XV: Music is a fantastic art of oblivious counting.

XV : Music is a fantastic art of oblivious counting.
         [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 17^th 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

 The high pressure area represented as the peaks of the graph. The low pressure areas are depicted as valleys. The physical distance between two consecutive peaks or valleys is referred to wave length in case of a transverse wave.

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 8^th 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.  

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.

                                                                         Tanpura.

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

       

              

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 5^th (Pa) and the 3^rd (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 1^st, 3^rd and the 5^th 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

(https://www.geogebra.org/m/SxNZa3Q2)

Pic. N0 3o.Tanpura.Attribution:Creative Cosmos  (https://commons.wikimedia.org/wiki/Fi )