L V Nagarajan

Introduction

Solid geometry has always fascinated me and has fired up my imaginative power. I was introduced to Solid Geometry by my elder brother (L V Sundaram) when I was barely 10 years old. In his university days, he had a habit of reading aloud and explaining to himself as a part of his study process. When I am around as a young boy, he will do the explaining to me, not caring how much of it I understood. But he was rather surprised that I understood a lot of what he told me especially in geometry. As I learnt more of plane geometry, I developed a habit of extending the results of the theorems to the third dimension. Subsequent to my earlier blog on Polyhedrons, I want to now share my research on trihedral angle, as it is the basis of all solid angles

Normal Plane Angles

I am re-stating some fairly obvious facts about plane angles, since we need the same to compare with the three dimensional angle, known as Trihedral Angle. In a plane if two straight lines are drawn, they intersect each other at some point. But when they are parallel to each other, they maintain constant distance between themselves all along their infinite lengths in both directions. Hence they never intersect each other. Two non-parallel straight lines in a plane, intersecting each other, have a rotational distance between them. When one of the straight lines is rotated about the point of intersection, it will coincide with the other straight line after some rotation. This rotational distance between the straight lines is known as Angle. One full rotation is divided into 360 parts for the purpose of quantifying the amount of rotation, each part known as a degree. Consider a circle with O as the centre and R as the radius. As the radius rotates a complete round it completes a circumferential distance of 2πR. Hence a full rotation of 360 degrees is associated with 2π Radians. Any arc of the circle of length L subtends at the centre O an angle of L/R radians. When two straight lines in a plane intersect each other they actually create four angles, with opposite angles of equal magnitudes. It is easy to see that the adjacent angles add up to 180 degrees or half rotation. These four angles are actually the four internal angles of a parallelogram! We may also easily see that two intersecting straight lines in a plane divide the plane into four quadrants. If the straight lines are perpendicular (90 degrees) to each other, they divide the plane into four equal quadrants. The lines themselves become X-axis and Y-axis of the coordinate system.

Trihedral Angle

Now let us move to the third dimension. Consider two planes parallel to each other. They maintain a constant distance between them all along their infinite surface. They never intersect each other. But two non-parallel planes do intersect each other, at some place, along a straight line. They do have a rotational distance between them. This is similar to plane angle and this rotational distance is known as dihedral angle, as the angle between two planes, varying from 0 to 360 degrees (or 2π radians). Now let us consider a third plane non-parallel to either of the earlier two planes. Let this plane intersect each of the earlier two planes along two different and non-parallel straight lines. All these three planes (and their three straight lines of intersection) will meet at a common point. At this point is formed a solid angle, known as trihedral angle. In general an angle represents a rotational distance. Similarly, the trihedral angle also is a measure of the sum of rotational distances, in two directions, i.e., rotations about two of the edges of trihedral intersections. Hence some time it is called as square-angle or square degrees. Trihedral angles vary from 0 to 720 square degrees. Consider a sphere with O as the centre and R as the radius. As the radius rotates a complete round in a, say, vertical plane, we get a vertical circle. When this circle rotates a complete round about any diameter as axis, we get a complete sphere with a surface area 4πR², covered by rotation. Hence a full rotation of 720 degrees is associated with 4π Stradians. Now consider a solid angle subtended by a partial spherical segment at the centre. It is given as surface area of the segment divided by radius squared. When three planes intersect each other in space, they actually create eight trihedral angles. These eight trihedral angles are actually the eight solid angles of a parallelepiped! We can also see that three intersecting planes divide the 3-dimensional space into eight octants. If these three planes are perpendicular each other, they divide the 3-dimensional space into eight equal octants. The lines of intersections, some time called as edges, form the X-Y-Z axis of the 3-demensional co-ordinate system.

Other angles of trihedral angle

Though we are talking of three planes making a trihedral angle, we feel comfortable to talk of trihedral angle as formed by three (non-planar) straight lines meeting at a point. Hence we talk about vertex angle as the plane-angle between any two edges at the vertex of the trihedral angle. Dihedral angles are the angles between adjacent planes making up any trihedral or polyhedral angles. The solid angle (or any polyhedral angle), the dihedral angle and the vertex angle are normally denoted as, Ω, Φ and θ. It is interesting to see below, how these angles relate to each other, say, for the case of a trihedral angle.

From the theory of spherical excess, it has been derived and stated that, the solid angle at O as enclosed by vectors A, B and C as above, is given by Ω = Φa + Φb + Φc – π

Let us consider the solid angle Ω1, at O, enclosed by OA, OB and OC. The dihedral angle Φa is as shown. This is the angle between two semi circular faces of the orange-like dice ABA₁CA. This piece of spherical surface is sometimes known as di-angle. The spherical surface area of this dice is Φa/2π times the whole surface area of the sphere. Hence the solid angle subtended by this area at the centre O is Φa/2π times 4π, i.e., 2Φa.

Hence supplementary solid angle Ω2, enclosed by OA₁,OB and OC is given by 2Φa –  Ω1.

We may infer similar results with OB₁ and OC₁ and get

 Ω2 = 2Φa – Ω1

 Ω3 = 2Φb – Ω1

 Ω4 = 2Φc – Ω1

It is easy to see that Ω1+ Ω2 + Ω3 + Ω4 = a hemisphere = 2π

 i.e. 2(Φa + Φb + Φc – Ω1) = 2π

i.e. Ω = Φa + Φb + Φc – π

This proves the result we got earlier. Ω1, Ω2, Ω3 and Ω4 are the four solid angles of the parallelepiped obtained from OA,OB, OC. The other four solid angles will be equal to their opposite angles. These eight solid angles form the octants in the three dimensional space as mentioned earlier.

Trihedral Angle Ω, in terms of vertex angles α, β and γ

In a parallelogram described by sides ‘a’ and ‘b’ and angle θ, the area is given by (ab sin θ). Referring to the diagram below:

We can write tan θ/2 = b sin θ/ (b + b cos θ)

i.e., tan θ/2 = ab sin θ/ (ab + ab cos θ)

                    = Area of Parallelogram/ ab (1+cos θ)

i.e., tan θ/2 = (a x b)/(ab + a.b)

We have a similar relationship for a parallelepiped described by vectors and angles:  a, b, c, α, β and γ,  given by Oosterom and Strackee (2) as:

Tan(Ω/2) = (a . b x c) / [ abc + (a.b)c + (b.c)a + (c.a)b]

It is easy to see that (a . b x c) represents the volume of the parallelepiped a, b, c.

Volume V = (a . b x c)

     = Det (a / b / c), with  vectors a, b, c represented as row vectors.

     = Det (a , b , c), with  vectors a, b, c represented as column vectors

 i.e., V = a . b x c = Det [a1,a2.a3/ b1,b2,b3/c1,c2,c3]

                                = Det [a1,b1,c1/a2,b2,c2/a3,b3,c3]

Hence, V*V = Det [a.a, a.b, a.c/ b.a, b.b, b.c/ c.a, c.b, c.c]

It will reduce to

V*V =

    a² b² c² (1 + 2 cos α cos β cos γ  – cos²α – cos² β – cos² γ )

The solid angle Ω does not depend on magnitudes of vectors a, b, c and hence assuming them to be unit vectors, we get the expression for solid angle Ω, as

tan (Ω/2)

 = (√ (1 + 2 cos α cos β cos γ  – cos²α – cos² β – cos² γ)) / (1+cos α+cos β+cos γ)

Now we have expressed Ω in terms of vertex angles α, β and γ. We have also earlier expressed Ω in terms of dihedral angles Φa, Φb, Φc. Later in this write-up, we will also find how α, β and γ relate to Φa, Φb and Φc.

‘Tetragonometric’ ratios of trihedral angle

When we talk of tan (Ω/2) as above, we are not talking of actual trigonometric ratio of trihedral angle Ω. We are only treating the square-degree Ω, as normal angle of the same magnitude and then expressing its trigonometric ratio in the above formula. But do we have ‘trigonometric ratios’, like Sine, Cosine, Tangent, for trihedral angles also? (May be, we should call them as ‘tetragonometric ratios’)

Let us go back to our comparison with a parallelogram.

Area A of the parallelogram a, b, θ, is given by ab sinθ

So we say, Sin θ = A/ab. i.e., Area/(product of adjacent lengths)

Similarly, in case of parallelepiped, (a, b, c, α, β, γ), G-sin Ω has been defined as,

G-sin Ω = V*V/ (Product of adjacent areas)

i.e., G-sin Ω  = V*V/  (ab sin γ)(bc sin α)(ca sin β);

We know V*V =

     a² b² c² (1 + 2 cos α cos β cos γ  – cos²α – cos² β – cos² γ )

Hence

G-Sin Ω =

(1+2 cosα cosβ cosγ -cos²α -cos²β -cos²γ) /(sinα  sinβ sinγ)

Volume V1 of a unit parallelepiped with angles α, β, γ and with (a=b=c=1) is given by,

V1 = √(1+2 cos α cos β cos γ  – cos²α -cos² β -cos² γ)

Hence, we can say,

G-Sin Ω = (V1*V1)/ (sin α sin β sin γ);

It is  easy to see that all the eight solid angles at the eight corners of the parallelepiped will have the same G-sine value, as the same three parallelograms are enclosing them, but in a different order. This is exactly as in the case of a parallelogram where all the four angles have the same sine value.

This G-sine value appears to signify nothing, especially when you see this value not being consistent. Even as two sets of α, β, γ produces same solid angle Ω, their G-sine values will vary. For a particular value of Ω, G-sine will be maximum when α = β = γ.

Let us approach G-sine from a different perspective. We know trigonometric ratios of any angle are derived from a right angled triangle that includes this angle. Please refer to the following diagrams.

Sin θ = BC/AC; Cos θ = AB/AC; Tan θ = BC/AB

Any trihedral angle at O can be represented as a part of a semi-right-angled tetrahedron OABC as above.

Angles OCA and OCB are right angles. i.e. OC is perpendicular to plane ACB.

Angle AOC (= β) and angle COB (=α) are independent and, along with the dihedral angle Φ, completely defines the trihedral angle at O. Angle AOB (=γ) and angle Φ depend on each other. Let OC = c. Now from the definition of G-sin we get,

G-Sin O = V*V/ Product of adjacent areas

V (of parallelepiped) = (AC*BC Sin Φ) * OC

Product of adjacent parallelograms

                      = (OC*AC sin β)*(OC*BC sin α )*(OA*OBsin γ)

(as α =β = 90 degrees);          = (AC*BC*OC*OC)*(OA*OBsin γ)

G-Sin O = AC*BC Sin² Φ / (OA*OB sin γ)

            = (2*Area ABC/ 2*Area ABO) Sin Φ

i.e. G-Sin O = (Area ABC / Area ABO) * Sin Φ

i.e G- Sin O = (Opposite area/ Hypotenuse area) * Sin Φ

In the special case when Φ = 90 degrees, we get,

G sin O = (Opposite area/ Hypotenuse area)

This is very similar to trigonometric ratio for plane angle. It is easy to see we require a special set of angles α, β, γ (with Φ = 90 deg) to produce a right angled tetrahedron. In all other cases, the relationship,

G- Sin O = (Opposite area/ Hypotenuse area) * Sin Φ, holds good.

For a right angles tetrahedron, it can be easily proved that, the sum of the squares of adjacent areas will be equal to the square of hypotenuse area. This is very similar to Pythagoras theorem of right-angled triangles. Using this property, we can define other ‘tetragonomertic’ ratios for trihedral angles as below:

G-Cos O = (√(Sum of the squares of the two adjacent areas)) / Hypotenuse area.

G-Tan O =  Opposite area / (√(Sum of the squares of adjacent areas))

We can also write G-Sin² O + G-Cos² O = 1

We can also note:

G-Sin² O + G-Sin² A + G-Sin² B = G-Sin² C = 1.

Please note that this right tetrahedron is right-solid-angled at C (i.e. π/2 stradians or 90 sq. degrees).

Other interesting relationships

We started this section with the fact that any trihedral angle can be represented as a part of semi-right-angled tetrahedron as below.

We derived the relationship

G- Sin O = (Opposite area/ Hypotenuse area) * Sin Φc

Now (AC = OA Sin AOC = a sin β) and (BC = OB Sin BOC = b sin α)

Therefore Area ACB

    = (1/2)AC.BC sin Φc = (1/2)(a sin β) (b sin α) Sin Φc

Similarly Area AOB = (1/2)OA.OB. Sin γ = (1/2)a.b. Sin γ

G-Sin O = (Area ACB / Area AOB)* Sin Φc

                 = (Sin α. Sin β. Sin² Φc) / Sin γ

By extending this logic, we can write,

G-Sin O    = (Sin α. Sin β. Sin² Φc)/Sin γ

                   = (Sin β. Sin γ. Sin² Φa)/Sin α

                   = (Sin γ. Sin α. Sin² Φb)/Sin β

Thus we have established a comprehensive relationship for any trihedral angle in terms of vertex angles and dihedral angles.

Using the above along with the expression for G-Sin O,

G-Sin O = (V1*V1) / (sin α sin β sin γ); we get,

Sin Φa = V1/( sin β sin γ),  Sin Φb = V1/( sin γ sin α) and

      Sin Φc = V1/( sin α sin β)

Remember V1 is volume of unit parallelepiped α, β, γ, with a = b = c =1 given by:

V1 = √(1 + 2 cos α cos β cos γ  - cos²α – cos² β – cos² γ )

Once the values of Φa, Φb, Φc are calculated as above, we can find each of the eight solid angles in the eight octants formed by α, β, γ planes. Of course these are same as the eight solid angles of the parallelepiped (a,b,c, α, β, γ). We already know that

Ω1 = Φa +  Φb + Φc – π  = Ω

Ω2 = 2Φa –  Ω1

Ω3 = 2Φb –  Ω1

Ω4 = 2Φc –  Ω1

These solid angles are repeated again as the diametrically opposite angles, thus forming all the eight trihedral angles of the parallelepiped. It may be observed that the above four trihedral angles add up to 2π. All eight trihedral angles add up to 4π.

As stated earlier, G-Sine of all the eight solid angles of the parallelepiped are equal.

G-Sin Ω1 = G-Sin (2Φa –  Ω1)

      = G-Sin (2Φb –  Ω1) = G-Sin (2Φc –  Ω1).

Now let us tabulate all these angles for different types of trihedral angles.

Types of trihedral angles are as listed below

Conclusion

The sites given in the references had triggered my interest in this topic of trihedral angle. However most of the concepts developed in this write-up are the results of my own research. I hope this will be useful for students of geometry and for others who are doing further research on the application of solid angles.  

References:

1. www.goiit.com

2. A. VAN OOSTEROM, J. STRACKEE: A solid angle of a plane triangle. – IEEE Trans. Biomed. Eng. 30:2 (1983); 125-126.

3. http://mathdl.maa.org/images/upload_library/22/Ford/CarlBAllendoerfer.pdf

Solid geometry has always fascinated me and has fired up my imaginative power. I was introduced to Solid Geometry by my elder brother (L V Sundaram) when I was barely 10 years old. He had a habit of reading and explaining to himself as a part of his study process. When I was around as a young boy, he would do the explaining to me, not caring how much of it I understood. But he was rather surprised that I understood a lot of what he told me especially in geometry. As I learnt more of plane geometry, I developed a habit of extending the results of the theorems to the third dimension.

Starting from triangle, quadrilateral and pentagon, there are infinite number of polygons. Polygons with all sides and all internal angles equal are called regular polygons and there are infinite numbers of regular polygons with 3 to infinite numbers of sides. In the limit the regular polygon becomes a circle.

Extending to the third dimension, there are infinite numbers of polyhedrons, starting from tetrahedron.

Two non-parallel straight lines in a plane make an angle. Three straight lines in a plane which are non parallel to each other make a triangle, which is a polygon of least order – 3 sides and 3 angles.

Similarly, two non-parallel planes make an angle all along the common edge. Three planes which are non parallel to each other make a solid angle at a unique point of intersection of all the three planes. This solid angle is referred to as trihedral angle, as they are made by three planes. The solid angle at the vertex of a typical polyhedron is known as a polyhedral angle, made by three or more planes. But three intersecting planes still do not make a closed polyhedron. A fourth non-parallel plan is needed to get a polyhedron of least order – 4 faces, 6 edges and 4 vertices; this is known as a Tetrahedron or sometimes as triangular pyramid. We may also think of a polygonal pyramid with the base as a polygon of n-sides. For a regular polygon of n-sides, this polygonal pyramid will become a circular cone in the limit.

Regular polyhedrons are those whose faces are all regular polygons, congruent to each other, whose polyhedral angles are all equal and which has the same number of faces meet at each vertex. An interesting fact is, there are only five regular polyhedrons: the Tetrahedron (four triangular faces), the Cube (six square faces), the Octahedron (eight triangular faces—think of two pyramids placed bottom to bottom), the Dodecahedron (12 pentagonal faces), and the Icosahedron (20 triangular faces). (as below)

Polyhedrons are classified and named according to the number and type of faces. A polyhedron with four sides is a tetrahedron, but is also called a Triangular pyramid. The six-sided cube is also called a hexahedron. A polyhedron with six rectangles as sides also has many names—a rectangular parallelepiped, rectangular prism, or box. However there are infinite numbers of semi-regular and non-regular polyhedrons which closely approximate to a hollow sphere in the limit. Imagine cutting off the corners of a cube to obtain a polyhedron formed of triangles and squares, for example. Other common polyhedrons are best described as the same as one of previously named that has part of it cut off, or truncated, by a plane.

Euler characteristic

The Euler characteristic χ relates the number of vertices V, edges E, and faces F of a polyhedron:

For a simply connected polyhedron, χ = 2.

For regular polyhedrons as below

Cube ;     χ = 8-12+6 = 2

Tetrahedron     χ = 4 -6 +4 = 2

Octahedron     χ = 6 – 12 + 8 = 2

Dodecahedron χ = 20 – 30 + 12 = 2

And so on ….

Duality

For every polyhedron there exists a dual polyhedron having:

• faces in place of the original’s vertices and vice versa,
• the same number of edges
• the same Euler characteristic and orientability.

Dodecahedron and Icosahedron are the dual of each other. Same is true with Cube and Octahedron. The dual of a Tetrahedron is another smaller tetrahedron.

The five regular polyhedra were discovered by the ancient Greeks. The Pythagoreans knew of the tetrahedron, the cube, and the dodecahedron; the mathematician Theaetetus added the octahedron and the icosahedron. These shapes are also called the Platonic solids, after the ancient Greek philosopher Plato; Plato, who greatly respected Theaetetus’ work, speculated that these five solids were the shapes of the five fundamental components of the physical universe.

Are we sure there are only five regular polyhedrons? By definition the faces of regular polyhedrons should all be regular polygons. Here are the possibilities:

• Triangles. The interior angle of an equilateral triangle is 60 degrees. Thus on a regular polyhedron, only 3, 4, or 5 triangles can meet at a vertex. If there were 6 or more, their angles would add up to 360 degrees or more and hence not admissible. Consider the possibilities:
  • 3 triangles meet at each vertex. This gives rise to a Tetrahedron.
  • 4 triangles meet at each vertex. This gives rise to an Octahedron.
  • 5 triangles meet at each vertex. This gives rise to an Icosahedron
• Squares. Since the interior angle of a square is 90 degrees, at most three squares can meet at a vertex. This is indeed possible and it gives rise to a hexahedron or cube.
• Pentagons. As in the case of cubes, the only possibility is that three pentagons meet at a vertex. This gives rise to a Dodecahedron.
• Hexagons or regular polygons with more than six sides cannot form the faces of a regular polyhedron since their interior angles are at least 120 degrees.

Hence convex polyhedral angles of regular polyhedrons can only be of three types: Trihedral (Tetra-, Hexa- and Dodeca-hedrons), Tetrahedral (Octahedron) and Pentahedral (Icosahedon). Hence we can say there can be only five regular polyhedrons as above.

The following are the patterns that may be cut off from paper or a card board and may be used to make 3-dimensional models of the regular polyhedrons.

My next blog on the subject will concentrate only on Tetrahedrons.

 Ref: Tamil Virtual Academy –  http://www.tamilvu.org/

Kamba Ramayanam, as popularly known, is composed by Kambar, who is dated between 855 and 1185 AD. The name given to the work by the poet is RAmAvathAram. He has followed the original work of Valmiki Ramayanam closely. It is revered as one of the major pieces of Tamil literature along with Thirukkural and SilappadhikAram. For more information the website above may be referred to.

Many thousands of Tamils would have enjoyed the emotions of the story of Rama as told by Kambar. Many hundreds of Tamils would have enjoyed, in addition, the beauty of Tamil language as handled by Kambar. Many people would have also enjoyed the poetic beauty and imagination of Kambar. It is as though, the nectar (of Ramayana) is stored in a gold cup (in the form of  Tamil language) which itself is decorated with diamonds (of beautiful Tamil Poetry). Myself, being just an insect, happened to squat on the edge of this dazzling cup and to lick a few traces of this nectar. In the following series of articles, I propose to share my delight with others.

In this first post, four verses are presented. They are describing the way Sita was decorated in preparation for her marriage with Lord Rama,. The verses and the summary in Tamil were taken from the above quoted site. The summary in English is mine.

Thanks to http://www.suratha.com/reader.htm for enabling conversion of the Tamil Texts to Unicode.

பால காண்டம்

கோலம்காண் படலம் 

1223 – தோழிமார் சீதையைச் சூழ்தல் 

பரந்த மேகலையும் கோத்த

                                         பாத சாலகமும் நாகச்

சிரம் செய் நூபுரமும் வண்டும்

                                          சிலம்பொடு சிலம்பி ஆர்ப்பப் 

புரந்தரன் கோல் கீழ் ஆனோர்

                                          அரம்பையர் புடைசூழ்ந்து என்ன 

வரம்பறு சும்மைத் தீம் சொல்

                                           மடந்தையர் தொடர்ந்து சூழ்ந்தார்

பரவின மேகலையாகிய இடையணியும், கோக்கப்பட்ட பாத சாலகம் எனும் காலணியும், பாம்பின் தலைவடிவில் முகப்புடைய நூபுரம் என்னும் காலணியும்,  கைவளையல்களும் காற்சிலம்பும் ஒலித்து ஆரவாரமுண்டாக்க; இந்திரன் ஆட்சிக்குட்பட்ட அரம்பை  முதலிய தேவப் பெண்கள் பக்கங்களில் சூழ்ந்துவந்தாற் போல; அளவு கடந்தவர்களாய், இனிய ஒலிக் குதலைச் சொல்லினராய். தோழிப் பெண்கள் பலர், பிராட்டியின் மருங்கே இடையீடின்றிச் சூழ்ந்து சென்றனர். 

Resembling the dancers of the heaven in King Indra’s court, the friends of Sita were surrounding her all the time. They were wearing broad waist bands, bejeweled footwear, cobra shaped armlets and beaded anklets, all jingling together along with their sweet giggles. 

1204 – சீதையைத் தாதியர் அணிசெய்தல் 

அமிழ் இமைத்துணைகள் கண்ணுக்கு

                                       அணியென அமைக்குமா போல்

உமிழ் சுடர்க் கலன்கள் நங்கை

                                       உருவினை மறைப்பது ஓரார் 

அமிழ்தினைச் சுவை செய்து என்ன

                                      அழகினுக்கு அழகு செய்தார் 

இமிழ் திரைப் பரவை ஞாலம்

                                      ஏழைமை உடைத்து, மாதோ! 

(கண்கள்) மறைதற்குக் காரணமான இரண்டு  இமைகளும் அக்கண்களுக்கு அழகென்று (இறைவனால்) அமைக்கப்பட்டிருப்பது போல;  (சீதைக்கு  அணிசெய்யும்  மகளிர்) ஒளிவீசும் ஆபரணங்கள் சீதையின் (அழகிய) வடிவத்தை மறைக்கும் என்னும் உண்மையை உணராதவர்களாய்;  (இயல்பாகவே, அத்துணைச் சுவைகளையும்    பெற்றிருக்கிற) அமிழ்துக்கு. (கற்கண்டு சர்க்கரை முதலியவை  கொண்டு) மேலும் சுவை கூட்டும் முயற்சியைப் போல; (இயற்கையிலேயே பேர்  அழகுவாய்ந்த சீதையெனும்) அழகிற்கு (மேலும்  புதிதாக) அழகு  செய்யத் தொடங்கினார்கள்;  அலை ஒலிக்கும் கடல்சூழ்ந்த இந்நிலவுலகத்து  மக்கள் (நன்மை புரிவதாக  நினைத்துப் பிழைபுரியும்) அறியாமையுடையவர்களே.     . 

Though the eye-lids cover the eyes, they are considered ornamental to the eyes. Same way the sparkling jewels hide her youthful body. Totally unaware of this, just like the futile attempt of adding taste to the already sweet divine nectar, the maids are trying to beautify, the already beautiful (Sita). It was appalling to see the ignorance of the people living in this world covered by the seas. 

1208 – கழுத்தணி அணிதல் 

கோன் அணி சங்கம் வந்து

                                    குடி இருந்த அனைய கண்டத்து

ஈனம் இல் கலன்கள் தம்மில்

                                   இயைவன அணிதல் செய்தார்

மான் அணி நோக்கினார்தம்

                                  மங்கலக் கழுத்துக்கு எல்லாம் 

தான் அணி ஆன போது,

                                 தனக்கு அணி யாது மாதோ?

தலைவனான  திருமால் தாங்கும் சங்கு (இங்கு) வந்து  குடியிருந்தாற் போன்ற (சீதையின்) கழுத்தில் குற்றமற்ற ஆபரணங்களில் பொருந்துவனவற்றைப் பூட்டினார்கள்; மானின் விழிகளையுடைய (உலகத்து) மங்கையர் மாங்கலிய நாணோடு கூடிய கழுத்துகளுக்கு எல்லாம்;  (திரு என்னும் அடையோடுகூடித் திருமாங்கல்யம் என்று) தானே ஓர் ஒப்பற்ற அலங்காரமாய் இருக்கும்போது;  தன் கழுத்துக்கு ஏற்ற அணியாவது எது? (ஒன்றும் இல்லை.)

Her neck looks like it (already) wears the white conch normally worn by Vishnu. (Still) more of such choicest of the spotless jewels were worn, by maids, on her neck. The women of this world, with beautiful deer-like glances, wear around their necks. Sita’s images as holy ornaments. Then what else can be an ornament for her (neck)?

1219 – பிராட்டிக்குக் காப்பிடுதல் 

நெய் வளர் விளக்கம் ஆட்டி, 

                                நீரொடு பூவும் தூவித், 

தெய்வமும் பராவி, வேத

                               பாரகர்க்கு ஈந்து செம்பொன்

ஐயவி நுதலில் சேர்த்தி,

                               ஆய் நிற அயினி சுற்றிக், 

கைவளர் மயில் அனாளை

                               வலம் செய்து காப்பும் இட்டார். 

(அணிசெய்து முடித்த தோழியர் பிராட்டிக்குக் கண்ணேறு படும் என்று) நெய் நிறைந்த விளக்கொளியைச் சுற்றிக்காட்டி, புனித நீரினோடு கூடிய பூக்களையும் முன்னால் இறைத்து; தெய்வத்தினையும் வழிபட்டு, வேதங்களைக் கரைகண்டறிந்த  அந்தணாளர்க்குச் தானம் அளித்து;  செம்பொன் போன்ற சிறுவெண் கடுகின் பொடிகளை நெற்றியில் தீற்றி, ஆராய்ந்தெடுத்த செந்நிற ஆலத்தி நீரைச் சுழற்றி; கைகளினால் எடுத்து வளர்க்கப்பட்ட மயிலைப் போன்ற பிராட்டியை வலப்புறஞ் சுற்றிக் காப்பு நாணையும் இட்டனர். 

The maids showed ghee-lamp to her. They showered on her, flowers and scented water. They prayed to God. They gave charity to brahmins steeped in vedic knowledge They sprayed golden dust on her fore-head. They surrounded her with saffron colored arati. Went around the peacock-like Sita and cast the evil eyes away by putting saffron on her forehead.

I am not sure whether my English summary did justice to the beauty of Kambar’s poetry. Here is one more attempt to bring out its beauty. Smt Gowri Nagarajan set these verses to music as a part of a dance opera enacted by students of Sri S. Natarajan of Melattur Bhagavata Mela. (I need to write about Sri S. Natarajan and Melattur Bhagavata Mela in a separate post). The verses have been tuned into popular ragas, Vasantha, Kalavathi and Maduvanthi. You may listen to the musical version of these poems by clicking below.

Azhakinirku

CellA idatthu china(m)theedhu cellidatthum

il adhanin theeya pira.

CellA Idatthu – in places beyond your reach or control

China(m) theedhu – Anger is harmful

Cellidatthum – even in places under your reach/control

adanin theeya pira – more harmful than it (i.e. anger)

il – one cannot find

One should not get angry in places not under one’s reach or control, since the same will not be tolerated. It may recoil on you. In addition, you will lose your purpose and also your goodwill for the future. This is easy to see. But, one should also understand that, even in places under one’s reach and command, there is nothing more harmful than anger. It has the danger of being highly disproportionate and will create ill will in your own camp.

In high places, wrath recoils, you know; 

In your own den, it’s a wild fire, you blow.

 Let us start at some of the values we were taught by schools, parents and community, at least till ten years back. 

-        We were taught not to lie. Then there was some moderation. Lying for a good cause was accepted. Still lying is not illegal unless it leads to large scale cheating. But lying under oath is illegal. Lying as a habit, should not be encouraged, irrespective of any thing else said earlier

-        We were taught not to smoke. Smoking is not illegal unless it is done in areas where it is prohibited. Neither it is immoral. But, it is not advisable to smoke in crowded areas and in front of elders and ladies. Irrespective of legality or otherwise, smoking as a habit should be discouraged at all times.

-        We were taught not to have sex before marriage. Having sex with a consenting partner is not illegal. But still pre-marital sex, though not illegal, is not something that should be widely encouraged. 

-        Being a transgender is not illegal. It is a biological defect which deserves all the sympathy and support from the community. But we should encourage such people to correct such biological defects early enough, with modern medical aids. We should never encourage them to establish themselves as a third gender.

-        Homo-sexual behavior may become ‘not illegal’, but this is not to widely encourage such a behavior. Homo-sexuality should be recognized as a behavioral problem and such people should be actively encouraged to correct the same. Treating them as new-heros and idols of liberation and celebration, will cause great damage to our society. It can never be a substitute for institutions of marriage and family. It is natural for anybody to discourage such homo-sexual behaviors and they should not be termed as retrogrades.

-        Sex is a natural part of our life. Our sexual life is very different from that of animals. Our sexual action results in families, unlike among animals. Sex within a bloodline is generally prohibited for a variety of reasons. We can say, as humans, we have an organized sex-life, which leads to an organized community with duties and responsibilities. For these reasons one-man-one-woman concept was accepted in our value system. Sex with multiple partners may not be illegal, but definitely is not to be encouraged. There is nothing ‘progressive’ in promoting ‘free-sex’.

-        Enjoying the close company of the other sex is not illegal, as long as it is done away from public glare. Laws of decent behavior protect the community from harmful exposures. Hence such exposures should be curtailed to reasonable limits.

The police will not interfere on activities which are within legal limits. But, as listed above, there are many other activities, though within legal limits, needs to be checked, or shall we use the bad word, ‘policed’. This kind of policing, generally referred to as ‘Moral Policing’, is done by community leaders, religious gurus, social activists and in some extreme cases by law enforcing agencies like police.

This kind of Moral-Policing is:

         Secular as it is practiced by all religious groups

         Democratic as it is adopted by all communities

         Non discriminative as it is supported by both genders

         Not anti-social as it is approved by all economic sections.

This moral policing is despised only by a very small minority of the new intellectuals and pseudo-secularists. We need Moral Policing for the community values to survive for our future generations.

Note: You may read my earlier blog on this subject http://lvnaga.wordpress.com/2009/02/16/moral-policing/

By John Horgan (2003)

I read this book a few years back and had prepared a set of quotes from this book. Having read and reflected on these quotes several times, I thought of sharing the same with my friends through this blog.

1. If you say you are advancing towards God or the Absolute, and (you) are not growing in love and charity towards your fellow persons, you are just deluding yourself. – by Huston Smith, Ph.D., University of Chicago.

2. Religious institutions have caused a great deal of harm, particularly when they insist that “believers are superior and non-believers are inferior or evil”. That is where religion becomes a damaging force. – by Michael Persinger, Psychologists, Laurentian University, Sudbury, Canada (1999)

3. God is unnecessary for my living or loving or being comfortable. I am amazed by the marvels of nature surrounding me, but I feel no need to attribute them to a divine creator. We can decide that this is God-inspired. Or, we can decide that this is some kind of happening that is ‘beyond belief’. – by James Austin, Neurologist, University of Colorado, Health Sciences Centre. Denver (1998).

4. Why should I be afraid of dying? I was not afraid of being born – by James Austin (1998)

5. You have this divine source that has the propensity to create and project itself out. Then when you end up here as a separate unit, there is trouble, suffering and so on. All these propel us back towards the source. These two opposite forces are called ‘hylotropic and ‘holotropic’, (i.e.) centrifugal and centripetal. – by Stanislov Grof, Psychiatrist, John Hopkins University.

(LVN’s Comment: It is like a man-made fountain, where the same water circulates from the tank to the fountain and back to the tank.)

6. Many psychological problems are ‘spiritual emergencies’ that stem from a deep rooted yearning for spiritual meaning and consolation. Properly treated, these spiritual crises offer a tremendous opportunity for growth. – by Mrs Christina Grof, Patient, student and wife of Stanislov Grof (referred above).

(LVN’s comment: Can we call the Tamil Poet, Subramania Bharati, as a classic example)

7. Ancient Hindu texts, the Upanishads, claim that the only reality is the formless, infinite and eternal void, from which all things emerge and to which they return. All else, including your mortal self is unreal. We cannot die because we do not exist in the first place. The Upanishads promise us that when we really know this fundamental truth, we will achieve Nirvana, Bliss or Heaven. To ordinary men like me, seeing the world and myself as unreal, itself, felt more like hell. Those who are enlightened, blissfully enlightened, must somehow sidestep or push past this dreadful state. BUT HOW? – by John Horgan, Author of Rational Mysticism.

8. The ‘mysterium tremendum’ (the inner sanctum of reality) is not a thing we can possibly identify with, let alone become. It is not a deity, nor a force, principle, spirit nor a ground of being – it is not a thing at all. It is ‘wholly other’, ‘nothingness’ and the opposite of ‘everything that is and can be’. It is absence not presence. Our encounter with ‘mysterium tremendum’ can strike us chill and numb and fill us with an utmost grisly horror and shuddering. (the state that could be called ‘mysterium tremens’ ) Religions do not reveal the ‘mysterium tremendum’ so much, as they shield us from direct confrontation with it. – By Rudolf Otto, German theologian.

9 You may call it infinite or call it God, Allah, Brahman, Void or ’mysterium tremendum’. It is the nothingness from which we came and to which we must return. You may feel compelled to ‘guess the riddle’, to explain how a ‘finite human something’ emerged from ‘infinite inhuman nothing’. You may respond to such a vision with joy, madness, terror, love, gratitude, hilarity – or all the above at once. You may delight in the world’s astonishing beauty or despair at its fragility and insignificance. – by John Horgan, Author of Rational Mysticism.

10. If a miracle is defined as an infinitely improbable phenomenon, then our existence is a miracle, which no theory natural or supernatural will explain. Science can never answer the ultimate question: How did something came from nothing? Neither theologies can. Honest physicist will admit that they have no idea why there is something rather than nothing? – by John Horgan.

11. You know there is no reason for you to exist. The odds seem over-whelming, our miniscule human hubbub will be swallowed up by the emptiness whence we came. But the flip side of this mystical terror is joy. We should not be here. Yet we are here. How lucky we can get? – by John Horgan.

12. In my kitchen, we put garbage in bag that come in boxes of twenty. After I yank the last bag from the box, the box becomes garbage and goes inside the last garbage bag. There is a paradox lurking within this ritual.- by John Horgan.

13. FREE WILL – Our belief in free will has a social value. It provides us with metaphysical justification for ethics and morality. It forces us to take responsibility for ourselves rather than entrusting our fate to Jehova, Allah or Tao. We must accept that things will get better and better, only as a result of our efforts, not because we are fulfilling some pre-ordained supernatural plan. If ‘free will’ is an illusion, it is the one I need – that I need, even more than God. I have no choice but to choose free will. – by John Horgan, Author of Rational Mysticism

Please read and reflect.

It was very interesting to read a paper by R.Fetea and A. Petroianu (of University of Cape Town), on the above subject. (refer: http://www.el.angstrom.uu.se/kurser/water05/Reactive_Power.pdf).

Reactive Power as a concept is really strange, but is very necessary for power system management. So we have to live with it. But we need some kind of reconciliation with many valid points raised by the authors of the above paper. Let me try the same in the following paragraphs. 

Let us start from their first two equations:

v = Vmax Cos(ωt)

 i = Imax Cos(ωt – θ)

Actual Instantaneous power = vi = Vmax Cos(ωt) Imax Cos(ωt – θ)

Let us now take the expression for current:

 I = Imax Cos(ωt – θ)

    = {Imax cos θ cos ωt + Imax sin θ sin ωt}

    = {Imax cos θ cos ωt – Imax sin θ cos (ωt – π/2)}  

We may now recognize the two terms in the above expression as:  in-phase and 90-deg-lagging components of current, I.

We say,

Active component:      Ia = Imax cos θ cos ωt

Reactive component: Ir = Imax sin θ cos (ωt – π/2)

Let,

Root Mean Square value of v, V= (Vmax/√2)

Root Mean Square value of  i,  I = (Imax /√2)

Then,

(instantaneous) Active power, p = v Ia

 i.e.,    p =  Vmax cos(ωt) Imax cos θ cos ωt

                 = (Vmax/√2) (Imax/√2) cos θ (2cos² ωt)

                 = VI cos θ (2cos² ωt)

                 = P (1 + cos 2ωt),        where P = VI cos θ

This is a positive sinusoidal function with an average value of P, the active power, transferred through the circuit.

(instantaneous) Reactive Power, q = v Ir

i.e.,  q = Vmax cos(ωt) Imax sin θ sin ωt

              = (Vmax/√2) (Imax/√2) sin θ (2 sin ωt cos ωt)

              = VI sin θ (sin 2ωt)                                        

              = Q (sin 2ωt), where Q = VI sin θ

This is a sinusoidal function with an average value of zero. But still we say a reactive power of value Q is transferred through the circuit!! Why at all?

Assume a power source and a load connected as below

If the load R is purely resistive,

            V —————————- R

there will be only active power flow through the circuit, as voltage and current will be in phase.

Suppose we add an inductive load X_L. Now the current will lag the voltage by an angle, and hence there will be some reactive power flow also.  

            V —————————- R+ X_L

Though average of this reactive power will be zero, the source will still have to ‘supply’ this reactive power flow also.

Suppose we now add a capacitative load Xc, to exactly compensate this inductance.

            V —————————- R + X_L +  Xc

Then the source need not ‘supply’ any reactive power, as the same is ‘compensated’ by the capacitance. (i.e, the lag by inductance is nullified by the lead created by capacitance). However there will be reactive power flow between X_L and Xc. Sometimes it is said, Xc ‘produces’ lagging reactive power to ‘supply’ X_L. 

It is exactly in this way, we handle the reactive power Q, even though its average is always zero.

Actual instantaneous Power

                          = Active power p + Reactive power q

                          = P (1 + cos 2ωt) + Q (sin 2ωt)

                          = P + P (cos 2ωt) + Q (sin 2ωt)

The average values of both second and third terms above are zeros over a voltage cycle. Hence P become actual average power transmitted over the circuit.

As phase angle θ varies from 0 to 90, the value of ‘active’ power P reduces from maximum value of VI to zero and ‘reactive power’, Q increases from zero to VI. However all through, the magnitudes of AC voltage and AC current remain the same and hence the value of VI. For this reason VI is known as apparent power, S.

 S² = VI²(cos² θ + sin² θ)

       = (VI cosθ) ² +  (VI sinθ) ²  = P² +  Q²  

S = √ (P^2 + Q^2) = VI, is an important parameter known as VA which is widely used for specifying power ratings of electrical devices such as generators, transformers and even major loads. Now, in phase component of S, S cosθ is same as active power P. The cross phase component of S, S sinθ is called reactive power, Q, which does not appear very strange now.

Now we may use complex algebra to revisit the same concepts as above. 

Phasor Representation

By using Euler’s Formula, any sinusoidal function can be expressed as
V = Vmax cos(ωt- θ) = Real Part of [Vmax e^{j(ωt- θ)}]

                                           = Re[Vmax e^-jθ  e^j(ωt) ]

[Vmax e^-jθ] is known as V’, the phasor representation of V(ωt), represented as V∟-θ

Now, V = Re [V’ e^j(ωt) ]

The time-dependency has been effectively factored out, in the phasor representation as it deals with only the static quantities of amplitude and phase angle.

By phasor representation as above, we may write,

V’ = V∟0 and I’ = I∟-θ

Apparent Power S’= V.I* = V.{I e^-jθ }* = VI∟θ

Alternately, if V’ = V∟-θ1 and I’=∟-θ2

Apparent Power S’= V.I* = V e^-jθ1.{I e^-jθ2 }*

                                      = VI∟(θ2- θ1) = VI∟θ,

where θ = (θ2- θ1), the actual phase difference.

 (i.e)    S’ = VI e^-jθ   = VI (cos θ + j sin θ)

                  = VI cos θ + j VI sin θ

                   = P + j Q

As we see the real part of S’ is ‘Active Power’, VI cos θ; and imaginary part of S’ is Reactive Power, VI sin θ. In this representation the Reactive Power does not seem as strange as earlier. 

Let us again consider only a resistive load, R. As in this case current will be in phase with voltage, θ = 0. Hence,

 S’ = V’. I’* = V∟0 . I∟0 = VI cos 0 + j VI sin 0

(i.e)      S∟0 = P + j0, where P = VI

Impedance Z’ = V∟0 / I∟0  = V/I cos 0 + j V/I sin 0

                             = R + j0

Now consider only an inductive load of X_L   In this case the current will lag voltage by an angle of 90 degrees, i.e., π/2. Now,

S’ = V’ I’* = V∟0. I∟π/2  = VI cos π/2 + j VI sin π/2

(i.e)      S∟π/2 = 0 + jQ, where Q = VI

Impedance Z’ = V∟0 /  I∟-π/2  = V/I cos π/2 + j V/I sin π/2

                              = 0 + j X_L

Now consider an combined load of R and X_L . In this case there will be a phase difference of, say, θ, lagging. Hence,

S’ = V’ I’* = V∟0. {I∟- θ}*  = VI cos θ + j VI sin θ

(i.e)       S∟θ = P + j Q, where P = VI cos θ and Q = VI sin θ

Impedance = Z’ = V∟0 /  I∟- θ  = V/I cos θ + j V/I sin θ

                                 = R + j X_L

               We recognize R = V/I cos θ and X_L = V/I sin θ    

When a capacitor is added to the load, we have

                Impedance Z’ = R + j X_L – j Xc

Here also we observe the mutually nullifying effect of X_L and Xc. Hence we are able to represent combined resistive and ‘reactive’ loads conveniently as a phasor or a complex number, known as impedence, Z.

It appears that Reactive Power concept, though somewhat strange, is very useful. By this concept, complicated trigonometric functions have been reduced to simple(!) complex algebra.

Yet another set of strange things about reactive power is its direction of flow and sign. If you still have appetite for further confusion you may refer to my blog:

 http://lvnaga.wordpress.com/2010/02/19/electrical-power-and-power-factor/   and

http://lvnaga.files.wordpress.com/2010/02/direction-of-flow-of-active-and-reactive-power.doc

We may meet again later.

L V Nagarajan

1. What defines music?

Music is any sound that is pleasant.

2. What is sound?

Sound is a set of vibrations produced in the air. It may be produced by clapping two solids together, or a flowing liquid, or a blowing air. The vibrations are a set of several frequencies with which air vibrates. Contrary to this there are sounds which dominantly have only one frequency, like plucking the strings of a Tanpura or Vina. There are sounds which are of instant duration (clapping your hands) compared to others of longer duration (striking a gong). There are sounds which are produced repeatedly on regular intervals (running train) or irregular intervals (car horns during traffic jams). From the above we instantly recognize the sounds that are pleasant – a) sounds which are dominant on single frequency, b) sounds which are of some discernable duration and c) sounds which occur at regular intervals. These are the sounds known as notes and beats in music. They give rise to melody and rhythm.  

3. When does the sound become music?

Many times we listen to sounds from one, two or more sources, simultaneously or in quick succession. If the two sounds are of same dominant frequency, they produce unison (resonance); if they are of two ‘agreeable’ frequencies they produce harmony (consonance); if they are of two ‘acceptable’ frequencies they produce assonance (musical); otherwise they produce dissonance (unmusical, noise). In Indian musical parlance, we may say, Vadi, Samvadi, Suswaram, Apaswaram, respectively. A sound with a dominant frequency is generally referred to as a ‘swara’ or ‘note’. A melody is produced if a set of ‘acceptable’ notes are sung or played in any instrument, in an ordered sequence. A harmony is produced if a melody is accompanied by another one or more in a way to produce a consonance. Both melody and harmony can be further enhanced by setting them to a rhythmic pattern of beats produced by other instruments. 

4. What is a Raga?

Raga in Indian Musical system is a defined melodic pattern of notes. Indian classical music heavily depends on melody unlike western classical music which depends on harmony. There are many aspects which define a Raga. They are typically:

-         acceptable set of notes (loosely called as scale)

-         Rules of usages of these notes in ascent and dissent

-         Frequent or rare usages of a few of these notes

-         Staying on and/or glancing over a few specific notes

-         Use of a ‘foreign’ note in a highly specific sequence

-         Flatness of a note or gliding and waving over specific notes   

These characteristics of ragas are evolved over a long period of practice of these ragas. Many of these definitions cannot be codified in any form except by listening and absorbing. Hence these ragas continually evolve over time. In carnatic music, we are lucky that these raga forms and definitions are preserved in the form of thousands of compositions. In major ragas there are umpteen numbers of compositions which preserve these ragas very effectively over centuries.

5. What is a scale?

As our ancient music evolved there were attempts to integrate them under a common grammar. Our first grammarian was probably Bharata Muni, though there could have been other contemporary grammar texts like Silappadikaram of Tamil origin. But Bharata Muni in his Natya shastra could have integrated other existing texts also in his treatise. Several of these texts mention 22-sruthis as the basis for our music. (http://www.naadhabrahmam.com/marga_desi.asp). These 22 shrutis were grouped into 7-unequal intervals in two different ways called as gramas, namely shadja grama, and madhyama grama:
Shadja grama:  S  4  R  3  G  2  m  4 P  4  D  3  N  2
Madhyama grama:  S  4  R 3  G  4  m  2  P  4  D 3  N  2

The melodies existing in those times were probably classified into these basic groupings by adopting modal shifts known as murchanas. These gramas may be loosely called as scales. But they are very different from present day scales.

The concept of present day scales has been imported from the western music. In the West an octave of eight notes are defined starting from a note of a specific frequency to its resonant note of double the frequency. This octave (set of eight notes) was divided into 12 equal intervals to give us 12 semitones as they call it, thus sacrificing some amount of consonance and assonance between notes. This system was called ‘equi-tempered’ as opposed to another system called ‘just-tempered’ which takes care of consonance ratios. These notes are denoted as

C, D1, D2, E1, E2, F1, F2, G, A1, A2, B1, B2, C

Still we have only seven notes; C, D, E, F, G, A, B, C. Except the 1^st and 5^th note (C and G), all other notes have variations. Using these notes scales were created. Western musical instruments are tuned accordingly.

Somewhere in the 17^th century this concept of scale was adopted by theoreticians of carnatic music. They were also influenced by the keyed instruments of western music like piano and harmonium. Hence attempts were made to fit our ancient musical system into the keys of the above instruments. At this point scales were introduced in carnatic music, to group all the notes used in a raga.    

It is in this period that a new factor called ArOhaNa-AvarOhaNa came to be incorporated as a characteristic of a rAga. Prior to this the ArOhaNa-AvarOhaNa, in the manner of a formula, had not been mentioned in any works. The incorporation of ArOhaNa-avarOhaNa as a lakshaNa of a rAga influenced the definition of scales. Though our ancient musical system is based on 22 Sruthis (or even more), the scales were defined using only the 12 notes as above as:

Sa, ri1, ri2, ga1, ga2, ma1, ma2, Pa, da1, da2, ni1, ni2, Sa.

Our ancient ragas like Varali and Nattai used notes that could not be accommodated in this scheme. Hence, as an approximation, 16 swarasthanas were defined among these 12 notes as:    

Sa, ri1, (ri2/ga0), (ri3/ga1), ga2, ma1, ma2, Pa, da1, (da2/ni0), (da3/ni1), ni2, Sa.

The swaras ga0 and da3 were required to accommodate ancient ragas, like Varali and Nattai respectively. Using these 16 swarasthana, 72 basic scales were ‘created’ in carnatic music as ‘Melakarthas’, though most of these scales did not exist in actual practice. The ragas existing prior to this period were forcibly fitted into this scheme. New ragas were also created. Saint Thyagaraja and Mutthu Swamy Dikshatar composed  krithis to popularize these newly ‘created’ scales/ragas.

Thus in carnatic music theory, we obtained 72 basic scales.

(For more on this please refer to FAQ-1 on the subject by clicking on the following link)

http://lvnaga.wordpress.com/2008/09/15/swaras-swarasthanas-and-sruthis/ 

6. Will you say that scale is not important for a raga?

Scale, if it is meant to list the notes used in a raga, it is useful. But it can never be a unique definition of a raga. There may be two or more ragas using the same scale: (eg) mayamalavagowla and nadanamakriya. These ragas are very different in their moods and ranges. Conversely there are ragas which cannot be bound to any specific scale as we know now: (eg) Anada Bhairavi and Natakurinji. There are some features of ragas which cannot be codified into any scale. Any attempt to do so will only restrict their range and application. Such defining features are: 

Vakra prayoga: where notes of a scale are used in a non serial fashion – like pGmrs in raga Kanada.

Varjya proyogas: Skipping over a note of a scale in some phrases – (eg) skipping ma in Todi

Bhashanga swaras: Taking in a foreign note outside the scale in some prayogas – (eg) both Nishadas in Begada. 

7. What do you say about the ragas created based on the 72 scales?

Even though initially these 72 scales were called melakarthas, gradually people started setting them up as ragas. Similarly new ragas were created from these scales using sampoorna/shadava/audava combinations of 7/6/5 notes in the arohana/avarohana. All these ragas are heavily dependent on the scale and many times sound similar in some phases. Most of them have failed to evolve their own unique characteristics apart from the notes they use. However, at least some of these ‘created’ ragas have evolved into unique raga patterns.

I quote Professor N.Ramanathan ( http://www.musicresearch.in/)  

•            rAga is not a tune, melody or meTTu. There may be more than two tunes in a rAga.
•            The notion of rAga is not based merely on the svara-s or on the svarasthAna-s.
•            The behaviour or the manner in which the svara-s move about and their sancAra-s, bestow the individuality to a rAga.
•             A rAga consists of svara-s some of which are very strong or profuse in the melodic movements while some others are relatively weak or rare.  

8. I have gone through your earlier note on Swaras, Swarathanas and Sruthis. I have somewhat understood the derivation of 22 Sruthis as proposed in the above note. Do you mean to say the Melakartha scheme has ended our concept of 22 Sruthis? 

No. Not exactly. Neither that was the intention. Melakartha scheme has actually achieved quite a few things: i) it has enabled our classical music to be played on keyed instruments like harmonium, though not perfectly, ii) it has enabled new scales/ragas to be developed and evolved, iii) it has enabled better codification of our musical scripts. 22-sruthi concept was expected to survive this development, at least through the ancient (pre-melakartha) ragas existing before this development (16th Century).

9. What are these pre-melakartha ragas? Do they retain there original forms? 

As per Mr. Ramanathan, there were only about 67 numbers of naturally evolved ragas existing at the turn of 15th Century. Some of them have evolved from Tamil musical system of PANN’s; some more of them from other hymns and folk music systems. Some of them existed in Hindustani system. A few came from other countries. Many of them still retain their original forms, thanks to the multitude of compositions in these ragas. Some of these ragas have changed forms but still retaining there micro tones (nutpa sruthis). But lighter forms of these ragas used in devotional and light music are now following the 12-notes pattern as dictated by keyed instruments like harmonium. This may eventually lead to some loss of nutpa sruthis, which is unfortunate.

 10. What about the ragas invented based on melakartha scheme? 

Prof Ramanathan calls them as created ragas. As per him, many of them are yet to evolve their distinct features. These created ragas are all defined more by their scales than by their sancharas and prayogas. 

Let us continue our discussions in the next blog.

Says P. Sainath, a Magsaysay Award winner and a chronicler of the unending human tragedy that’s unfolding in rural India, “There are 311 billionaires in India, a survey says this is the fourth happiest country in the world, and we had not one but two fashion meets this year.’’ The scathing sarcasm is laced with anger: “It’s very clear who the government exists for. When the sensex fell a few years ago, it took two hours for the then finance minister to come to Bombay by a special flight to hold the hands of weeping billionaires. It took ten years for the prime minister to visit farm households in a state where over 40,000 farmers have committed suicide since 1995 according to government data.’’

Now to visuals of a hungry child asking his mother for food, now to cattle patiently ploughing the soil:

Mute calves from Warhad are we

Watching the plunder of mother’s milk

Drenching the earth drop by drop

With our sweat yielding pearls

Yet our babies in hunger fret.

(Warhad is a village in Maharashtra)

 
There’s a clip of Sainath’s address to Parliament where he talks of how farmers were forced to kill themselves because they couldn’t get 8,000 rupees at a decent rate of interest. “And after covering such cases, I come back to my house and get a letter from my bank offering me a loan for a Mercedes at six per cent interest, no collateral required. What kind of justice is there in such a society? What kind of justice is this?’’ he demands agitatedly as Rahul Gandhi and Mani Shankar Aiyer look visibly uncomfortable. 

Deepa closes her film by cutting back to the question raised at the beginning: Who were Nero’s guests? Sainath then relates the true story of Nero, the notorious Roman emperor who, faced with a paucity of lighting at a grand party, provided it by emptying his prisons and burning undertrials at the stake. “The guests at the party were the elite of Rome, and to the best of our knowledge, nobody protested,’’ Sainath says. “I always wonder what sort of mindset it would require to pop one more grape as another human being bursts into flames.’’ Parallel drawn, he pauses for a second, and then continues: “We can differ on how to solve this problem, on even our analysis of the problem. But maybe we can make one starting point: we can all agree that we will not be Nero’s guests.’’

You can revisit your conscience by logging on to www.nerosguests.com. You may by the DVD and Part of the proceeds will go to farmers’ families in distress.

Many practicing electrical engineers, some even in the utility, do not have a clear  understanding of the concepts of Active and Reactive Powers and the Lagging and Leading Power Factors in electrical supply lines. Many do have an implicit knowledge of them, adequate under any normal circumstances. In this note an attempt is made to derive these concepts from basic principles of Ohm’s law and I²R power. This will also lead to a better understanding of quality issues of electrical power as supplied to customers.

2.0   Basics

Power in an electrical circuit is commonly understood as the product I²R of resistance and current-squared. By Ohms law, it is also expressed as VI or V²/R where I, V and R are the usual representations for Current, Voltage and Resistance. The above expressions remain largely true as long as we consider direct current (DC) circuits. When you consider alternating currents, the input voltage is alternating between a positive and a negative voltage, as a sine wave, (normally) at a frequency of 50 or 60 cycles per second. In this dynamic situation, two other major elements of the circuitry gain importance, namely, the Inductance (L) and the Capacitance (C). They are together called as Reactance (X) and they, along with the Resistance (R), affect the flow of current in a circuit profoundly. When a voltage is applied to a circuit with reactance (X), it takes some time for the current to get established to a steady state condition, due to induced voltage across the inductance and due to charging up of the capacitance. Even in the case of AC voltage input, the resulting alternating current reaches a steady state condition, but due to the effects of induced voltage and capacitance charging, there is a displacement between the current and voltage waveforms. This displacement is known as phase angle between the AC-voltage and the current. Coming back to our discussion on electrical power, V*I is still the power, but in this case, it is an alternating power. Initially let us consider an AC circuit only with a resistive load. As before I²R is the power consumed in the circuit. As the current is alternating the power also will be ‘alternating’. So the average power in the circuit will be R multiplied by the average of I² over a cycle of the alternating current. This average of I² over a cycle is knows as the Mean Square value. The square root of this current is known as the Root Mean Square value or I_RMS . Same way, we can define a V_RMS for the voltage wave form. Without going into rigours of mathematics, Power in a AC circuit with a resistive load, can be expressed as:

Power, P = I_RMS².R =   V_RMS²/R =  V_RMS * I_RMS .

For a pure sinusoidal waveform, RMS value = Peak Value/ √2

3.0   Complex Power

Now let us consider an AC circuit with Resistance (R) and Reactance (X). To represent resistance and reactance together, we have a term known as Impedance (Z). As discussed earlier, power can be expressed as I²Z or V²/Z. To enable AC circuit analysis, all these parameters are expressed as vectors or complex numbers, as below:

Voltage V= V e ^jo= V + j0  —- (Reference)     

Current C = I e ^{- jØ}   =  Ia – jIr

Impedance Z = Z e ^jØ = R + jX

Total Power = V* C^*  = V * I e ^jØ = P + jQ

[where Ø = arctan(X/R)]

The Total Power as mentioned in the above expression is normally known as Apparent Power, S, expressed in units of Volt-Ampere (VA). In Z, if reactance X is zero, then Ir will be zero, hence Ir is known as reactive current. Same way if R is zero, Ia will be zero, hence it is known as resistive current, or more commonly known as active current.

Now we have, from above,

S = V*(I cos Ø + j I sin Ø) = P + jQ = V*Ia + jV*Ir

This angle Ø is immediately recognised as the phase displacement between voltage and current waveforms introduced by the presence of reactance X in the circuit. At the instant, when ‘V’ achieves its peak value of the sine wave from, ‘I’ will lag behind and will have a value of only I cosØ. Active power, P, is the actual Active Power in the circuit, whereas Q is the imaginary power generated by the induced emf in the inductance (and the charging emf in the capacitance), as a reaction to the (sinusoidally) varying applied voltage. Hence Q is termed as Reactive Power, expressed in units of Volta-Ampere-Reactive (VAR). 

Now we are ready to write the full expressions for Power in the AC circuit with resistance and reactance as,

The Magnitude of Apparent Power |S| =  V_RMS . I_RMS       (VA)    

Active Power       P  =  V_RMS . I_RMS cos Ø,     (Watt)

Reactive Power    Q = V_RMS . I_RMS sin Ø        (VAR)

 The term ‘cos Ø’ is known as the Power Factor.

4.0   Effects of Frequency and Distortion

Another important factor is that the value of reactance X is frequency dependant.  The inductive reactance X_L increases directly as frequency whereas capacitive reactance X_C decreases inversely as frequency. The modern power systems have consumers whose loads include many more dynamic elements in addition to L and C in the form of rectifiers, non linear loads and switched mode power supplies for electronics circuitry, etc. These loads tend to distort the current and voltage wave forms away from a pure sinusoid. To analyse such circuits, the current and voltage forms are considered to have several harmonics components superimposed over the basic sine wave of 50 or 60Hz. The power calculations get further complicated if these harmonics are considerably high. Even in DC circuits the so-called ripples create similar ill-effects on power calculations.

5.0   3-Phase Power

So far we have confined our discussions to single phase AC circuits. Now let us move on to 3-Ph AC.

From now on, V and I mean only RMS values unless otherwise specified.

Trivially we may write for 3-ph AC,

P = 3 (V I) cosØ

However we should specify that both V and I are per-phase values. In a normal situation voltage between phases (known as line voltage) is more important than voltage of each phase, (Phase Voltages). In a 3-phase system,

V = V(line) = √3 * V(Phase), and hence,

P = √3 (V I) cosØ, and Q = √3 (V I) sinØ

6.0   Lagging and Leading Phase Angle

Ø is already recognised as the angular displacement between the voltage and current sinusoids of the circuit. This displacement is the result of the presence of inductance and/or capacitance in the circuit. The induced voltage across the inductance makes the current to lag behind voltage by a phase angle Ø, whereas the delay in charging up the capacitance  makes the current to lead the voltage by a  phase angle Ø. Accordingly the phase angle Ø will be (+) positive  or (-) negative. The active power P remains positive in either case, whereas the reactive power Q changes sign as per the inductance or capacitance in the circuit. It may be observed that the lagging reactive power Q is rendered as positive in the earlier expressions for complex power. Lagging Q is considered as consumption of lagging reactive power. The leading reactive power is negative and is sometimes considered as generation of lagging reactive power.

7.0   Active and Reactive Power.  

Active power is the real power resulting in actual work done. Reactive power is a necessary nuisance. The inductive load requires a higher current for the same amount of power and thus the power source also needs to supply this increased current. As this increased current does not result in any actual work done, it is termed as reactive current, Ir. The current I in the circuit is resolved into two components: one component Ia is in phase with Voltage and another component, Ir, with a phase angle of 90 degrees lag with the Voltage. This lagging reactive power requires to be compensated by the source, by ‘generating’ this reactive power. This is done dynamically by the following process: with active power remaining same (say), if reactive load increases, it results, (a) in a demand for higher current, (b) which drops the voltage all along, (c) voltage regulator at the generator end senses this, (d) generator terminal voltage gets picked up automatically or manually (for essentially the same output power), (e) phase angle between voltage and current increases, resulting in higher generation of reactive power as required by the system. But generators in the system have capacity limitations on reactive power generation and total volt-ampere generation. These may ultimately result in lower voltages all through the system, when the system reactive power requirement exceeds the total reactive capacity of generators in the system. Generation of reactive power is comparatively cost free. But to generate the same at the generator end and then to transmit it to the load end where it is required, is costing the power utility in terms of higher transmission losses. Hence the reactive power compensation is more effectively done at the load end, by using shunt capacitor banks. We know that capacitors act as leading-reactive loads. But in this context, we use them as lagging-reactive source. In general, in a utility power system, – just like we balance the active power requirement by active power generation by using frequency as our index -, reactive power requirement is balanced by reactive power generation by using system voltage as the index. In this process, in addition to generators, the shunt capacitors also contribute as lagging reactive sources. For voltage/reactive control of power systems, utilities also use a device known as Synchronous Condensers, loosely described as AC generators-without-a-prime-mover, which can generate only reactive power, both leading and lagging.

8.0   Direction of flow of Active and Reactive Power

Even though the AC current flows alternatively in both the directions, the direction of AC current is always rendered positive in the direction of power flow. In power balance calculations at any node in the power system, by a convention adopted by most of the utilities, the outgoing power from the node is taken as positive and incoming power as negative. For a detailed discussion on directions of active and reactive power flow please refer to the link below

Direction of flow of Active and Reactive Power 

The link also includes a drawing showing the quadrant principle of power factor.

9.0   Power Factor Monitoring

The power factor has already been defined earlier as cosine of phase angle between voltage and current wave forms in an AC electrical circuit. This is an important parameter that affects the quality of power supply and also the performance of the power system. Hence power factor requires to be monitored at all the important nodes in a power system and also at all bulk power supply points. But what is a power factor? It is just a measure of reactive power requirement as demanded by the various types of connected loads. In a three phase AC power utility system, the power factor is rather an ambiguous measurement for the following reasons – the phase angle between current and voltage wave forms is very likely to differ significantly among the three phases – both current and voltage waveforms may not remain strictly sinusoidal due to the presence of harmonics thus affecting the phase angle and power factor. In a way to solve some of these ambiguities in the power factor as defined (called some times as displacement power factor), another term, true power factor is defined as the ratio of total active power to total apparent power. Utility penalties and other decisions to improve performance of the power system are based on this true power factor.

In addition there are problems in online monitoring of the power factor. Power factor varies in the range of 0 to 1. The value as such does not say whether it is lagging or leading.  Some utilities use a range of ‘-1 to 0 to +1’ for power factor to go from lagging to leading PF ! In this representation the middle range of, say, -0.5 to +0.5, is a non acceptable range. The ends of this range, -1 and +1, are essentially same representing unity PF with no phase lag or lead. Such a representation for power factor as a mesurment appears ridiculous. (even the limits for LOLO, LO, HI and HIHI conditions cannot be defined for this parameter).

Some energy meter manufacturers use a range of 0 to 100 to 200 for pf; 0 to 100 representing ‘lagging pf 0 to 1’ and 100 to 200 representing ‘leading pf 1 to 0’. Many utility engineers are not comfortable with this usage. The author of this note has solved this problem in an Indian utility by defining two pfs, namely ‘Leading pf’ varying from 0 to 1 and ‘Lagging pf’ varying from 0 to 1. Both were derived as calculated points from the actual measurement of pf.

Furthermore, pf is a not an easily measurable parameter and it is a highly fluctuating parameter.  For all the above reasons, the author of this note feels pf may not serve well as a parameter to monitor and we may think of other ways to achieve monitoring of reactive power requirement in a system. May be tan Ø, instead of cos Ø, will serve this function better. Tan Ø varies from -(infinity)  to 0 to +(infinity) , as Ø varies from -90 to 0 to +90. It gives the ratio of reactive power to active power and hence may be termed as ‘Reactive Factor’. Utility penalties and other decisions to improve performance of the power system can be based on this reactive factor. This reactive factor can be easily monitored. This is only a suggestion for further consideration by power system operators and experts.

10.0   Conclusion

An attempt has been made in the above note to resolve some of the ambiguities as felt by many practicing utility and industrial electrical engineers in understanding the concepts of Reactive Power and Power Factor. The effect of high reactive requirement on the utility system and the need to penalise low pf consumers are also explained. I will be glad to receive suggestions and comments.