XV : Music is a fantastic art of oblivious counting.
[Contd. A Journey to the Wonderland of Math.by Ajay Kumar Chaudhuri ]
[Contd. A Journey to the Wonderland of Math.by Ajay Kumar Chaudhuri ]
Is there any relation between music and math whatsoever?
Common notion of many people about math and music is that, these are placed completely
is two separate categories. Even it is assumed that people who are good at math
and science are averse to art and music and vice-versa, as if the two could not be placed together logically. But in reality, math and music are
indeed related and we commonly use numbers and math to describe and teach
music. It is known that many mathematicians play some instruments. Also there
are books about music which use a lot of math in- order to express music.
According to Gottfried Leibnitz, the famous German mathematician and
philosopher of the 17th century, “Music is a secret exercise in
arithmetic of the soul, unaware of its act of counting.”
Musical pieces are read much like we read math symbols.
These symbols represent some bit information about the piece.
Music is an art, entertainment, pleasure and medicine for
the soul and body. Music is one of the few activities that involve using the
whole brain. It is intrinsic to all cultures and has surprising benefits not
only for learning language imposing memory and focusing attention, but also for
physical coordination and developments.
But one thing is to be kept in mind that not all types of
music have favourable effects. Music can be distracting if it is too loud or
too jarring thereby disrupting our attention and concentration. But for the
most part, exposure to classical music has beneficial effects.
Let us see some of those surprising effects of music on our
mind and body:
Music may be helpful in relieving pain in many cases and
that is why music therapy is increasingly used in hospitals to reduce the need
for medicines during childbirth, to decrease post-operative pain and complement
to anesthesia during surgery. Using music therapy during childbirth decreases post-natal
anxiety and pain, increases the satisfaction of childbirth and reduces the
likelihood of postpartum depression.
If you are prone to hypertension or have any heart ailment
or simply wish to keep your heart healthy, then listen music for some time in everyday,
The speed of the music, called tempo (the speedometer of the music!) may make
your heart throbbing with joy and keep it robust.
Unfortunately, if somebody falls prey to heart attack, music
may come to his or her relief; for melodies of music can speed recovery from
debilitating strokes.
Are you suffering from nagging headache or migraine? Music
may help you to reduce frequency and intensity of your headache.
Even for hearing problem (tinnitus) music thereby in an
early stage can prevent it from becoming a chronic condition.
As healthy life style increases our immunity and longevity
so also can do a particular type of music. It helps contribute to a reduction
in the factors responsible for illness.
Our brain is one of the largest and most complex organs in
our human body. It is made up of more than 100 billion nerves that communicate
in trillion connections, called synapses. The brain is also made up of many
specialized areas that work together. Really surprising and unbelievable! Music
activates many regions of our brain which of course, has beneficial effects.
Listening music is good playing is even better. Certain
pieces of music can change our mood, get us motivated, improving concentration
and attention, decrease depression, reduces stress and induce sleep.
Furthermore, music improves memory performance, enhance
intelligence, learning and IQ.
Athletes can also be benefitted from music, as it improves
body movements and coordination.
Music may make our work joyful and more productive.
Listening to upbeat music can be a great way to find some extra energy. Music
can effectively eliminate exercise-induced fatigue and fatigue symptoms caused
by monotonous work.
So, summarily the effects of music on our body, mind and
soul is amazing wonderful, miraculous, magical and something more and
surprisingly math lies at its core.
Now if I ask what does actually music mean? It seems an easy
question to answer, because we all are familiar with music but being familiar
with something is not the same thing as knowing what it is. In fact, when we
seek an answer of this question, we are seeking its’ scientific aspects, that
relates music to our scientific understanding of the world.
I am sure that no one else has yet answered this question
clearly and explicitly. The production of new music that people want to listen
to be more an art than a science which means that even people making music does
not really know what it is.
Yet perhaps there is no dispute if I say, music is an art
which solely relies on sound. In real life we hear all sorts of sounds like
noises, screaming, shouting, laughing and some pleasing musical sounds. This is
not just restricted to humans. Animals also make sounds but these are
distinctly different from the human voice. Does a drum make the same sound as
flute? Or does a bird’s song resemble that of ours? So what is the difference?
To understand this we should be acquainted with some basic characteristics of sound.
What is a sound? It is a form of energy, just like
electricity, heat or light etc., each of these energies have some
characteristic features including sound. But how this sound energy is
generated? Take an example, when you strike a bell, it makes loud ringing
noise. Now put your finger on the bell when it is ringing. What is your
feeling? You must feel the bell is shaking. This movement or shaking that is,
the to and fro motion of the body is termed as vibration. Sound originates from
this vibration and propagates through a medium, like air, water or any other
material whatsoever by alternately contracting and expanding parts of the
medium it is travelling through in the form of wave. This compression and
expansion (rarefaction) create a minute pressure difference that we perceive as
sound.
To understand sound in general and musical sound in
particular, we should be acquainted with some basic characteristics of sound
like “wave length”, “frequency”, “amplitude” and “timber” as musical notes of
cultures of all races, whatsoever, are built on these attributes of sound.
As sound reaches to us from the source in the form of a
wave, so we must at first know what is a wave. A wave is a disturbance or
variation which travels through a medium. The medium through which the wave
travels may experience some local oscillations or to-and-fro motion, as the
wave passes, but the particles in the medium do not travel with the wave.
How many types of waves we observe? There are three categories
of waves: Transverse, Longitudinal and Surface waves. Of these the transverse
and longitudinal waves are very important to us; for energies like heat, light,
electricity, sound etc. are transferred by these waves from the source to reach
us which we perceive through our five senses.
A ripple on a pond or a wave on a string is easily
visualized by us as waves which are transverse in character. For transverse
waves the displacement of the medium is perpendicular to the direction of
propagation of the wave.
Transverse waves may occur on string, on the surface of a
liquid and throughout a solid. Transverse waves cannot propagate in a gas or
liquid because there is no mechanism for driving motion perpendicular to the
propagation of the wave.
Have you ever noticed the wave in a slinky? (a spring like a simple toy made of soft and clinging
material). It is a good visualization of wave, called “longitudinal”. In
longitudinal waves the displacement of the medium is parallel to the
propagation. Sound waves in air are longitudinal waves. The earthquake spread
from its origin or epicenter in the form of this longitudinal wave. Can you
remember the horrific tsunami of December 26, 2004 in Indian Ocean having
origin in Indonesia? The devastation
spread in the form of longitudinal waves.
But what about the third type of waves ---- the Surface
waves? We visualise it in our cup of tea or on the surface of sea waves. In
such waves, the particles of the medium travel in a circular motion. The waves
occur at interfaces.
Now let us be acquainted with few basic characteristics of
transverse and longitudinal waves, namely, wave length, amplitude, frequency
and timber which are closely related to our musical sound where math has a
subtle role to play. The pictorial form of transverse wave and that of
longitudinal wave [Pic. No. 28] will give us a fair impression of some of those
basic characters.
Pic. No. 28
On the other hand, in case of a longitudinal wave a
compression shows high pressure area whereas a rarefaction signifies low
pressure area. The distance between two consecutive rarefactions or two
consecutive compressions comprises a wavelength. A sound wave is made of areas of high pressure alternated by areas of low pressure.
We perceive sounds some as loud, some are normal while
others as feeble, but why? The reason behind it is a characteristic of sound,
namely, “Amplitude” which refers to the distance of the maximum vertical
displacement of the wave from its mean position, as we can see from the
picture. Larger the amplitude higher is the energy. In sound, amplitude refers
to the magnitude of compression and expansion experienced by the medium in
sound wave in travelling through. High amplitude is equivalent to loud sound.
We have seen before that a vibrating body emits sound. The
rate at which it vibrates is termed its “frequency” and also called “pitch”.
Because of this parameter of sound we can differentiate the female voice from
that of a male voice. It is found that men have lower pitched voice than women.
The frequency of a wave means how often the particles of the
medium vibrate when a wave passes through the medium. We measure the frequency
of a wave by taking into account how many times of the particles of the medium
vibrate when a wave passes through the medium. A commonly used unit of
frequency is “Hertz” and is abbreviated as HZ, and 1 HZ equals 1 vibration per
second. For example, if a particle of air undergoes 500 vibrations in one
second, then its frequency will be 500 HZ.
The ears of a human, as well as of other animals are
sensitive detectors capable of detecting the fluctuations in air pressure that
impinge up on the eardrum. But there is a limit of perception of hearing for
all living beings. Human ear is capable of hearing sounds of frequencies
ranging approximately from 20 HZ to 20,000 HZ. Animals like dogs, cats, bats
etc. have far more range of hearing.
A sound with
frequency less than 20 HZ, that is, below our audible range is called
“Infrasound” and above the limit 20,000 HZ known as “Ultrasound”. Dolphins can
detect sounds of frequencies as high as two lakhs HZ and dogs, cats, bats and
dolphins have an unusual ability to detect ultrasound.
A musical sound has another unique character or quality by which
we can distinguish two sounds having the same frequency or pitch and amplitude
or intensity created by two instruments or voices. For example, say, a piano
and a flute are playing musical notes of the same frequency and amplitude .You
can feel that those two sounds are very different. For this, timbre which
distinguishes different types of sound production is also called “tone colour” or
“tone quality” is the reason.
But what is ‘sound of music’? – Which is so appealing, such
amazing, wonderfully serene and something more pertaining to the artist’s
contributions!
A sound will be musical if it is produced in a controlled
manner relying lots on pitch (frequency) and timbre (quality) of the voice of
the singer.
For instruments the factors like frequency, intensity and
also resonance of the sound matter. ‘Resonance’ of sound means the quality in a
sound being deep, full and reverberating. For a better idea of it you can
notice for the stringed instruments like violin, guitar, sitar or percussion
instruments such as drum like instruments. There is a big hollow at the end
of each of these instruments. The vibrations of the strings or the diaphragms
cause the enclosed air in the hollow to vibrate with more vigour or larger amplitude
at a specific frequency making the music further distinct to the instruments.
Now let us explore the wonderfully magical role of math in
music. But, before that we should be familiar with some of the terms related to
music which are frequently used. Those terminologies are: Naada (musical sound
or tone), shruti (microtone) and swara (note). Here we have taken Indian music
for an example and reference.
Naada is a musical sound generated by a series of regular
vibrations in a medium like air. If the vibrations are irregular, it will
create unpleasant sound or noise.
We know every sound has a frequency of its own. A sound
having twice this frequency will sound the same but ‘higher’ or louder. The
later sound is called the octave of the first. For example, if we take sound
(or tone) having a frequency of 100 HZ, the sound whose frequency is 200 HZ
will be the octave of 100 Hz. The number of sounds in between these two
frequencies which human ear can hear is infinite.
Here is an interesting relationship of math with the
frequencies of sound expressed in number of units, like Hertz. Mathematicians
have proved conclusively that in between any two numbers, however big or small
the numbers may be, there are an infinite number of numbers. So in between the
numbers denoting the frequency of the first note (here 100 HZ) and its Octave (here
200 HZ) there are an infinite number of numbers and hence an infinite sounds
having frequencies in relation to each of these infinite numbers. But
unfortunately, out of these infinite number of sounds, we can hear, recognise,
differentiate or grasp is only 22. These are called Shrutis (microtones,
intervals or steps).It is mainly determined through fine auditory perception.
Out of these 22 shrutis 7 are selected to form a musical
scale. But how and where from we start? To begin with we may choose any audible
sound having a certain frequency as our point of reference. It is called the
“tonic” or “key” or “fundamental tone”.
For convenience, let us take the sound represented by 100 HZ
as their tonic. So the sound of twice this frequency (here 200 HZ) will be its
Octave and there will be 22 ratios or shrutis in between 100 HZ and 200 HZ.
After choosing the fundamental tone, 6 more shrutis are chosen obeying some
definite rules to form a 7 – ladder musical scale. These 7 sounds or tones are
called “Swara”s (or notes).
The first tonic, as we choose,
is named as “Sa”. This sa would be followed by 6 more notes, 7 in all. The 8th
note sound like sa, but it would sound “higher” or louder. These 7 notes form
the “Saptaka” of Indian music. These 7 notes are suitably named as:
“Shadja” or ‘Sa” for short,
symbol S;
“Rishabha” or ‘Re’, symbol R;
“Gandhara”, ‘Ga’ symbol G;
“Madhyama”, or ‘Ma’, symbol M;
“Panchama”, or ‘Pa’ symbol P;
“Dhaibata” or ‘Dha’ symbol D;
And “Nishada” or ‘Ni’ symbol N.
Music is universal and each
community has their own system of music. In modern days it is observed that the
Western music has impact on non-western music including on the Indian music
system. So, it will be amusing to mention the Western musical notes along with
the Indian notes.
The Indian Fundamental note Sa corresponds C in Western
system; similarly, D for Re, E for Ga, F for Ma, G for Pa, A for Dha and B for
Ni.
One remarkable thing is that the first and fifth notes,
namely C (Sa) and P (pa) are regarded as immutable or unchangeable(named
“achala” in Indian system). The remaining 5 notes have two systems each. Thus
we have 12 notes in an octave.
Indian music is known for its complex use of microtones. But
for convenience of notations and explanation we divide an octave into 12
semitones. We use a movable scale, which means that an octave can start from anywhere
we like. The starting point is the root of any one’s octave, all the other
notes are defined in relation to the root. Each of the 12 notes in the octave
has a unique identity which is shown in the following table.
The names and designated symbols
of these 12 notes used both in Indian and Western music are in this table to
have a better glimpse.
Indian Names
|
Indian Symbols
|
Western Names
|
Western Symbols
|
1. Sa (Suddha)
|
S
|
Unison
|
C
|
2. Re (Komal (flat)
|
r
|
D flat (Minor second)
|
Db
|
3. Re (suddha) (Natural)
|
R
|
E flat (Augmented second)
|
D#
|
4. Ga komal (flat)
|
g
|
F (Diminished third)
|
Eb
|
5. Ga (Suddha) (natural)
|
G
|
E (Major third)
|
E
|
6. Ma (Suddha (natural)
|
m
|
F (Perfect fourth)
|
F
|
7. Ma (teevra) (sharp)
|
M
|
F sharp (Augmented
fourth)
|
F#
|
8. Pa (suddha) (natural)
|
P
|
G (Perfect fifth)
|
G
|
9. Dha komal (flat)
|
d
|
A flat (Minor sixth)
|
Ab
|
10. Dha (suddha) (natural)
|
D
|
A (Major sixth)
|
A
|
11. Ni komal (flat)
|
n
|
B flat (Minor seventh)
|
Bb
|
12. Ni (suddha)
(natural)
|
N
|
B (Major seventh)
|
B
|
Sa (octave)
|
S’
|
Unison (Octave)
|
C#
|
In this table, Komal or flat notes
in Indian and flat or minor notes in Western music means notes of lower pitch
than that of natural in Indian and major in Western music.
For the note at Ma-sharp in Indian and its counterpart
augmented fourth in Western music denotes of higher pitch than natural and
perfect fourth at those two systems respectively.
Wise people, who used to see the reflection of nature’s
diversity in their spirit, felt that inner being of human respond to nature’s
various calls like those of birds, animals, change of seasons, different times
of the day. Thus emerges the knowledge of Indian musical notes and infinite
ragas or moods.
It will be very interesting to
know that the seven basic swara or notes have their origin in nature’s
elements. Such as:
The last swar, Nishad or Ni,
derives from the majestic call of the elephant.
The sixth sound Dhaibat or Dha
comes from the sound of the frog.
The fifth swar Pa, the short
form of Pancham has its origin in the sound of the coo coo bird.
The middle sound Madhyama or Ma
is actually comes from the sound of the crane.
The third Gandhar or Ga
represents the sound of a goat (sheep)
The second sound Rishabha or Re has
come from the sound of the bull.
The first sound Shadja or Sa, the fundamental note
represents the sound of the peacock.
But another interesting thing is that apart from these
sources of Indian musical notes, they may also be believed to emanate from the
different energy centres of our human body, called “Chakra” meaning “wheel”.
These centres are named as such spinning energy centres which exist in our
subtle etheric body, the non-material energetic counterpart of the physical
body.
There are seven main chakras and they are located along the
spine extending out the front and back of the body. Each chakra has a specific
quality that corresponds to the refinement of energy from the base-level
material- self-identity. Each chakra is associated with a certain part of the
body and a certain organ which it provides with energy it needs to function. [Pic.
No.29]
Pic. No.29
Seven chakras in human body.
One widely popular schema of chakras is as follows from top to bottom.
No. Name of the chakra. Location in the human body.
1 Sahasrara. Crown.
2 Ajna. Between eyebrows.
3 Vishuddha. Throat.
4 Anahata. Heart.
5 Manipura. Navel.
6 Svadhisthana. Root of sexual organs.
7 Muladhara. Base of spine.
No. Name of the chakra. Location in the human body.
1 Sahasrara. Crown.
2 Ajna. Between eyebrows.
3 Vishuddha. Throat.
4 Anahata. Heart.
5 Manipura. Navel.
6 Svadhisthana. Root of sexual organs.
7 Muladhara. Base of spine.
Perhaps all of us love melody, love music, but most of us
are little aware or least interested to know the role of musical notes behind
any music, although a very few are inquisitive to know how a music is composed.
The simplest answer is that any music is essentially mathematical game of 12
notes. For example, look at piano. There are only 12 notes in between any
octave. After those 12 notes, the octaves simply repeat all over again.
While there are 12 unique notes in all music, there are
hundreds, if not thousands of combinations that can be formed from these 12
notes. Thanks to math for helping in these “combinations”. These notes can
combine in fascinating ways to create wonderful harmonies and melodies
reflecting the mood of the composer.
Don’t indulge in thinking that the composition of music is
not at all an easy job. Because there are 26 letters in the English alphabet,
39 in Armenian, 74 in Cambodian, 51 in Bengali alphabet etc. and surprisingly
Chinese writing can contain as many as 40,000 characters, but still only 12
notes in musical alphabet.
It is a matter of ever wonder that some note combinations
sound pleasing to our ears, while other makes us disgusting. Long ago, the
Greeks realized that sounds which have frequencies in rational proportions are perceived
as harmonious. For example, a doubling of frequency gives an octave, that is,
sounds the same but louder. A tripling of frequency gives a perfect fifth (Pa)
one octave higher. But they didn’t know this in terms of frequencies; for
characteristics of sound were not known to them as we know today. They realized
it in terms of length of vibrating strings. In this way, they discovered the
confluence of two pleasantly flowing streams of music and math.
Now let us
see the musical note of math in the frequencies (or pitch) of 12 notes which
are the main component of all music. We will establish these relations conclusively
not theoretically but experimentally. For this purpose we shall take the help
of an instrument called “frequency meter” for measuring frequencies of a tone
or pitch. This meter shows tones in Hertz or HZ in short with which we have
already been acquainted. Also let us take another instrument for our experiment
--- Tanpura, an Indian stringed instrument. [Pic. No.30]
Pic. No. 30.
Just like Sitar, the Tanpura is
one of the long neck lutes. It is usually stringed with 4 or 5 metal strings
(rarely with 6 strings) and it is a basic note instrument and as such an
important component in Indian classical music. They are tuned to the basic note(S)
and its fifth(P) and octave(S'). The strings are only plucked. A characteristic
feature of Tanpura is its sound that is very rich in overtones, and the special
rich sound effect that produced by the constant playing of the individual
strings which has an intensive effect on the listener. The Tanpura is thus not
only an ideal accompaniment for traditional Indian music, but also an ideal
companion for musical meditation, overtone singing or model.
Now let us start our experiment with a Tanpura for
calculating the frequencies of 12 notes:
Pluck the fourth string of the Tanpura. This is the lower
most sounding thick copper string. It should sound firm and strong. It is not
required to be tuned in any particular frequency. Let us take its pitch as our
‘Sa’ or S as tonic or fundamental tone. If you are an experimenter, it is
supposed that you should have good sensitive ears for music.
After Sa your sensitive ears also pick up two more easily
grasped clear, ringing notes Pa and Ga. You can hear other notes too. But ignore
them for the moment and focus your attention on Sa, Pa and Ga only. Here one
thing is to be remembered that Sa is the plucked note whereas Pa and Ga are
self-generated or ‘swyam-bhoo’ as in the language of Sanskrit, the Indian
classical language.
Now with the help of frequency meter, measure the
frequencies of those notes Sa, Pa and Ga, say in the Major Scale. But what is a
scale in music? A scale in music is a series of notes which we desire as
correct or appropriate for a song. Generally it needs to define the series
within an octave and the same series will be used for all for all other
octaves.
Perhaps most musicians need two scales: The Major and the
Minor scale. Both of these scales have 7 notes per octave.
In fact, there are many scales and a musician has liberty to
choose any one of them as he likes. After all, scales are just a series of
notes. Different cultures have developed different scales because they found
some series of notes more pleasing than others.
It is a matter of surprise that
the ratios of Sa, Ga and Pa in this Major scale always bear a fixed ratio of 4
: 5: 6, irrespective of the frequency of the fundamental tone Sa chosen. For
example, if the Tanpura string so tuned that the frequency meter shows it to be
vibrating at a frequency of 100 HZ for Sa, then the meter will show Ga to be at
125 HZ and Pa at 150 HZ. Then clearly the ratio is Sa : Ga : Pa =100 : 125 = 4
: 5 : 6 (Dividing each by 25 to get them at their lowest term).
Similarly, if the frequency
meter shows Sa to be tuned at 240 HZ it will show the self-generated Ga and Pa
to be at 300 HZ and 360 HZ respectively. Here again
Sa : Pa : Ga = 240 : 300 : 360 =
4 : 5 : 6 (dividing each by 60)
So it is amply Clear that
whatever be the frequencies of those notes, their frequency ratio remains 4 : 5
: 6. Hence it is those ratios, also called intervals, are more fundamental than
the frequencies measured in HZ.
Let us take the frequency of Sa
at 240 HZ as a reference or starting point to find the frequencies of all the
12 notes in the Major scale.
We have already found that the
frequencies of Sa = 240 HZ, Ga = 300 Hz and Pa =360 HZ. The frequencies of the
remaining notes can be calculated* by using rudimentary arithmetic and knowing
the ratios of the notes they maintain among themselves found from the
experiments.
Taking the
frequency of the fundamental tone Sa = 240 HZ as reference the frequencies of
the remaining 11 notes will be as follows. The frequencies of the notes are
given in HZ along with their symbols, mentioned within bracket.
Sa (S) = 240 Komal
Re (r) = 256
Suddha
Re (R) = 270 Komal
Ga (g) = 288
Suddha Ga (G) = 300 Suddha Ma (m) = 320
Teevra
Ma (M) = 337.5 Pa (P)
= 360
Komal Dha (d) = 384
Suddha Dha (D) = 400
Komal
Ni (n) = 432 Suddha
Ni (N) = 450
The frequency of Sa of the next octave will = 480 HZ and is
represented as S’ symbolically.
[It is from our experimental data that Pa, Ni and Re stand in the same
relationship as Sa, Ga and Pa. Using the symbols of notes for convenience, we
may write
P : N : R = S : G : P
= 4 : 5 :6
Now let us proceed to find the
frequencies of Suddha Ma (m) and Suddha Dha (D) with the data on notes we have
already determined. Let us assume, for the time being, that Suddha Ma in our concerned
octave now is our Sa (for Sa may be assigned any frequency as the fundamental
tone). The Ma, Dha and SA’, the SA in the next octave, stand in the
same relationship as Sa, Ga and Pa.
That is Ma : Dha : SA’ = Sa
: Ga : Pa
Or m : D : S’
= S : G : P
=
4 : 5 : 6s'
Next we try to find the frequency of
Komal Ni (n) by utilising the relation that Komal Ga (g) and
Komal Ni (n) have the same ratio of Sa and Pa
Komal Ni (n) have the same ratio of Sa and Pa
So, it is now clear how
we get the frequencies of 12 notes as shown before.]
So, we may conclude, the frequencies of musical notes
maintain close mutual relation through mathematical bonds. Also combinations
and arrangements of musical notes (called Combination and Permutation in the
language of math) give birth to innumerable magical melodies.
Perhaps you can remember the series consisting of the numbers
(0, 1, 1, 2, 3, 5, 8, 13 ….) invented by Leonard Fibonacci of Italy and related
with it the famous ratio phi (Φ
= 1.618), the golden ratio, we have already seen that the number of this series
and the golden ratio are manifested in nature and certain works of art like
painting, sculpture, architecture and even in beauties of living beings including
human and more particularly that of beauties of a girl.
The nature around us is indeed
beautiful and we found with surprise the amazing role of Fibonacci numbers and
that the golden ratio in it. But most of us are not aware that those numbers
and that ratio amazingly form the foundation of music and making musical
instruments. If we look minutely at the different aspects of music and some
musical instruments like piano, guitar, violin etc. we will be surprised to
discover the presence of Fibonacci numbers and the golden ratio in them.
Some striking features are as:
There are 13 notes in the span of an octave.
A scale is composed of 8 notes
of these 8 notes the 5th (Pa) and the 3rd (Ga) notes
create the basic foundation of all chords. As we found during our experiments
on frequencies of notes with a Tanpura where Sa, Ga and Pa that is, the 1st,
3rd and the 5th note played leading role in determining
the frequencies of the other nine semitones in a scale.
A scale is based on 2steps and 1 step from the root note,
means the first note (Sa in this case) of the scale.
If we look at the piano key board scale at an octave of 13
keys, we find 8 white keys and 5 black keys, split in the group of 3 and 2.
So in music the number 1, 2, 3, 5, 8, 13 frequently appear.
Are these numbers nothing but Fibonacci numbers?
The Fibonacci composition of music reveals the inherent
aesthetic appeal of this mathematical phenomenon.
Now, can we say that the music
is mathematical and math has a magical role in music?
The answer
is a big emphatic “Yes”.
[To continue]
Reference Internet:All the images (except otherwise stated ) of this article are downloaded from Public Domain.
Pic. No28--Longitudinal and transverse waves: Attribution--GeoGebra
Pic. No28--Longitudinal and transverse waves: Attribution--GeoGebra
(https://www.geogebra.org/m/SxNZa3Q2)
Pic. N0 3o.Tanpura.Attribution:Creative Cosmos (https://commons.wikimedia.org/wiki/Fi )
