Musimatics and Fibonacci! December 20, 2009Posted 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.
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.
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.