Art-Appreciation-UNIT-7-Lesson-13-and-141.pptx

UNIT VII
Lesson 13: Mathematical
Structure of Music
(Formalism in Music)

“There is geometry in the humming of
the strings, there is music in the spacing
of the spheres.”
— Pythagoras

04
01
02
At the end of the lesson, students must have:
Explained the interrelationship of Art, Music, and
Mathematics;
03
Objectives
Discussed the meaning and importance of music in one’s life
with focus on WVSU Hymn and March;
Showcased through drills the mathematical structure of
WVSU Hymn and March as a musical piece;
Memorized, performed, and recorded a Video Performance of
University Hymn and March.

Things to Ponder:
 Mathematics and music have traditionally
been intricately connected. The seventeenth
century has been seen by historians as a
crucial turning point, when music was
changing from science to art, and science was
moving from theoretical to practical. Many
connections between science and music can
be traced for this period. In the nineteenth and
twentieth centuries, the development of
science of music, and of mathematical
approaches to composition further extended
the connections between two fields.

 Musical pieces are read much like you would read
math symbols. The symbols represent some bit of
information about the piece. Musical pieces are
divided into sections called measures or bars. Each
measure embodies an equal amount of time.
Furthermore, each measure is divided into equal
portions called beats. These are all mathematical
divisions of time.

 Fractions are used in music to indicate
lengths of notes. In a musical piece, the
time signature tells the musician information
about the rhythm of the piece. A time
signature is generally written as two
integers, one above the other. The number
on the bottom tells the musician which note
in the piece gets a single beat (count). The
top number tells the musician how many of
this note is in each measure. Numbers can
tell us a lot about musical pieces.

 Notes are classified in terms of numbers as well. Each note has a
different shape to indicate its beat length or time. There are whole
notes (one note per measure), half notes (two notes per measure),
quarter notes (four notes per measure), eighth notes (eight notes
per measure), and sixteenth notes (sixteen notes per measure).
These numbers signify how long the notes last. That is, a whole
note would last through the entire measure whereas a quarter note
would only last 1/4 of the measure and thus there is enough time
for four quarter notes in one measure. This can be expressed
mathematically since 4 × 1/4 = 1.

 It was Pythagoras, the Ancient Greek
Philosopher and mathematician, who realized
that different sounds can be made with different
weights and vibrations. This led to his discovery
that the pitch of a vibrating string is proportional
to and can be controlled by its length. Strings that
are halved in length are one octave higher than
the original. In essence, the shorter the string, the
higher the pitch and the longer the string, the
lower the pitch. He also realized that notes of
certain frequencies sound best with multiple
frequencies of that note.

 The math-music connection shines in the
field of education as well. Research shows
that children who learn their academics
through music and dance retain the
information better than children who learn the
same concepts by verbal instruction.

 Probably the closes connection between music
and math is that they both use patterns. Music
has repeating choruses and sections of songs
and in math patterns are used to explain and
predict the unknown. Mathematics in the study
of patterns, and you can study everything in
music from different mathematical perspectives,
including geometry, number theory,
trigonometry, differential calculus, and signal
processing. Research has even shown that
certain pieces of music end up being more
popular due to their mathematical structure.

 Mathematics and music, subjects that some people perceive as
opposites, are creative and vibrant endeavors concerned with beauty
and elegance. That may be hard to believe about math, but in fact
mathematicians are motivated to search for beautiful results supported
by elegant proofs, and their journey towards this goal frequently
involves a good deal of improvisation.
 A mathematician and Jazz musician, Rob Schneiderman, says: Every
day, musicians and mathematicians are bringing new music and
mathematics into the world. Mathematical research frequently involves
mathematicians working together engaged in thematic development,
dealing with mistakes, taking tangential explorations, exchanging lead
and accompaniment roles in real time, and spontaneously generating
constructive thoughts. All these dynamics occur as well in a small group
jazz performance.

Music and Math: The
Genius of Beethoven

Beethoven, the composer
of some of the most
celebrated music in
history, spent most of his
career going deaf.

Mathematical Ways
of Listening to Music

 Lyric content or dance generally tends to interact strongly
with accompanying musical statements, and when music
is presented with video, the music will likely play a
subservient role (and in such a setting the power of
sound to generate its own images has been
compromised).
 In the case of mathematically oriented music theory, it is
usually tacitly assumed that an awareness of any
“explanatory” mathematical notions will improve the
musical experience.

It is remarkable how in spite of the strong link
between music and its ambient culture of origin,
appreciation of music can bridge wide cultural
gaps.
The effects of imposing conscious listening
techniques often appear in the setting of music
pedagogy.

The Correlation Between
Math and Music

 Many people are proven to excel in both music and
math, suggesting a strong relationship between the
two.
 According to the American Mathematical Society, applications
of math in music include rhythm, intervals, patterns, time
signatures, pitch, and other elements of music composition.
 Concepts in physics, a course that explores both science and
math, delving into waves and sound, such as the harmonic
series － a chronology of the frequency integer multiples of the
fundamental (first harmonic) － show evidence in musical
notes.

Few Ways That Music and Math Work
Together:
Counting: Music is divided into measures,
which are counted in beats. There are a
certain number of beats in each measure.
These beats and measures set the pace and
help musicians to play by themselves and
with each other.

 Fractions: Fractions are all over the place in music. As
mentioned above, music is divided into measures,
which are counted in beats. Each beat has a note (or
rest) and each of those notes/rests has a value (the
length of time the note is held).

 Patterns: Music features all sorts of patterns
“Twinkle Twinkle Little Start”
Twinkle, twinkle little star (A)
How I wonder what you are (B)
Up above the world so high (C)
Like a diamond in the sky (C)
Twinkle, twinkle little star (A)
How I wonder what you are (B)

• Scales: Music is made of repeating patterns
called scales. The scale can start on any
note, but it follows the same pattern each
time.
• Rhythm: Rhythm describes the repeating
pattern of how the music fits into the beats
and measures.

• Visual: There are some visual patterns in
music.

How Composers Use Fibonacci
Numbers & Golden Ratio |
Composing with Fibonacci

The Golden Section, also known as
Golden Ratio or Golden Mean, is the
division that cuts a fixed length in such a
way that the shorter portion bears the
same ratio to the longer portion bears to
the whole length the actual ratio is about
1.618.

UNIT VII
Lesson 14: Art and
Golden Ratio

— Luca Pacioli
“Without mathematics there
is no art.”

04
01
02
At the end of the lesson, students must have:
Discussed the meaning and importance of Golden Ratio in
Art, Beauty, and nature;
03
Objectives
Explained how the Golden Ratio is applied in quantifying body
proportion, art composition, and judgment of what is beautiful.
Expressed personal feelings and judgment about computer mapping,
technology or other body modification technique as form of art and beauty;
Compiled photos of any artwork or kinds of nature which
showcase the golden ratio or proportion.

Things to Ponder:
 The ratio of approximately 1 to 1.618.
 Known by the Greek letter Φ , the number phi
is a mathematical concept that people have
been referencing since the time of the ancient
Greeks; an irrational number.
The golden ratio is:

 The entire basis of the human face. Particularly, the
head forms a Golden rectangle with the eyes at its
midpoint. The mouth and nose are each placed at
Golden sections of the distance between the eyes and
the bottom of the chin. The proportions of the length of
the nose, the position of the eyes and the length of the
chin, all conform to some aspect of the Golden ratio.

 Prevalent in Da Vinci’s artworks such as
The Annunciation, Madonna with Child and
Saints, The Mona Lisa and St. Jerome. He
was famous for using the Golden ratio in his
works. The Mona Lisa, a well-known portrait
of a woman with a coy smile, is embedded
with Golden rectangles. The sketch, An Old
Man, has many rectangles on it.

 Used in painting, to define the horizon, to place points
of interest and to create balance in what would appear
to be a very active scene.
 Used in more elegant ways to create aesthetics and
visual harmony in any branch of the design arts.

Some Applications of Golden Ratio
Girl with a pearl
earring
The Great Wave
by Katsushika
Hokusai

The Relation of Golden Ratio,
Mathematics and Aesthetics

 Mathematics and art are considered as two
distinct disciplines, two disciplines as two
extremes. However, both mathematics and fine
art are the output of human consciousness that
strive to express not only the physical reality but
also the metaphysical one.

 Mathematics and painting are related in several ways.
Mathematical geometry is used to analyze paintings (both
figurative and abstract) and architect in terms of shapes such
as points and lines, circles, triangles, cubes etc.
 Mathematics and visual arts share common aspects in both
form and function. One of such relations is explored as Golden
ratio. Examples of Golden section can be seen throughout
nature such as in shells, plants, flowers and animals,
considered as the strongest and the oldest ties between
mathematics and visual arts.

 The Golden Ratio when used in cubic geometry
is called the Golden section. The Golden
rectangle refers to a rectangle with a short to
long side ratio of 1: 1.618. An interesting aspect
of the Golden rectangle is that if one cuts out a
square starting from one of the short sides of
the Golden rectangle one will have another
Golden rectangle.
 When an isosceles triangle has the ratio of the
perpendicular a to the b in the Golden Ration, it
is called a Golden Triangle. Similarly, there is
the Golden spiral which grows logarithmically.

 The concept of Golden ratio has close connection
to the Fibonacci sequence. Golden ratio ties into
this sequence through the ratio between the
numbers. This can be seen in the following:
2/1=2.0 3/2=1.5
5/3=1.67 8/5=1.6
13/8=1.625 21/13=1.615
34/21=1.619 55/34=1.618
89/55=1.618 …. …. …

 The idea behind the Golden ratio is: if
a line is divided into two parts, the ratio
of longer part and smaller part should
be equal to the ratio of whole length
and longer part. This makes the
Golden ratio as in Fig. 1.

 Equivalently, they are in the golden ratio
if the ratio of the larger one to the smaller
one equals the ratio of the smaller one to
their difference as follows:
 After simple algebraic manipulations, i.e., multiplying
equation (1) by a/b, or equation (2) by (a − b)/b, both of
these equations are equivalent to which is the quadratic
expression, and hence the Golden Ratio is

 It is believed that the Egyptians were well aware of it and used in
building pyramids and later they taught this concept to Greeks. So the
ratio is very connected to the Greeks as it was named after Phidia, the
sculptor of Athena, who made most of the decoration on Parthenon.
 Legend says the Golden ratio was discovered by Pythagoras and that
it was through him that the true knowledge of this ratio began to be
understood. It is believed that the Greek Philosopher Pythagoras
discovered the concept of harmony while listening to the different
sounds produced as the blacksmiths’ hammers hit their anvils.

 Euclid, a Greek mathematician who is known as
the Father of Geometry, was the first person to
write a definition of the Golden ratio: “A straight
line is said to have been cut in extreme and
mean ratio when, as the whole line is to the
greater segment, so is the greater to the less”.
 The “extreme and mean ratio”, was not referred to as “golden” until
the early 16th century AD. Luca Pacioli equated the Golden ratio
with God in his book, La Divine Proportione (The Divine
Proportion). Leonardo Da Vinci contributed several drawings to La
Divine Proportione and made reference to the “section aurea” (Latin
for “golden section”).

Golden Triangle
 The Golden triangle can be characterized as an isosceles triangle ABC with the
property that bisecting the angle C produces a new triangle CXB which is a similar
triangle to the original.

3.2 Golden Rectangle
Golden rectangle can be sketched in the following steps as shown in the Fig. 3:
Step 1: Begin with a square ABCD. Find the midpoint M of AB.
Step 2: Draw a circle with radius MC, centered at M. Produce the side AB through B
until it intersects the circle at the point E.
Step 3: Draw a line EF perpendicular to AE. Produce DC through C up to the point
F. The rectangles AEFD and BEFC are known as Golden Rectangles.

Golden Spiral
Golden Spiral is drawn in the following steps as sketched in Fig. 4:
Step 1: Begin with Golden Rectangle. Construct a square by drawing a
circle whose radius is the height of the Golden Rectangle.
Step 2: Repeat this process inside the smaller Golden rectangles.
Step 3: Inscribe quarter circles in each square to create the spiral.

Golden Pentagram
The regular pentagon consists of a number of wonderful figures, which are widely used in
works of art. The Pentagon and Pentagram have Golden relationships as shown in Fig. 5.
The ratio of the side of a regular pentagon to its diagonal is φ. In the condition, the
pentagram is inscribed within the pentagon, many of the ratios between segments are
also φ. If a pentagon is divided by diagonals from one vertex, the resulting triangles are
known as Golden triangles. The middle triangle is an acute Golden triangle and the other
two are obtuse Golden triangles.

4. Golden Ratio and Nature
Plant and animal worlds hold the abundance of Golden symmetry in their form, internal and
external both. For example, animal horns grow only from one end resulting on the equiangular
spiral. It is proved that among different kinds of spirals showing in horns of rams, goats,
antelopes and other horned animals, the Golden spirals meet most often.
Some of the evidence of pentagonal symmetry
apparent in nature are shown in Fig. 7.

The next noticeable surprise in nature is the screw symmetry in
organism which is formulated as the stringent mathematical laws.
The pattern of phyllotaxis exists in screw arrangement of leaves on
plant stems (branches on trees, petals in racemes and so
on). Arrangement of leaves on the plant stem conforms the
phyllotaxis phenomena, particularly the screw axis of symmetry (Fig.
8).

5. Golden Ratio and Human Body
A human body and all its parts are based on the principle of the
Golden proportion. A harmonic human body is divided by the navel
into the golden section.

6. Golden Ratio and Architecture
The use of Golden ratio is prevalent in ancient and modern architecture.
The creation of the goddess Athens temple, the magnificent Parthenon, is
outcome of joint efforts of architects and sculptors of ancient Greece.
The base angle of the triangle in the Yantra is seen to be around 51ᵒ, the
same value that was attributed to the base of the great Pyramid of Giza.

7. Golden Ratio and Art
Critics have found the wide use of the proportion of the Golden section in the compositional
structures of certain well-known artworks,
Michelangelo (pentagram), Rafael Santi (Golden triangle), Iva Shishkin, Konstantin
Vasil’ev (Golden rectangle). The Golden ratio is prevalent in Da Vinci’s The
Annunciation, Madonna with Child and Saints, The Mona Lisa and St. Jerome. He was famous fo
using the Golden ratio in his works.
Golden Ratio in Art
The sacrament of the last supper

 The golden ratio creates harmony and organic compositions in pieces of art,
describing the perfectly balanced relationship between two proportions.
 The most wide-spread criterion of beauty is one unique mathematical
proportion called the Golden ratio which is termed as Divine proportion or
Golden section or Golden number or Golden mean. Examples of well-known
works, which exhibit this proportion, are Khufu’s Pyramid of Egypt, the
Parthenon in Athens, Greek sculpture, the “Mona Lisa” by Leonardo da Vinci,
paintings by Rafael, Shishkin and the modern Russian artist Konstantin
Vasiljev, music of Beethoven and Mozart, (Thapa, G.B., 2018).
 Truth, Beauty and Goodness are interrelated, an artist searches for the
Truth in the Beauty, and a scientist searches for the Beauty in the Truth.
The flip side of beauty is goodness which tempts human heart to yearn and
explore beauty in creation. Ancient Greeks developed the science of
esthetics as a way to analyze beauty, believing that harmony is its basis,
(Thapa, G.B., 2018).

Body modification often has multiple meanings; its primary aim is
sometimes aesthetic and erotic enhancement. Scars enhance an
individual’s best traits. A man with ‘good legs’ (full calves and prominent
heels) will have scars applied to draw attention to them. Often rubbed with
oil and red dye until they ‘glow’, scars have a strongly erotic attraction.
In the West, body modification and decoration are often viewed as
primarily a female and feminized activity. However, Non-Western body
modification and theories of the body have been in flux due to
colonialism, religious conversions, and other forces of
globalization. Beauty is equally a male concern.

 Advanced mathematical concepts paradoxically lose some relations to
the physical world from which they are derived. The intensely
scrutinized complex numbers are drawn from the real numbers; yet do
not conform to the experience of real numbers to the real world.

Mathematics, like art stands alone as an object of free perception.
Since art is not obliged to make reference to outside world,
mathematics is also motivated by wonder and imagination. “The great
questions of mathematics- the kind that draw people to math in the
first place are called great not because they may lead to applications,
but because they captivate the imagination. They inspire wonder and
delight. One could say they are beautiful”. The closer the dimensions
of the face fit to the Golden ratio, the more beautiful the person is
perceived by others.
Conclusion

● The architects explored connection between geometrical design and
artistic beauty when incorporating the Golden ratio into their
construction.
●
Golden ratio is not just lofty mathematical theory; it shows up all the
time in the real world. Likewise, graphic designer can use Fibonacci
sequence as a general guideline and creative tool to make the design
perfect.
●
People may argue that the Golden ratio probably does not have any
mystical powers of beauty drawn from archetypal fabric of the
transcendental world. But it is more likely that this ubiquitous pattern
has some aesthetically appealing properties suggesting a sense of
natural balance and visual harmony.

CREDITS: This presentation template was created by Slidesgo,
including icons by Flaticon, infographics & images by Freepik
THANKS!

SALVADOR, Arienne Isabel
TANGENTE, Krysmar Joy
TOMAGAN, Cherybel
TUBLE, Dennis II
GROUP
7

Art-Appreciation-UNIT-7-Lesson-13-and-141.pptx

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