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Musimatics and Fibonacci! December 20, 2009

Posted by Ragesh G R in Physics and Maths, Uncategorized.

Music and Maths! these two have always enamoured me, and more so when both of them are inter-linked.

One can write an epic on the mathematical fundamentals of music! But I want to focus on some of the interesting patterns in music.

For the benefit of people who don’t know let me touch upon the basics.

The Basics:

Let us take the 12 notes in the musical scale. If you play all twelve one after the other on the Piano, does n’t it “feel” like they are arranged like a stair-case, with equal intervals between steps? Yes, that’s exactly how your ear perceives it, “as if” it is an Arithmetic progression. But the truth is ou ears auditory response is logarithmic. Hence to give you a linear feel, their logs have to be in A.P. Hence actually, these notes are in Geometric Progression. That is, the ratio between each note is the same. What is this ratio?

Let’s look at 13 notes (inclduding the start of the next octave), say

C2, C2#, D2, D2#, E2, F2, F2#, G2, G2#, A2, A2#, B2, [C3… ] OR

Sa, Ri1, Ri2/Ga1, Ri3/Ga2, Ga3, Ma1, Ma2, Pa, Da1, Da2/Ni1, Da3/Ni2, Ni3, [Sa^…]

The 12 notes are arranged in octaves, in way that, the same note in the next octave is twice its frequency. So C3 is twice the frequency of C2, OR Sa^ is the twice the frequency of Sa. Hence all the corresponding notes in the next octave is in phase with the notes in the current octave, because the notes in the next octave are overtones of the current octave tones.

Therefore, when two people sing Sa and Sa^ together, you won’t feel much difference, because every second cycle, the wavelengths synchronise.

Now we have 12 notes in GP and Freq(Sa^) = 2*Freq(Sa).

Therefore each notes is separated by (12th root of 2) or Antilog((1/12 )* log2) = 1.0594 ~~ 1.06

So every note is 1.06 times the previous note, and so (1.06)^12 = 2.

The Patterns:

Now, I told you all these because, there are many interesting patterns based on it.

If you see, Sa and Pa are stable , that is they dont have any derivative frequencies, we can call them the base. That is why most musicians sing Sa, Pa, Sa^ before commencing the concert, so as to set the reference. But the beauty is still to come.

If you take the octave as flanked by the 13 notes from Sa to Sa^, Pa comes as the 8th note. Hmm, 13 and 8 ? does it ring a bell? yes our Fibonacci. 8 and 13 are both consecutive Fibonacci numbers.

One more!, typically not all 12 frequencies are used in a Scale,  only 7 are used, that is  chosing only one Ri, one Ga etc. So then our Octave becomes 8 notes in length. Now here Pa is the 5th note. 5 and 8 ? does it ring a bell? Oh yes, again consecutive Fibonacci numbers.

So POSITION Wise, Pa divides the Octave in the Golden Ratio, (5:3 Or 8:5)

What about frequency?

Since Pa is the 8th frequency in an Octave. So if the frequency of Sa is say X.

Pa is (1.06)^7 = 1.5 !

So, Sa, Pa, Sa^ has a frequency of X, 1.5X, and 2X respectively.

Notice anything? Yes! they are in Aithmetic Progression! So Pa divides the Octave in the Golden ratio position wise, and divides the frequency in to an Arithmetic Progression!

I can over-ambitious and say that even Frequency-wise Pa divides the Octave ALMOST in Golden Ratio (1.5 ~ 1.6). LOL! But I don’t want to :-p.

See, how many beautiful patterns emerge, even from a simple and basic analysis!

Disclaimer: If you are a music connoisseur, you might have known this already.


Easiest/Laziest formula for Nth Fibonacci number December 12, 2009

Posted by Ragesh G R in Physics and Maths, Uncategorized.

There are many formulae for finding the Nth Fibonacci number, from recursive to non-recursive. Let me try to find the easiest laziest formula.

We know that most of the Fibonacci numbers ( from the 7th )  are in the Golden ratio. Let Golden Ratio (1.618…) be Phi.

Then F(n) = F(n-1)*Phi = F(n-2)*Phi^2 and so on.

So generally F(n) = F(n-x)*Phi^x

But we can’t write F(n) = F(1)*Phi^(n-1), because this will give you inaccurate results , because the 1st 7 numbers do not follow the Golden Ratio.

So the highest value x can take for a fairly accurate determination is n-7.

So our formula becomes F(n) = F(n-(n-7))*Ph^(n-7)

Therefore F(n) = F(7)*Phi^(n-7)

Fibonacci series goes as 1,1,2,3,5,8,13

So F(n) = 13*Phi^(n-7).

This will give a you a fairly accurate value for Nth Fibonacci.

You can get even lazier and say F(n) = 3*Phi^(n-4)

But this will magnify the error for 3 non compliant numbers and give you inaccurate resulsts, but if only it was accurate, this would be better  from a laziness perspective if you are programming, because there are only 4 special cases. Of course we can argue that computation of Phi^(n-4) will take fractionally more time than computation of Phi^(n-7), but we can neglect that 🙂

Disclaimer : I am sure this already in some literature, but I got this idea when I was in the restoom (a link to my previous post LOL! ), so its mine , he he ! and I could not find similar text by “lazy” googling

R-Buddies December 10, 2009

Posted by Ragesh G R in Uncategorized.

I have been working in a corporate office for 2.5 years now and have made many real friends. But this is not I am about tell you now. I am going to tell about a set of pseudo buddies you make because you cannot control your body’s metabolism. Yes! Restroom buddies!

You still may think it is some people you befriend in the restroom. No! these are pseudo! Over the past 2.5, I have seen a pattern, the same set of people seem to be present in the restroom when I enter the restroom, regardless of what time I enter, be it morning, noon, afternoon, 3 PM 4 PM whatever odd time, they are there! And even on the same day, they are there everytime you go! Then you realise, that these are the people whose bodies may probably function similar to yours, may be they have the same rate of metabolism, may be they have the same bodily cycles. It makes me get a friendly feeling towards them. After all, who does not like similar people!? Something like a stockholm syndorme. But the friendly feeling is not because you see them everyday, it is because deep down you know that you may be physically similar people.

And this similarity, cannot be masked by any artifical means, it is the most basic of similarity. This is where all humans become equal. The equality here is unbelievable. I see all people from Entry level trainees to Senior Managers everyday, different ages, different ethnic groups, different regions.

It cannot get more equal than this. Irrespective the class / strata / race you are, it only means you may have similar functioning bodies, and no one can deny that or no one can change that. It gives a sense of belonging, that you are not alone.

It might seem gross , does it? Think about it! Where can you find more equality? All protocols cease to exist here. and it is great to know that “similar” people to you exists in all walks of life.

I am writing this because, I was curious to know if anyone else too felt the same way, about meeting the same people in restroom everyday and wondering if that is because these set of people have a similar metabolism pattern as you!