In response to the interest expressed here in medieval polyphony as a
topic of compositional study, I would like modestly to attempt
addressing one aspect of the learning process by making available some
relevant passages from one very important primary source: Johannes de
Grocheio or Grocheo (c. 1300) on the technique of composition for
three and four voices.

Grocheio's treatise from around the end of the 13th century, known as
_De Musica_ or _Theoria_, is centered on a kind of sociological survey
of the kinds of music used in Paris, and the social milieus and
functions served by these different genres. Ecclesiastical plainsong,
secular music for voices and instruments, and measured polyphony are
all part of this fascinating picture.

As a lover of part-music, and especially the motet as the most subtle
and sophisticated manifestation of this art calling for the most
discerning audiences, Grocheio describes the technique of composing in
three or voices with a focus on the theme of complete vertical
concord.

Specifically, he teaches that _three_ voices are required in order to
realize _consonantia perfectissima_ ("most perfect concord") through a
sonority consisting of outer octave, lower fifth, and upper fourth.
He describes this sonority, through a theological analogy to the
Trinity, as manifesting _trina harmoniae perfectio_, the "threefold
perfection of harmony" -- thus the modern English word "trine." 

Such a complete harmony or trine is formed, for example, by the notes
G3-D4-G4 (with C4 representing middle C), which includes the outer
octave G3-G4; the lower fifth, G3-D4; and the upper fourth D4-G4.

The following passages might be read in connection with a presentation
on common 13th-century sonorities and progressions, for example, my
page <http://www.medieval.org/emfaq/13c.html> at the Early Music FAQ
Site of the Medieval Music and Arts Foundation, edited by Todd McComb. 
There are also some concluding remarks after the texts and translations.

Here the Latin text is from E. Rohloff, _Der Musiktraktat des Johannes
de Grocheo_, E. Rohloff, ed., _Media Latinitas Musica II (Leipzig,
1943); for an alternative translation, see A. Seay, _Johannes de
Grocheo Concerning Music (De musica)_, Colorado College Music Press
Texts/Translations I (Colorado Springs, 1967). The translation here is
my own version, informed by Seay and others:

                             * * *

Dicamus igitur, quod omnium sublimis creator a principio in sonis
trinam harmoniae inseruit perfectionem, ut in eis suam bonitatem
ostenderet et per illos nomen suum laudaretur, unde David: _Laudate
dominum in sono tubae etc._, et etiam ut nullus possit se excusare a
laude divina, sed onmis lingua in sonis nomen gloriae fateatur. Et
forte, sicut est in trinitate gloriosa, ita quodammodo in hac
experientia docetur.

Est enim una prima harmonia quasi mater, quae _diapason_ ab antiquis
dicta est, et alia quasi filia, in ista contenta, _diapente_ dicta, et
tertia ab eis procedans, quae _diatessaron_ appellatur. Et istae tres
simul ordinatae consonantiam perfectissimam reddunt. (Rohloff at 44)

Let us say therefore, that the sublime Creator has from the beginning
instilled a threefold perfection of harmony [_trinam harmoniae
perfectionem_], so that in them [Her/His] goodness may be shown and
through them [Her/His] name may be praised, wherefore David: _Praise
the Lord with the sound of the tuba, etc._, and even so that none
might excuse themselves from divine praises, but every tongue would
confess in sound the name of glory. And perhaps, just as it is in the
glorious Trinity, so in this it is taught to some degree by
experience.

Indeed there is a first harmony like a mother, which was called by the
ancients _diapason_ [the octave], and another contained within this
one like a daughter, called _diapente_ [the fifth], and a third
proceeding from these, called _diatessaron_ [the fourth]. And these
three arrayed simultaneously yield most perfect consonance. (Compare
Seay at 6)

                             * * *

Sed alii ad tres consonantias perfectas attendentes cantum ex tribus
compositum, uniformi mensura regulatum invenerunt, quem _cantum
praecise mensuratam_ vocaverunt. Et isto cantu moderni Parisiis
utuntur. (Rohloff at 53)

But others attending to the three perfect consonances have invented a
song composed with three voices, regulated by uniform measure, which
they have called _precisely measured song_. And the modern Parisians
use this song.

                             * * *

Motetus vero est cantus ex pluribus compositus, habens plura dictamina
vel multimodem discretionem syllabarum, utrobique harmonialiter
consonans. Dico autem _ex pluribus compositus_, eo quod ibi sunt tres
cantus vel quattuor, _plura_ autem _dictamina_, qua quilibet debet
habere discretionem syllabarum tenore excepto, qui in aliquibus habet
dictamen et in aliquibus non. Sed dico _utrobique harmonaliter
consonans_, eo quod quilibet debet cum alio consonare secundum aliquam
perfectarum consonantiarum, puta secundum diatessaron vel diapente vel
diapason, de quibus superius diximus, cum de principiis tractabamus.
(Rohloff at 56)

A motet truly is a song composed of many voices, having many texts or
a varying choice of syllables, everywhere sounding together
harmoniously. I say moreover _composed of many voices_, because there
are three songs or four; moreover _many texts_, because each ought to
have a variation of syllables except the tenor, which in some pieces
has a text and in others not. But I say _everywhere sounding together
harmoniously_, because each ought to sound together with the other
according to some of the perfect consonances, that is according to the
fourth or fifth or octave, of which we have spoken above, when we
treated of the basics. (Compare Seay at 25)

                             * * *

Partes autem istorum plures sunt, puta _tenor_, _motetus_, _triplum_,
_quadruplum_ et in hoquetis _primus_, secundus et ultimo eorum
_duplum_.

Tenor autem est illa pars, supra quam omnes aliae fundantur
quemadmodum partes domus vel aedificii super suum fundamentum. Et eas
regulat et eis dat quantitatem, quemadmodum ossa partibus aliis.

Motetus vero est cantus ille, qui supra tenorum immediate
ordinatur. Et in diapente ut plurimum incipit et in eadem proportione,
in qua incipit, continuatur vel in diapason ascendit. Et in hoquetis
ab aliquibus dicitur _magistrans_, ut in hoqueto, qui dicitur _Echo
montis_ [Rohloff suggests the motet _Ego mundus_].

Triplum vero est cantus ille, qui supra tenorem in diapason
proportione incipere debet et in eadem proportione ut plurimum
continuari. Dico autem _ut plurimum_, qua aliquoties in motetum vel
diapente descendit propter euphoniam, quemadmodum motetus aliquando in
diapason ascendit.

Quadruplum vero est cantus, qui aliquibus additur propter consonantiam
perficiendam. Dico autem _aliquibus etc._, quia in aliquibus sunt
tantum tres et ibi sufficiunt, cum perfecta consonantia ex tribus
causetur. In aliquibus vero quartus additur, ut dum usus trium pausat
vel ordinatim ascendit vel duo ad invicem se truncant, quartus
consonantiam servet. (Rohloff at 57)

The parts of these [motets and hockets] are many, that is, _tenor_,
_motetus_, _triplum_, _quadruplum_, and in hockets, _primus_,
_secundus_, and in the last of these, _duplum_.

The tenor, moreover, is that part above which all the others are
founded, in the same manner as the parts of a house or building above
their foundation. And it regulates these and gives them their measure,
just as bones do for the other [anatomical] parts.

The motetus truly is that song which is placed immediately above the
tenor. And it most often begins at the fifth and in this proportion,
in which it begins, continues, or ascends to the octave. And in
hockets it is called by some _magistrans_ [the "leading" or "guiding"
part], as in the hocket which is called _Ecce montis_ [Rohloff
suggests a known motet entitled _Ego mundus_].

The triplum truly is that song which ought to begin at the proportion
of the octave above the tenor and in this same proportion for the most
part continue. I say moreover _for the most part_, since sometimes it
descends to the motetus or fifth for the sake of euphony, just as the
motetus now and then ascends to the octave.

The quadruplum truly is a song which is added to some piece so that
the consonance may be perfected. I say however _some, etc._, because
in some there are there are only three and these suffice, since
perfect consonance is caused by three [voices]. In some, truly, a
fourth part is added, so that when one of the three pauses or duly
ascends, or two truncate each other [as in hocket], the fourth
preserves the consonance. (Compare Seay at 26-27)

                             * * *

Primus vero in hoquetis est, qui primo truncare incipit, sed secundus,
qui secondo post primum truncat. Duplum vero est, qui supra tenorem
minutam facit abscisionem et cum eo aliquoties in diapente consonat et
aliquando in diapason proportione, ad quod multum iuvat bona discretio
decantantis.

Volens autem ista componere primo debet tenorem ordinare vel componere
et ei modum et mensuram dare. Pars enim principalior debet formari
primo, quoniam ea mediante postea formantur aliae, quemadmodum natura
in generatione animalium primo format membra principalia, pua cor,
hepar, cerebrum, et illus mediantibus alia post formantur. Dico autem
_ordinare_, quoniam in motellis et organo tenor ex cantu antiquo est
et prius composito, sed ab artifice per modum at rectam mensuram
amplius determinatur. Et dico _componere_, quoniam in conductibus
tenor totaliter de novo fit et secundum voluntatem artificis
modificatur et durat.

Tenore autem composito vel ordinato debet supra eum motetum componere
vel ordinare, qui ut plurimum cum tenore in diapente proportione
resonat et propter sui harmoniam aliquoties ascendit vel descendit.

Sed ulterius debet istis triplum superaddi, quod cum tenore ut
plurimum debet in diapason proportione resonare et propter sui
harmoniam potest in locis mediis sistere vel usque ad diapente
aliquoties descendere.

Et quamquam ex istis tribus consonantia perficiatur, potest tamen eis
aliquoties decenter addi quadruplum, quod, cum alii cantus descendent
vel ascendent ordinatim vel abscisionem facient vel pausabunt,
consonantiam resonabit.

The primus truly is in hockets the voice that first begins to
truncate, but the secundus, that which truncates second after the
primus. The duplum truly is the voice above the tenor which least cuts
up and sounds together with it [the tenor] sometimes at the fifth and
sometimes in the proportion of the octave, in which the good judgment
of the singer much assists.

Wishing moreover to compose in this genre, one should first order or
compose the tenor and give it [rhythmic] mode and measure. Indeed the
more principal part ought to be formed first, since by means of this
the others are formed afterwards, just as nature in the generation of
animals first forms the principal members -- that is the heart, liver,
brain -- and by means of these afterwards are formed the others. I say
moreover _to order_, since in motets [motellis] and organa the tenor
is made from an old and pre-existent song, but by art is delimited
further through mode and right measure. And I say _to compose_,
because in conductus the tenor is made totally anew and according to
the will of the crafter is measured and takes its set form.

Moreover, the tenor having been composed or ordered, the motet should
be composed or ordered above it, so that most often it resonates with
the tenor in the proportion of the fifth, and according to the harmony
sometimes ascends or descends.

But last should the triplum be added also to these, which ought most
often to resonate with the tenor in the proportion of the octave, and
according to the harmony may stay in middle ranges or sometimes
descend to the fifth.

And although consonance may be perfected by these three, nevertheless
it is possible sometimes fittingly to add a quadruplum, so that, when
other voices duly descend or ascend, or make a cutting up, or pause,
it will cause [full] consonance to resound. (Compare Seay at 27-28)

                             * * *

Please let me emphasize that this is a quick attempt. The most
important point is that Grocheio, like Johannes Lippius some 300 years
later (1610, 1612), views a perfect _three-voice_ sonority as at once
manifesting the mystery of the Trinity in music and providing a key to
the craft of practical composition.

Grocheio's presentation of the forms of 13th-century polyphony focuses
on the goal of maintaing complete harmony or _consonantia
perfectissima_ wherever possible. Thus the tenor typically serves as
the foundation, with the duplum or motetus at the fifth, and the
triplum at the octave -- with an upper fourth between these latter two
parts. (Grocheio, in his theologically oriented description of the
_trina harmoniae perfectio_ or "threefold perfection of harmony," says
that the fourth proceeds from both the "mother" octave and "daughter"
fifth -- just as the Holy Spirit in a Latin view proceeds in the
Trinity from both Father and Son.)

His remarks about the purpose of adding a quadruplum are especially
interesting: he explains that the idea is to "preserve the consonance"
if two voices should be in unison (i.e. by "descending or ascending
together"), or should truncate themselves as in hocket, or if one of
the voices should take a rest. Various four-part compositions seem
often to fit this pattern, and Grocheio's role for the optional
quadruplum reminds one of Gioseffo Zarlino's explanation in a
Renaissance context (1558) of the role of the alto in a four-part
texture, which likewise "supplements the other parts and perfects
their harmony, which could not be perfect without it." See Gioseffo
Zarlino, _The Art of Counterpoint: Part Three of Le Istitutione
harmoniche, 1558_, trans. Guy A. Marco and Claude V. Palisca (New
York: W. W. Norton, 1976), Chapter 58, at 180.


Most appreciatively,

Margo Schulter
mschulter@calweb.com