---------------------------------------------------------------
Texts and translations:
Jacobus on closest approach, semitones, and _musica falsa_
Excerpts from _Speculum musicae_,
Book II, Chapter 80; Book IV, Chapter 11; Book VI, Chapter 66
With an Appendix: A parallel chapter in
Philippe de Vitry's _Ars nova_
---------------------------------------------------------------
Introduction and Acknowledgements
---------------------------------
The purpose of this very modest series of texts and translations is to
make available some striking passages with a focus on _musica
practica_ or practical music in the _Speculum musicae_ of Jacobus.
Here the focus is on what Jacobus has to say, writing perhaps
sometimes in the 1320's, about what both he and later writers of the
14th and 15th centuries -- and indeed of the 16th and early 17th
centuries also -- describe as "closest approach" resolutions, here
especially the progression from a minor third to a unison by stepwise
contrary motion (Book II, Chapter 80); and from a major third to a
fifth (Book IV, Chapter 11).
A preference for these progressions from an unstable interval to a
stable one by contrary motion where one voice moves by a tone and the
other by a semitone is interestingly common ground for Jacobus, who
champions his beloved _Ars Antiqua_ or _Ars Vetus_ of the 13th
century; and the Ars Nova moderns such as Philippe de Vitry and Johannes
de Muris, whose rhythmic and especially notational innovations Jacobus
famously decries in Book VII as at best unnecessary, and at worst,
"monstrosities." Yet it turns out that these adversaries, when it
comes to notation and evidently also other nuances of polyphonic
style, are much in agreement in describing, accepting, and indeed
embracing the principle of closest approach and some of its
ramifications.
Jacobus gives considerable attention to two of those ramifications
also relevant to his Ars Nova colleagues. The first is that the
preferred directed resolutions by stepwise contrary motion involve
melodic motion by a semitone in one of the voices -- or, often, more
than one in a multi-voice setting -- causing a focus on the vital role
of the semitone both in melody and in the everyday singing of these
favored two-voice resolutions.
The second is that singers can and will as a matter of course make
inflections -- often _musica falsa_ inflections outside of the
standard or _musica recta_ gamut -- in order to obtain desired closest
approach progressions (Book II, Chapter 80; Book IV, Chapter 11); or
also to obtain stable concords such as a fifth (e.g. B3-F#4) that the
regular steps of the gamut could not provide (Book VI, Chapter 66)
without the help of such "false mutation."
On this last point, the use of _musica falsa_ to obtain a fifth such
as B3-F#4, the passage in Jacobus is strikingly parallel to a chapter
_De semitonio_ in Philippe de Vitry's _Ars nova_, both using the
example of an upper voice seeking the fifth above the step B-natural,
and using the same Latin verb _oportet_ ("it is necessary, proper,
fitting") to explain why this singer would use the _musica falsa_ step
of f-mi (in later terms, F#) in order to obtain the desired fifth at
its concordant ratio of 3:2.
Because the discussions of Philippe de Vitry and Jacobus are so
strikingly parallel, I have added a text and translation of the
relevant chapter from _Ars nova_ as an appendix.
This set of excerpts from the _Speculum musicae_ is meant to
complement the first release in this series, including the two final
and critically important chapters from Book IV: Chapter 50 on
_cadentia_ or two-voice resolutions; and Chapter 51 on _partitio_ or
multi-voice sonorities. Those two chapters include unique material
relevant to 13th-century and 14th-century styles alike.
.
These excerpts focus mainly on the theme of _cadentia_ or the
resolution of two-voice intervals, explicating in more detail the
preference for closest approach progressions that Jacobus expresses in
Book IV, Chapter 50 (and more specifically his preference for a minor
third before a unison, and a major third before a fifth). His approach
is strikingly pragmatic, focusing on what singers are observed to do,
consistently and as a matter of course, at least as he sees and hears
it.
Since Book II is dedicated to a thorough investigation of the various
intervals of music, and Book IV to the related theme of concord and
discord, we should not be surprised to find directed resolutions and
the accidental inflections they may often involve addressed in these
books. Book IV, Chapter 11, discusses both melodic intervals larger
than a fifth whose use in plainsong was unknown to Guido, but is found
in more recent chants; and the use of _musica falsa_ in polyphony to
obtain a major rather than minor third immediately before a fifth.
This polyphonic practice of singers is presented as analogous to an
intriguing phenomenon on which Jacobus remarks: the stretching of a
minor seventh to a major seventh when a singer wishes directly to
touch upon the octave. Is this a reference to polyphony or possibly to
plainsong, and is the seventh in question a successive or simultaneous
interval? These may be open questions.
Book VI addresses ecclesiastical chant or plainsong, with Chapter 66
touching on another intriguing theme -- maybe a bit of a digression
from the main focus on plainsong, but at another level a connection to
the larger plan of the _Speculum_ -- of how _musica falsa_ is foreign
to plainsong, but not to measured music. Here we meet the discussion
so strikingly parallel to that of Philippe de Vitry in _Ars nova_ of
how singers seek out concordant intervals such as the fifth at places
in the gamut requiring "false" inflections which are in fact fitting
and necessary.
As the next release in this series of chapters and excerpts from
Jacobus, I plan to compile some excerpts from the _Speculum musicae_
on multi-voice sonorities that complement and expand upon some
sonorities and topics addressed in his chapter on partition.
-------
Here an acknowledgement is especially due to Karen Desmond, whose
germinal study Behind the Mirror: Revealing the contexts of Jacobus's
_Speculum Musicae_ (PhD dissertation, Department of Music, New York
University, 2009) helps those of us on "factfinding missions" focusing
on questions of 13th-14th century polyphony in theory and practice to
consider also the larger intellectual and spiritual perspective of
this monumental achievement, and thus herself shares in its enduring
value which she makes more accessible and helps us better appreciate.
Also, I would like to thank Rob Wegman for his inspiring example as a
dedicated and diligent translator, however imperfect my emulation; and
also for the creativity and generosity of spirit shown in his
scholarship, raising questions about topics such as the origins of
counterpoint in the earlier 14th century that at once provide a
fertile starting point and engage the imagination to envision a
multitude of paths which one might take from there.
Of scholars such as these, it may be said that when they touch on a
given area, they leave behind them a redolent and inviting aroma of
curiosity to inspire whoever chances to come upon the scene; or, to
borrow a metaphor from Luminita Florea in her article on Jacobus (see
n. 1 below), they season the topics which they address with the spices
of catalytic insight and cautious but bold hypotheses, thus whetting
the appetite of anyone who accepts their invitation to join in the
feast of learning.
Margo Schulter
2 September 2015
-------
The Latin text of the portions of the _Speculum musicae_ used here for
these excerpts are 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. 3/2 ([Rome]: American Institute of
Musicology, 1961), 128-231 (Used by permission).
,
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).
,
THESAURUS MUSICARUM LATINARUM, School of Music, Indiana University
Bloomington, IN 47405 Source: Jacobi Leodiensis Speculum musicae,
ed. Roger Bragard, Corpus scriptorum de musica, vol. 3/6 ([Rome]:
American Institute of Musicology, 1973), 161-317 (Used by permission).
------------------------
From Book II, Chapter 80
------------------------
(Here the topic is the ditonus or major third at a ratio of 81:64,
equal precisely to two 9:8 tones. The discussion compares the degree
of concord in this interval to that of the semiditonus or minor third
at 32:27, addressing in the process the vital role of the regular or
minor semitone at 256:243, and then notes the preference of singers to
sing a minor rather than major third when two voices converge on a
unison by stepwise contrary motion.)
Utuntur autem aliqui suis in discantibus consonantia hac, quia voces
eius non omnino discordant, nec tamen perfecte concordant, ut de
semiditono dictum est, cuius voces, ut videtur aliquibus, plus ad
concordiam attingunt quam ditoni, quia, licet minus semitonium per se
discordiam magnam includat, tamen sine ipso iuncto tono vel tonis
nulla est perfecta concordia, quia ex puris tonis nulla est integra
melodia. Et nonne, si duo simul cantent, unus la la la sol, alius
la fa fa sol, descendens in fa facit unam falsam? Non ditono, sed
potius utitur semiditono, quia voces eius magis placent auditui, et,
faciens unam tertiam sub vel supra diapason, quae est una decima,
semiditonum accipit. Item si duorum diapente voces tenentium extremas
per aliquam secundam unus descendat et alius ascendat, ut in tertia,
in unisonum, ad quem tendunt, cadunt. Semper semiditono funguntur,
non ditono, puta si quis descendendo dicat la sol, alius ascendendo
re mi, ut in fa unisonent et perfecte concordent, quia imperfecta
concordia tendit ad perfectam, vel si quis dicat sol fa, et alter
ut re, ut in mi cadant in unisonum; illi, hic et ibi, semiditono
utuntur qui est inter re et fa et inter mi et sol. Et idem observari
videtur cum diapente tritonum includat cum minore semitonio, ut est
inter voces .E. quintae et .[sqb]. nonae secundae, quia, ascendens de
mi in fa, de minore semitonio tonum facere videtur, una falsa utens,
ut, cum alio descendente in .a., septimae clavi, in .G. gravi, vocibus
| [F, 60r in marg.] unisonent. Nec obstat quod ditonus in maiore
consistit proportione quam semiditonus, ut est probatum, quia,
secundum hoc, tritonus meliorem redderet concordiam quam semiditonus,
quam ditonus, et quam diatessaron.
Moreover, some use this interval [the ditone or major third, 81:64] in
their discants, because its voices do not entirely make discord, nor
yet perfectly concord, as has [also] been said of the semiditone [the
minor third at 32:27], whose voices, as it seems to some, attain more
to concord than those of the ditone, since, although the minor
semitone [i.e. the regular diatonic semitone at 256:253] contains
great discord, yet without this same interval joined to the tone
[9;8], or [some number of] tones, there is no perfect concord, since
out of pure tones alone there is no complete melody.[1]
And does it not happen, if two sing at the same time, one
la la la sol, and the other la la fa sol, that the one descending into
fa makes a false [note]?[2] They use not the ditone, but rather the
semiditone, whose voices more greatly please the hearing, and, when
making a third below or above the diapason [2:1 octave], which is a
tenth, they take the semiditone.[3]
And likewise if of the two voices holding the outer notes of the
diapente [3:2 fifth], one descends and the other ascends by a second,
so that they fall into the third and then into the unison, toward
which they tend. They always perform the semiditone, not the ditone:
for example, if someone descending pronounces la sol, and another
ascending re mi, so that in fa they may make a unison and perfectly
concord, since an imperfect concord tends toward a perfect one; or if
someone pronounces sol fa, and another ut re, so that in mi they may
fall into a unison; here and there, they use the semiditone which is
between re and fa, and between mi and sol.[4]
And just so it is seen to be observed when the diapente includes a
tritone [729:512] plus a minor semitone [256:243], as it is between
the notes E the fifth step and b-natural, the second version of the
ninth step[5], since, ascending from mi into fa, [the singer] seems to
make from a minor semitone a tone, using a false [note], so that, with
the other [singer] descending into a, the seventh clef [or step], into
low G, the voices may make a unison.[6] Nor does it hinder that the
ditone consists of a greater proportion than the semiditone, as is
proved, because, according to this, the tritone would give better
concord than the semiditone, than the ditone, and than the diatessaron
[the 4:3 fourth].[7]
-----
Notes
-----
1. Jacobus says similarly of the semiditone, see Book II, Chapter 76,
_etsi eius voces non perfecte concordant, etiam nec perfecte
discordant_, "although its voices do not perfectly concord, yet
neither do they make perfect [i.e. thorough, complete] discord." And
he likewise says of the 9:8 tone, Book II, Chapter 38, _Tonus autem
nec perfectam concordiam, nec perfectam dicit discordiam, sed, quasi
medio modo, se habet inter haec_. "The tone, moreover, speaks neither
perfect concord nor perfect discord, but as if in a middle manner,
holds itself between these."
The perception that the Pythagorean semiditone or minor third at 32:27
(294.135 cents) is somewhat more concordant or mild than the ditone or
major third at 81:64 (407.820 cents) is one that I share with Jacobus.
He then offers a fascinating observation that, more generally,
although the Pythagorean minor or regular diatonic semitone at 256:243
(90.225 cents) contains in itself strong discord, yet only when the
minor semitone is joined to some number of 9:8 tones can perfect
concord result. Examples he might have cited are the 4:3 diatessaron
or fourth (from two 9:8 tones plus a minor semitone), and the 3:2
diapente or fifth (from three 9:8 tones plus this semitone). He
further observes that _integra melodia_ whole or complete melody,
cannot result purely from steps of a tone, but only when semitones are
also included.
For a 14th-century Ars Nova source sharing the view of Jacobus, see
Luminita Florea, "A Feast of Senses: Grinding Spices and Mixing
`Consonances' in Jacques of Li`ege's Theoretical Works," _Philobiblon:
Transylvanian Journal of Multidisciplinary Research in Humanities_
Vol. XVII (2012), No. 1, pp. 15-49, at pp. 16-17 and n. 5. .
Philippe de Vitry, in the treatise _Ars nova_, declares, _Semitonium
ut dicit Bernardus est dulcedo et condimentum totius cantus et sine
ipso cantus esset corrosus transformatus et dilaceratus._ As Florea
translates, "The semitone, as St. Bernard says, is the sweetness and
condiment of all song, and without it the song would be gnawing
[harsh, corrosive], changed, and dismembered." Id., p. 16, n. 5. For
the text of de Vitry's _Ars nova_, see
or .
These passages from Jacobus and de Vitry may have a certain affinity
with remarks from Zarlino like the following in 1558: "Guido placed
the semitone at the middle of each hexachord, considering this to be
the most dignified and honored position, the seat of virtue, as they
say. Such is its excellence and nobility that without it every
composition would be bitter and unbearable to the ear; no perfect
harmony could exist without it." 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 19,
pp. 38-39.
The statement in de Vitry's _Ars nova_ about St. Bernard and the
semitone opens a discussion about accidentals and _musica falsa_, see
my Appendix below, of which part is closely paralleled in Jacobus,
Book VI, Chapter 66, with text and translation below.
2. Jacobus is describing a passage like this, with C4 as middle C, or
as Jacobus says elsewhere in this chapter the tenth step of the gamut
(taking A2 as the first step):
la la la sol
A3 A3 A3 G3
la fa fa sol
A3 F#3 F#3 G3
The progression is from a unison on la (A3) to the cadential minor
third pronounced fa-la (F#3-A3) resolving to a unison on sol (G3).
Jacobus observes that the singer of the lower voice will sing _una
falsa_, F#3, so that a minor third F#3-A3 rather than a ditone or
major third F3-A3 precedes the goal of a unison on G3. The step F#3 is
_musica falsa_ in the sense of being outside _musica recta_ or the
regular gamut (including the fixed steps C-D-E-F-G-A plus the two
versions of B, Bb and B-natural); and also in the sense that the
interval fa-sol (F#3-G3) is here actually a minor semitone (normally
mi-fa) rather a usual tone.
3. This discussion of the minor tenth, or semiditonus cum diapason
(semiditone plus octave, 64:27), may relate to the preference
described by Jacobus in Book IV, Chapter 50, his discussion of
_cadentia_ or the tendency of unstable intervals to seek apt
resolutions to stable ones, for a progression by stepwise contrary
motion from the minor tenth to the octave. For text and translation,
see .
4. These two examples feature the very common progression by stepwise
contrary motion from a fifth to an unstable third to a unison, with
Jacobus noting that in each instance the minor third is preferred
(here occurring between unaltered steps of the regular gamut, so that
_musica falsa is not necessary).
la sol fa || sol fa mi
A3 G3 F3 G3 F3 E3
re mi fa ut re mi
D3 E3 F3 C3 D3 E3
These popular progressions and similar ones in the conductus and other
genres of 13th-century polyphony are summed up in the anonymous
_Tractatus de discantu_, Coussemaker's Anonymous II, from the genera
era of around 1300, which identifies as imperfect concords
_semiditonus et ditonus_ (the minor third and major third), _quae sunt
bonae veniendo a diapente in diapento, vel a diapente ad unisonum, et
e converso_ ("which are good for proceeding from fifth to fifth, or
from the fifth to the unison, or conversely").
.
Jacobus here focuses on the progression "from the fifth to the
unison," but with a specific preference for the minor third before the
unison not expressed by the author of the _Tractatus de discantu_, who
generally recommends either the major or minor third as apt for such
progressions.
These two examples of Jacobus illustrate resolutions of minor third to
unison with ascending semitonal motion (with the minor third mi-sol or
E3-G3 resolving to a unison on fa or F3, with the semitone step mi-fa
in the lower voice); and descending semitonal motion (with the minor
third re-fa or D3-F3 resolving to a unison on mi or E3, with the step
fa-mi in the upper voice). As he observes, the singers "use the
semiditone which is between re and fa" (in the second example) "and
between mi and sol" (in the first example).
Jacobus, in describing these two progressions, says that the voices at
the minor third _cadunt_ or "fall" into the unison, a usage related to
his concept of _cadentia_ (n. 3 above). An analogous idiom in English
might be: "Everything will fall into place," or "The marchers fell
into step." I agree with David Maw, "Redemption and Retrospection in
Jacques de Li`ege's Concept of _Cadentia_," _Early Music History_
Vol. 29 (2010), pp. 79-118 at p. 92, on this point: "Any `falling'
conveyed by the term is conceptual: the passage from less to more
blended sonority may as well be climbing as falling."
5. Here the two notes forming a fifth are the "fifth step" of the
gamut at E-mi or E3, starting on A2 as the first step; and the "second
version of the ninth step" of the gamut, would be B-mi at B3 (German
H3), a major ninth higher, the "second version" this step Bfa-Bmi, or
Bb3/B3 (German B3/H3).
6. As Jacobus notes, the fifth E3-B3 can be divided into the minor
semitone E3-F3 (256:243) plus the tritone or augmented fourth F-B
(729:512), equal to three 9:8 tones F-G-A-B. The progression from the
fifth through the third to the unison proceeds as follows:
mi la sol
B3 A3 G3
mi fa sol
E3 F#3 G3
Jacobus describes how the singer of the lower part, "ascending from mi
into fa, seems to make out of a minor semitone a tone" -- that is, the
solmization syllables mi-fa, normally indicating a semitone, rendered
as E3-F#3, a 9:8 tone. The Latin _videtur_ could be translated either
"seems" or "is seen" to make a minor semitone into a tone. This
example, like the earlier one in this chapter addressed in n. 2 above,
illustrates the directed resolution of the minor third F#3-A3 to the
unison G3-G3, with the singers applying _musica falsa_ as a matter of
course.
7. Jacobus is evidently arguing that the fact that the major third at
81:64 or 408 cents has a larger ratio or greater size than the minor
third at 32:27 or 294 cents does not prevent the minor third from
being the smoother or better concord. If larger meant more concordant,
he argues, then the tritone at 729:512 (612 cents), a perfect or most
acute discord to Jacobus (as in much 13th-century theory), would give
better concord than the 32:27 semiditone (a middle or mildly unstable
concord); the 81:64 ditone (also a middle concord, although not so
good as the semiditone); or indeed the perfectly concordant
diatessaron or fourth at 4:3 or 498 cents!
------------------------
From Book IV, Chapter 11
------------------------
(Jacobus offers some fascinating observations on the use of melodic
intervals larger than a fifth in more recent ecclesiastical chants, as
compared to the practice reported by Guido d'Arezzo; and also reports
that singers sometimes seem to prefer, evidently also in plainsong, a
melodic ditonus cum diapente or major seventh [243:128] rather than a
semiditonus cum diapente or minor seventh [16:9], as the step
approaching an octave. He compares this last tendency to the
preference of singers in polyphony to sing a ditonus or major third
[81:64] rather than a semiditonus or minor third [32:27] when two
voices expand by stepwise contrary motion to a fifth.)
Et est notandum quod, praedictarum consonantiarum, sex sunt quae in
frequentiore sunt usu praeter unisonum, hae scilicet: diesis, tonus,
semiditonus, ditonus, diatessaron et [-22-] diapente. Et, de istis sex
loquens, Guido dicit quod sex sunt vocum regulares coniunctiones in
depositione et in elevatione, hoc est in descensu vel ascensu. Ceteras
autem vocum inaequalium coniunctiones, quae scilicet superant
diapente, vocat irregulares, de quibus dicit quod nusquam reperiuntur.
And it should be noted that, of the intervals mentioned above, six are
those which are in more frequent use as well as the unison: the diesis
[minor semitone, 256:243], the tone [9:8], the semiditone [minor
third, 32:27], the ditone [major third, 81:64], the diatessaron
[fourth, 4:3], and the diapente [fifth, 3:2]. And, speaking of these
six, Guido says that six are the regular conjunctions of notes in
deposition and in elevation, that is in descending and ascending.
Other conjunctions of unequal notes, however, namely those which
exceed a diapente, he calls irregular, of which he says that they are
never found.
Hoc autem, etsi pro tempore ipsius Guidonis verum erat, non tamen pro
moderno tempore. Reperiuntur enim nunc aliae vocum inaequalium
coniunctiones in aliquibus cantibus ecclesiaticis; unde, in prosa vel
sequentia, quae sic incipit: Veni Sancte Spiritus, et emitte coelitus
Lucis tuae radium, et cetera, reperitur coniunctio vocum, inter quas
est tonus cum diapente, et, earum, inter quas est semiditonus cum
diapente, earum etiam, inter quas est diapason. In multis etiam
cantibus invenitur coniunctio vocum, inter quas est diesis cum
diapente; unde, in antiphonis tertii modi vel toni incipiunt
Saeculorum ad sextam vocem supra finem antiphonae, ut infra videndum
est libro sexto. Ditonus autem cum diapente tunc, secundum rem
invenitur, cum quis de semiditono cum diapente cupit immediate duplam
seu diapason tangere; tunc enim semiditonus cum diapente sic intendi
videtur ut fiat ditonus cum diapente qui proximior est ipsi diapason,
sicut cum quis dicit: re re ut et alius, simul cum illo, dicit: re fa
sol; quamvis regulariter inter re fa sit semiditonus, sic tamen
intendit fa, ut faciat ibi ditonum et amplius accedat ad diapente.
This, however, even if it was true in the time of Guido, nevertheless
is not in modern times. There are now indeed found other conjunctions
of unequal voices in certain ecclesiastical songs, whereupon, in the
_prosa_ or sequentia, which thus begins: _Veni Sancte Spiritus, et
emitte coelitus Lucis radium_, et cetera, there is found a conjunction
of voices, between which is a tone plus diapente [tone plus fifth or
major sixth, 27:16]; and also a [conjunction] of notes between which
is a semiditonus plus diapente [semiditone or minor third plus fifth,
the minor seventh, 16:9], and indeed of notes between which is a
diapason [the octave, 2:1]. In many songs indeed the conjunction of
voices is found, between which is a diesis plus diapente [i.e. a minor
semitone plus fifth, the minor sixth at 128:81]; whereupon, in
antiphons of the third mode or tone the Saeculorum begins at the sixth
voice [or note] above the end of the antiphon, as is to be seen in the
Sixth Book [Chapter 40].[8]
The ditone plus diapente [major third plus fifth or major seventh,
243:128] is moreover encountered according to circumstance, when
someone from the minor seventh desires immediately to touch upon the
_dupla_ ratio or diapason [the 2:1 octave]; then indeed the minor
seventh seems to stretch so as to become a major seventh which is
closer to this same octave; even as it is when someone pronounces re
re ut, and another, at the same time with that [singer], pronounces re
fa sol; although between re fa there is regularly a semiditone, so
nevertheless the singer raises fa, so as to make there a ditone and
more closely approach to the fifth.[9]
-----
Notes
-----
8. The comparison Jacobus makes between the chant practice described
by Guido, with melodic intervals no larger than a fifth, and more
recent chants using the major sixth and minor seventh, suggest that
while he very like much dislikes some of the stylistic and especially
rhythmic and notational innovations of the Ars Nova, he approves of
and indeed takes an active interest in other modern developments in
both plainsong and polyphony, both addressed in the course of this
chapter. In the area of plainsong, he thus notes that the major sixth
and also the minor seventh, exceeding Guido's largest melodic interval
of the fifth, are to be found in more recently composed chants such as
the _Veni Sancte Spiritus_ ("Come, Holy Spirit"); and that the setting
of the _saeculorum, amen_ in many chants in the third tone or mode
have this portion beginning on a note a minor sixth above the final or
note ending the chant. As Jacobus notes, he addresses these chants
further in Book VI on plainsong, Chapter 40, where he quotes the chant
_Tertia dies est quod haec facta sunt_, and notes how this concluding
portion of the chant begins on C4, at a distance from the ending on
the final of E3 of _semitonio cum diapente_, a (minor) semitone plus
fifth or a minor sixth.
9. An intriguing question is whether Jacobus, in describing how a
minor seventh [16:9, 996.090 cents] may "according to circumstance"
[_secundum rem_] stretch so as to become a major seventh [243:128,
1109.775 cents] when the singer desires immediately to proceed from
there to the octave, is addressing plainsong as well as measured
music. If he is, then this would be a description of what seems like
an instance of _musica falsa_ in plainsong; but the analogy he draws
to an example of two-voice discant might suggest that he is also
addressing measured music here. Let us first consider his analogous
example involving a progression from major third to fifth by stepwise
contrary motion, which may lead to one possible interpretation of his
observation concerning the altering of the minor seventh to a major
seventh when directly seeking the octave:
re fa sol
D3 F#3 G3
re re ut
D3 D3 C3
Here the progression starts at a unison, with the upper voice leaping
up by a third -- which the singer makes major by choosing F#3 -- with
the resulting major third D3-F#3 then expanding to the fifth C3-G3. As
Jacobus observes, "although between re fa there is regularly a
semiditone" or minor third, (i.e. re-fa as D3-F3), here the singer of
the upper voice "stretches" or "raises" this melodic interval to
D3-F#3 in order to arrive at the vertical or simultaneous ditone
D3-F#3 between the voices, which "more closely approaches to the
fifth." From major third to fifth is, of course, a standard
14th-century progression embraced by the Ars Nova "moderns" also, with
both schools sharing the perception that the major third is favored in
practice and theory because it "more closely approaches" the stable
interval, here the fifth, that is the goal of the resolution.
Could Jacobus, in referring to the practice "according to
circumstance" of a minor seventh being stretched to a major seventh
when a singer wishes immediately to touch upon the octave, have in
mind the use of a vertical major seventh? If so, there is a relevant
passage in Book II, Chapter 95, devoted to this interval: _Ponitur
autem in numero perfecte discordantium consonantiarum si voces eius
absolute proferantur, et non petant diapason, vel unisonum_. "It is
placed moreover in the number of the perfectly discordant intervals if
its voices were brought forth at the same time, and it were not to
seek the octave, or the unison."
Jacobus, like Franco, regards the major seventh as one of the most
tense or "perfect" discords, and excludes it from his catalogues of
directed two-voice resolutions and of acceptable multi-voice
sonorities (see Book IV, Chapter 50 and 51, and n. 3 above). In
further developing and refining Franco's scale of concord/discord, he
does recognize the major seventh in his new terminology as an
"imperfect discord," along with the semitritone or diminished fifth
(e.g. B-F, or German H-F, at 1024:729 or 588.270 cents, rather less
discordant than the tritone or augmented fourth, e.g. F-B, at 729:512
or 611.730 cents), and also the minor sixth at 128:81 or 792.180
cents. Yet even these mildest of the traditional 13th-century "perfect
discords" are excluded from a regular place in measured music.
However, doubtless deferring to 13th-century tradition, Jacobus
remarks in his chapter on the major seventh just quoted that this
interval would be numbered among those that are "perfectly discordant"
(using the familiar terminology of Franco, and also Johannes de
Garlandia) -- except if it seeks the octave, which would fit the
scenario we are now considering in Book IV, Chapter 11 -- or resolves
to the unison. The focus on these two progressions may reflect a
13th-century theoretical tradition, also reported by Garlandia, that
resolving a "discord" to the purest concords of the unison or octave
was considered most "proper," although "improperly" (I might say "less
canonically") in practice resolutions to the stable concords of the
fourth and fifth often occur also. See Johannes de Garlandia, _De
mensurabili musica_, Chapter 10.
.
Indeed the resolution from major seventh to octave is a noted feature
in Leonin and Perotin -- but are there examples fitting the remarks of
Jacobus of how a singer "stretches" (_intendit_) a minor seventh to a
major seventh immediately seeking to touch the octave, as Jacobus
tells us the major seventh can do and so avoid "discord"? One possible
illustration of this effect, and a stunning one, comes at the opening
of Perotin's organum triplum _Alleluia: Posui adjutorium_ as
transmitted by the Montpellier Codex, see 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, #14-15, p. 31, mm. 1-2:
F#4 G4
C#4 D4
G3
Here the triplum or highest voice at a major seventh G3-F#4 involving
F#4 (outside the regular gamut) immediately resolves to the octave
G3-G4 just as Jacobus describes, while the duplum or middle voice
likewise has a tritone or augmented fourth G3-C#4 involving C#4 that
directly resolves to the fifth G3-D4, an oblique resolution in which
each upper voice ascends by a minor or regular diatonic semitone.
Whether or not this is what Jacobus had in mind, his mention of the
major seventh evokes a truly great moment in his beloved heritage of
Ars Antiqua polyphony.
There is also a possibility that Jacobus is referring to a melodic
interval of a major seventh, of which he comments in Book II, Chapter
95, _Voces extremae consonantiae huius immediate succedentes rarissime
reperiuntur in cantibus planis vel mensuratis_. "The outer voices of
this interval directly following [one another] are very rarely found
in plain or measured songs." Here _rarissime_ may suggest, "very
rarely, but yet it is known to occur." Thus see the end of Book II,
Chapter 80 (following the portion quoted and translated above), where
Jacobus takes note of the next interval he will discuss, the
_semitritonus_ or diminished fifth at 1024:729 (e.g. B3-F4), remarking
that _et ipsam vidi, etsi raro, in aliquibus ecclesiasticis planis
cantis_, "and I have seen it, even if rarely, in certain
ecclesiastical plainsongs."
------------------------
From Book VI, Chapter 66
------------------------
Sed cum duplex sit musica, plana et mensurabilis, irregularis vel
falsa musica vadit contra planam musicam quia musica plana tali
mutatione et cantu, qui ad aliam sequitur, non utitur, cum obviet
regulari dispositione monochordi quam observare nititur. Sed tacta
irregularis mutatio non sic vadit contra musicam mensurabilem quae
respicit plures voces simul prolatas et aliquam concordiam inter se
habentes, uti si [-187-] quis a mi de bfa[sqb]mi supra vult habere
vocem diapente resonantem, oportet ut tacta falsa utatur mutatione
ponendo mi cum vocibus ipsius .f. acutae et ibi cantando quasi per
.[sqb]. durum et mutando semitonium in tonum, quia voces ipsius
.f. acutae, quae sunt fa et ut, per semitritonum distant a mi de
bfa[sqb]mi. Superat autem diapente semitritonum quantum tonus superat
minus semitonium, idest in semitonio maiore, et consimiliter est in
aliquibus aliis ipsius gammatis locis. Ut ergo possit ubique haberi
diapente de quinta ad quintam sub vel supra, expedit tactas facere
mutationes. Illae igitur etsi dicantur falsae quantum ad planam
musicam, non tamen quantum ad musicam mensurabilem.
But since musica is twofold, plain and measurable, irregular or false
music goes against plain music because plain music does not use such
mutation and singing, which pursues a different [kind of music], since
it is meet for [plainsong] to strive to observe the regular
arrangement of the monochord. But the touched upon irregular mutation
does not so go against measurable music which has regard for many
voices brought forth at the same time and a certain concord obtaining
among them: as when someone wishes from the step mi of bfa-bmi [that
is, e.g. B3 or German H3, as opposed to B-fa, i.e. Bb3 or German B3]
to have a resonating note at the fifth above, it is necessary that
[this singer] may use this mentioned false mutation, placing mi with
the syllables [_vocibus_] of this same high F and there singing as if
by a square or hard B [B-natural, German H] and changing a semitone
into a tone [i.e. F#4 in place of F4], since the [regular] syllables
of this same high F, which are fa and ut, are distant by a semitritone
from mi of bfa-bmi [i.e. B3-F4 at a semitritone or diminished fifth,
1024:729]. The diapente [fifth] moreover exceeds the semitritone as
much as the tone [9:8] exceeds the minor semitone [256:243], that is
by a major semitone [the apotome at 2187:2048, here as at Bb3-B3 and
also F4-F#4], and it is similar in certain other places of this same
gamut.[9] So that therefore it is possible in all places for the
diapente [at 3:2] to be had from fifth to fifth below or above, to
make these said mutations is expedient. These therefore, although they
may called false inasmuch as it relates to plain music, are
nevertheless not so inasmuch as it concerns measured music.[10]
Ideo, in aliquibus | [P1, 242v in marg.] instrumentis, ut in organis,
quasi ubique tonus dividitur in duo semitonia inaequalia, ut plures
ibi fieri possint concordiae.
For that reason, in certain instruments, as in organs, nearly
everywhere the tone is divided into two unequal semitones, so that
there it may be possible for there to be made many concords.[11]
-----
Notes
-----
10. Jacobus holds that while _musica falsa_ or the use of irregular
steps and mutations is generally foreign to plainsong, it is not so
alien but rather routinely accepted in measured music or polyphony,
where concords between two or indeed often among many voices must be
considered. While the previous chapters translated here focus on
_musica falsa_ inflections to obtain closest approach resolutions in
polyphony, specifically minor third to unison (Book II, Chapter 80)
and major third to fifth (Book IV, Chapter 11). this chapter focuses
on another main motivation of obtaining stable concords such as the
fifth at locations in the gamut where they would not otherwise be
available. Jacobus evidently accepts both motivations as frequently
arising and causing singers to make the indicated inflections as a
matter of course.
In his example, the singer of an upper voice wishes to sing a diapente
or perfect fifth above the note B-mi, say B3-F#4, and therefore raises
the usual step F4 fa ut (which might be the fourth step fa of a
natural hexachord on C4-D4-E4-F4-G4-A4, or the first step ut of a soft
hexachord F4-G4_A4-Bb4-C5-D5) by a major semitone to F-mi, or F#4 as
we write it. This produces the pure 3:2 fifth B3-F#4, which is larger
than the semitritone B3-F4 by the major semitone or apotome F4-F#4, or
2187:2048, the same amount by which the 9:8 tone E4-F#4 exceeds the
256:243 minor semitone E4-F4.
In the following diagram, the minor or regular diatonic semitone is
called by its Greek name of limma to save space. As shown, the desired
fifth B3-F#4 is equal to a 4:3 fourth (B3-E4) plus a 9:8 tone (E-F#4):
while the semitritone or diminished fifth at B3-F4 is equal to a 4:3
fourth (B3-E4) plus the minor semitone E4-F4 at 256:243.
diapente or fifth
3:2, 702.0 cents
|-----------------------------------------------------------|
B3 F#4
| diatessaron or fourth tone
| 4:3, 498.0 cents 9:8, 203.9 cents
|--------------------------------------|--------------------|
B3 E4 F4 F#4
|--------------------------------------|---------|----------|
limma apotome
256:243 2187:2048
90.2 cents 113.7 cents
|------------------------------------------------|----------|
B3 F4 F#4
|------------------------------------------------|
semitritone or diminished fifth
1024:729 or 588.3 cents
In this context, Jacobus uses the term _vox_ to mean not only a
"voice" or tone of a melodic or simultaneous interval, but also an
applicable solmization syllable to a given step on the gamut. Thus the
regular step F4 has the "voices" or syllables fa, ut; but here, to
obtain a perfect fifth above B-mi (B3), receives the _musica falsa_
syllable of F-mi (F#4), an apotome or major semitone higher than
either of the regular syllables pertaining to F4.
It is noteworthy that this example of using _musica falsa_ to find a
perfect fifth above B3, F#4, is identical to that presented by
Philippe de Vitry in the treatise _Ars nova_, see n. 1 above, in the
chapter entitled De Semitonio. Using the same verb form _oportet_ ("it
is necessary, fit, proper") as Jacobus, de Vitry remarks of the B3-F#4
example, _et oportet quod ubi est dyapente ab una voce ad aliam, ibi
sit bona et vera consonantia_, "and it is necessary (or proper) that
where there is a fifth from one voice to another, there should be a
good and true consonance." This discussion in _Ars nova_ leads to the
famous assertion that such an inflection is _non falsa, sed vera et
necessaria_, "not false, but true and necessary." Here Jacobus seems
very much in agreement.
For the relevant passages in de Vitry's _Ars nova_, see the Appendix
below.
11. This discussion of organs suggests that early 14th-century
keyboards had their usual diatonic 9:8 tones (C-D, D-E, F-G, G-A, A-B)
"nearly everywhere" divided into the unequal steps of 2187:2048 major
semitone or apotome and regular 256:243 minor semitone. This
translation assumes that _quasi ubique_ means "nearly (or almost)
everywhere." The following arrangement is one that would include the
accidentals sometimes occurring in usual Ars Antiqua sources such as
the Montpellier Codex, including the steps F# and C# which composers
such as Adam de la Halle and Petrus de Cruce (or, at least, those
musicians transmitting their pieces) seem to favor in order to obtain
closest approach progressions of the kind addressed by Jacobus, with
intervals above C shown as ratios and cents:
113.7 294.1 611.3 996.1
2187/2048 32/27 729/512 16/9
C# Eb F# Bb
C D E F G A B C
1/1 9/8 81/64 4/3 3/2 27/16 243/128 2/1
0 203.9 407.8 498.0 702.0 905.9 1109.8 1200
In this possible scheme with 11 notes per octave, we have the eight
_musica recta_ notes of the gamut plus Eb, F#, and C#. Thus the only
diatonic tone not "divided" in the manner described by Jacobus is at
G-A. By around the time Jacobus was writing, perhaps the 1320's, the
desire for a closest approach progression involving an ascending
semitonal motion from the major third E-G# to the fifth D-A, may have
provided a motivation for one way of completing this division: by
adding the step G#, and arriving at one version of the 12 notes per
octave which have mostly remained standard on European keyboards
during the almost seven centuries since:
113.7 294.1 611.3 815.6 996.1
2187/2048 32/27 729/512 6561/4096 16/9
C# Eb F# G# Bb
C D E F G A B C
1/1 9/8 81/64 4/3 3/2 27/16 243/128 2/1
0 203.9 407.8 498.0 702.0 905.9 1109.8 1200
Here each of the five accidental steps, including the _musica recta_
step B-fa or Bb, divides one of the usual diatonic tones at 9:8 into
two unequal semitones: thus C-C#-D, D-Eb-E, F-F#-G, G-G#-A, and
A-Bb-B.
-------------------------------------------------------------------
Appendix: Philippe de Vitry, _Ars nova_, on _musica falsa_
Compare Jacobus, Book VI, Chapter 66 (above)
-------------------------------------------------------------------
This version of the Latin text is from TML:
.
De semitonio
On the semitone
Semitonium, ut dicit Bernardus, est dulcedo, et condimentum totius
cantus, et sine ipso cantus esset corrosus, transformatus, et
dilaceratus. Boetius autem determinat de semitonio per solutionem
cujusdam questionis. Nam ita est quod aliquando per falsam musicam
facimus semitonium ubi non debet esse; nam in mensurabili musica illud
vidimus quod tenor sive biscantus alicujus motecti vel rondelli stat
in b fa, [sqb], mi, dicendo per [sqb] durum, tunc accipientem in
dyapente superius suum biscantum oportet dicere mi in f acuta, et sic
per falsam musicam; nam facere dyapente a mi in fa non est bona
concordantia, eo quod ab ipsa [sqb] quadrata usque ad ipsum f acutum
sunt duo toni, et duo semitonia, quorum conjunctio nulla est
consonantia, et oportet quod ubi est dyapente ab una voce in aliam,
ibi sit bona et vera consonantia.
The semitone, as St. Bernard says, is the sweetness and condiment of
all song, and without it the song would be gnawing [harsh, corrosive],
changed, and dismembered.[12] Boethius moreover settles the matter of
the semitone by the solution of a certain question. Now as to this it
is because sometimes by _musica falsa_ we make a semitone where one
ought not to be; for in measurable music we see it because when the
tenor or discant of a certain motet or rondellus stands in BfaBmi,
B-natural, mi, with pronouncing through square or hard B (German H),
then it is proper for someone taking the diapente (fifth, 3:2) above
in their discant to pronounce mi in high F, and this through _musica
falsa_; for to make a fifth from mi to fa is not good concord, by
which from this square B (B-natural or German H) even to the same high
F there are two tones, and two semitones, the conjunction of which is
in no way a consonance, and it is necessary that wherever there is a
fifth from one voice to another, there should be good and true
consonance.[13]
Et ideo oritur questio ex hoc videlicet, que fuit necessitas in musica
regulari de falsa musica sive de falsa mutatione, cum nullum regulare
debeat accipere falsum, sed potius verum. Ad quod dicendum est quod
mutatio falsa, sive falsa musica non est inutilis, imo est necessaria
per bonam consonantiam inveniendam, et malam evitandam. Nam sicut
dictum est, si velimus habere dyapente, de necessitate oportet quod
habeamus tres tonos cum semitonio, ita quod si aliqua figura sit in b
fa, [sqb] mi, sub [sqb] quadrato, et alia sit in f acuta per naturam,
tunc non est ibi consonantia, quia ibi non sunt tres toni cum
semitonio, sed tantum duo toni cum semitonio duplici; verumtamen fieri
potest ibidem quod per falsam musicam appellamus, scilicet quando
facimus de semitonio tonum vel e converso; non tamen est falsa musica,
sed inusitata.
And for that reason there arises a question from this, namely, that
there has been a necessity in regular music of _musica falsa_ or of
false mutation, since nothing regular ought to be taken for false, but
rather for true. As to which it should be said that false mutation, or
_musica falsa_ is not useless, but to the contrary is necessary for
finding good consonance and avoiding the bad. For just as has been
said, if we wish to have a fifth, of necessity it is fitting that we
may have three tones plus a semitone, so that if a certain figure is
in BfaBmi, B-natural or square-B (German H), and another is in high F
by nature (e.g. B3-F4, or German H3-F4), then there is not there a
consonance, because there are not three tones plus a semitone, but
only two tones plus a double semitone; notwithstanding, it is possible
for [consonance] to be made in that very place through that which we
call _musica falsa_, namely when we make from a semitone a tone, or
the converse; yet it is not false music, but rather unusual (or
unfamiliar).[14]
Unde notandum est quob b molle non est de origine aliarum clavium; hoc
autem cognoscitur per signum [sqb] quadrati vel b rotundi, in loco
inusitato locati, ita quod dicamus mi durum in f acutam cum signo
[sqb] quadrati; vel si b rotundum ponamus in b fa, [sqb] mi, vel in
consimilibus, ita quod sit in toni proportione, et tunc erit cum
dyapente consonantia; et ideo falsa musica est necessaria quandoque,
et etiam ut omnis consonantia seu melodia in quolibet signo
perficiatur.
Wherefore it should be noted that soft-B (Bb) is not from the origin
of other steps (of the gamut); this sign however is known by the sign
of square-B (B-natural, German H) or rounded-B (Bb, German B), located
in an unusual (or unaccustomed) place, and accordingly we should
pronounce hard-mi for a note in high f with the sign of square-B; or
if we should place a rounded-B at B-fa (Bb, German B), or in similar
places, thus that it may be in the proportion of a tone, and then
there will be a fifth with consonance; and for that reason _musica
falsa_ is sometimes necessary, and even so that every consonance or
melody wherever you please may be perfected by the sign.[15]
Igitur scire debes secundum dictum, quod duo sunt signa false musice,
scilicet b rotundum et ista alia figura [sqb]; et talem proprietatem
habent, videlicet quod b rotundum habet facere de semitonio tonum,
tamen in descendendo, et de semitonio in ascendendo habet facere
tonum. Et e converso fit de alia figura ista [sqb], scilicet quod de
tono descendente habet facere tonum. Tamen in illis locis ubi ista
signa requiruntur, et, ut superius dictum est, non falsa, sed vera et
necessaria, quia nullus moctetus, sive rondellus sine ipsa cantari non
possunt, et ideo vera, quia id quod falsum est, sequitur quod non sit
verum, sed hoc non est falsum, ergo.
Therefore you ought to know according to what has been said, that
there are two signs of _musica falsa_, namely rounded-B (Bb, fa-sign)
and this other figure square-B (B-natural, mi-sign); and they have
such a property, namely that rounded-B has to make from a semitone a
tone, yet in descending, and in ascending has to make from a tone a
semitone.[16] And it happens conversely with this other figure
square-B (B-natural, mi-sign), namely that from a descending tone it
has to make a semitone[17]. Yet in those places where these signs are
needed, and as has been said above, there are not false, but true and
necessary, because no motet, or rondellus without the same would be
possible to sing, and for that reason true: because that which is
false, it follows that it may not be true, but this is not therefore
false.
-----------------
Notes to Appendix
-----------------
12. This translation of the first sentence in de Vitry is borrowed
from Florea, "A Feast of the Senses" (see n. 1 above), p. 16, n. 5.
13. The statement that from B-natural to uninflected F above is only
"two tones and two semitones" -- as opposed to the concordant ratio of
the 3:2 diapente or fifth with a size equal to three 9:8 tones plus a
minor or regular diatonic semitone at 256:243 -- may be illustrated as
follows, with intervals shown as ratios and cents:
90.2 203.9 203.9 90.2
256:243 9:8 9:8 256:243
B3 C4 D4 E4 F4
|----------------------------------|
This passage also includes the observation _nam facere dyapente a mi
in fa non est bona concordantia_, "for to make a fifth from mi to fa
is not good concord," a statement of what will become the oft-repeated
precept that "mi contra fa" is to be avoided in perfect consonances,
with _musica falsa_ (or _musica ficta_) as a frequent remedy in such
situations.
14. The following diagram shows how the _musica falsa_ step f-mi or F#
brings about the desired concord of the diapente or fifth at B3-F#4
(German H3-F#4), equal to precisely three 9:8 tones plus a minor
semitone at 256:243.
90.2 203.9 203.9 203.9
256:243 9:8 9:8 9:8
B3 C4 D4 E4 F#4
|---------------------------------------|
15. Here the idea may be that, for example, when B-fa or Bb is used,
then the usual F above it will yield the desired consonance of a
fifth, again with three 9:8 tones and a 256:243 minor semitone:
203.9 203.9 203.9 90.2
9:8 9:8 9:8 256:243
Bb3 C4 D4 E4 F4
|---------------------------------------|
16. This version of the Latin text literally says that the rounded-B
or fa sign (corresponding to a modern flat) has such a property so as
_facere de semitonio tonum, tamen in descendendo_ "to make from a
semitone a tone, yet in descending," which is correct; but then _et de
semitonio in ascendendo habet facere tonum_ "and has [a property] in
to make from a semitone, in ascending, a tone." My translation assumes
that the descending or ascending motion is to rather than from the
altered note, so that with the B-fa or flat sign it is as follows:
From F to Eb From D to Eb
"Descending to b-fa sign: "Ascending to b-fa sign:
From a semitone, a tone" From a tone, a semitone"
203.9 203.9
9:8 tone 9:8 tone
|-------------------| |------------------|
Eb----------|-------F D-------|----------E
E Eb
2187:2048 256:243 256:243 2107:2048
113.7 90.2 90.2 113.7
Likewise, the text literally states, that, conversely, with the
square-B or mi-sign (corresponding, depending on the location, to
modern natural or sharp sign), _de tono descendente habet facere
tonum_, "from a descending tone it has to make a tone." Again, if the
descending motion is to the altered note, then here from a descending
tone is made a semitone; and in ascending to the same note, a semitone
is made into a tone:
From G to F# From E to F#
"Descending to b-mi sign: "Ascending to b-mi sign:
From a tone, a semitone" From a semitone, a tone"
203.9 203.9
9:8 tone 9:8 tone
|-------------------| |------------------|
F-----------|-------G E-------|----------F#
F# F
2187:2048 256:243 256:243 2187:2048
113.7 90.2 90.2 113.7
Another version of the text of _Ars nova_ already linked to in n. 1 above,
,
seems neatly to fit this interpretation as far as the square-B or mi
sign is concerned: _Et e converso fit de alia figura ista [sqb]
scilicet quod de tono descendente habet facere semitonium, et de
semitonio ascendente habet facere tonum_. "And conversely it happens
with this other figure the square-B (B-natural, mi-sign), namely that
from a tono in descending it has to make a semitone, and from a
semitone in ascending it has to make a tone."
Margo Schulter
mschulter at calweb dot com
2 September 2015