FLURA #2010 - 13

Platonic and Pythagorean Ratios in the Formal Analysis of 15th Century Music

Paul Kinsman

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Abstract:

In the fifteenth century European composers did not explicitly use the formal

vocabulary that we use today to define musical structure. There were no sonata forms

and no rondo form and certainly no piece named as such; those terms had not been

invented yet. Without any documentation from composers, today we ask: How did they

structure their music? The answer is believed to lie in two other artistic fields of the time,

art and architecture, where artists and architects used mathematical proportions to

structure their works. This study attempts to find a common ground between these two

artistic fields and 15th C music in terms of compositional proportion. In other words, the

techniques used to achieve symmetry in a cathedral were also used in the paintings and

music housed within. We know how painters and architects used proportions in their

work because they extensively wrote about their creative process and the virtues of

beauty. Since no composers wrote treatises overtly describing the compositional process

for writing a mass and no composer left a description of his training, we are left to do a

little detective work with primary sources. Therefore we will examine the intellectual

community of the church, the possible avenues of education, and finally the music to

infer that church composers had an identical ear for proportion as did the eyes of painters

and architects.

SECTION 1: Evidence that Composers knew and used proportions

In order to give this study perspective, one must first understand how a 15th

century composer might come to know these proportions. Compared to the creative

process today, it would seem unlikely that musical composers would be concerned with a

highly mathematic, proportional form in their compositions. However, the intellectual

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climate in Europe at that time would have created a climate where the use of proportions

was ideal. Historical data and primary sources from the 15th C show that a composer’s

primary school education, university education, religious training, and musical training

would contribute to their understanding of proportions.

In the 15th C intellectual rebirth swept Europe reigning in a new era, the

Renaissance. Ottoman attacks on Constantinople starting in 1396 forced Byzantine

scholars to flee to Italy, taking with them their most important ancient Greek literature.

Thereafter, the Greek language was taught to Italian scholars who either went east to

collect more writings or stayed in Italy to translate those writings into Latin. Therefore,

15th century Western Europeans experienced resurgence in the values of Greek literature,

namely what the Europeans called “studia humanitatis” or Humanism (Norton 151).

The emphasis on restoring Greek learning ideals lead to a Neo-Platonic era which

took Plato’s writing at the forefront of ancient Greek literature and reexamined their

significance. Specifically, Plato’s Timeaus was an extremely influential source on 15th C

scholars because it details the creation of the universe through mathematic relationships.

In the Library of Rome there was said to be fifteen copies of Timeaus. Given the cost of

paper and the difficulty in publishing during that time it is apparent that this book was of

notable significance. In the Timeaus, Plato’s demiurge, or Supreme Being’s, presence

within the physical world is related to the reader through mathematics thus linking

mathematics with celestial harmony (Donald Zeyl).

Along with the restoration of Platonic ideals came an overall reverence for Greek

academics. Pythagoras, a Greek mathematician from the 5th century B.C. is responsible

for initiating the unification of our Western music tuning system. The Pythagorean

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tuning system is derived from Pythagoras’s use of an instrument that he was said to have

invented called the monochord. This instrument consisted of a single string tightly

spanning two fixed planks and was used in teaching, tuning, and experimentation. The

monochord string sat atop a movable bridge which divided the string into two sections.

The division of the monochord used the numbers 6, 8, 9, and 12 with relation to Plato’s

arithmetic and harmonic means (discussed in Timeaus). The ratios of 12:6 (2:1), 9:6 or

12:8 (3:2), 8:6 or 12:9 (4:3), and 9:8 translated to the musical intervals of an octave, fifth,

fourth and major second, respectively. The accuracy of these intervals and other

considerations were fiercely debated on by music theorists throughout the fifteenth

century. Theorists like Boethius, Tinctoris, and Garfurius wrote numerous treatises on

the process of creating intervals through the use of whole number proportions (Adkins).

The 15th C architect Alberti said “the numbers by means of which the agreement of

sounds affect our ears with delight, are the very same which please our eyes and our

minds…We shall therefore borrow all our rules for harmonic relations from the

musicians to whom this kind of numbers is well known.” (Margret Varbell Sandresky

109).

The way in which music was notated in 15th Europe was also mathematically

based, like the tuning systems. The equivalent to time signature in modern music was

called mensuration in the 15th century. Rhythmic notation stemmed from division into

twos or threes. The longest duration, “long”, could be further divided into a “breves” and

“semibreves” much like a whole note today can be divided into half notes and quarter

notes. The division of the long was called the modus, division of the breve was called

tempus, and of the semibreve was called prolatio. Each of these values could be divided

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into groups of two or three which were called imperfect and perfect division,

respectively. With this in mind, 15th century time signatures can be equated to our

modern time signatures as such:

Perfect tempus, Perfect prolation- 9/8 Perfect tempus, Imperfect prolation- ¾ Imperfect tempus, perfect prolation- 6/8 Imperfect tempus, imperfect prolation- 2/4

These perfect and imperfect divisions had religious implications as well. Perfect division

represented the Christian trinity (Norton 120). As we have seen, music at its most

fundamental is extremely mathematic. While some of this information is common

knowledge to us today, these considerations were fresh in the minds of 15th C scholars.

We know this because of treatises and the education system in 15h C Europe.

With regard to the education that a composer may have received, we will start

from the ground up. Music aside, knowledge of proportions was essential for everyday

use. Simply stated a proportion is four terms where the ratios are equal. In a composer’s

primary education, proportions were known as the “Rule of Three” and played a large

part in a 15th century person’s ability to function in commercial life. Finding a third

value with the knowledge of two other terms had application in farming, bartering,

pasturage, brokerage, discount, currency exchange, and many other issues of trade. The

application for this rule became widespread during the fifteenth century with the

Enlightenment’s increase in commerce and trade. Of course the commercial use of the

“Rule of Three” did not originate in the fifteenth century or in Europe. The earliest forms

of the golden rule are oriental and traveled to Indian, Arab nations and later European

nations via trade routes. The Rule of Three, also known as the “Golden Rule” and

“Merchants Key” is universal arithmetic tool that was used to determine proportional

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relationships using two terms to find the third. Piero della Francesco, a 15th C painter

that we will revisit later, details the rule as such:

The Rule of Three says that one has to multiply the thing one wants to know about by the thing that is dissimilar to it, and one divides product by the remaining thing. And the number that comes from this is of the nature of that which is dissimilar to the first term; and the divisor is always similar to the thing which one wants to know about (Baxandall 95).

Knowledge of proportions was essential in everyday commercial life, and

remains important in everyday education. The university education system, to which 15th

century composers would have attended, would have included the study of music within

the scientific and mathematic disciplines. With the resurgence of Platonic ideas, Greek

academies centered on the humanistic seven liberal arts including grammar, dialect,

rhetoric, geometry, arithmetic, astronomy, and harmonics (music). The first three verbal

arts became known as the “Trivium” and the last four mathematic disciplines became

known as the “Quadrivium” (Norton 40). Fifteenth century European universities taught

with a division between these two main schools. Today we think of music as a creative

field which aligns itself more with literature and therefore the Trivium, however in the

15th C, music was a mathematical field of study. As we saw with mensuration and

Pythagorus, the music curriculum would have focused more on the math behind the

music. This is proved with 15th C mathematicians, Cardano, and Johannes Taisnier,

along with many others who wrote treatises on music harmonics to compliment their

mathematic writing (Carpenter 129).

Before we examine the actual proportions and the artists and architects of the 15

C, it is important to know that all these creative luminaries worked for the same patrons.

Most composers were hired as church clerics and fulfilled their composing careers from

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within in the church. The 15th Christian church was the center of nearly all aspects of

society, including an intellectual center. Church’s commissioned music pieces as well as

buildings and pieces of artwork. We have extensive evidence that painters and architects

relied on proportions in their works. Therefore, under the same employment, it is safe to

assume that the mingling of ideas within the same walls could have lead to a composer’s

knowledge of these mathematics.

SECTION 2 Platonic-Pythagorean Ratios

The case for composers knowing about proportions is well stated. It becomes

clear that if music composition did not employ proportions, it was indeed the only thing

that didn’t. What proportions would have been significant and why? The first, and

arguably most important numerical relationship is called the “Golden section”, the

Golden Ratio, the Golden mean, the Greek letter Phi, or as Leonardo DiVinci called it in

the 16th century “De Divina Proportione” (The Divine Proportion). The Golden section is

an uneven proportion, approximately 1:1.1618 that is seen as a unifying proportion in

nature, astronomy, and physiology. The Golden Section for example is found in the

concentric spirals of a seashell, the construction of the human body, and the distance of

the planets in our solar system. The Golden section is equally ubiquitous in design and

scholarly discussion. The Egyptians used it to construct the Pyramids and the Greeks to

build the Parthenon, Plato wrote about it in the Timeaus, and Leonardo Fibbonacci

constructed a number series (the Fibbonacci series 0, 1, 1, 2, 3, 5, 8 ect.) that carries the

Golden Ratio. The golden section expressed algebraically is as follows: if a is to a and b

is to a plus b then a²=b² +ab. (Newman W Powell 233) In the 15th C, with Neo-Platonic

resurgence, the divine proportion was accepted as one mysterious relationship that binds

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the microcosm and macrocosm alike; it was a type of “Rennaissance String Theory”

(Witkower 104). Due to its seeming omnipresence in nature, the Golden Section in the

15th C acquired a religious significance that Plato previously described in the Timeaus.

Artists and scholars alike believed that the golden section was seen as the math with

which manifests God’s presence in our world.

Another series of proportions that would have been used would have been derived

from the work of Pythagoras. As stated, the Pythagorean tuning system is derived from

whole number proportions. Out of all the intervals: the fourth, fifth, and unison were

seen as perfect consonances or “symphonies”. The term “symphonies” is reserved for

intervals that are formed when a string length is divided using the numbers 1-4.

Therefore, the octave or unison (2:1, 1:1), fourth (4:3), the fifth (3:2) all fit this criteria

(Claude V. Palisca). Plato uses these whole number rations to construct his “world-soul”

in the Timeaus and believed that the planets’ were arranged from one another in the same,

simple ratios. Therefore, Pythagorean numbers also took on a religious and cosmological

significance in the 15th C.

We know that the use of proportions in 15 century life was ubiquitous, but why

would composers use them in their composition. It is not likely that they were

incorporated for the sake of perception. Cognitive psychologists agree that our brains are

designed to recognize patterns and intuitively pick up on them. Jonathan Kramer states

that proportions in music are perceived as a means of cumulative listening, or non-linear

listening (Kramer 336). The human ear and brain are unable to hear and appreciate

formal proportions in music as it is being performed. Therefore it is safe to say that

proportions in music were not meant to be heard. That being said, it would seem that in

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something as deliberate as music composition, the use of proportions (particularly simple,

whole number proportions) had to have been a conscious decision made by the composer.

It is understood that proportions in the 15th century had significance with regard to

the construction of the universe and the mathematic relationships with which God used to

create the world. Evidence strongly suggests that the composers were using these

proportions in much the same way that artists and architects were using them to echo a

celestially valid harmony (Wittkower 8). In addition to their cosmological importance,

these proportions were practical for creating a formally beautiful end product. For

example, the Golden section, when derived from within the Fibbonacci series, was useful

for architects to could construct endless chains of identical ratios (Powell 233). These

proportions became an aesthetic preference in art and architecture. Now let us frame this

study of 15th music within architecture and painting.

SECTION 3. Architecture

Use of proportions in 15th century architecture is well established. We know why

architects used proportions, we know which proportions they used, and we know what

the metaphorical or allegorical connotations associated with the various proportional

systems. There are many important architects from the 15th C, however three key figures,

Filippo Brunelleschi, Alberti and Palladio, must be considered within this study because

of their explicit work with proportions in their building plans.

The Italian architect and sculptor Filippo Brunelleschi b. 1377-1446 is called the

father of Renaissance architecture and was solely responsible for initiating the

rediscovery of Roman architecture. Brunelleschi embraced classical orders in his

construction (“orders” being like the architectural vocabulary of a building) and

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developed his own proportional system with which he integrated classical order into his

buildings (Harold Meek). Brunelleschi is given credit for the development of

Renaissance perspective theory despite having never committed his findings to writing.

Brunelleschi’s perspective theory rests on the problem that calculated proportions cannot

be perceived as one walks through a building (the architectural equivalent to Kramer

linear and non-linear listening). The architect’s ideas on perspective remedies this

problem and suggests that with regard to proportioning, there should by no difference

between a painted building and the real thing. Leonardo da Vinci more clearly describes

how this is achieved by saying “In the case of equal things, there is the same proportion

of size to size as that of distance to distance from the eye that sees them.” In numerical

terms, one’s perspective of four objects of equal size that are placed at one, two, three,

and four meters away will be perceived as 1/1, ½, 1/3, 1/4 the size, respectively (Padovan

214).

(Figure 1)

Brunelleschi’s most important work was the Dome of Florence Cathedral (Figure

1) and was finished in 1436. This building is important to our study because it implies a

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direct connection between the music and architecture fields with regard to proportion.

The early 15th C composer, Guillaume Dufay was commissioned to write a motet to be

performed at the cathedral’s opening which he titled “Nuper Rosarum Flores”. When

Craig Wright analyzed Dufay’s motet and the cathedral’s architectural design, he found

that both compositions are found to be referencing the proportioning of Solomon’s

temple in the Bible. Solomon’s temple measures 60 cubits long, 20 cubits wide, and 30

cubits high. The length of the temple is divided between the house of prayer which was

40 cubits long and a smaller holier place, the “sanctum sanctorum” which was 20 cubits

long. Therefore simple proportions are evident on the biblical end. The house of prayer

which measures 20 x 40 cubits reduces 1:2 and the inner sanctum 20 x 20 cubits reduces

1:1. Brunelleschi’s house of prayer is 72 braccias wide (braccia is a term measurement, 1

braccia= 1 arm’s length) and 144 braccia long, 2:1, and the sanctum sanctorum is 72 x 72

braccias, 1:1. Dufay’s motet,” Nuper Rosarrum Flores” is structured to be broken down

in sections that relate one to another 6:4:2:3. These numbers mimic the exact dimensions

of Solomon’s temple. Although this theory is somewhat controversial within both the

fields of music theory and architecture, it provides an interesting example for

consideration (Craig Wright 402-420).

Andrea Palladio (1508-1580) was a late renaissance architect who was the last

great architect to derive contemporary buildings from ancient architecture. Palladio is

most known for his designs of Villas where his mastery of symmetry and proportion was

applied to private residences. In his Four Books on Architecture Palladio echos Alberti

and Plato by suggesting that the lengths of a room should be the harmonic or geometric

mean of the height and width of that room. Wittkower argues that Palladio used only the

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proportions of musical consonances when determining the dimensions of his buildings

however studies have shown that this is not always the case. (Deborah Howard 116).

Another architectural luminary, Leon Battista Alberti (1404-1472) was a multi-

talented scholar of the 15th C, who is most known for his contributions to the field of

architecture. Alberti, like Brunelleschi, contributed to the renewed interest in the

architecture of antiquity. In his ten books on architecture, De Re Aedificatoria, Alberti

focuses on making clear and reinterpreting the architectural wiritings of Vitruvius, the

first century Roman architect. Alberti aligned his architectural ideals with those of

Vitruvius which can be summarized

Just as the head, foot, and indeed any member must correspond to each other and to all the rest of the body of an animal, so in a building, and especially in a temple, the parts of the whole body must be so composed that they all correspond one to another, and any one, taken individually, may provide the dimensions of the rest (Padovan 213).

In other words, a building’s proportioning is the most important consideration in its

design.

Perhaps more explicitly than architects before and of his time, Alberti defines

beauty and describes the difference between beauty and ornament. Alberti states an

object’s beauty is determined as being “the harmony and the concord of the parts

achieved in such a manner that nothing could be taken away or altered except for the

worse” (Wittkower 33). An object’s perfection in terms of beauty is taken in the object

as a whole whereas ornamentation can only add to the beauty. In his book “Architectural

Principles In the Age of Humanism” Rudolph Wittkower argues that the harmonies in

which Alberti is talking about are indeed the proportions which Pythagorous found to be

consonances.

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(Figure 2., Copyright The Alberti Group)

The first building that we will examine is Alberti’s San Novella Cathedral (Figure

2.). The design of this building employs simple Pythagorean ratios that echo the

musical interval of the unison. In the façade of the San Novella cathedral, Alberti uses

simple symmetry. The length and height of the building are both 48 units long.

Exactly in the middle of the length is an arched doorway which partitions the building

in half, or shows a 1:1 ratio in the length. Similarly the height is partitioned halfway

between floors. Further 1:1 subdivisions occur. Lengthwise, the second story

outermost column divides the bottom floor in two 12 unit halves. Height-wise the

second story buttresses end after 12 units, therefore a 1:1 subdivision. It seems

obvious that one side of a building should mirror the other, but this is merely a jumping

off point for the more complex relationships that expand upon Pythagorean

consonance.

SECTION 4. Painting

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Painters during the 15th century will be considered in the same way as the

architects. There will be a brief overview of artists and their concentrations and one

example of proportional analysis. Fifteenth century artists received their training at

painter’s guilds where master painters were the instructors to the next generation. It is

well documented who studied under whom, and where these painter’s guilds were

located. This wealth of knowledge allows today’s scholars to draw distinct connections

between painters. The painters that this study concerns are: Leonardo da Vinci, Albrecht

Durer, Raphael, Luca Pacioli and Piero della Francesca.

One well documented area of a painter’s education was that of learning to

accurately depict the human body. A key aspect of guild training involved developing a

working understanding of the human body’s proportioning. The study of human

proportioning dates back to Vitruvius and was studied by all the master painters

throughout the 15th century and onward. Vitruvius’s body proportioning system was

based on the ratio 1:10 where the length of the face represented 1/10 of the entire body.

From this logic, other parts of the body could be derived form the length of the face. For

example, the length of the torso was three face lengths. In fact, most of the

measurements were derived from divisions of three which falls in line with the important

religious connotation of the trinity. Furthermore Vitruvius famously described the

proportioning of a man whose arms and legs, when extended, form a circle with a

circumference defined by the fingertips and the soles of the feet. The height and

wingspan were said to be equal as well, this forming a square. This description was

recreated by Leonardo da Vinci in the famous picture of the Vitruvian man as seen below

(Firgure 3.) (Padovan 167).

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(Figure 3)

Some artists such as Alberti and Albrecht Durer continued onward trying to find

consistently equal measurements of the human body. In fact there are numerous

Vitruvian depictions besides Leonardo’s including those of Cesariano, Francesco di

Giorgio and Fra Giocondo. Durer for example arrived at different body types which each

had a different set of proportions. Leonardo da Vinci although creating the most

recognizable Vitruvian man, became more concerned with aesthetic proportioning instead

of Durer’s more scientific approach. (S. Braunfels-Esche).

Luca Pacioli was a 15th century mathematician, scientist, and Franciscan monk.

Although he was not a painter or architect he is valuable to this research to show how

interconnected all of these painters, architects, and musicians were. In 1471, Luca

Pacioli lived with Alberti and from there he came into contact with Piero della Francesca.

We know this because Pacioli is depicted in Francesca’s S Bernardino altarpiece. From

1496-1499 Luca Pacioli collaborated with Leonardo da Vinci on his treatise “De Devina

Proportion”, a treatise entirely focused on the golden section. Leonardo did the

illustrations (Peter Boutourline Young).

On last example that shows the importance of proportioning in painting is found

in one of Rapheal’s most famous works, The School of Athens. In this work, Rapheal

depicts a large collection of the most important thinkers with Plato and Aristotle at the

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center. In this picture, there is drawing of the Pythagorean musical scale represented on a

table. The existence of this table in this picture suggests the continued importance of

importance of musical consonances in other forms of composition. Also, the fact that

Rapheal was a painter in the late renaissance allows for further assumptions that these

considerations remained important throughout the 15th century (Wittkower 44b).

The artist Piero della Francesca’s was certainly a notable painter but he was also a

mathematician who wrote numerous treatises on proportions and perspective including

De prospectiva pingendi (‘On perspective for painting’), Trattato d’abaco (‘Abacus

treatise’) and De quinque corporibus regularibus (‘On the five regular bodies’). Pierro’s

work as a painter and his work as a theorist are inseparable (Frank Dabell). Unlike

Brunelleschi’s, Piero’s work on ratios and perspective were not centered on using the

ratios in Pythagorean musical consonances. Instead, Piero was more concerned with a

rule of perspective where the relationships of equally sized objects at different distances

from the eye form a harmonic progression. In this type of progression the diminutive

objects are spaced in a progression where the series of the numerator if constant and the

denominator form an arithmetical progression: A/B, A(A+B) , A/(A+2B),

A/(A+3B)…(Padovan 216).

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(Figure 4.)

Piero’s painting “The Baptism of Christ” (Figure 4.) is structured in this way with

a rectangular bottom and half circle coming out of the top. If the half circle is continued

and another circle drawn to overlap the painting seems to be showing the Vesca Pices.

The Vesca Pices, visually depicted as two overlapping circles, functions symbolically to

represent the intersection of the heavenly world of our human world by the intersection of

two circles. In this painting the exact intersection of these two circles is Christ in the

process of baptism. With regard to proportions, this painting is built upon simple ratios

based on the threes. In the picture a tree trunk divides the picture 2/3 lengthwise.

Vertically a dove signals the end of the rectangle bottom which comes at a 2/3 division.

This unequal ratio approximates the 1.617 golden section.

SECTION 5. MUSICAL ANALYSIS

Albeit brief, the examination of architecture and painting in the 15th C is useful

for establishing the artistic climate. We will now move on to the musical analysis after

an overview of the methodology.

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All of the subsequent musical examples are taken from the liturgy. The

composers who wrote for the church were the most important and most educated

musicians of their time, thus giving this research the highest caliber example. Also while

studying the origin of these proportions it became clear that their use had religious

implications. Furthermore, the musical examples have been taken from the beginning,

middle, and end of the 15th C. This allows for a representative look at how the

importance of these proportions carried through the century. Finally, with regard to

choosing the repertoire, two voice motets provide the clearest view of a piece’s structure.

With two voices, musical points of punctuations become very visible and easier to

discern. With more than two voices, divisions in form become harder to see. The

methods for finding points of articulation in 15th century music is very similar to those

used in formal study for any type of music. The tools: cadences, motions to cadences,

changing note values (longer to shorter or vice versa), and contour determine where

primary and secondary cadences exist.

Finally, there are two aspects to this music which I do not take into account

during my analysis, first of which is the text. With the exception of the Lasso motet, all

of the subsequent examples are taken from the mass ordinary. In practice, composers of

masses would write the music with no instruction as to how the texts should be set to the

music. Therefore publishers of this music have taken the liberty of apply the text to their

best interpretation of the music. Unfortunately this does not make the beginning/ending

of lyric a valid point of articulation. Again, the Lasso motet is different from the others

because the composer himself wrote the music with the text. Similarly, performance

practice dictates that we should not include the final cadence in a piece or mass section in

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the calculations for this study. In practice, the final sonority, called the longus, of a piece

or section was an indefinite time marker. A conductor of a 15th choir would hold the last

note to his liking. Therefore it cannot be taken as a credible note value to factor into the

study.

Ockeghem, Christe, Missa Prolationum

Johannes Ockeghem was the most important composer of the Franco-Flemish

school in the 15th C and arguably his most important composition is his Missa

Prolationum (Prolation mass). This mass typifies a musical composition whose structure

suggests a link with the type of celestial mathematics that pervades 15th C composition.

The Prolation mass consists entirely of mensuration canons, or prolations, which employ

a single melody to be imitated by one or more voices after a duration of time. In

Ockeghem’s original mass only two voices are notated and the other two voices must be

derived from the notated. When the two notated voices perform with the other two

derived voices, a double prolation canon ensues. Ockeghem places the upper pair of

original and derived voices in the prolatio major while the lower voices in prolatio minor

and one voice in each of the pair sings in perfect time while the other sings in imperfect

time. In Missa Prolationum the interval of imitation increases throughout the mass from

the unison up to the octave, using all of the traditionally “imperfect” intervals as well.

Two-hundred years later Bach would do the same thing in his Goldberg Variations. This

intricate mass is one of Ockeghem’s most intricate pieces and in turn one of the most

complex Renaissance compositions. Beside the intricate prolation canon, the actual

structure of the mass references divine mathematics.

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In the Kyrie section of Missa Prolationum, the Christe section divides the

movement. In this section only one voice from each voice pair sings and the intervallic

imitation occurs from one phrase to another instead of from one voice to another. This

drastic composed retard, or slowing down, to the form could signify the impact of Christ

on the Christian religion. In the Christe section, golden ratios or the 3:2 approximation of

1.168 are found when counting and comparing the number of measures or tactus. Golden

sections are found on a large scale analysis of the entire Kyrie movement and these same

proportions fill in lower tiers of structure. In order to isolate these relationships I will

start with numerical relationships that arise on the level of the entre Kyrie movement and

focus the examination from there.

The most obvious 2:3 relationship exists in the masse’s prolation canon. In the

top pair of voices, the derived voices imitate the original voice by one dotted whole note

to the whole. This translates into the top and lower voices relating to one another with

respect to duration by 2/3. This is also seen in the lower two voices only the direct

relationship is that of 6/9 from the top to the bottom voice.

The whole Kyrie movement is 534 tactus long. As mentioned in the methodology

the final sonorities in each phrase were not included because those sonorities had no fixed

duration in practice. The first Kyrie section, Kyrie I, is 100 tactus long, the Christe, 320,

and Kyrie II is 114. The combined Kyrie sections form approximately .40 of the entire

movement and the Christe fills in the remaining .60.

The argument for the importance of this ratio in the compositional process is

legitimized when one examines the smaller segments of the piece and finds these

divisions. The Christe section provides this proof on many levels. The segmentation of

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the Christe section is as follows: two phrases (P1 and P2) that are repeated up a major

second (P1’ and P2’), the third phrase (P3) is repeated in a similar fashion up a major

second (P3’). The bipartite division of the Christe section occurs between the P1’ and

P2’ and the first P3. On one side of the bipartite division, the original and repeated P1

and P2 total 200 tactus, and on the other side of the division the P3 and P3’ equal 120

tactus. Against the entire Christe section the bipartide division comes at a .62 division

which approximates the 2:3 relationship that is seen throughout the movement, and is the

precise golden section number. One further 3:2 relationship is seen in the relationship

from P1 to P2 (and again in P1’ and P2’). The P1 phrase has 60 tactus and the P2 phrase

has 40 tactus.

Figure 5.

The 3:2 relationship is therefore seen on four levels in this movement, in the

prolation, in the Kyrie I-Christe-Kyrie II division, in the Christe sections bipartide

division, and in the phrase segmentation in the Christe section. Since the ratio 3:2

approximates the golden section, the Kyrie’s unity with regard to this proportion suggests

FLURA #2010 - 13 21

that the entire mass was composed with this in mind and provides a strong case for the

use of proportions in 15th C music as a compositional device.

Josquin-Benedictus- Missa L’Homme Arme’ Sexti Toni

Josquin’s Benedictus from his Missa L’Homme Arme’ Sexti Toni is a three

phrase mass segment where the first and last phrases are identical. Each phrase is

performed by two voices, the first and third phrases by the Superius and Tenor voices and

the middle paring is the Altus and Bassus. In the first phrase, the both voice’s music can

be characterized by melodic arches. The first melodic arch is in the Superius and is

followed immediately by a second arch which rises and falls with melisma

(ornamentation). The tenor, performing in canon, enters after six breves and performs the

Superius’s first melodic arch. At the point of the tenor’s entrance, a rest in the music

signifies the end of the Superius’s first rise and fall. The tenor performs in canon to the

Superius’s first five bars by playing the first arch.

Again, without counting the longus at the end of the measure, the bipartite

division of this phrase occurs with the Tenor’s entrance at the seventh breve. This point

of articulation can be justified with the presence of rest in the Superius and the entrance

of another voice. It is the end of the first arch. With six breves forming the first part of

the phrase the latter is 10 breves. Justification for this grouping is seen with the entrance

of the tenor, and the falling phrase which is immediately spun out into melisma.

Therefore the bipartite division shows a primary 5:3 ratio (10 breves and 6 breves

respectively) in the superius and a 3:5 ratio in the tenor. Secondary divisions occur in the

opening rise and fall figure in the superius which is divided by a rest into an uneven

proportion of 6:4 breves, or 3:2.

FLURA #2010 - 13 22

The second phrase breaks down into the same proportions as the first. This can

serve as justification for the first phrases proportioning of a primary 3:5 and a secondary

3:2. The Altus and Bassus enter together and perform in counterpoint for 7 and 6 breves

respectively. The presence of a rest after these first six breves suggests a point of

articulation and is indeed the place of this phrase’s bipartite division. The remaining 10

breves form the 3:5 ratio which was seen in the previous phrase. Identical to the first

phrase, the longer division breaks into a 3:2 split. This bipartite division of this 3:2 can

be justified with the movement towards cadence as a point of articulation. Faster note

values and melodic rising shows that this portion of the phrase has developed direction.

Figure 6

Josquin L’homma arme’ sexti toni- Pleni sunt

This portion of the Sanctus consists of two identical phrases performed by the two

voice parings of Superius and Altus followed by Tenor and Bassus. The top voice leads

and the second voice performs in canon after four tactus. The main division of the first

phrase occurs after the first set of counterpoint finishes and if followed by a string of

FLURA #2010 - 13 23

quick imitation which leads to the cadence. This primary division occurs after twenty-

four breves with 12 remaining. Therefore there is a simple 2:1 symmetry. Secondary

subdivisions can be derived, but none are as conclusive as the primary.

Josquin Desprez- Super Voces Benedictus

The Benedictus in this mass consists of three sections. Each sections is performed

by a voice pair: two Bassus’ two Altus’ and two Superius’, respectively. The first section

uses simple 1:1 symmetry to partition its two phrases. Again, not counting the longus at

the ends of phrases the primary division of the first phrase is signaled by chromatic

resolution. The presence of a leading tone G# leading to A in the bottom voice shows

that the bassus melody is ending while on the same beat of resolution the top voice takes

over the melody. There are 16 breves on either side of this musical moment therefore

1:1. A secondary proportion on the left side of the primary division occurs on the eighth

breve where the top voice’s melody is at its highest point and the bottom voice is at its

lowest point.

Identical to the first phrase, in the second phrase the primary division occurs at

the mid point, the twelfth breve. A secondary division occurs at the sixth breve, a further

1:1 division. This secondary point of articulation is justified by the two Altus voices

entering together on the sixth breve. In this section, the two Altus voices only come

together on major arrival points, the bipartite division and the final cadence.

In the third and final phrase of this Benedictus the primary division comes at the

eighteenth breve at which point a phrase is ending. From this division to the longus at the

end there are eighteen breves on either side. A secondary 1:1 symmetry comes seven

beats after the primary division. This arrival point is signified by the bottom voice

FLURA #2010 - 13 24

resting and the top voice starting a new phrase. The symmetry of seven breves on the

other side of this is seen with the chromatic resolution in the bottom voice. There are

four breves after this but it would seem that they are part of the final longus.

The large scale proportion of this mass section is a golden section division. The

entire mass section is 46 breves and the first two Benedictus sections form 28 breves.

Therefore 28:46 comes out to approximately .61.

Figure 7.

Di Lasso Oculus non Vidit

This example is taken from the latter part of the 15th century and it shows how

these forms were utilized on a small scale. These motets were written for personal use as

opposed to the previous examples which were for use in the Mass. Performances of these

motets were probably done by amateurs.

The piece features two voices, Cantus and Altus performing in canon with the

leader of the canon switching constantly. Major cadences come at the eighteenth breve

(signaled by chromatic motion in the Altus and another cadence at the fortieth breve

FLURA #2010 - 13 25

again with chromatic motion in the Altus. The piece in its entirety is sixty breves long

and therefore the bipartite division at the 40:60 point approximates a golden section. The

secondary division to this primary 2:3 division occurs at the end of the eighteenth breve

where the first phrase ends. This point approximates a 1:1 simple symmetry.

Figure 8.

This study raises several questions. With regard to the audience, it is unlikely that

these proportions were used for the sake of perception. However, music cognition

scientists are discovering new methods by which the brain processes sound and music. It

is interesting to think that on some level proportions can be perceived peripherally. Are

these proportions an unconscious determinant as to how and why music is affects us on

an emotional level or how it remains beautiful for performers and audience members

hundreds of years later? These questions will be examined with further study.

Proportions fulfilled many roles in design in the 15th C. It seems their function ranges

from use as a technical tool leading to a well-balanced harmonious design to a more

esoteric function where they were used as cosmological symbol or metaphor for

contemplation of the divine order of creation. By finding evidence that composers were

aware of and actively using proportions we allow for the development of a common

FLURA #2010 - 13 26

design grammar based on use of similar systems of proportion. In turn this may lead to

the performance of these pieces with a new dimension of precision.

FLURA #2010 - 13 27

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