Wednesday, 6 April 2016

Fibonacci Colour and Raaga

The Fibonacci sequence is given as

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181,

where the m-th term Fm = Fm1 + Fm2, for m 3, is the sum of the previous two terms with F1 = 0 and F2 = 1. Note that F3k+1 is even, for k {0}. The other numbers are always odd. This can be easily observed by dividing each Fibonacci number Fm by 2 and writing down its remainder. Thus, we get

0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1,.

The remainder zero corresponds to even terms and remainder one corresponds to odd terms. Note that the pattern repeats itself as 0, 1, 1, a set of three elements in modulo 2. It so happens that this property is true if we divide by any natural number k instead of 2. In that case we get a repeated pattern in modulo k. The periodicity with which the pattern repeates is called Pisano period. The pisano period was first introduced by Lagrange while addressing the question of in what frequency does the last digits of Fibonacci numbers repeat itself. Answering this question is equivalent to dividing each Fm by 10 and considering the remainders. It turns out that in modulo 10 the periodicity is 60.

1 Modulo n

The set of whole numbers

0, 1, 2, 3,

are infinite but they are smallest of the infinity. They are listable or what mathematicians call denumerable or countable, in contrast to, all real numbers which are infinite but not denumerable. Consider a n and identify all its integral multiple, i.e, 0,n, 2n, 3n,, as one element denoted as [0]. Similarly, we identify all elements which leave the same remainder k when divided by n, i.e., k,n + k, 2n + k, 3n + k,, as one element denoted as [k]. Note that k takes value in {0, 1,,n 1}. The modulo n set is defined as [n] := {[0], [1],, [n 1]}. Thus, we have divided the infinite set of whole numbers in to a set of exactly n elements, a finite set. Mathematically, the integer modulo n concept can be made precise using an equivalence relation. Given n , we define the equivalence relation a b, called as a equivalent to b, if a b is an integral (positive or negative) multiple of n. This equivalence relation will divide {0} into n disjoint equivalence classes, where each equivalence class contains numbers which have the same remainder. In this sense, the Fibonacci sequence elements in [2] are

[0], [1], [1], [0], [1], [1], [0], [1], [1], [0], [1], [1], [0], [1],.

Henceforth, we shall drop [] symbol in the sequence, for simplicity, and write it as

0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1,.

Note that the sequence is periodic (or repeating) of period three (after every third term). This period of the resulting periodic sequence is called Pisano period. The Pisano period of the Fibonacci sequence in [n] is given in the table below for the first 30 numbers.





[1]1
[2] 3
[3] 8
[4] 6
[5] 20
[6] 24
[7] 16
[8] 12
[9] 24
[10] 60
[11] 10
[12] 24
[13] 28
[14] 48
[15] 40
[16] 24
[17] 36
[18] 24
[19] 18
[20] 60
[21] 16
[22] 30
[23] 48
[24] 24
[25] 100
[26] 84
[27] 72
[28] 48
[29] 14
[30] 120




It is known that the resulting sequence of modulo n has Pisano period at most n2 1, for n 2. However, it is still an open problem to find a general formula for Pisano period in terms of n.

2 Fibonacci Colours

Note: This section and the next are my own creation/imagination. As on this day, to my knowledge, there is no such thing as Fibonacci Colour or Raaga. This is created only as an experiment, purely motivated by pleasure and leisure.

Let us associate Red, Green, Blue (RGB) colours corresponding to the Fibonacci sequence. Since we have chosen three colours, the possible rearrangement of these colours is 3! = 6. Thus, we can obtain six Fibonacci colours. For 3 colours the Pisano period is 8 and, hence, we get the following proportions for the colours:

















011 2 0 2 2 1








I IIIIIIIIIIIIIIII
















Thus, the three colours have to be mixed in the following proportion: I colour should be 2 8 = 25%, II and III colour should be each of 3 8 = 37.5%. Though we said there are six possible Fibonacci colours for three choice, the fact that two of them have to be mixed in same ratio reduces the possibility to 3 = 3! 2!. This is because the number of choices for the 25% proportion is three.

For instance, if we choose RGB we need to mix in the percentage 37.5% each of Blue and Green, and 25% of Red. This combination will give us the colour resembling dark cyan whose colour code is #406060. Note that this combination is same as RBG. If we choose BGR we need to mix in the percentage 37.5% each of Red and Green, and 25% of Blue. This combination will give us the colour resembling dark yellow whose colour code is #606040. Note that this combination is same as BRG. The only other possibility is GBR which is same as GRB. This combination will give us the colour whose colour code is #604840.

Suppose we choose four colours, viz., Cyan, Magenta, Yellow, Black (CMYK) then the possible rearrangement is 4! = 24, thus, giving us 24 Fibonacci colours. For four colours the Pisano period is six and, hence, we get the following proportions for the colours:













011 2 3 1






I IIIIIIIIVII












Thus, the four colours have to be mixed in the following proportion: First, third and fourth colour in 1 6 = 16.7% and second should be in 3 6 = 50%. Though we said there are 24 possible Fibonacci colours for four choice, the fact that three of them have to be mixed in same ratio reduces the possibility to 4 = 4! 3!. This is because the number of choices for the 50% proportion is four.

For instance, if we choose the order CMYK we need to mix in the percentage 16.7% each of CYK, and 50% of Magenta. This combination will give us the colour resembling strong magenta whose colour code is #AA55AA. The other three colours are MYK with Cyan, CMK with Yellow and CMY with Black.

If we choose the seven rainbow colours VIBGYOR then the possible rearrangement is 7! = 5040, thus, giving us 5040 Fibonacci colours. For seven colours the Pisano period is 16 and, hence, we get the following proportions for the colours:

































011 2 3 5 1 6 0 6 6 5 4 2 6 1
















I IIIIIIIIVVIIIVIIIVIIVIIVIVIIIVIIII
































In this case, one has to mix the colours in the following proportions: First, third, sixth colour in 12.5%, second and seventh colour in 25%, fourth and fifth colour in 6.25%. Though we said there are 5040 possible Fibonacci colours for seven choice, the fact that some of them have to be mixed in same ratio reduces the possibility to 210 = 7! 3!2!2!. For instance, if we mix VIBGYOR we get a colour resembling strong pink or strong raspberry whose colour code turns out to be #94305B. If we mix ROYGBIV we get the colour code #AE5055.

3 Fibonacci Raaga

The basic notes or swaram of carnatic music are seven, viz., Sa, Ri, Ga, Ma, Pa, Dha, Ni. Of these Sa and Pa are called Prakriti or nature swaras and the remaining five swaras are called vikriti or artificial swaras. It is believed that the prakriti swaras are given directly by God through Sama veda and the vikriti swaras are man-made. These seven notes are present in the nature in the sounds of peacock (Sa), bull (Ri), goat (Ga), dove/crane (Ma), cuckoo (Pa), horse (Dha) and elephant (Ni), respectively. In western music this corresponds to the following notes: C, D, E, F, G, A, B.

An octave or sthayi is the collection of swaram with different tones (or volume) such that between any pair of similar swara with different tone, the higher swaram is twice more than the lower swaram. The tones of Sa and Pa remains constant, Ma takes two tones while the other four swaras take three tones. This makes a total of 16 notes:

Sa, Ri1, Ri2, Ri3, G1, G2, G3, M1, M2, Pa, D1, D2, D3, N1, N2, N3.

I do not know if it is a strange coincidence that the 7 swaras and 16 notes matches with the pisano period 16 for modulo 7. Of the sixteen notes there are only 12 scales or volumes:

Sa, Ri1, Ri2 = G1, Ri3 = G2, G3, M1, M2, Pa, D1, D2 = N1, D3 = N2, N3.

The raagas are a combination of swaras. The raagas are classified into two categories: the parent/Janaka/Melakarta raagas and the Janya raagas. There are 72 Janaka raagas and Janya raagas are derived from the parent raagas. The Janaka raagas must have all the 8 swaras either in an ascent (aarohanam) or in descent (avarohanam). The 72 raagas are actually 36 pairs where one contains M1 and the second contains M2. One of each pair has 6 chakras (cycles) and each chakra has 6 raagas. The 6 raagas in each chakra differ only in one of the vikriti swara.

Consider the Fibonacci sequence in [7]

0, 1, 1, 2, 3, 5, 1, 6, 0, 6, 6, 5, 4, 2, 6, 1.

Note that its Pisano period is 16. Let us associate a raaga corresponding to the Fibonacci sequence, as follows:

































0 1 1 2 3 5 1 6 0 6 6 5 4 2 6 1
















SaRi1Ri1Ga1Ma1Da1Ri1Ni1SaNi1Ni1Da1PaGa1Ni1Ri1
















C D D E F A D B C B B A G E B D
































One may appropriately replace different notes corresponding to Ri, Ga, Ma, Da, Ni.