20 th CENTURY EXPERIMENTAL MUSIC AND MATHEMATICS
From the beginning of the history
of humanity the relations between
music and mathematics had been
always perplexing and puzzling for
the people. The music described
as the art form and the amalgam
of harmony, destiny, rhythm and
melody. The people used it for
having rest and make their life
more peaceful. Furthermore, in
ancient world the relation between
music and mathematics has been
attracted to the attention, because
since the beginning of humanity mathematics used for finding
patterns and make complex society in their countries. Like
every branch of science mathematics used for finding the logic
behind all the subjects. The musicians comprised thee sounds
that we hear are combination of a pattern with logic behind
them. In early 6 th century, the beauty of music and mathematics
became more striking with the discoveries of Pythagoras Equal
Tempered Scale. As every mathematician who is well- known
with their successes in mathematical discoveries however,
many mathematicians are known with their musical studies.
Additionally, like Ludwig von Beethoven many musicians used
the mathematical patterns and functions in order to make
musical compositions.
Ludwig von Beethoven used the relations between the
mathematics and music in geometrical way in order to
convey the emotions in a certain way. In one of his most
popular composition which is “Moonlight Sonata” he has
uses mathematical patterns by
creating Geometrical sense in
the musical pieces. He explained
mathematics as the music of the
reason. The mathematicians for
instance Pythagoras used the
ratios between an octave and
he created a chart by that way
Pythagorean Equal Tempered
Scale used for proving of
mathematic behind the music.
In Pythagorean group the
students are divided into groups
which are called as inner and outer cycles. The inner group
includes the “Mathematikoi” in this group the people believes
the importance of music and they all studied for mathematics
and the other science subjects. They think that music makes their
life more enjoyable and help them to sleep easily. Furthermore,
Pythagoras continued to develop his studies about music and
found the ratios between an octave. He examined ratios between
two octaves as 1:1, 2:1, 3:2 and 4:3. As musicians described the
sound as the sound produced by the instruments are the basic
sounds and they are accompanied with different overtones. By
the original fundamentals of the overtones their frequencies
differ.. Between the notes of A and D he found the value of 2:1
which is one of the octave values. According to ratios between
the notes B to C he get the value of 12:9 and 4:3 and which made
equivalent, so that he called this ratios as perfect fourth and
perfect fifth. Thus, they produce consonance and dissonance.
He noticed that all the
values are related with
the length of the strings.
In 20th century the
people shows the ratios
with guitar. Nowadays
many musicians use the
chromatic scale which is
a scale that divides the
octaves into semitones
in order to make a
pattern by dividing the tones into 12 semitones. Additionally,
the composers began to use “12 tone matrix” which is a
technique first used by Amold Schoenberg who created musical
compositions which involved chromatic scale. In method
of creating 12 tone matrix using 12 x 12 grid square and by
creating a diagonal with “0” from right to left or left to right. By
giving numbers to notes and converting the tables into notes
and obtain the numbers which are smaller than 12 for each
square. They have played
it in guitar and it called as
12 tone music since the
beginning of 20th century.
The fundamental idea of
12 tone music consists
of serialism which is a
method of composition
in which the musicians
used to control various
types of music elements
by providing harmony
and melody. The main idea of 12 tone music is repeating the
same serial order over and over. Nowadays many popular
musicians use 12 tone
music in their musical
compositions
The logic behind all the discoveries and relations between
music and mathematics consist of having a pattern. Thus, the
musicians can create a pattern by using the same technique
with 12 tone music an create a grid square. By giving numbers
to each 8 tone they may have numbers with their formula of
patterns. By dividing them into 8 and calculating the remainder
of them like 12 tone matrix method they will get numbers which
are less than 8. By converting them into the notes that are given
from the beginning they may play them in different instruments
with different rhythms.
Bibliography:
Shah, S. (2010). An Exploration Of The Relationship Between
Mathematics and Music. Manchester: University of Manchester .
serialism. (2016). In Encyclopædia Britannica. Retrieved from http://
www.britannica.com/art/serialism
How to make a 12-tone matrix,2011(video file),Availabe from: [11 February 2016]
Yaren YENİAY
10-B
MATHEMATICS
DEPARTMENT
THE CLAPPER 2015 - 2016 37