F4 E4
D4 E4 Bb3 A3
G3 A3 or G3 A3
Machaut also uses these directed progressions, sometimes prominently,
as in his _Hoquetus David_ and Motet 16, _Se j'aim mon loyal ami/
Lasse! comment oublieray/Pour quoy me bat mes matris?_.
7. The three-voice sonority of the 2:1 octave partitioned into a 3:2
fifth below and a 4:3 fourth above is, as Jacobus observes, Book II,
Chapter 36, the best way a measured song may end. In Book IV, Chapters
6-8, Jacobus delves at some length into the reasons why, although the
4:3 fourth is a concord in its own right or when placed below the
fifth, it sounds best when placed above it in the partition of the
octave.
The partition into, in descending order, a minor third, tone, minor
third, and major third would require five voices, and result, for
example, in a sonority of G3-B3-D4-E4-G4. A possible complication of
this sonority, from the perspective of Jacobus, is the interval of a
discordant minor sixth between the two upper voices at B3-G4. If we
omit the octave, however, then we are left with a four-voice sonority
that occurs in 13th-century music, and invites a superbly efficient
directed progression:
E4 F4
D4 C4
B3 C4
G3 F3
Here the two-voice resolutions by stepwise contrary motion are Maj6-8
(G3-E4 to F3-F4), Maj3-5 (G3-B3 to F3-C4), min3-1 (B3-D4 to C4-C4),
and Maj2-4 (D3-E4 to C4-F4). This sonority also passes muster for a
proper partition as defined by Jacobus, since all four unstable
intervals (middle concords Maj3, min3; imperfect concords Maj2, Maj6)
have some degree of concord, as do the perfectly concordant fifth
(G3-D4) and fourth (B3-E4).
8. The partition of the major ninth into two ideally concordant fifths
(e.g. G3-D4-A4) is favored by composers such as Perotin and Machaut,
with Jacobus elsewhere noting how these fifths make the outer voices
of the ninth "seem to concord better" (Book IV, Chapter 55). The
partition into octave and tone (e.g. G3-G4-A4) also occurs in
13th-century practice. Another partition proper under the rules of
Jacobus, and which occurs in Machaut, for example, is into a major
sixth plus a fourth, e.g. G3-E4-A4. Thus his Ballade 32, _Ploures
dames_, has a sonority of D3-A3-E4 (the partition of the major ninth
into two euphonious fifths endorsed by Jacobus) at measure 4; and
D3-B4-E4 at measure 37, as the opening sonority of the refrain. See
Christian Berger, "Machaut's Balade _Plures dames_ (B32) in the Light
of Real Modality," originally published in Elizabeth Eva Woodbridge,
_Machaut's Music: New Interpretations_, Studies in Medieval and
Renaissance Music 1 (Boydell Press: 2003), pp. 193-204.
.
As Berger observes, p. 203, these two occurrence of the major ninth
"cannot be justified by contrapuntal rules," i.e. the new rules of
counterpoint and discant formulated around the second quarter of the
14th century. However, both partitions are acceptable under the
guidelines of Jacobus, since all of their intervals have some degree
of concord or compatibility, although the first is more "agreeable" or
"suitable" (_conveniens_).
9. Here the obvious partition for a minor tenth, or "minor third plus
octave," is into these named parts, e.g. G3-G4-Bb4 or G3-Bb3-Bb4. The
suggested four-voice division into three fourths, e.g. G3-C4-F4-Bb4,
seems not so common (although it momentarily occurs in Machaut's Mass,
_Ite missa est_); but more common, and possibly open to interpretation
as a subset of this partition, is the three-voice sonority of lower
minor seventh and upper fourth, e.g. G3-F4-Bb4, which I have seen in a
transcription of the Las Huelgas repertoire, and also in Machaut. A
directed resolution has the minor tenth contracting by stepwise
contrary motion to an octave, and the minor seventh to a fifth, e.g.
Bb4 A4
F4 E4
G3 A3
Jacobus cautions against a partition of the minor tenth involving the
fifth, e.g. G3-D4-Bb4, since the remaining interval would be a minor
sixth; or likewise, conversely, G3-Eb4-Bb4.
(Notice to readers: a discussion of a rather intricate mathematical
question follows in the rest of this note, which is certainly
optional, but provides an opportunity to query a small detail in one
of the most monumental and valuable works of all time on Western
European music practice and theory.)
His final caution against a partition of the minor tenth involving the
major third, e.g. B3-Bb4-D5 (with the middle voice a major third down
from the upper voice of the tenth), is quite correct in warning that a
dissonant interval would result -- but, if I am right, a different one
than that specified by Jacobus, _tetratonus cum tono minore_, i.e. the
tetratone or augmented fifth equal to precisely four 9:8 tones, plus a
minor tone or diminished third equal to two regular diatonic semitones
at 256:243. Here I will first calculate the ratio of this specified
interval, and then the actual ratio for the interval B3-Bb4 which
would remain if the minor tenth is partitioned by a middle voice at a
major third from the higher voice of the tenth.
The tetratone is equal to four 9:8 tones, e.g. C-D-E-F#-G#, and thus
also to two 81:64 ditones or major thirds, e.g. C-E-G#, at 6561:4096.
A modern amenity to simplify or verify these calculations is the
logarithmic unit of the cent, equal to 1/1200 octave. We find that a
9:8 tone is equal to 203.910 cents, an 81:64 major third thus to
407.820 cents, and a tetratone or augmented fifth, e.g. C-G#, at
6561:4096 to 815.640 cents.
Jacobus, if I read him correctly, specifies that the remaining
dissonant interval of the partition would be equal to this tetratone
or augmented fifth plus a minor tone, an interval we could also call a
diminished third, since it is equal to two regular or diatonic
semitones at 256:243 (90.225 cents), e.g. G#-A-Bb, or 65536:59049
(180.450 cents). In contrast, a regular tone like G#-A# at 9:8 would
include a regular diatonic semitone (e.g. G#-A) plus an apotome or
chromatic semitone a Pythagorean comma (531441:524288 or 23.460 cents)
larger, or 2187:2048 (113.685 cents), thus G#-A-A#.
However, even without doing the math (with ratios and/or the modern
shortcut of cents), if we spell out the tetratone and minor tone of
Jacobus starting at a convenient note such as C, we get C-G# for the
tetratone or augmented fifth, and then, adding two regular diatonic
semitones: C-G#-A-Bb -- we arrive at the acceptable imperfect concord
of a minor seventh, C-Bb at 16:9 or 996.090 cents. Multiplying the
ratios 6561:4096 and 65536:59049 (815.640 cents + 180.450 cents), we
likewise find that a regular minor seventh results.
How about the actual interval that results between the lower pair
voices in the sonority B3-Bb4-D5 that Jacobus warns us against? Here
we can note that B3-D5 is a minor tenth at 64:27 (1594.135 cents), and
Bb-D5 a major third at 81:64 (407.820 cents). Dividing the first ratio
by the second, or subtracting the size in cents of the second ratio
from that of the first, yields a ratio for B3-Bb4 at 4096:2187 or
1086.315 cents -- a diminished octave, or 2:1 octave less an apotome
or chromatic semitone (Bb4-B4) at 2187:2048 or 113.685 cents.
As it happens, while Jacobus discusses the apotome as a part of the
9:8 tone (and as a strong discord in itself), his very extensive list
of intervals doesn't seem to include the octave-less-apotome or
diminished octave (which he would, no doubt, also consider a
discord). However, his apparently mistaken description of this
remainder when a minor tenth is partitioned by a middle voice at a
major third from one its outer voices (here B3-Bb4-D5, or also
D3-F#3-F4 with the major third below and the diminished octave above)
mentions a _tonus minor_, the minor tone or diminished third. Is there
a way of deriving the diminished octave that includes this interval?
For one answer, let's consider B3-Bb4-D5, and its lower interval of
B3-Bb4. We can derive this interval from a regular major sixth (27:16,
905.865 cents) plus two regular diatonic semitones at 256:243 or
90.225 cents each, thus B3-G#4-A4-Bb4. If Jacobus had described this
interval as a major sixth (_tonus cum diapente_, a 9:8 tone plus a 3:2
fifth) plus his mentioned minor tone, he would have been exactly
right.
10. In addition to the obvious three-voice partition of the major
tenth, also known as _ditonus cum diapason_ or "major third plus
octave," into these specified intervals, Jacobus mentions two
four-voice partitions, the second of which has great practical use in
the 14th century, but requires that the adjacent intervals be placed
in the right order to avoid an excluded discord.
His first four-voice partition, "a tone with two fifths," suggests to
me sonority where the tone appears between a lower and upper fifth,
thus inviting a wonderful directed progression to my taste, although I
have not encountered this in 13th-14th century sources (at least yet):
B4 C5
E4 F4
D4 C4
G3 F3
Directed resolutions by stepwise contrary motion include Maj10-12
(G3-B4 to F3-C5); Maj6-8 (Gb3-E4 to F3-F4; D4-B4 to C4-C5); and Maj2-4
(D4-E4 to C4-F4). The lower pair of voices, and likewise the upper
pair, move smoothly in parallel fifths.
The second partition of Jacobus, "a fourth, ditone [major third], and
fifth," calls for a caveat: the ditone or major third and the fifth
should not be adjacent, or else a major seventh (_ditonus cum
diapente_, i.e. a major third plus fifth) would result, a discord
which Jacobus does not authorize for partitions. For example,
literally following the order of a fourth, major third, and fifth, we
would get G3-C4-E4-B4 (with the major seventh C4-B4); or, in the
converse order, G3-D4-F#4-B3 (with the major seventh G3-F#4). However,
the order of major third, fourth, and fifth gives this superb and
much-favored four-voice sonority of the 14th century, e.g. Machaut:
B4 C5
E4 F4
B3 C4
G3 F3
Here we have the directed resolutions by stepwise contrary motion
of Maj10-12 (G3-B4 to F3-C4), Maj6-8 (G3-E4 to F3-C4), and G3-B3 to
F3-C4. There are smooth parallel fifths (E4-B4 to F4-C5), octaves
(B3-B4 to C4-C5), and fourths (B3-E4 to C4-F4). This progression also
serves as a fine example of the way that the new 14th-century
prohibitions in discant and counterpoint against parallel fifths and
octaves apply to simple two-voice writing, but not consistently to all
pairs of voices in multi-part textures.
11. The partition of an eleventh or _diatessaron cum diapason_
("fourth plus octave") often involves these two intervals,
e.g. G3-G4-C5 or G3-C4-C5. The partition of "fifth, fourth, and
fourth," if arranged with the fifth as the middle interval, would
yield, e.g., D3-G3-D4-G4, with a fourth, octave, and eleventh above
the lowest note -- a sonority which, in the early organum of the
germinal _Musica enchiriadis_ and _Scolica enchiriadis_ treatises of
the 9th century, would result from organum at the fifth by the two
middle voices (e.g. at G3-D4), with the lower voice doubled at the
octave above, and the upper voice at the octave below.
An incomplete three-voice variation on fifth-fourth-fourth in that
order (say G3-D4-G4-C5) which I have encountered, e.g. in Machaut, is
the eleventh partitioned into fifth and minor seventh, e.g. G3-D4-C5,
where the upper interval, as Jacobus notes, may sometimes be called
_bis diatessaron_ or a "double fourth" (with the minor seventh at 16:9
equal precisely to two 4:3 fourths). With the division of "minor
third, tone, and octave" (e.g. G3-G4-Bb4-C5 or G3-Bb3-C4-C5), I am not
aware of having encountered this in the 13th-14th century repertoire,
although it might be interesting to look for it.
12. The partition of the twelfth or _diapente cum diapason_ ("fifth
plus octave") into a lower octave plus upper fifth, e.g. F3-F4-C5,
becomes increasingly popular around 1300 as three-voice pieces often
lean to a wider range between outer voices. In terms of string ratios,
this "tripla sonority" as it might be called (since the ratio of the
twelfth is a pure 3:1, or tripla) is expressed as 6:3:2, a harmonic
division, since the differences between the terms of 2:1 or 6:3 octave,
(6 - 3) or 3, and those of the 3:2 fifth, (3 - 2) or 1, have the same
ratio of 3:1 as the outer terms of 6:2 defining the perfect twelfth.
The ideal partition of the 2:1 octave into a lower 3:2 fifth and upper
4:3 fourth, e.g. 12:8:6, likewise has this property that the ratio
between the differences of the adjacent terms, (12 - 8):(8 - 6) or 4:2
or 2:1, is identical to the ratio between the outer terms defining the
octave, 12:6 or also 2:1. See _Speculum musicae_, Book I, Chapters
67-68. The converse ordering of fifth and octave (e.g. D3-A3-A4) also
occurs in practice.
The second alternative, calling for four voices, "two fifths and a
fourth," may be implemented as a stable sonority with
fifth-fourth-fifth (e.g. D3-A3-D4-A4), majestically combining the
harmonic divisions of the 2:1 octave into lower fifth and upper fourth
(12:8:6 or 6:4:3), and of the 3:1 twelfth into lower octave and upper
fifth (6:3:2), thus 6:4:3:2. In the later 13th century, this is the
preferred sonority in the improvisational technique described by Elias
Salamon; and it famously occurs in Machaut's Mass. In the early
organum of the Enchiriadis treatises (see n. 11 above), this sonority
would result from organum at the fourth (e.g. with a sonority of
A3-D4) where the lower voice is doubled at the upper octave, and the
upper voice at the lower octave.
A mildly unstable option with this same alternative of Jacobus is
fifth-fifth-fourth, e.g. D3-A3-E4-A4, the sonority of a major ninth
"split" into two fifths which he recommends, plus a twelfth (string
ratios 9:6:4:3). This sonority opens a phrase in the motet In
salvatoris nomine/Ce fu en tres doux tens de mai/In veritate
comperi/Veritatem, m. 71, in Gordon A. Anderson, ed., and Elizabeth
A. Close, text ed. and trans., _Motets of the Manuscript La Clayette_,
Corpus Mensurabilis Musicae 68 (American Institute of Musicology,
1975), pp. 32-34 at 34. It would also be possible to interpret _in bis
diapente et diatessaron_ as referring to the undivided interval of
_bis diapente_, a "double fifth" or major ninth, and an upper fourth,
producing a three-voice subset of the last option, e.g. D3-E4-A4
(9:4:3).
I am not sure about the last alternative of Jacobus, "a major third
and three fourths," e.g. F3-A3-D4-G4-C4 or D3-G3-C4-F4-A4 -- maybe a
bit academic in this form, since five-voice textures seem uncommon in
this era.
13. Jacobus generally counsels that simultaneous intervals much wider
than a twelfth [3:1] are too large to be practical, e.g. in his
discussion of _cadentia_, Book IV, Chapter 50.
14. This is an important warning that _all_ intervals of a proper
partition or multi-voice sonority must be acceptably concordant to
some degree, a requirement we might call "pancompatibility," since
"panconsonance" has taken on a different meaning. The latter term
relates to 15th-century styles where intervals other than the accepted
concords of by-then-established discant and counterpoint theory --
essentially the perfect unison, fifth, and octave, plus imperfect
thirds and sixths, and the octave extensions of these -- are decidedly
restricted. The pancompatibility of Jacobus, in contrast, embraces
major seconds, minor sevenths, and major ninths as legitimate
intervals both in two-voice resolutions (often by either directed
contrary motion) and in multi-voice sonorities or partitions.
However, like some 14th-century theorists with a narrower view of
"concord," Jacobus directs that all pairs of voices should form
legitimate concords: thus the caution against adjacent intervals of
major third and fifth, both acceptable, producing an unacceptable
outer major seventh or _ditonus cum diapente_ (e.g. F3-A3-E4); or two
minor thirds producing a dissonant diminished fifth or _semitritonus_
(e.g. B3-D4-F4).
Differences of view arise between Jacobus and some other 14th-century
theorists not because of disagreements regarding mathematics, or over
the precept that all intervals in a multi-voice sonority should be
acceptably concordant, but over judgments of taste. Thus Jacobus
recommends the partition of an outer major ninth (9:4) into two 3:2
fifths (9:6:4) both because he finds even the simple major ninth not
without some "concord" or compatibility; and because he finds that
adding a middle voice splitting this ninth into two fifths makes it
"seem to concord better."
For other theorists such as Walter Odington in the first decades of
the 14th century or the author of _Ars discantus secundum Johannem de
Muris_ toward the middle of the century, a major ninth evidently seems
more clearly "discordant" in itself, and not so attractive even when
"split" into two concordant fifths. _De gustibus non est disputandum_:
matters of taste are often beyond logical dispute, and our best
response may be to appreciate the incommensurate but awesome
possibilities both of 13th-century discant as consummately celebrated
by Jacobus, and of the new 14th-century counterpoint-based technique.
Machaut, characteristically, seeks out and achieves his own "new
forge" or creative synthesis of these traditions. He often relishes at
once the taste of the "moderns" for freer use of thirds and also
sixths; and also the traditional Ars Antiqua liberties for sonorities
and resolutions involving major seconds, minor sevenths, and major
ninths. A knowledge of Jacobus generally, and more specifically of his
chapters at the end of Book IV on _cadentia_ and partition, is
especially valuable, not only for understanding 13th-century styles,
but for one critically important perspective on aspects of Machaut's
style that do not seem to fit so well with modern 14th-century
counterpoint theory.
15. The first part of this passage is an important reminder about the
terminology used by Jacobus: a _consonantia_ is an interval with the
two notes sounded together, i.e. a simultaneous or vertical as opposed
to melodic interval, whether concordant or discordant. Therefore
_consonantiae_ which are discordant should be avoided in legitimate
partitions.
The second part is a helpful clarification by Jacobus both for the
musicians of his own time, and for 21st-century readers: partitions
require three or more voices sounding at the same time, with two
singers sounding the outer voices of the interval being partitioned or
"split," and at least one other singer if this interval is to be split
into two adjacent intervals, as in the example of the "split fifth."
Jacobus, evidently reckoning the steps of the gamut with low A as the
first step, places the lower voice of the fifth at C ut, the "third
step"; its upper voice at G sol, the "seventh step"; and the added
voice that does the splitting at E mi, the "fifth step." Thus the
fifth is split into the preferred arrangement of Jacobus for this
partition into two thirds, with the ditone or major third (81:64)
below, and the semiditone or minor third (32:27) above, thus C3-E3-G3,
or in string ratios 81:64:54.
After using this mildly unstable "split fifth" as an example, Jacobus
remarks that it is the same when the octave is split into fifth and
fourth, an ideal stable sonority, e.g. D3-A3-D4; in this example the
singers of the outer voices would pronounce D3 and D4, with a third
singer adding the mediating A3 at a fifth above the lowest voice and a
fourth below the highest (string ratios 12:8:6 or 6:4:3).
If the outer interval is to be split into more than two adjacent
intervals, then Jacobus notes that more singers will be required. A
fine example would be the partition of a major tenth into major third
or ditone, fourth, and fifth, e.g. G3-B3-E4-B4, with three adjacent
intervals and four voices needed (string ratios 81:64:48:32), here
inviting a resolution to F3-C4-F4-C5.
In short, Jacobus confirms that all the intervals of these partitions
are meant to sound simultaneously, with three or more singers required
to realize this intention.
16. Indeed Jacobus, in his discussion of partition, provides valuable
material both for composers seeking out a guide to preferred or
acceptable multi-voice sonorities; and to singers improvising discants
based on some known two-voice framework, for example.
Margo Schulter
August 23, 2015