“There is geometry in the humming of the strings. There is music in the spacings of the spheres.”
“Numbers rule universe”
“Numbers in space is Geometry, Numbers in time is Music, Numbers in space and time is Astronomy”
— Pythagoras (580-500 BC)
If you are reading this blog, I am presuming you are interested in learning a bit about music – perhaps specifically Indian Classical music. So what are these quotes by a mathematician – especially the one who most of us remember for his trigonometric theorem related to the nightmarish memories of high school geometry.
As it turns out, music has a lot to do with mathematics.
Human ear is supposed to be capable of hearing frequencies between 20 Hz and 20000 Hz. As we grow older, that range becomes more like 50 Hz to 12000 Hz. If you think about it, there are millions of frequencies in that spectrum. So why is it that some frequencies sound sweet and not others?
However, one frequency or note in itself does not convey anything. Music is formed when two or more notes are perceived relative to each other, either together (e.g. as a chord) or sequentially (e.g. as a melody).
In Indian Classical system, each note has a name – Sa, Re, Ga, Ma, Pa, Dha, Ni – which corresponds to Western Do, Re, Mi, Fa, So, La, Ti. The notes referred here are actually acronyms – Sa stands for Shadja, Ga stands for Gandhar, Pa stands for Pancham and so on.
The word Shadja in sanskrit literally means the father of the (other) six. In other words, Sa gives birth to the other six notes. The note Sa is the anchor in any Indian melody and is prominently played as a backdrop in form of tanpura drone.
We know from our high school physics that when the frequency of a note doubles it sounds similar to the base note and is called the Octave note. So if Sa is x Hz, higher Sa will be 2x Hz. And all other notes have to be between the frequency band between x and 2x. That still leaves millions of possible notes.
The answer to finding all notes lies in yet another important concept – known as Shadja-Pancham bhaav or the relationship between Sa and Pa- the fifth note.
As it happens, Pa falls exactly in the midpoint of Sa and octave Sa. In other words, if Sa is x Hz, Octave Sa is 2x Hz then Pa will be 3/2 x or 1.5x Hz. Great, then that gives us one more note. What about others.
Why else is Shadja Pancham Bhaav important concept? Look at the piano keyboard below. If you start with C in the first octave as Sa and play the fifth pure note (8th key if you count half steps or komal notes), then you reach Pa. If you then count 8 keys from Pa (in other words Pa of Pa), you reach Re. If you then count 8 keys from Re (i.e. Pa of Re) then you reach Dha and so on. The amazing thing is that you then come back through the cycle after 12 notes and reach back on Sa.
This is called the Circle of the Fifths.
It will then be possible to calculate the exact frequencies of all these notes. For example –
Sa = 100 Hz
Pa = 3/2 x 100 = 150 Hz
Re = 3/2 x 3/2 x 100 Hz = 225 Hz. This gives us Re in the second octave. Therefore, we can divide it by 2 and get Re in base octave as 112.5 Hz.
Dha = 3/2 x 3/2 x 3/2 x 100 Hz and so on.
So, we managed to mathematically derive the 12 notes through Shadja-Pancham bhaav and the circle of the fifths. But I am sure, you are wondering what is so special about the Shadja Pancham bhaav. The answer to that question in one of the subsequent editions of my blog.
Oh, and by the way, here’s a parting mathematical riddle –
We got back to Sa through the circle of fifths by stepping through 12 steps. Which means that if the starting Sa had the frequency of f Hz, the Sa we got after 12 steps should have the frequency of 3/2 x 3/2 x 3/2 … (12 times) x f Hz. Which is same as 3^12/2^12 x f Hz or 129.7463 f Hz.
However, as you can see from the keyboard above, that Sa is actually the 7th Octave of the base Sa. Or 2^7 f Hz. Or 128 Hz.
How can we reach the same note with two consistent mathematical calculations with two different results?
If you have the answer, I would love the hear from you.