Not everyone is well-versed with numbers, and there are definitely billions of people who wish that Mathematics be erased from the general curriculum in any academic level in the industry of education. What people fail to realize is that Mathematics is a very significant field of study that has provided a very important contribution to world civilization and even human development. It may be hard to think that Mathematics is needed in everyday life, but in reality, order in this world would be almost impossible without numbers. It has to be established in order for the world and its inhabitants, to be aware that almost everything that is seen around us is connected to Mathematics. Astronomy and science will not be well understood, and will never be explained, to begin with, without numbers. Arts and music have developed through time with the help of Mathematics. Art and music is appreciated the way people do, because Mathematics provided the tools to better understand the art behind Arts and Music. All these, no matter how many twists and turns there could be, are all closely related to each other.
Paintings of flowers, animals and human beings must first be examined. Simple painting of a woman standing in the middle of an empty street is highly influenced by geometry. During the primitive times, the first human beings used to paint on the walls of their caves and their habitats in general. Geometry existed even before humans started to paint, and even before they recognized it (Wilson, 2003, p.333).
The question then, is in what way is geometry and painting related together? The physical form of whatever a human’s eyes can see is symmetrical in shape. The shape of a human’s face, or an animal’s form, is symmetrical. There are symmetrical proportions that define the form of a work of art. Artists like Albrecht Dürer and Leonardo da Vinci recognize this very well, because these artists are highly concerned with proportions and symmetry especially when they create artworks with human forms as their subjects. The recognition of artists to Mathematics is evident because of the development of art and Mathematics itself through time. Geometry as a part of art was developed and was carried to the Renaissance period as evidenced by the works of Michelangeki, Alberti and Brunelleschi (Wilson, 2003, p.333).
The Proponents of Geometry Who Influenced Art
Euclid, the father of Geometry and a Greek mathematician, came up with the concept of rigor, number theory, spherical geometry, conic sections and perspectives. He has published several works that talked about these discoveries. Renaissance artists used Euclid’s geometry of perspective to come up with detailed analysis on art works. Rene Descartes is another mathematician and philosopher who became the proponent of coordinate systems. He was one of the main personalities behind analytical geometry (Wilson, 2003, p.334).
These mathematicians contributed in the realization of designs and repeating patterns. Proof of this is the various properties of Islamic art which relies highly on different shapes like octagon, hexagon, squares, triangles, and other geometrical shapes. These shapes are measured and defined within the context of coordinates by Descartes (Wilson, 2003, p.334).
Mathematics Vis-à-vis Art and Music
Music is also a form of art. That being said, it is logical to conclude that even Mathematics made music possible, especially when it comes to the type of music people create and hear these days. The different elements that make music possible are derived mainly from the properties of Mathematics and it cannot be denied that this fact is being taken for granted. Mathematics and music are closely connected to each other and this is evidenced by the compositions of Johann Sebastian Bach. He has published two books namely Fugues and 24 Preludes. His music are all defined by the specific type of scale called the equal-temperament scale. Other musicians these days also refer to this scale as the well-tempered Clavier (Assayag, et.al., 2002, p.49).
Pythagoras, a Greek mathematician who was also the proponent of Pythagorean Theorem, created significant works the helped musicians make their music better. Pythagoras influenced music through his discovery of the vibration of strings and of musical scale. A perfect example of the connection between musical scales and Mathematics is see through Mersenne, a French Mathematician, from whom equal-temperament scale originated. Musicians themselves admit that they owe a lot to Mathematics. Even mathematicians wish they were musicians, too. An artist named Paul Klee was able to come up with excellent compositions with his violin because of Mathematics (Assayag, et.al., 2002, p.49).
Music Meets Mathematics
Some combinations of notes are pleasant to listen to, while some are not. This remains a mystery to a lot of people, and this is exactly where mathematics comes in. To begin, it has to be understood that a musical instrument is created to make different wave patterns. A guitarist plays the guitar by plucking a string of a guitar. By plucking, a vibration happens back and forth. Through waves, mechanical energy is triggered to travel, which is exactly the reason why there are wave patterns are produced as guitar strings are plucked (Heimiller, 2002). As these waves hit the human ear, a measurement is made. The number of times the waves hit the ear is called frequency, and there unit of measure is called Hertz. Now, as the waves per second increase, the pitch increases, too (Heimiller, 2002). To better understand the relationship between math and music, here is an illustration:
Figure 1 (Heimiller, 2002)
Figure 1 will help us understand that some combinations of note will always sound better than other combinations because of these constant wave patterns. The wave patterns shown in Figure 1 represent two notes. The two notes used in the figure are G and C. Figure 1 represents a combination of two notes that sound good together. Now, the next figure shows the combination of two notes that do not sound good together (Heimiller, 2002).
Figure 2 (Heimiller, 2002)
The intervals in this figure are not regular and do not follow the suggestion of order by Mathematics. Even chords need to follow the very basic foundations and concepts of Mathematics to come up with a sound that is pleasant to the ears. An example of the frequencies that make notes sound good when combined with others are as follows:
G – 392 Hertz
E – 329.6 Hertz
C – 261.6 Hertz
This example shows the ratio between E and C, which suggests that there is approximately 5/4ths of ratio. This means, too, that the fifth wave of G will sound good when combined with the fourth wave of the chord C. The same is true with the ratio of G and E, where G sounds good with the second wave of C. The frequency of each note combines well with each other, if and only if, a regular interval is used as pattern (Heimiller, 2002).
The secret then, to making art and music pleasant to us, is to follow properties of Mathematics that we can never underestimate.
Assayag, G., H G. Feichtinger, J. Rodrigues and the European Mathematical Society.
Mathematics and Music: A Diderot Forum. Springer.
Heimiller J. (2002). Where Math Meets Music. Music MasterWorks. Retrieved February 3, 2009 from http://www.musicmasterworks.com/WhereMathMeetsMusic.html.
Novelli, J. (1998). Using Caldecotts Across the Curriculum: Reading and Writing Mini-Lessons, Math and Science Spin-Offs, Unique Art Activities, and More!. Scholastic, Inc.
Wilson, S. (2003). Information Arts: Intersections of Art, Science, and Technology. MIT