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THE MATHEMATICS OF

MUSICAL RATIOS IN

RENAISSANCE ARCHITECTURE

GROUP FIVE:

Addie Davis, Haley Brown, & Liana Garcia-Osborne

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ABOUT THE AUTHORS

HALEY BROWN is a sophomore English major with a Still Photography minor from

a small town in southern Connecticut. When she is not reading 19th century literature

or taking photographs of obscure objects, she enjoys

knitting, listening to music, and eating. Her other dream

college major would be Culinary Arts, where she would

open a sustainable organic bookstore/restaurant on the

coast. She is also a member of the IC Equestrian team

(yes, we have an equestrian team) and spends copious

amounts of time playing in the dirt with her horses. She

loves to travel, and has worn the same pair of Adidas

Sambas in four different countries. This picture was taken

in Hazard, KY after taking 50 high-school students on a 4am trip to Huddle House for

breakfast. The meal? Chicken-fried steaks.

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ADDIE DAVIS is a sophomore English major with minors in Honors and Art

History. She grew up in Guilford, Connecticut, a small town on Long Island known

for its historic town center and its big apple farm. (Yes, thats right: its big apple

farm.) When at school, Addie is an active participant in student theatre, including IC

Players, IC Underground, and the Acahti Longform Improv Troupe. She also works

as a tutor at the Writing Center. At home, Addie teaches sailing through the local

YMCA. She has never been and never will be any good at writing biographies.

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LIANA GARCIA-OSBORNE is a Junior English major at Ithaca College. Raised in a

small town outside of Philadelphia, Pennsylvania, she spent all of her free time on her

friends horse farm, riding, working and training horses. Her summers were mostly

spent attending a program in Northern Ontario to participate in whitewater canoe

trips. The farm girl life style carried over into her college life, where she is a now

member of the Ithaca College Equestrian Team (Yes. IC does have an equestrian

team). Singing and acting have been called important to her and listed along with

other passions, but competitive riding soon became more important than the theatre,

and now there is little room for anything else in her life. She plans to graduate early

from Ithaca College in order to put her education to use sooner rather than later, as

she has been known to move through certain walk of life with leisure.

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ABSTRACT

Our paper will discuss the historical significance of Pythagoras and Platos

musical ratio systems in architecture. Once a foundation in the relationship between

the Pythagorean system & music theory has been established, we will discuss how

that applied to Renaissance architects, who incorporated musical ratios into their

designs so that the building we are in partakes of the vital force which lies behind all

matter and binds the universe together (Wittkower 27). As an ultimate incorporation

of these concepts, we will explore the potential for using Geometers Sketchpad to

create a design incorporating these ratios. Using the Santa Maria Novella of

Florence, Italy as inspiration, we use GSP to design a architectural faade structured

by musical ratios.

Addie will discuss the mathematics behind finding musical ratios, and discuss

the processes of both Pythagoras and Plato. A historical overview of Renaissance

architects Leon Battista Alberti and Filippo Brunaleschi, who both incorporated

musical ratios into their designs, will be provided by Haley. Liana will explore the

application of these concepts by mimicking the faade of the Santa Maria Novella,

using GSP. A further exploration of the potential incorporation of musical ratios

within that design will be provided by Liana and Addie.

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INTRODUCTION The first mention of music and architecture appears in Greek mythology with

the story of Amphion:

Amphion, son of Zuess, king of the gods, and Antiope, the queen of Thebes,

was raised by shepherds with his brother, Zethus. The gods gave him a

lyre and he became a prodigy; his brother became a master huntsman.

Once they grew up, the took over Thebes. Triumphan, Amphion played his

lute; his music made the stones arrange themselves into the outer

defensive walls of the city.

Figure 1: Jansz, A. (1665). Plate 43: Amphion building the walls of Thebes with his music [Illustration]. Retrieved from ARTstor database.

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Since the beginning, architects have searched for a system of ratios that would

simplify the creation of beautiful structures. In particular, Renaissance architects, like

other artists of the period, were obsessed with the idea of finding ideal beauty through

the manipulation of mathematics. This quest for an indisputable, universal systematic

approach to beauty led to the belief that a rule that applied to the aesthetics of one

sense would apply to any other. Francesco Giorgi, the Renaissance philosopher,

wrote in 1525 that The proportions of the voices are harmonies for the ears; those of

measurements are harmonies for the eyes (2, pg. __). Using the mathematics of

musical intervals defined by the ancient Greeks, Renaissance architects established a

universal system: buildings designs were structured in such a way that, were their

dimensions translated to sound, the resultant interval would be harmonious and

pleasing to the ear.

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THE MATH

PYTHAGORAS MUSICAL RATIOS

Greek mathematician Pythagoras (570-490 BCE) [fig. 2] was the first to

define the musical ratios that Renaissance architects would later find so invaluable

(1). Legend says that his interest was

piqued when walking by a

blacksmiths shop; he stopped when

he heard the that pounding of the

anvils was harmonious. Upon further

investigation, he realized that anvils

with sizes of certain proportions made

more harmonious noises when struck

at the same time.

Pythagorean ratios are derived

by comparing sizes; it is perhaps best

understood in terms of two vibrating strings. Two vibrating strings of the same size

will produce the same pitch (or a unison); what, then, will happen if one of the

strings lengths is reduced or increased? Rudolf Wittkower, chairman of the Art

History department at Columbia University from 1956 to 1969, describes the

phenomena in his book, Architectural Principles in the Age of Humanism:

Figure 2: Portrait bust of Pythagoras of Samos [Bust sculpture]. (n.d.). Retrieved from ARTstor database.

ARTstor database.

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If two strings are made to vibrate under the same con-

ditions, one being half the length of the other, the pitch of

the shorter string will be one octave (diapson) above that of

the larger one. If the lengths of the strings are in the

relation of two to three, the difference in pitch will be a

fifth (diapente), and if they are in the relation of three to

four, the difference in pitch will be a fourth (diatessaron).

(2, pg. __).

In other words, it is the ratio of one strings length to the others that defines the

resulting interval of both strings played simultaneously. Pythagoras experimented

with different ratios until he uncovered the following relationships (3, pg. __):

Musical Interval

Numerical Ratio

Unison 1:1

Major Second 9:8

Major Third 5:4

Major Fourth 4:3

Major Fifth 3:2

Major Sixth 5:3

Major Seventh 15:8

Perfect Octave 2:1

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These ratios are confirmed through analysis of the frequencies of pitches within a

musical scale. Take, for example, the key of C Major. Middle C, the root of the

scale, has a frequency of 264 hertz. The C pitch one octave above middle C has a

frequency of 528 hertz; their ratio, then, is 2:1, as predicted by the Pythagorean

system. The same rule applies to other intervals, as well. For example, the G above

middle Cwith the interval of a perfect fifthhums at 396 hertz; the resulting ratio

is 396:264, or 3:2, as predicted using strings of varying lengths. A chart of the entire

scales ratios confirms an exact correlation between frequency and string length:

Note Above

Middle C Frequency

(Hertz) Interval Ratio to Middle C

Simplified Ratio

D 297 Major Second 297:264 9:8

E 330 Major Third 330:264 5:4

F 330 Perfect Fourth 352:264 4:3

G 356 Perfect Fifth 396:264 3:2

A 440 Major Sixth 440:264 5:3

B 495 Major Seventh 495:264 15:8

C 528 Perfect Octave 528:264 2:1

PLATOS LAMDA

Plato (429347 B.C.E.) based his manipulations of musical ratios on the belief

that the creator made the world soul out of ingredients, and formed it into a long

strip (2, pg. 103). The strip was then marked into intervals that corresponded to the

consonances of intervals that the ancient Greeks recognized: the progression 1:2:3:4,

or octave (1:2), fifth (2:3), and fourth (3:4) (2, 103). From that simple progression

came Platos lamda, named for the greek letter it resembles:

1

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2 3 4 9

8 27 The creation of the lamda starts with placing the progression at the top of the eventual

lamda shape, like so:

1 2 3

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Then, we follow a pattern that the numbers have already established for us: the third

row, once completed, becomes the squares of the numbers above. The third row

becomes the cubes. The final shape, then, looks like this:

(monad) 1

(first even & odd number) 2 3

(squares) 4 9

(cubes) 8 27

Some of the more prominent ratios are already visible within the lamda: the

unison, octave, fifth, fourth, and major second are obvious. The rest, Plato filled in

through a manipulation of both harmonic and arithmetic means (3, pg. 22).

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HISTORY OF MUSICAL RATIOS IN ARCHITECTURE:

While Plato, Pythagoras, and even Leonardo da

Vinci were able to conceive of the mathematics behind

musical ratios, one of the first people to use these ratios in

architecture was Filippo Brunelleschi. Born in 1377 to

lawyer Brunellesco di Lippo and Giuliana Spini,

Brunelleschi got his education at the Silkmakers Guild

where he learned to work with bronze, gold, and silver. He achieved the honor of

master goldsmith in 1398, after which he submitted a set of bronze doors as part of a

contest to be used in the Baptistery in Florence. Ultimately, he ended up losing the

contest to one Lorenzo Ghiberti, the man who ended up as his mentor and partner for

future projects. Brunelleschis first commissioned work was the Ospedale Degli

Innocenti (Foundling Hospital) in Florence, which was built from 1419 through 1445

(2, p. 1). This buildings architecture showed great reference to classical antiquity in

style, exemplifying Brunelleschis classic architectural methods that continue to be

admired today. Brunelleschis other works include the Sagrestia Vecchia, also in

Florence, as well as the famous Cathedral of Florence Santa Maria del Fiore that

sports el Duomo (2, p. 1), one of the most famous destinations in Italy.

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Santa Maria del Fiore, better known as the Cathedral of Florence, is one of

Italys largest churches and thus is one of the biggest tourist attractions in Florence.

Its dome, built by Brunelleschi, can be seen for miles around Florence and has

become a key element in the

Florentine skyline. The dome has

been described as a miracle of

design and engineering (1, p. 1)

and was built using innovative

design techniques that did not

require the use of scaffolding to

hold up the sienna-colored bricks

(1, p. 1). It remains, to this day, to be the tallest building in Florence, a marvel that

Leon Battista Alberti, another Renaissance architect who took his inspiration from

Brunelleschis work, thought was vast enough to cover the entire Tuscan population

with its shadow (1, p. 1).

Not only are architects inspired by the work of musicians in designing their

pieces, but also musicians frequently are inspired by

the same musical ratios that are used in architecture.

Guillaume Dufay, a Belgian Renaissance composer,

wrote his motet Nuper rosarum flores to echo

Brunelleschis architecture. The same architectural

elements that Brunelleschi utilized in creating his

hallmark Renaissance buildings are seen in this

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motet, including the homogeny of wall, space, light, and articulation, all of which can

be attributed to certain ratios. In this piece, Dufay also used what is called the

proporzioni musicali, a ratio that is seen throughout Brunelleschis work that is

achieved by taking the square root of an octave ratio 2:1 and multiplying it by the

sides of the number of square utilized and then rounding to the nearest whole number.

The proporzioni musicali always ended up as 6:4:2:3:1, a proportion that Dufay

mimicked all through his motet. Nuper rosarum flores was not the only piece where

Dufay used the proporzioni musical, nor the only piece that took inspiration from

Brunelleschis work. He also composed a piece called the Magnanimal gentes where

the canon doubles the length of the first section, changing the proporzioni musicali to

12:4:2:3:1. Numbers were incorporated into Dufays pieces in many ways, but one

unique way he bridged the gap between mathematics and art is in his favor of the

number seven, especially in sacred pieces. The number seven is a holy or special

number, and so in using this number he reiterates the sacred motif of his particular

pieces. Dufay also believed in the musical space (9) of a building, asserting that the

way a building is shaped should be taken into consideration when composing a piece

of music that was to be performed in that particular building. For example, in a

church that contained extremities like a cross, the composition of the vocal sections

should be done in a way that allows the sounds to travel throughout the extremities,

making the most out of the building itself and utilizing the space to its highest ability.

Another architect fascinated by musical ratios, and perhaps

the most well known Renaissance architect who did so, was Leon

Battista Alberti. Alberti was born February 18, 1404 in Genoa, Italy.

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From an early age, Alberti was involved in political activity when his family was

exiled from their native city because of their political loyalties the Republican

government run by the Albizzi family did not tolerate any family of a different

political standing and so right away, Alberti was thrown into the world of the

intellectual and the political. He received his education at the University of Bologna

in law, and quickly became not only a well-known lawyer but also an author and an

abbreviator at the Papal Curia. His most famous written works include On the

Advantages and Disadvantages of Letters, a Latin comedy entitled Phiolodoxus, and

De Pictura (Della Pictura), which was published in 1435 and was his most famous

literary work. While drafting papal briefs in the Papal Curia, he joined the priesthood

and quickly became the papal inspector of monuments in Rome and supervised

numerous building projects throughout the city. From here, his interest in architecture

emerged. Some of his better-known buildings include churches like San Sebastiano

(1460), San Andrea (1470), and the faade of the Palazzo Rucellai (1446), but his

most famous works of art include the Tempio Malatestiano in Rimini, Italy (1447)

and the Santa Maria Novella in Florence (1448) (4, p. 1).

Albertis Santa Maria Novella is

considered to be a landmark of Renaissance

architecture (4, p. 1 ) and was built from 1448

through 1470. Its faade includes a 2:1 octave

ratio, and its

mosaic effect

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gives it a sense of rhythmic, geometric unity(3, p. 112). The faade also is

circumscribed by the form of the perfect square, which was Albertis preferred form

(3, p. 112) and is seen on many of his other works including the Tempio

Malatestiano. The Tempio Malatestiano, also known as San Francesco was built in

1447 and was Albertis first church commission and, like Dufay, used the sacred

number seven in its composition. There are seven chapels in the church, all with the

tombs of famous Riminese citizens (4, p. 1). The church was also composed of whole

number multiples in 50, 40, 36, 30, and 18 Roman feet, and these measurements are

evident in the composition of the outside faade of the structure (6, p. 75).

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PRACTICAL APPLICATION USING RENAISSANCE ARCHITECTURE AS INSPIRATION

Within the Santa Maria Novella church of Florence, there are numerous

musical ratios that can be located in the structure. These ratios are, as Alberti

suggests, both musically and visually harmonious and therefore pleasing to look

upon. The faade of the Santa Maria Novella is one such part of the structure where

these musical ratios can be seen and easily identified.

The primary ratio found in the faade of the church is the octave, or 2:1

ratio. In other words, the bottom, or ground layer of the church has two perfect

squares that act as exact reflections of each other, while the second story is the same

square duplicated, but just once. The first story represents 2, the second 1, with the

combined effect of the octave ratio, or 2:1. The octave ratio applies within the first

level, as well; the base:height ratio is that of 2:1. Other musical ratios are present in

the structure of the church as well, such as the ratio of 2:3. The central bay of the

upper level of the church is a square of units. Similarly, two squares measuring

units enclose the pediment portion of the faade. The front bay entrance to the church

contains the ratio of a (or 2/4) unit to a unit, together creating a ratio of 2:3.

These ratios give the church a beautiful, holistic appearance, in complete

agreement with the architectural Renaissance perspective of harmonious musical

congruities and their connection to mathematics and therefore architecture.

Directions to create a faade using GSP: mimicking the Santa Maria Novella

faade

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1. Create a perfect square and grid to divide the square into three parts,

utilizing the mid-point application in the GSP program

2. Create the lower door frame, again by creating a mid-point and extending

a vertical line up to meet the horizontal line, and then by creating a second

mid-point, close the door frame by extending a vertical line upwards to

meet the horizontal line.

3. Repeat this process to create the larger, central left-hand side of the door.

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4. Then highlight the entire structure and reflect it, thereby creating the entire

lower level of the facade

5. Next, add decorative circles to give an arch effect to the doors

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6. The last step is to create the final top level of the faade. Extend a line

through the top of the square to mark the top of the triangular cap (hide

when finished). Add window lines and decorative arch circles.

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POTENTIAL FOR FURTHER INCORPORATION OF MUSICAL RATIOS The faade structure provides an infinite amount of possibilities for the

incorporation of musical ratios within the design. Consider the possibility of altering

the height:width ratio of the doors, window panes, and even decorative accents on the

sides. Every dimension can be manipulated to create a more fully integrated system

of ratios.

Also, consider the potential for a musical congruity; manipulating ratios in a

certain way could produce a similar effect to that of Dufays motet, but in reverse. In

other words, rather than translating the architecture into music, translate the music

into architecture. For example, starting with a unison ratio, an incorporation of a

major third ratio (5:4) [using the unison length as a starting length] combined with the

incorporation of the major sixth ratio (5:3) sets up an entire minor chord. In that way,

music is not only the foundation for the ratios, but is actually written in to the

design. The potential for different chord structures throughout the piece is

obviously not limited to a combination of those ratios; using different starting points

would likewise alter the end chord result.

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CONCLUSION The incorporation of musical ratios served as essential part in the progression

of architecture in the Renaissance period. Though they have since fallen out of

fashion, musical ratios were the foundation of architects work throughout the era.

The search for a universal aestheticone that was appealing to all sensesspeaks to

the idea of a fundamental order in the natural world, the existence of which

Renaissance intellectuals constantly strove to prove. The use of musical ratios within

an architectural context lends itself to the belief in the natural order of things and a

rhythmic, geometric unity throughout the universe. An investigation of these

concepts reveals the potential for a mathematical understanding of every facet of the

physical world: aural and visual are both linked by a mathematical standard.

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Works Cited

1. Huffman, Carl. Pythagoras. The Stanford Encyclopedia of Philosophy.

November, 2009. 2. Wittkower, Rudolf. Architectural Principles in the Age of Humanism. New York:

W.W. Norton, 1971 3. Calter, Paul A. Squaring the Circle: Geometry in Art and Architecture. 3. "Brunelleschi's Dome." Brunelleschi's Dome - BRUNELLESCHI'S DOME. Web.

28 Apr. 2010. . 4. "Filippo Brunelleschi." Wikipedia, the Free Encyclopedia. Web. 26 Apr. 2010.

. 5. Gadol, Joan. "Art, the Mirror of Nature." Leon Battista Alberti: Universal Man of

the Early Renaissance. Chicago: University of Chicago, 1969. 108-17. Print. 6. "Leon Battista Alberti." Wikipedia, the Free Encyclopedia. Web. 25 Apr. 2010.

. 7. Oron, Aryeh. "Guillaume Dufay (Composer) - Short Biography." Bach Cantatas

Website - Home Page. Feb. 2006. Web. 28 Apr. 2010. .

8. Tavernor, Robert. On Alberti and the Art of Building. New Haven: Yale UP, 1998.

44+. Print. 9. Charles W. Warren. Brunelleschis Dome and Dufays Motet. The Musical

Quarterly, Vol. 59, No. 1 (Jan., 1973), pp. 92-105. Oxford University Press. Stable URL: http://www.jstor.org/stable/741461