Why does Sa-Pa feel "right"? The maths hiding inside my tanpura

Amma says I started singing before I started talking. I've been learning Carnatic music since I was six, so I've sung the same varnam about four hundred times. And only this year, in the waves chapter of physics, I realised that the thing I do every day is just maths in a costume. Dramatic, I know. Let me explain.

A string and its halves

Pluck a string and it vibrates at a fundamental frequency. Press the string at its middle and pluck again: the vibrating length is halved and the frequency doubles. For the same string and tension, $f \propto 1/L$. A frequency ratio of 2 : 1 is an octave: the same note, higher. In my music this is Sa and the upper Sa. They feel like the same note because in a way they are.

Now stop the string at two-thirds of its length and the frequency is $3/2$ times the original. A ratio of 3 : 2 is the perfect fifth. In Carnatic music it's Sa to Pa. That is why the tanpura is tuned to Sa and Pa (or sometimes Ma instead of Pa): after the octave, these have the simplest ratios, and they sound the smoothest.

Why smooth? A note is never one frequency. A string vibrating at fundamental $f_1$ also produces overtones at $2f_1, 3f_1, 4f_1, \ldots$, which is $f_n = n f_1$ (for a string fixed at both ends, $f_n = \dfrac{n v}{2L}$). Take a Sa with fundamental $f$. Its third harmonic is $3f$. Now take Pa, with fundamental $\tfrac{3}{2}f$. Its second harmonic is $2 \times \tfrac{3}{2}f = 3f$. Same frequency! The two notes share overtones, so when you sound them together nothing fights. That's why the constant tanpura hum feels like the floor under the whole raga.

Another thing I love: in our system there is no fixed Sa frequency. Every singer picks their own Sa (their shruti), and everything else is a ratio relative to it. My Sa is higher than my brother's. Neither is "correct". Ratios, not numbers.

Some ratios I use every day

(This is just intonation, the "pure" version. In real performance, singers shade notes with ornaments.)

Interval	Ratio	In my practice
Octave	2 : 1	Sa to upper Sa
Perfect fifth	3 : 2	Sa to Pa
Perfect fourth	4 : 3	Sa to Shuddha Madhyama
Major third	5 : 4	Sa to Antara Gandhara
Major second	9 : 8	Sa to Chatusruti Rishabham

The problem: the circle of fifths doesn't close

If you climb up by perfect fifths twelve times, you should land back on an Sa, seven octaves up. Check it:

$\left(\tfrac{3}{2}\right)^{12} = 129.746\ldots \qquad 2^7 = 128$

They don't match. The ratio $\dfrac{(3/2)^{12}}{2^7} = \dfrac{531441}{524288} \approx 1.0136$. That small leftover is called the Pythagorean comma. And it can never be fixed by cleverness, because $3^a = 2^b$ has no solutions in positive whole numbers (unique prime factorisation, the same idea behind the √2 proof Tushar wrote about). So any instrument with fixed notes has to compromise.

Equal temperament: the compromise

A piano or harmonium divides the octave into 12 equal steps. Each semitone is a frequency ratio of $2^{1/12} \approx 1.05946$. Twelve steps give $\left(2^{1/12}\right)^{12} = 2$ exactly, so octaves are perfect. A fifth is seven semitones: $2^{7/12} \approx 1.4983$ versus the pure $1.5$. Your ear forgives that (it's about a tenth of a percent). A major third, four semitones, is $2^{4/12} \approx 1.2599$ against the pure $1.25$, and that one is more noticeable.

A quick check with real numbers: concert A is 440 Hz, so the E a fifth above it is $440 \times 2^{7/12} \approx 659.3$ Hz in equal temperament, whereas the pure ratio would give exactly 660 Hz.

Carnatic music doesn't run on equal temperament. A singer, a violinist or a veena player places each note where it sounds right and can slide into it with gamakas, the oscillations and slides that give a raga its identity. A harmonium's pitches are fixed, so it can't do that, which is why many Carnatic musicians avoid it in concerts. Which is also why my cousin (harmonium fan) and I fight at every family function. He says convenient, I say honest. In fairness, plenty of beautiful music, bhajans and North Indian music included, uses a harmonium happily. It's about what the music needs.

Why twelve?

Twelve isn't magic. It's a good compromise because seven semitones happen to land very close to 3 : 2, the most important interval. Other cultures slice the octave differently. Indian theory also talks about 22 shrutis, and how those map onto actual sounded pitches is a long, friendly argument among musicians. I'm not going to settle that here.

What I take away

I used to think of music as feeling and maths as a subject. They're the same event seen from two sides: the feeling is the maths arriving at your ear. You can check it yourself with a ruler and a rubber band stretched between two fingers. Twang it, then press at the midpoint and twang again. Same note, one octave higher. You just did the first line of this blog.